{"m1001":{"id":"m1001","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Infinitely many solutions with two constants","tier":1,"traps":["partial"],"answer":"-1","accept":["-1"],"stem":"In the equation 52​(15x−10)+kx=9x+c, k and c are constants. If the equation has infinitely many solutions, what is the value of k+c?","explain":"Distribute: 6x−4+kx=9x+c, so (6+k)x−4=9x+c. Infinitely many solutions means both sides are the same expression: 6+k=9 and c=−4. So k=3 and k+c=−1. Trap: stopping at k=3."},"m1002":{"id":"m1002","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"No solution with a constant in two places","tier":2,"traps":["setup"],"answer":"D","stem":"In the equation a(3x−2)=12x+5a−14, a is a constant. For what value of a does the equation have no solution?","choices":["2","12","There is no such value of a.","4"],"explain":"Expand: 3ax−2a=12x+5a−14. No solution means equal x-coefficients but unequal constants. 3a=12 gives a=4; then the constants are −8 and 6, which differ, so there is no solution. Trap: setting the constants equal gives a=2, but then the coefficients are 6 and 12, so there is exactly one solution."},"m1003":{"id":"m1003","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"No solution with fractions and two constants","tier":3,"traps":["concept","partial"],"answer":"15/4","accept":["15/4","3.75"],"stem":"In the equation 3x−a​+5x+a​=k2x​, a and k are constants and a=0. If the equation has no solution, what is the value of k?","explain":"Combine the left side over 15: 155(x−a)+3(x+a)​=158x−2a​. A linear equation has no solution when the x-terms match but the constant terms don't. The x-coefficients match when 158​=k2​, so k=415​. The left side then has constant term −152a​, which isn't 0 because a=0, while the right side has constant term 0, so there is no solution. Enter 15/4 or 3.75. Trap: assuming the −a and +a cancel; after weighting by 5 and 3 they leave −2a."},"m1004":{"id":"m1004","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Piecewise pricing in context","tier":1,"traps":["partial"],"answer":"717","accept":["717"],"stem":"A phone plan charges a flat $32 each month plus $0.05 for each text message sent beyond the first 400 that month. In March, Dana's bill was $47.85. How many text messages did Dana send in March?","explain":"Only texts beyond 400 cost money: 32+0.05(t−400)=47.85. Then 0.05(t−400)=15.85, so t−400=317 and t=717. Trap: 317 is the number of texts she paid for, not the number she sent."},"m1005":{"id":"m1005","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Constant from a given solution","tier":1,"traps":["sign"],"answer":"-2","accept":["-2"],"stem":"In the equation k(x+3)−4(x−k)=2x+26, k is a constant. If x=−5 is a solution to the equation, what is the value of k?","explain":"Substitute x=−5: k(−2)−4(−5−k)=−10+26. So −2k+20+4k=16, which gives 2k=−4 and k=−2. Check: with k=−2 the equation becomes −6x−14=2x+26, whose solution is −5."},"m1006":{"id":"m1006","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Conditions for no solution","tier":2,"traps":["concept"],"answer":"C","stem":"In the equation ax+b=3(x−2)+7x, a and b are constants. If the equation has no solution, which of the following must be true?","choices":["a=10 and b=−6","a=10 and b=−6","a=10 and b=−6","a=3 and b=−2"],"explain":"The right side simplifies to 10x−6. A linear equation has no solution when the x-coefficients match and the constants don't: a=10 and b=−6. Choice A gives infinitely many solutions, and B gives exactly one. D compares against 3(x−2) alone and forgets the 7x."},"m1007":{"id":"m1007","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Successive fractional changes","tier":2,"traps":["setup"],"answer":"180","accept":["180"],"stem":"A number is decreased by 31​ of itself, and the result is then increased by 41​ of that result. The final value is 6 more than 54​ of the original number. What is the original number?","explain":"Call the number x. After the decrease it is 32​x; increasing that by a quarter of itself gives 45​⋅32​x=65​x. So 65​x=54​x+6, which gives 301​x=6 and x=180. Trap: adding the fractions, 1−31​+41​, ignores that the second change applies to the new value."},"m1008":{"id":"m1008","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Infinitely many solutions with cross-multiplication","tier":2,"traps":["partial"],"answer":"-10/3","accept":["-10/3","-3.333"],"stem":"In the equation 43x+a​=6bx−5​, a and b are constants. If the equation has infinitely many solutions, what is the value of a?","explain":"Cross-multiply: 6(3x+a)=4(bx−5), so 18x+6a=4bx−20. For every x to work, 18=4b and 6a=−20. So a=−310​ (enter -10/3 or -3.333)."},"m1009":{"id":"m1009","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Solving for an expression that hides the constant","tier":2,"traps":["setup"],"answer":"-7","accept":["-7"],"stem":"In the equation 3(2x−k)=5x+4k−7, k is a constant. What is the value of x−7k?","explain":"Expand: 6x−3k=5x+4k−7. Subtract 5x from both sides and add 3k: x=7k−7. So x−7k=−7, whatever the value of k. Trap: trying to find x and k separately; one equation can't pin down both, but it does pin down x−7k."},"m1010":{"id":"m1010","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Mixture","tier":2,"traps":["setup"],"answer":"50/9","accept":["50/9","5.555","5.556"],"stem":"A chemist mixes x liters of a 12% acid solution with (20−x) liters of a 30% acid solution to make 20 liters of a 25% acid solution. What is the value of x?","explain":"Track the acid: 0.12x+0.30(20−x)=0.25(20). So 0.12x+6−0.30x=5, giving −0.18x=−1 and x=950​≈5.556. Enter 50/9, 5.555, or 5.556."},"m1011":{"id":"m1011","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Perimeter preserved after two changes","tier":3,"traps":["setup","partial"],"answer":"104","accept":["104"],"stem":"The length of a rectangle is 3 inches more than twice its width. The length is decreased by 25% and the width is increased by 4 inches, and the new rectangle has the same perimeter as the original. What is the area, in square inches, of the original rectangle?","explain":"Equal perimeters mean the length and width have the same sum before and after. The length loses 0.25L and the width gains 4, so 0.25L=4 and L=16. Then 2w+3=16 gives w=6.5, and the original area is 16×6.5=104. (Setting up 0.75(2w+3)+(w+4)=(2w+3)+w leads to the same w=6.5.) Trap: answering with the new rectangle's area, 12×10.5=126."},"m1012":{"id":"m1012","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"A constant that removes every solution","tier":2,"traps":["concept"],"answer":"3","accept":["3"],"stem":"In the equation 3x+a​=ax−2​, a is a nonzero constant. For what value of a does the equation have no solution?","explain":"Cross-multiply: a(x+a)=3(x−2), so ax+a2=3x−6. There is no solution when the x-coefficients match (a=3) and the constants don't (9=−6). So a=3."},"m1013":{"id":"m1013","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Recovering a linear function from a composition","tier":3,"traps":["setup"],"answer":"-13/3","accept":["-13/3","-4.333"],"stem":"For the linear function f, f(2x−3)=8x+1 for all values of x. If f(k)=k, what is the value of k?","explain":"Let u=2x−3, so x=2u+3​. Then f(u)=8(2u+3​)+1=4u+13. Setting f(k)=k gives 4k+13=k, so k=−313​. Enter -13/3 or -4.333. Trap: treating f(x) as 8x+1, which gives k=−71​."},"m1014":{"id":"m1014","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Difference equation for a line","tier":2,"traps":["setup"],"answer":"-18","accept":["-18"],"stem":"For the linear function g, g(x+2)−g(x)=7 for all values of x, and g(4)=3. What is the value of g(−2)?","explain":"An increase of 2 in x raises g by 7, so the slope is 27​. From x=4 to x=−2 is a change of −6, so g changes by 27​(−6)=−21. Then g(−2)=3−21=−18. Trap: using 7 as the slope gives −39."},"m1015":{"id":"m1015","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Slope with a constant in both points","tier":2,"traps":["sign"],"answer":"8/5","accept":["8/5","1.6"],"stem":"In the xy-plane, a line with slope 2 passes through the points (k,3k) and (2k,8), where k is a nonzero constant. What is the value of k?","explain":"Slope =2k−k8−3k​=k8−3k​. Setting this equal to 2: 8−3k=2k, so k=58​=1.6."},"m1016":{"id":"m1016","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Missing values in a table","tier":3,"traps":["setup"],"figure":"
xf(x)
2a
519
113a
","answer":"19/3","accept":["19/3","6.333"],"stem":"The table shows three values of the linear function f, where a is a constant. What is the value of f(0)?","explain":"A linear function changes at a constant rate. From x=2 to 5 the rate is 319−a​; from 5 to 11 it is 63a−19​. Set them equal: 2(19−a)=3a−19, so a=557​. The slope is 319−57/5​=1538​, and f(0)=f(2)−2⋅1538​=557​−1576​=319​. Enter 19/3 or 6.333."},"m1017":{"id":"m1017","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Two tanks, two conditions","tier":3,"traps":["setup"],"answer":"32","accept":["32"],"stem":"Tank A starts with 1,200 liters of water and drains at a constant rate. At the same moment, tank B starts empty and fills at a constant rate. After 15 minutes, tank A holds 3 times as much water as tank B. After 25 minutes, the two tanks hold the same amount of water. At what rate, in liters per minute, does tank A drain?","explain":"Let tank A drain at a liters per minute and tank B fill at b liters per minute. At 25 minutes, 1,200−25a=25b, so b=48−a. At 15 minutes, 1,200−15a=3(15b)=45(48−a)=2,160−45a. Then 30a=960 and a=32. Check: at 15 minutes tank A holds 720 liters and tank B holds 240; at 25 minutes both hold 400. Trap: answering with tank B's rate, 16."},"m1018":{"id":"m1018","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Perpendicular line through two points with a constant","tier":3,"traps":["sign","setup"],"answer":"-10","accept":["-10"],"stem":"Line ℓ passes through the points (2,a) and (a,8), where a is a constant, and is perpendicular to the line 3x−4y=12. At what x-coordinate does line ℓ cross the x-axis?","explain":"The line 3x−4y=12 has slope 43​, so line ℓ has slope −34​. Using the two points: a−28−a​=−34​, so 3(8−a)=−4(a−2), which gives 24−3a=−4a+8 and a=−16. Line ℓ passes through (2,−16): y+16=−34​(x−2). At y=0, x−2=−12, so x=−10. Trap: using slope 34​ (the reciprocal without the sign change)."},"m1019":{"id":"m1019","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Self-composition with a sign condition","tier":3,"traps":["sign"],"answer":"8","accept":["8"],"stem":"The function h is defined by h(x)=ax+b, where a and b are constants and a<0. If h(h(x))=9x−16 for all values of x, what is the value of b?","explain":"h(h(x))=a(ax+b)+b=a2x+ab+b. Matching: a2=9, and since a<0, a=−3. Then ab+b=b(a+1)=−2b=−16, so b=8. Trap: taking a=3 gives 4b=−16, or b=−4."},"m1020":{"id":"m1020","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Undoing a tax on a linear cost","tier":2,"traps":["percent"],"answer":"120","accept":["120"],"stem":"The function C(m)=0.32m+54 gives the cost, in dollars, of renting a car for one day and driving it m miles, before tax. A customer paid $97.02 for one day, which included a 5% sales tax on the cost. How many miles did the customer drive?","explain":"The taxed total is 1.05(0.32m+54)=97.02, so 0.32m+54=92.40, 0.32m=38.40, and m=120. Trap: subtracting 5% of $97.02 instead of dividing by 1.05 gives a slightly different, wrong pre-tax cost."},"m1021":{"id":"m1021","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Reading a line from a graph","tier":2,"traps":["partial"],"figure":"
\"A
","answer":"29/3","accept":["29/3","9.666","9.667"],"stem":"The graph of the linear function y=f(x) is shown. If f(k)=−211​, what is the value of k?","explain":"The marked points (−3,4) and (5,−2) give slope 5−(−3)−2−4​=−43​, so f(x)=4−43​(x+3). Set 4−43​(k+3)=−211​: 43​(k+3)=219​, so k+3=338​ and k=329​. Enter 29/3, 9.666, or 9.667."},"m1022":{"id":"m1022","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Fixed fee from relationships, not points","tier":3,"traps":["setup"],"answer":"160","accept":["160"],"stem":"A kayak rental costs a fixed fee plus a constant charge per hour. A 7-hour rental costs $40 more than a 3-hour rental. A customer rents a kayak for 5 hours and uses a coupon for 20% off the total cost, paying $8 more than the fixed fee. What is the fixed fee, in dollars?","explain":"Let the fixed fee be F and the hourly charge r. Four extra hours cost $40, so r=10, and a 5-hour rental costs F+50. With the coupon the customer pays 0.8(F+50)=0.8F+40. Setting this equal to F+8: 0.2F=32, so F=160. Trap: applying the discount to the hourly charges only."},"m1023":{"id":"m1023","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Points with swapped coordinates","tier":2,"traps":["concept"],"answer":"B","stem":"A line in the xy-plane passes through the points (a,b) and (b,a), where a=b, and also through the point (2,9). Which of the following is an equation of the line?","choices":["y=x+7","x+y=11","x+y=7","y=−x+9"],"explain":"The slope through (a,b) and (b,a) is b−aa−b​=−1 for any a=b. A line with slope −1 through (2,9) is y−9=−(x−2), or x+y=11. D has the right slope but misses (2,9)."},"m1024":{"id":"m1024","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Constant that fixes slope and intercept","tier":2,"traps":["partial"],"answer":"14/5","accept":["14/5","2.8"],"stem":"The function f is defined by f(x)=(k−3)x+2k, where k is a constant. The graph of y=f(x) passes through the point (2,1). What is the x-intercept of the graph?","explain":"f(2)=2(k−3)+2k=4k−6=1, so k=47​. Then f(x)=−45​x+27​. Setting f(x)=0 gives x=27​⋅54​=514​, or 2.8. Trap: entering k=47​."},"m1025":{"id":"m1025","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Totals with a difference between quantities","tier":2,"traps":["partial"],"answer":"180","accept":["180"],"stem":"A café sells small coffees for $2.75 and large coffees for $3.50. One morning it sold 30 more large coffees than small coffees and collected $573.75 from coffee sales. How many coffees did it sell in all that morning?","explain":"With s small coffees, there were s+30 large: 2.75s+3.50(s+30)=573.75, so 6.25s+105=573.75 and s=75. Large coffees: 105. In all: 75+105=180. Trap: entering 75, the number of small coffees."},"m1026":{"id":"m1026","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Intercepts with a constant","tier":1,"traps":["setup"],"answer":"6","accept":["6"],"stem":"In the xy-plane, the graph of 3x+ky=18, where k is a positive constant, has an x-intercept that is twice its y-intercept. What is the value of k?","explain":"The x-intercept is (6,0) and the y-intercept is (0,k18​). So 6=2⋅k18​, giving k=6. Trap: reversing the relationship (k18​=2⋅6) gives k=1.5."},"m1027":{"id":"m1027","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Trading one quantity for another","tier":1,"traps":["sign"],"answer":"A","stem":"A club spends exactly $2,400 on basic calculators that cost $40 each and graphing calculators that cost $110 each, as shown by 40x+110y=2,400. If the club instead buys 4 fewer graphing calculators and still spends exactly $2,400, how does the number of basic calculators change?","choices":["It increases by 11.","It decreases by 11.","It increases by 4.","It increases by 44."],"explain":"Four fewer graphing calculators frees 4×110=440 dollars, which buys 440÷40=11 more basic calculators. D is the number of extra dollars divided by 10, not by 40."},"m1028":{"id":"m1028","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Intercepts with a constant and a point","tier":3,"traps":["setup"],"answer":"9/2","accept":["9/2","4.5"],"stem":"In the xy-plane, the graph of ax+by=12, where a and b are constants, has x-intercept (p,0) and y-intercept (0,2p) for some positive constant p. The graph also passes through the point (1,6). What is the value of a+b?","explain":"A line with intercepts p and 2p is px​+2py​=1, or 2x+y=2p. The point (1,6) gives 2p=8, so the line is 2x+y=8. To make the right side 12, multiply by 23​: 3x+1.5y=12. So a=3, b=1.5, and a+b=4.5."},"m1029":{"id":"m1029","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Intercept form","tier":2,"traps":["setup"],"answer":"18","accept":["18"],"stem":"In the xy-plane, the graph of ax​+by​=1, where a and b are nonzero constants, passes through the point (6,4) and has a slope of −31​. What is the value of a?","explain":"Solving for y gives y=−ab​x+b, so ab​=31​ and a=3b. Substituting (6,4): 3b6​+b4​=1, so b6​=1 and b=6. Then a=18."},"m1030":{"id":"m1030","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Whole-number solutions in context","tier":3,"traps":["setup"],"answer":"3","accept":["3"],"stem":"Student tickets to a science fair cost $4 and adult tickets cost $7. A group spent exactly $100 on tickets and bought at least one ticket of each type. How many different combinations of student and adult tickets could the group have bought?","explain":"Solve 4s+7a=100 in positive integers. Since 4s and 100 are multiples of 4, 7a must be too, so a is a multiple of 4: a=4,8,12 give s=18,11,4; a=16 would cost more than $100. That is 3 combinations."},"m1031":{"id":"m1031","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Sign of the slope and intercepts","tier":2,"traps":["sign"],"answer":"B","stem":"In the xy-plane, the graph of ax+by=c is a line, where a, b, and c are positive constants. Through which quadrant does the line NOT pass?","choices":["Quadrant I","Quadrant III","Quadrant II","Quadrant IV"],"explain":"In slope-intercept form, y=−ba​x+bc​: the slope is negative and the y-intercept is positive, and the x-intercept ac​ is positive. Such a line falls from upper left to lower right through Quadrants II, I, and IV, missing Quadrant III."},"m1032":{"id":"m1032","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Three points, one constant","tier":2,"traps":["concept"],"answer":"C","stem":"The points (−1,3a), (2,a+6), and (5,9) lie on the same line in the xy-plane, where a is a constant. Which of the following is an equation of the line?","choices":["x=2","y=x+7","y=9","y=3x+12"],"explain":"Equal spacing in x (steps of 3) means equal steps in y: (a+6)−3a=9−(a+6), so 6−2a=3−a and a=3. The points are (−1,9), (2,9), and (5,9): the line is horizontal, y=9. Trap: assuming the line must slant and picking an equation through just one of the points."},"m1033":{"id":"m1033","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"No solution with two constants","tier":1,"traps":["concept"],"answer":"D","stem":"In the system below, k and c are constants.

3x−ky=8 6x+10y=c

For which of the following does the system have no solution?","choices":["k=−5 and c=16","k=5 and c=16","k=5 and c=16","k=−5 and c=16"],"explain":"No solution means parallel, distinct lines. Doubling the first equation gives 6x−2ky=16. For the y-coefficients to match, −2k=10, so k=−5. The lines are distinct only if c=16. Choice A describes the same line twice (infinitely many solutions)."},"m1034":{"id":"m1034","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Infinitely many solutions with fractional coefficients","tier":2,"traps":["setup"],"answer":"-4/3","accept":["-4/3","-1.333"],"stem":"In the system below, a and b are constants.

32​x−21​y=4 ax+by=6

If the system has infinitely many solutions, what is the value of ba​?","explain":"For infinitely many solutions the second equation must be a multiple of the first. Since 6=23​⋅4, the second equation is x−43​y=6, so a=1 and b=−43​. Then ba​=−34​. (Shortcut: the ratio ba​ must equal −1/22/3​.)"},"m1035":{"id":"m1035","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Constant in both equations, no solution","tier":3,"traps":["concept"],"answer":"-3","accept":["-3"],"stem":"In the system below, k is a constant.

(k+1)x+4y=10 3x+(k−3)y=5

For what value of k does the system have no solution?","explain":"Parallel lines need proportional coefficients: 3k+1​=k−34​, so (k+1)(k−3)=12, k2−2k−15=0, and k=5 or k=−3. Check both. With k=5: 6x+4y=10 and 3x+2y=5 are the same line, which gives infinitely many solutions. With k=−3: −2x+4y=10 becomes x−2y=−5, and 3x−6y=5 becomes x−2y=35​. These are parallel and distinct, so k=−3. Trap: stopping at k=5."},"m1036":{"id":"m1036","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Sales with a relationship between counts","tier":2,"traps":["partial"],"answer":"165","accept":["165"],"stem":"A school store sold T-shirts for $12 each and hoodies for $30 each. Sales totaled $2,790, and the number of T-shirts sold was 15 less than 3 times the number of hoodies sold. How many items did the store sell in all?","explain":"Let h be hoodies; T-shirts are 3h−15. Then 12(3h−15)+30h=2,790, so 66h−180=2,790 and h=45. T-shirts: 3(45)−15=120. In all: 165. Trap: stopping at h=45."},"m1037":{"id":"m1037","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Adding equations with unknown coefficients","tier":3,"traps":["setup"],"answer":"8","accept":["8"],"stem":"In the system below, a and b are constants.

ax+by=12 bx+ay=4

If the solution (x,y) of the system satisfies x+y=2, what is the value of a+b?","explain":"Add the two equations: ax+bx+by+ay=16, which factors as (a+b)(x+y)=16. Since x+y=2, a+b=8. There isn't enough information to find a, b, x, and y individually, and none of them is needed."},"m1038":{"id":"m1038","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Exactly one solution is impossible","tier":2,"traps":["concept"],"answer":"A","stem":"In the system below, c is a constant.

y=2x+c 4x−2y=9

For how many values of c does the system have exactly one solution?","choices":["None","Exactly one","Exactly two","Infinitely many"],"explain":"The second equation is y=2x−29​, with the same slope as the first. Two lines with equal slopes are either parallel (no solution, when c=−29​) or identical (infinitely many, when c=−29​). They can never cross at exactly one point. Trap: B confuses \"infinitely many solutions for one value of c\" with \"exactly one solution.\""},"m1039":{"id":"m1039","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"A condition on the solution fixes a constant","tier":2,"traps":["setup"],"answer":"9","accept":["9"],"stem":"In the system below, k is a constant.

3x+ky=18 x−2y=1

If the solution (x,y) of the system satisfies x=3y, what is the value of k?","explain":"Use the equation without k first: x−2y=1 with x=3y gives y=1 and x=3. Substitute into the first equation: 9+k=18, so k=9."},"m1040":{"id":"m1040","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Wind speed from two trips","tier":2,"traps":["setup"],"answer":"60","accept":["60"],"stem":"A plane flies 1,800 kilometers with a steady tailwind in 3 hours. The return trip along the same route, into the same wind, takes 45 minutes longer. What is the speed of the wind, in kilometers per hour?","explain":"With the wind: p+w=31,800​=600. Against it, the trip takes 3.75 hours: p−w=3.751,800​=480. Subtracting gives 2w=120, so w=60. Trap: using 45 minutes as the whole return time, or 3.45 hours."},"m1041":{"id":"m1041","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Symmetric system with a solution off the line y = x","tier":3,"traps":["concept"],"answer":"4","accept":["4"],"stem":"In the system below, k is a constant.

4x+ky=20 kx+4y=20

The system has a solution (x,y) in which x=y. What is the value of k?","explain":"Subtract the second equation from the first: (4−k)x−(4−k)y=0, or (4−k)(x−y)=0. If k=4, every solution has x=y. So a solution with x=y requires k=4. Then both equations are 4x+4y=20, which has infinitely many solutions, such as (0,5). Trap: k=−4 makes the lines parallel, so there is no solution at all."},"m1042":{"id":"m1042","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Unknown coefficients from a known solution","tier":2,"traps":["partial"],"answer":"10","accept":["10"],"stem":"The solution to the system below is (2,1), where a and b are constants.

ax+by=12 bx+ay=18

What is the value of a+b?","explain":"Substitute (2,1): 2a+b=12 and 2b+a=18. Adding gives 3a+3b=30, so a+b=10, with no need to find a and b separately (they are 2 and 8)."},"m1043":{"id":"m1043","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Intersection with a coordinate condition","tier":2,"traps":["partial"],"answer":"-20","accept":["-20"],"stem":"The graphs of y=4x−1 and y=mx+23, where m is a constant, intersect at a point whose y-coordinate is 3 times its x-coordinate. What is the value of m?","explain":"On the first line, y=3x means 4x−1=3x, so x=1 and y=3. The point (1,3) is also on the second line: 3=m+23, so m=−20."},"m1044":{"id":"m1044","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"No solution after clearing fractions","tier":1,"traps":["concept"],"answer":"6","accept":["6"],"stem":"In the system below, a is a constant.

21​x−43​y=6 ax−9y=20

If the system has no solution, what is the value of a?","explain":"Multiply the first equation by 12: 6x−9y=72. The y-coefficient already matches the second equation, so parallel lines need a=6. The constants (72 and 20) differ, so the lines are distinct and there is no solution."},"m1045":{"id":"m1045","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Totals, then a derived quantity","tier":2,"traps":["partial"],"answer":"120","accept":["120"],"stem":"A warehouse packed 480 jars into 30 boxes. Each small box holds exactly 12 jars and each large box holds exactly 20 jars, and every box was filled. How many more jars were packed in large boxes than in small boxes?","explain":"With s small boxes: 12s+20(30−s)=480, so 600−8s=480 and s=15. Small boxes hold 15×12=180 jars; large boxes hold 15×20=300. The difference is 120. Trap: answering with a number of boxes."},"m1046":{"id":"m1046","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Intersection that is not on the grid","tier":2,"traps":["setup"],"figure":"
\"Two
","answer":"62/11","accept":["62/11","5.636"],"stem":"The graphs of two lines are shown. If the lines intersect at the point (a,b), what is the value of a+b?","explain":"Line 1 through (0,−2) and (3,4): y=2x−2. Line 2 through (0,5) and (4,2): y=−43​x+5. Setting them equal: 411​x=7, so a=1128​ and b=2⋅1128​−2=1134​. The sum is 1162​≈5.636. Reading the crossing point off the graph won't give this exactly; use the marked points."},"m1047":{"id":"m1047","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Rates meeting in time","tier":2,"traps":["setup"],"answer":"40/3","accept":["40/3","13.33"],"stem":"Two candles are lit at the same time. The first is 30 centimeters tall and burns 1.2 centimeters per hour. The second is 24 centimeters tall and burns 0.75 centimeter per hour. How many hours after they are lit will the candles be the same height?","explain":"Set the heights equal: 30−1.2t=24−0.75t. Then 6=0.45t, so t=340​≈13.33 hours. At that time both candles are 14 centimeters tall, so neither has burned out."},"m1048":{"id":"m1048","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Systems in context with a choice of model","tier":3,"traps":["setup"],"answer":"C","stem":"A school orders p packs of pencils and e packs of erasers. Pencil packs cost $3.50 and eraser packs cost $1.25. The school spends $126 and orders 4 times as many eraser packs as pencil packs, minus 8 packs. Which system represents this situation?","choices":["3.5p+1.25e=126 and p=4e−8","3.5e+1.25p=126 and e=4p−8","3.5p+1.25e=126 and e=4p−8","3.5p+1.25e=126 and e=4(p−8)"],"explain":"Cost: 3.50 per pencil pack times p plus 1.25 per eraser pack times e. \"4 times as many eraser packs as pencil packs, minus 8\" is e=4p−8. A swaps the roles; D subtracts 8 before multiplying. (The system gives p=16 and e=56.)"},"m1049":{"id":"m1049","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Capacity with a ratio between two box sizes","tier":2,"traps":["partial"],"answer":"16","accept":["16"],"stem":"A delivery van can carry at most 1,500 kilograms. On one trip it carries a driver who weighs 80 kilograms, small boxes that weigh 22.5 kilograms each, and large boxes that weigh 40 kilograms each. If it carries exactly twice as many small boxes as large boxes, what is the greatest number of large boxes it can carry?","explain":"With L large boxes and 2L small boxes: 80+22.5(2L)+40L≤1,500, so 85L≤1,420 and L≤16.7. The greatest whole number is 16. Trap: rounding up to 17 overloads the van by 25 kilograms."},"m1050":{"id":"m1050","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"A point with a parameter in a region","tier":2,"traps":["sign"],"answer":"A","stem":"For which of the following values of k does the point (k,2k+1) satisfy both y>3x−4 and y<−x+9?","choices":["2","3","4","5"],"explain":"First: 2k+1>3k−4 gives k<5. Second: 2k+1<−k+9 gives k<38​. Both hold only when k<38​, and of the choices only 2 qualifies. Choices 3 and 4 satisfy the first inequality but not the second."},"m1051":{"id":"m1051","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Constant that sets the region's leftmost point","tier":3,"traps":["concept"],"answer":"8","accept":["8"],"stem":"In the xy-plane, the solution region of the system y≥−x+k and y≤3x−8, where k is a constant, contains a point with x-coordinate 4 but no points with x-coordinate less than 4. What is the value of k?","explain":"A point (x,y) is in the region only if −x+k≤y≤3x−8, which requires −x+k≤3x−8, or x≥4k+8​. So the least x-coordinate in the region is 4k+8​, where the two boundary lines meet. Setting 4k+8​=4 gives k=8. Trap: substituting x=4 into just one of the inequalities."},"m1052":{"id":"m1052","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Two constraints, whole hours","tier":2,"traps":["setup"],"answer":"18","accept":["18"],"stem":"Asha earns $15 per hour tutoring and $11 per hour at a café. This week she wants to earn at least $400 while working no more than 30 hours in total. If she works only whole numbers of hours, what is the least number of hours she can spend tutoring?","explain":"Tutoring as little as possible means working the full 30 hours, with the rest at the café: 15x+11(30−x)≥400. So 4x≥70 and x≥17.5. In whole hours, she must tutor at least 18 hours (18 tutoring and 12 café hours earn $402)."},"m1053":{"id":"m1053","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Compound inequality with exactly six integer solutions","tier":3,"traps":["concept","partial"],"answer":"31","accept":["31"],"stem":"The compound inequality −3<2−5x≤k, where k is an integer, has exactly 6 integer solutions. What is the greatest possible value of k?","explain":"From −3<2−5x: x<1. From 2−5x≤k: x≥52−k​. The integers less than 1 are 0,−1,−2,…, so exactly six solutions means 0 through −5 work and −6 does not. That requires 52−k​≤−5, so k≥27, and 52−k​>−6, so k<32. The greatest integer k is 31. Trap: answering 32, which lets x=−6 in."},"m1054":{"id":"m1054","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Inequality with a parameter","tier":3,"traps":["sign"],"answer":"B","stem":"The inequality ax+6>3x+2a, where a is a constant, is true for every negative value of x. Which of the following must be true?","choices":["a>3","a<3","a=3","a>0"],"explain":"Rearrange: (a−3)x>2(a−3). If a>3, dividing gives x>2, which excludes negative x. If a=3, it reads 0>0, never true. If a<3, dividing by the negative number a−3 flips the sign: x<2, which includes every negative x. So a<3."},"m1055":{"id":"m1055","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Ratio limits in context","tier":2,"traps":["setup"],"answer":"4","accept":["4"],"stem":"A field trip must have at least one chaperone for every 8 students and no more than 4 chaperones for every 25 students. If 96 students go on the trip, how many different numbers of chaperones are possible?","explain":"At least one per 8 students: c≥896​=12. At most 4 per 25 students: c≤254​(96)=15.36. Whole numbers from 12 to 15 work: 12, 13, 14, 15. That is 4 values."},"m1056":{"id":"m1056","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"What must be true","tier":3,"traps":["setup"],"answer":"D","stem":"The points (0,0) and (2,5) are both solutions to the inequality y<ax+b, where a and b are constants. Which of the following must be true?

I. b>0
II. a>0
III. 2a+b>5","choices":["I only","I and II only","I, II, and III","I and III only"],"explain":"(0,0) gives 0<b, so I is true. (2,5) gives 5<2a+b, so III is true. II need not be: with b=10 and a=−1, both points work but a is negative."},"m1057":{"id":"m1057","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Two constraints in context","tier":2,"traps":["setup"],"answer":"A","stem":"A bakery's daily profit, in dollars, is P=2.4c−180, where c is the number of cookies sold. Each cookie uses 0.4 pound of flour, and the bakery has 150 pounds of flour each day. Which inequality gives all possible values of c for a day on which the bakery makes a profit of at least $600?","choices":["325≤c≤375","250≤c≤375","c≥325","c≤375"],"explain":"Profit: 2.4c−180≥600 gives c≥325. Flour: 0.4c≤150 gives c≤375. Together, 325≤c≤375. B forgets the 180 in the profit formula, and C and D each leave out one of the two conditions."},"m1058":{"id":"m1058","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Counting whole-number solutions","tier":3,"traps":["setup"],"answer":"19","accept":["19"],"stem":"A shop sells notebooks for $2 and binders for $3. Ravi has $12 and will buy x notebooks and y binders, where x and y are whole numbers (either may be zero), and spend no more than $12. Including buying nothing at all, how many different combinations (x,y) are possible?","explain":"Count by x. For each x from 0 to 6, the number of binders allowed is 0 through ⌊312−2x​⌋: that gives 5,4,3,3,2,1,1 choices. The total is 19."},"m1059":{"id":"m1059","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Region shown on a graph","tier":2,"traps":["sign"],"figure":"
\"Two
","answer":"B","stem":"The shaded region shown, including its boundary lines, represents the solutions to a system of two linear inequalities. Which system is it?","choices":["y≥−21​x+3 and y≥2x−4","y≤−21​x+3 and y≥2x−4","y≤−21​x+3 and y≤2x−4","y≤−2x+3 and y≥21​x−4"],"explain":"The lines are y=−21​x+3 (through (0,3) and (4,1)) and y=2x−4 (through (0,−4) and (4,4)). The origin is in the shaded region: 0≤3 and 0≥−4, matching B. D has the slopes swapped."},"m1060":{"id":"m1060","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"A constant pinned by one true and one false case","tier":3,"traps":["concept"],"answer":"D","stem":"Consider the statement \"5−2(x−3) is at least the sum of 3x and k,\" where k is a constant. The statement is true when x=2 but false when x=3. Which of the following could be the value of k?","choices":["−5","−4","2","0"],"explain":"The statement is 11−2x≥3x+k. True at x=2: 7≥6+k, so k≤1. False at x=3: 5<9+k, so k>−4. Of the choices, only 0 satisfies −4<k≤1. Trap: k=−4 makes the statement true at x=3, since 5≥5."},"m1061":{"id":"m1061","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Intercept and parallel condition together","tier":2,"traps":["sign"],"answer":"-10","accept":["-10"],"stem":"In the xy-plane, the graph of ax+by=12, where a and b are constants, passes through (3,0) and is parallel to the graph of 2x−5y=7. What is the value of b?","explain":"The point (3,0) gives 3a=12, so a=4. The line 2x−5y=7 has slope 52​, and ax+by=12 has slope −ba​=−b4​. Setting −b4​=52​ gives b=−10. Trap: taking the slope of 2x−5y=7 as −52​."},"m1062":{"id":"m1062","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Perpendicular condition with a parameter in both coefficients","tier":3,"traps":["sign"],"answer":"6/5","accept":["6/5","1.2"],"stem":"In the xy-plane, line k has equation (2m−1)x+(m+3)y=10, where m is a constant. For what value of m is line k perpendicular to the line y=3x+1?","explain":"Line k has slope −m+32m−1​, which must equal −31​. So 3(2m−1)=m+3, which gives 5m=6 and m=56​, or 1.2. Trap: setting the slope equal to 31​ (forgetting the negative) gives m=0."},"m1063":{"id":"m1063","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Comparing two pricing plans","tier":3,"traps":["partial"],"answer":"32","accept":["32"],"stem":"A caterer charges a fixed fee plus a constant price per guest. The total charge is $1,120 for 40 guests and $1,645 for 65 guests. A second caterer charges $30 per guest with no fixed fee. What is the least number of guests for which the second caterer charges more than the first?","explain":"Per guest: 65−401,645−1,120​=21 dollars, so the fixed fee is 1,120−40(21)=280. The second caterer costs more when 30g>280+21g, so 9g>280 and g>31.1. The least whole number of guests is 32. Trap: stopping at 31.1 and rounding down to 31, where the second caterer is still cheaper."},"m1064":{"id":"m1064","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Using a linear relationship inside exponents","tier":3,"traps":["concept"],"answer":"128","accept":["128"],"stem":"If 3x−2y=7, what is the value of 4y8x​?","explain":"Rewrite with base 2: 4y8x​=22y23x​=23x−2y=27=128. There's no need to find x or y, and they can't be found from one equation anyway."},"m1065":{"id":"m1065","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Whole-number purchases with a minimum","tier":2,"traps":["setup"],"answer":"35","accept":["35"],"stem":"Pens cost $1.50 each and notebooks cost $3.75 each. Kenji spends exactly $60.00 and buys at least one of each item. What is the greatest number of pens he could buy?","explain":"1.5p+3.75n=60 becomes 2p+5n=80 after multiplying by 34​. For p to be a whole number, 5n must be even, so n is even; the smallest allowed is n=2, giving 2p=70 and p=35. Trap: n=0 gives 40 pens, but he must buy at least one notebook, and n=1 gives p=37.5."},"m1066":{"id":"m1066","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Sign conditions on a solution","tier":3,"traps":["sign"],"answer":"C","stem":"The system 2x+3y=12 and y=kx, where k is a constant, has a solution (x,y) in which x and y are both positive. Which of the following could be the value of k?","choices":["−1","−21​","21​","0"],"explain":"Substitute: 2x+3kx=12, so x=2+3k12​, which is positive when k>−32​. Then y=kx is positive only if k>0. Of the choices, only 21​ works, giving (x,y)=(724​,712​). B makes x positive but y negative; D makes y=0."},"m2001":{"id":"m2001","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Recovering a hidden denominator","tier":2,"traps":["setup"],"answer":"-13","accept":["-13"],"stem":"For all x>5, x2+ax+bx2−25​=x+2x+5​, where a and b are constants. What is the value of a+b?","explain":"The numerator factors as (x−5)(x+5). For the fraction to equal x+2x+5​, the denominator must be (x−5)(x+2)=x2−3x−10. So a=−3, b=−10, and a+b=−13. Trap: using (x+5)(x+2), which would leave x+2x−5​."},"m2002":{"id":"m2002","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Rational exponents with two variables","tier":2,"traps":["concept"],"answer":"D","stem":"For positive a and b, which expression is equivalent to (4a−2b)1/2(8a6b−3)2/3​?","choices":["b3/22a3​","2b5/2a5​","2a5b3/2","b5/22a5​"],"explain":"Numerator: 82/3=4, a6⋅2/3=a4, b−3⋅2/3=b−2. Denominator: 41/2=2, a−1, b1/2. Divide: 24​a4−(−1)b−2−1/2=2a5b−5/2. A subtracts a exponents as 4−1; C adds the b exponents."},"m2003":{"id":"m2003","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Fractional exponent, then negative exponent","tier":2,"traps":["partial"],"answer":"B","stem":"If x2/3=9 and x>0, what is the value of x−1/2?","choices":["271​","93​​","31​","33​"],"explain":"Raise both sides to the 23​ power: x=93/2=27. Then x−1/2=27​1​=33​1​=93​​. D is x1/2; A is x−1."},"m2004":{"id":"m2004","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Matching a simplified form with two constants","tier":2,"traps":["partial"],"answer":"5","accept":["5"],"stem":"For all x=2 and x=−2, x−23​−x2−412​=x+ba​, where a and b are constants. What is the value of a+b?","explain":"Over the common denominator (x−2)(x+2): (x−2)(x+2)3(x+2)−12​=(x−2)(x+2)3x−6​=(x−2)(x+2)3(x−2)​=x+23​. So a=3, b=2, and a+b=5. Trap: stopping at x2−43x−6​ without canceling the common factor."},"m2005":{"id":"m2005","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Quotient given, coefficient and remainder unknown","tier":2,"traps":["setup"],"answer":"3","accept":["3"],"stem":"When 2x3+kx2−7x+6 is divided by x−2, where k is a constant, the quotient is 2x2+3x−1 and the remainder is r. What is the value of k+r?","explain":"The polynomial equals (x−2)(2x2+3x−1)+r=2x3−x2−7x+2+r. Matching coefficients: k=−1, and 2+r=6, so r=4. Then k+r=3. Trap: a sign slip on the x2 term: 3x2−4x2=−x2."},"m2006":{"id":"m2006","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Factoring with integer constants","tier":3,"traps":["setup"],"answer":"28","accept":["28"],"stem":"For all values of x, 6x2+kx−10=(2x+a)(3x+b), where a, b, and k are integers. What is the greatest possible value of k?","explain":"Expanding gives 6x2+(2b+3a)x+ab, so ab=−10 and k=3a+2b. Try every integer pair with ab=−10: (1,−10)→−17, (−1,10)→17, (2,−5)→−4, (−2,5)→4, (5,−2)→11, (−5,2)→−11, (10,−1)→28, (−10,1)→−28. The greatest is 28. Trap: checking only one factor pair, or putting the larger factor with the 2x."},"m2007":{"id":"m2007","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Matching coefficients","tier":2,"traps":["sign"],"answer":"-1","accept":["-1"],"stem":"For all values of x, (ax+3)(2x−b)=6x2+cx−15, where a, b, and c are constants. What is the value of a+b+c?","explain":"Leading terms: 2a=6, so a=3. Constants: 3(−b)=−15, so b=5. Middle term: −ab+6=−15+6=−9, so c=−9. The sum is 3+5−9=−1."},"m2008":{"id":"m2008","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Rewriting an exponential for a new time unit","tier":2,"traps":["concept"],"answer":"A","stem":"The function P(t)=800(1.21)t models a quantity t years after it was first measured. Which equivalent function shows, as a constant or coefficient, the percent increase every half-year?","choices":["P(t)=800(1.1)2t","P(t)=800(1.105)2t","P(t)=800(1.21)t/2","P(t)=800(1.4641)2t"],"explain":"1.21=1.12, so 1.21t=(1.12)t=1.12t, and 2t counts half-years. The base 1.1 shows a 10% increase per half-year. B halves the 21%, which ignores compounding. C isn't equivalent: it grows much more slowly."},"m2009":{"id":"m2009","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Rewriting as quotient plus remainder","tier":1,"traps":["sign"],"answer":"-2","accept":["-2"],"stem":"For x=−3, the expression x+32x+5​ is equivalent to a+x+3b​, where a and b are constants. What is the value of ab?","explain":"2x+5=2(x+3)−1, so x+32x+5​=2−x+31​. Then a=2, b=−1, and ab=−2."},"m2010":{"id":"m2010","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Simplifying a radical with variables","tier":1,"traps":["concept"],"answer":"C","stem":"For x>0 and y>0, which expression is equivalent to 50x3y5​?","choices":["25xy22xy​","5x2y32​","5xy22xy​","5xy22y​"],"explain":"50=25⋅2, x3=x2⋅x, and y5=y4⋅y. Take the square root of each perfect square: 25x2y4​=5xy2, leaving 2xy​ inside. A forgets to take the square root of 25; D drops the leftover x."},"m2011":{"id":"m2011","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Nested radical and fractional exponents","tier":2,"traps":["concept"],"answer":"1/2","accept":["1/2","0.5",".5"],"stem":"For x>0, the expression (x​x3​)2/5⋅x−1/2 is equivalent to xk. What is the value of k?","explain":"Inside the parentheses: x1/2x3​=x5/2. Raising to the 52​ power gives x1. Multiplying by x−1/2 gives x1/2, so k=21​, or 0.5."},"m2012":{"id":"m2012","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Chained ratios","tier":2,"traps":["setup"],"answer":"3","accept":["3"],"stem":"If yx​=4 and zy​=23​, where y and z are nonzero, what is the value of y+zx+y​?","explain":"Write everything in terms of y: x=4y and z=32​y. Then y+zx+y​=35​y5y​=3."},"m2013":{"id":"m2013","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Choosing a constant to remove a term","tier":2,"traps":["setup"],"answer":"4/3","accept":["4/3","1.333"],"stem":"The expression (x2+kx+4)(x−3), where k is a constant, is rewritten as a polynomial in standard form. If the polynomial has no x term, what is the value of k?","explain":"Only two products make an x term: kx⋅(−3)=−3kx and 4⋅x=4x. So −3k+4=0 and k=34​. Trap: eliminating the x2 term instead (k−3=0) gives 3."},"m2014":{"id":"m2014","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Exponents tied together by a linear condition","tier":3,"traps":["setup"],"answer":"2","accept":["2"],"stem":"If 2x−3y=5, what is the value of 16x−14x+1⋅8y​?","explain":"Write each power with base 2: 4x+1=22x+2, 8y=23y, and 16x−1=24x−4. The exponent of the whole expression is (2x+2)+3y−(4x−4)=−2x+3y+6=−(2x−3y)+6=−5+6=1. So the expression equals 21=2. Trap: rewriting 16x−1 as 24x−1."},"m2015":{"id":"m2015","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Which constant lets a fraction reduce","tier":3,"traps":["concept"],"answer":"B","stem":"The expression 2x2+kx−104x2−25​, where k is a constant, can be rewritten as a quotient of two linear expressions by canceling a common factor. Which of the following could be the value of k?","choices":["−9","1","−3","3"],"explain":"The numerator factors as (2x−5)(2x+5), so the denominator must contain one of these factors. With 2x+5: (2x+5)(x−2)=2x2+x−10, so k=1. (With 2x−5: (2x−5)(x+2)=2x2−x−10, so k=−1, which isn't a choice.) For −9, −3, and 3, the denominator has no rational zeros, so nothing cancels."},"m2016":{"id":"m2016","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Remainder theorem with a constant","tier":1,"traps":["setup"],"answer":"2","accept":["2"],"stem":"When the polynomial p(x)=2x3−5x2+kx−4 is divided by x−3, the remainder is 11. What is the value of k?","explain":"The remainder when dividing by x−3 is p(3): 54−45+3k−4=5+3k. Setting 5+3k=11 gives k=2. Trap: evaluating p(−3)."},"m2017":{"id":"m2017","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Two factors, two unknown coefficients","tier":2,"traps":["sign"],"answer":"-10","accept":["-10"],"stem":"The polynomial p(x)=x3+ax2+bx−6, where a and b are constants, has x+1 and x−2 as factors. What is the value of ab?","explain":"Factors give zeros: p(−1)=0 and p(2)=0. First: −1+a−b−6=0, so a−b=7. Second: 8+4a+2b−6=0, so 2a+b=−1. Adding gives 3a=6, so a=2 and b=−5; ab=−10. (Indeed, p(x)=(x+1)(x−2)(x+3).)"},"m2018":{"id":"m2018","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Two remainders, two unknowns","tier":3,"traps":["sign"],"answer":"2","accept":["2"],"stem":"When the polynomial p(x)=x3+ax2+bx+6 is divided by x−1, the remainder is 4. When p(x) is divided by x+2, the remainder is 16. What is the value of a?","explain":"By the remainder theorem, p(1)=4 and p(−2)=16. First: 1+a+b+6=4, so a+b=−3. Second: −8+4a−2b+6=16, so 2a−b=9. Adding: 3a=6, so a=2 (and b=−5). Trap: evaluating p(2) for the divisor x+2."},"m2019":{"id":"m2019","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Solving a formula for one variable","tier":2,"traps":["setup"],"answer":"C","stem":"Two resistors with resistances R1​ and R2​ connected in parallel have a combined resistance R, where R1​=R1​1​+R2​1​. Which equation gives R2​ in terms of R and R1​?","choices":["R2​=R−R1​","R2​=RR1​R1​−R​","R2​=R1​−RRR1​​","R2​=R−R1​RR1​​"],"explain":"R2​1​=R1​−R1​1​=RR1​R1​−R​. Take reciprocals: R2​=R1​−RRR1​​. B forgets to flip; D flips the subtraction (it would make R2​ negative, since R<R1​)."},"m2020":{"id":"m2020","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Isolating a rate inside a power","tier":3,"traps":["setup"],"answer":"D","stem":"The formula A=P(1+nr​)nt gives the value of an account. Which of the following correctly expresses r in terms of A, P, n, and t?","choices":["r=n(PA​)1/(nt)−1","r=(PA​)1/(nt)−n","r=n1​((PA​)nt−1)","r=n((PA​)1/(nt)−1)"],"explain":"Divide by P: (1+nr​)nt=PA​. Take the nt-th root: 1+nr​=(PA​)1/(nt). Subtract 1, then multiply by n. A multiplies by n before subtracting 1."},"m2021":{"id":"m2021","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Completing the square to match a form","tier":1,"traps":["partial"],"answer":"32","accept":["32"],"stem":"For all values of x, x2−10x+c=(x−a)2+7, where a and c are constants. What is the value of c?","explain":"(x−a)2+7=x2−2ax+a2+7. Matching −2a=−10 gives a=5, so c=25+7=32. Trap: entering a=5 or a2=25."},"m2022":{"id":"m2022","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Squaring an identity twice","tier":3,"traps":["partial"],"answer":"79","accept":["79"],"stem":"If x>0 and (x+x1​)2=11, what is the value of x4+x41​?","explain":"(x+x1​)2=x2+2+x21​, so x2+x21​=9. Square again: x4+2+x41​=81, so x4+x41​=79. Trap: stopping at 81, or at 9."},"m2023":{"id":"m2023","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Difference of squares to a sum of squares","tier":2,"traps":["partial"],"answer":"58","accept":["58"],"stem":"If x2−y2=40 and x+y=10, what is the value of x2+y2?","explain":"x2−y2=(x−y)(x+y), so 10(x−y)=40 and x−y=4. Then x=7 and y=3, so x2+y2=49+9=58. Trap: answering 100, which is (x+y)2."},"m2024":{"id":"m2024","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"A sum of equal powers","tier":2,"traps":["concept"],"answer":"7","accept":["7"],"stem":"If 3x+3x+3x=812, what is the value of x?","explain":"Three copies of 3x make 3⋅3x=3x+1. And 812=(34)2=38. So x+1=8 and x=7. Trap: treating 3x+3x+3x as 33x, which gives x=38​."},"m2025":{"id":"m2025","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Negative exponents in a complex fraction","tier":2,"traps":["concept"],"answer":"A","stem":"For positive x and y with x=y, which expression is equivalent to x−1−y−1x−2−y−2​?","choices":["xyx+y​","x+yxy​","x−1−y−1","x+y"],"explain":"Let u=x−1 and v=y−1: the expression is u−vu2−v2​=u+v=x1​+y1​=xyx+y​. B flips the result; D treats negative exponents as if they cancel."},"m2026":{"id":"m2026","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Two unknowns inside a difference of squares","tier":3,"traps":["setup","partial"],"answer":"4","accept":["4"],"stem":"For all values of x, (2x−a)2−(x+4)2=(x−7)(3x+b), where a and b are positive constants. What is the value of a+b?","explain":"The right side is 0 when x=7, so the left side is too: (14−a)2=112, giving a=3 or a=25. Expand to find b: the left side is 3x2−(4a+8)x+a2−16 and the right side is 3x2+(b−21)x−7b, so b=13−4a. With a=25, b=−87, which isn't positive. So a=3, b=1, and a+b=4."},"m2027":{"id":"m2027","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Rationalizing a denominator","tier":1,"traps":["concept"],"answer":"C","stem":"Which expression is equivalent to 7​−2​5​?","choices":["7​−2​","95(7​+2​)​","7​+2​","5​"],"explain":"Multiply by 7​+2​7​+2​​: the denominator becomes 7−2=5, which cancels the 5 on top. B adds the squares in the denominator (7+2) instead of subtracting."},"m2028":{"id":"m2028","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"A constant that makes division exact","tier":3,"traps":["concept"],"answer":"15","accept":["15"],"stem":"For x=21​, the expression 2x−16x2+kx−5​, where k is a constant, is equivalent to ax+b, where a and b are constants. What is the value of ab?","explain":"For the division to leave no remainder, 2x−1 must be a factor of the numerator, so the numerator is 0 at x=21​: 6(41​)+2k​−5=0, which gives k=7. Then 6x2+7x−5=(2x−1)(3x+5), so a=3, b=5, and ab=15. Trap: dividing with k left unknown and keeping a remainder."},"m2029":{"id":"m2029","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Identity with two unknowns","tier":1,"traps":["sign"],"answer":"3","accept":["3"],"stem":"For all values of x, (x−a)(x+3)=x2+bx−12, where a and b are constants. What is the value of a+b?","explain":"Expand: x2+(3−a)x−3a. Matching constants: −3a=−12, so a=4. Then b=3−4=−1, and a+b=3."},"m2030":{"id":"m2030","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Exponential rewritten with a coefficient","tier":3,"traps":["concept"],"answer":"4","accept":["4"],"stem":"The expression 52x+1−25x is equivalent to k⋅25x for all values of x. What is the value of k?","explain":"52x+1=5⋅52x=5⋅25x. So 5⋅25x−25x=4⋅25x, and k=4. Trap: 52x+1 is not 25x+1; that would give 24."},"m2031":{"id":"m2031","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Discriminant with a parameter","tier":2,"traps":["sign"],"answer":"19","accept":["19"],"stem":"In the equation −2x2+bx−50=0, b is a positive integer. If the equation has no real solutions, what is the greatest possible value of b?","explain":"No real solutions means a negative discriminant: b2−4(−2)(−50)<0, so b2<400 and b<20. The greatest positive integer is 19. Trap: b=20 makes the discriminant zero, which gives one real solution."},"m2032":{"id":"m2032","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Line tangent to a parabola","tier":2,"traps":["sign"],"answer":"-2","accept":["-2"],"stem":"In the xy-plane, the graphs of y=x2−4x+7 and y=2x+k, where k is a constant, intersect at exactly one point. What is the value of k?","explain":"Set the expressions equal: x2−6x+(7−k)=0. Exactly one intersection means a zero discriminant: 36−4(7−k)=0, so 8+4k=0 and k=−2."},"m2033":{"id":"m2033","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Tangent point coordinates","tier":3,"traps":["partial"],"answer":"-3","accept":["-3"],"stem":"In the xy-plane, the graphs of y=3x2+5x−1 and y=−x+c, where c is a constant, intersect at exactly one point (a,b). What is the value of b?","explain":"3x2+6x−1−c=0 has one solution when 36+12(1+c)=0, so c=−4. The double root is x=−2⋅36​=−1, and b=−(−1)−4=−3. Trap: entering c=−4 or a=−1."},"m2034":{"id":"m2034","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Find the constant, then check for an extraneous root","tier":2,"traps":["extraneous"],"answer":"D","stem":"In the equation ax+1​=x−1, a is a constant. If x=5 is a solution, what are all the solutions to the equation?","choices":["0 and 5","0 only","−5 and 5","5 only"],"explain":"Substituting x=5: 5a+1​=4, so 5a+1=16 and a=3. Squaring 3x+1​=x−1 gives 3x+1=x2−2x+1, or x(x−5)=0. Check x=0: the left side is 1​=1 but the right side is −1, so 0 is extraneous. The only solution is 5. A keeps the extraneous root."},"m2035":{"id":"m2035","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"A root that is always extraneous","tier":4,"traps":["extraneous","concept"],"answer":"12","accept":["12"],"stem":"In the equation x−3x​−x+3k​=x2−918​, k is a constant. What is the sum of all values of k for which the equation has no solution?","explain":"Multiply both sides by (x−3)(x+3): x(x+3)−k(x−3)=18, or x2+(3−k)x+3k−18=0. Substituting x=3 gives 9+9−3k+3k−18=0, so 3 is always a root of this quadratic, and it is always excluded from the original equation. Factoring, (x−3)(x−(k−6))=0, so the only possible solution is x=k−6. It fails exactly when it is an excluded value too: k−6=3 or k−6=−3, so k=9 or k=3. The sum is 12."},"m2036":{"id":"m2036","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Absolute value inside a square, with a shift","tier":3,"traps":["partial"],"answer":"21","accept":["21"],"stem":"In the equation (x−h)2=4∣x−h∣, h is a constant. The greatest solution to the equation is 11. What is the sum of all solutions to the equation?","explain":"Let u=∣x−h∣. Since (x−h)2=u2, the equation becomes u2=4u, so u=0 or u=4. That gives x=h, x=h+4, and x=h−4. The greatest is h+4=11, so h=7, and the solutions are 3, 7, and 11, with a sum of 21. Trap: dividing both sides by ∣x−h∣, which loses the solution x=h."},"m2037":{"id":"m2037","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Product of intersection coordinates fixes the line","tier":3,"traps":["setup"],"answer":"1","accept":["1"],"stem":"In the xy-plane, the line y=x+k, where k is a positive constant, intersects the circle x2+y2=25 at two points. The product of the x-coordinates of the two points is −12. What is the value of k?","explain":"Substitute y=x+k: x2+(x+k)2=25, or 2x2+2kx+k2−25=0. For a quadratic ax2+bx+c, the product of the roots is ac​, so 2k2−25​=−12 and k2=1. Since k>0, k=1. (The line y=x+1 meets the circle at (3,4) and (−4,−3).)"},"m2038":{"id":"m2038","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Exactly one solution with a nonzero constant","tier":2,"traps":["partial"],"answer":"4","accept":["4"],"stem":"The equation x2−kx+2k=0, where k is a nonzero constant, has exactly one real solution. What is that solution?","explain":"One solution means a zero discriminant: k2−8k=0, so k=8 (since k=0). The equation is x2−8x+16=0, or (x−4)2=0, so x=4. Trap: entering k=8."},"m2039":{"id":"m2039","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Sum of roots fixes a constant, then the product","tier":3,"traps":["setup","extraneous"],"answer":"-2","accept":["-2"],"stem":"In the equation xa​+x+13​=1, a is a constant. The equation has two solutions, and their sum is 4. What is the product of the two solutions?","explain":"Multiply both sides by x(x+1): a(x+1)+3x=x2+x, or x2−(a+2)x−a=0. The sum of the roots is a+2=4, so a=2, and the product of the roots is −a=−2. Neither root is 0 or −1 (they are 2±6​), so both are valid."},"m2040":{"id":"m2040","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"A sum of solutions that doesn't depend on the constant","tier":4,"traps":["concept"],"answer":"3","accept":["3"],"stem":"In the equation 9x−k⋅3x+27=0, k is a constant greater than 12. The equation has two real solutions. What is the sum of the two solutions?","explain":"Let u=3x, so 9x=u2 and the equation becomes u2−ku+27=0. For k>12 this has two positive roots u1​ and u2​, and their product is 27. Since u1​=3x1​ and u2​=3x2​, 3x1​+x2​=u1​u2​=27=33, so x1​+x2​=3 no matter what k is. Graphing with a particular k shows two roots that move as k changes; the product of the roots explains why their sum stays fixed."},"m2041":{"id":"m2041","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Symmetry of equal heights","tier":3,"traps":["setup"],"answer":"55","accept":["55"],"stem":"The height, in feet, of a ball t seconds after it is thrown is h(t)=−16t2+bt+6, where b is a constant. The ball is at the same height at t=0.5 and at t=3. What is the maximum height, in feet, that the ball reaches?","explain":"Equal heights at two times put the vertex halfway between them, at t=20.5+3​=1.75. The vertex of −16t2+bt+6 is at t=32b​, so b=56. The maximum height is h(1.75)=−16(3.0625)+56(1.75)+6=−49+98+6=55. Trap: stopping at b=56, or answering with the vertex time."},"m2042":{"id":"m2042","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Roots with a fixed difference","tier":2,"traps":["setup"],"answer":"21","accept":["21"],"stem":"The solutions to x2−10x+c=0, where c is a constant, are r and r+4. What is the value of c?","explain":"The sum of the roots is 10: r+(r+4)=10, so r=3 and the roots are 3 and 7. Their product is c=21."},"m2043":{"id":"m2043","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Tangency with a sign condition","tier":2,"traps":["sign"],"answer":"-2","accept":["-2"],"stem":"In the xy-plane, the graphs of y=x2+bx and y=2x−4, where b is a negative constant, intersect at exactly one point. What is the value of b?","explain":"Setting the expressions equal gives x2+(b−2)x+4=0. One intersection means a zero discriminant: (b−2)2=16, so b−2=±4, giving b=6 or b=−2. Since b is negative, b=−2."},"m2044":{"id":"m2044","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Quartic with a negative parameter","tier":3,"traps":["concept"],"answer":"B","stem":"In the equation x4−(k+4)x2+4k=0, k is a negative constant. How many real solutions does the equation have?","choices":["0","2","3","4"],"explain":"The left side factors: (x2−4)(x2−k)=x4−kx2−4x2+4k. So x2=4 or x2=k. Since k<0, x2=k has no real solutions, while x2=4 gives x=2 and x=−2. That's 2 real solutions."},"m2045":{"id":"m2045","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Counting constants that give four solutions","tier":3,"traps":["partial","concept"],"answer":"8","accept":["8"],"stem":"For how many integer values of k does the equation ∣x2−9∣=k have exactly four real solutions?","explain":"The graph of y=∣x2−9∣ is the parabola y=x2−9 with the part below the x-axis flipped up. It touches 0 at x=±3 and has a local peak of 9 at x=0. A horizontal line y=k crosses it 4 times when 0<k<9, 3 times when k=9, and 2 times when k=0 or k>9. The integers 1 through 8 work: 8 values. Algebraically, x2=9+k gives two solutions for any k>0, and x2=9−k gives two more only when 0<k<9."},"m2046":{"id":"m2046","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Greatest constant that allows two solutions","tier":4,"traps":["extraneous","concept"],"answer":"-3","accept":["-3"],"stem":"In the equation x+a​=x−3, a is a constant. What is the greatest value of a for which the equation has exactly two real solutions?","explain":"A solution must satisfy x≥3 (a square root can't equal a negative number) and x+a=(x−3)2, or x2−7x+9−a=0. Its roots are 27±13+4a​​. The larger root is at least 27​, so it is valid whenever the roots are real. The smaller root is valid only if 27−13+4a​​≥3, that is, 13+4a​≤1, which means a≤−3. Two different solutions also need 13+4a>0. So there are two solutions exactly when −413​<a≤−3, and the greatest such a is −3. (At a=−3 the solutions are 3 and 4.)"},"m2047":{"id":"m2047","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Product of roots without the middle coefficient","tier":2,"traps":["setup"],"answer":"-2/3","accept":["-2/3","-.6666","-.6667","-0.666","-0.667"],"stem":"In the equation 3x2+kx−12=0, k is a constant. If one solution of the equation is 6, what is the other solution?","explain":"The product of the roots of ax2+bx+c=0 is ac​=3−12​=−4, whatever k is. So the other root is 6−4​=−32​. (Or: substituting 6 gives k=−16, and 3x2−16x−12=(x−6)(3x+2).)"},"m2048":{"id":"m2048","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Tangent line to a circle","tier":3,"traps":["sign"],"answer":"10","accept":["10"],"stem":"In the xy-plane, the line y=2x+b, where b>0, intersects the circle x2+y2=20 at exactly one point. What is the value of b?","explain":"Substitute: x2+(2x+b)2=20, so 5x2+4bx+(b2−20)=0. One point means a zero discriminant: 16b2−20(b2−20)=0, so −4b2+400=0 and b=10. (Check: the distance from the center to the line is 5​10​=20​, the radius.)"},"m2049":{"id":"m2049","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Two distinct real roots","tier":2,"traps":["sign"],"answer":"B","stem":"In the equation x2+(k−2)x+9=0, k is a constant. Which of the following values of k gives the equation two distinct real solutions?","choices":["−3","9","0","7"],"explain":"Two distinct real solutions need (k−2)2−36>0, so ∣k−2∣>6: k>8 or k<−4. Only 9 qualifies. Check A: (−5)2=25<36."},"m2050":{"id":"m2050","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Rational equation with no solution, two cases","tier":3,"traps":["extraneous"],"answer":"7","accept":["7"],"stem":"In the equation x−2ax​=3+x−28​, a is a constant. What is the sum of all values of a for which the equation has no solution?","explain":"Multiply by x−2: ax=3(x−2)+8=3x+2, so (a−3)x=2. Case 1: a=3 gives 0=2, no solution. Case 2: otherwise x=a−32​, which fails only if it equals the excluded value 2: a−32​=2 gives a=4. The values are 3 and 4, with sum 7. Trap: missing the excluded-value case."},"m2051":{"id":"m2051","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"A known zero fixes the constant","tier":2,"traps":["partial"],"answer":"3","accept":["3"],"stem":"The equation x3−4x2+kx+4=0, where k is a constant, has x=1 as a solution. What is the sum of the other two solutions?","explain":"Substituting x=1: 1−4+k+4=0, so k=−1. Then x3−4x2−x+4=x2(x−4)−(x−4)=(x−4)(x−1)(x+1). The other solutions are 4 and −1, with a sum of 3. Trap: answering 4, the sum of all three solutions."},"m2052":{"id":"m2052","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Exactly three solutions after substitution","tier":4,"traps":["concept","setup"],"answer":"15/2","accept":["15/2","7.5"],"stem":"In the equation (x2−5x)2−2(x2−5x)−k=0, k is a constant. The equation has exactly three distinct real solutions. What is the sum of the three solutions?","explain":"Let u=x2−5x. The least value of x2−5x is −425​ (at x=25​), so x2−5x=u has two solutions if u>−425​, one if u=−425​, and none otherwise. The equation u2−2u−k=0 gives at most two values of u, so three solutions requires one root u=−425​ and the other greater than −425​. The u-roots sum to 2, so the other is 433​. The solutions are x=25​ and the two roots of x2−5x=433​, which sum to 5. The total is 215​, or 7.5."},"m2053":{"id":"m2053","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Line through the vertex, second intersection","tier":2,"traps":["setup"],"answer":"2","accept":["2"],"stem":"In the xy-plane, the line y=−x+c, where c is a constant, passes through the vertex of the parabola y=(x−3)2−4. The line intersects the parabola at one other point. What is the x-coordinate of that point?","explain":"The vertex is (3,−4), so −4=−3+c and c=−1. Set (x−3)2−4=−x−1: x2−5x+6=0, so x=3 (the vertex) or x=2."},"m2054":{"id":"m2054","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Perfect-square system with a sign condition","tier":3,"traps":["extraneous"],"answer":"18","accept":["18"],"stem":"If x2−2xy+y2=49, x−3y=3, and x>0, what is the value of xy?","explain":"The first equation is (x−y)2=49, so x−y=7 or x−y=−7. With x−3y=3: if x−y=7, subtracting gives 2y=4, so y=2 and x=9. If x−y=−7, then y=−5 and x=−12, which violates x>0. So xy=18."},"m2055":{"id":"m2055","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Exponential system","tier":2,"traps":["setup"],"answer":"27/8","accept":["27/8","3.375"],"stem":"If 2a⋅4b=64 and 9b3a​=271​, what is the value of ab?","explain":"Rewrite with base 2 and base 3: 2a+2b=26 and 3a−2b=3−3. So a+2b=6 and a−2b=−3. Adding: 2a=3, so a=23​, and b=49​. Then ab=827​, or 3.375."},"m2056":{"id":"m2056","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Parabola and horizontal line","tier":2,"traps":["setup"],"answer":"11","accept":["11"],"stem":"In the xy-plane, the graph of y=x2−6x+c, where c is a constant, intersects the line y=2 at exactly one point. What is the value of c?","explain":"A horizontal line meets a parabola once only at the vertex. The vertex is at x=3, with y=9−18+c=c−9. Setting c−9=2 gives c=11."},"m2057":{"id":"m2057","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Projectile meets a slope","tier":3,"traps":["setup"],"answer":"30","accept":["30"],"stem":"A ball is launched from a hill. Its path is modeled by y=−0.05x2+2x+10, and the hillside below it by y=0.5x+10, where x and y are in feet. At what horizontal distance x, in feet, does the ball land on the hillside?","explain":"Set the heights equal: −0.05x2+2x+10=0.5x+10, so −0.05x2+1.5x=0, or x(−0.05x+1.5)=0. x=0 is the launch point; the landing point is x=30."},"m2058":{"id":"m2058","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Horizontal lines that cross a folded parabola four times","tier":2,"traps":["concept"],"answer":"A","stem":"For which values of k does the graph of y=k intersect the graph of y=∣x2−4∣ at exactly four points?","choices":["0<k<4","k>4","0≤k≤4","k>0"],"explain":"The graph of y=∣x2−4∣ is the parabola y=x2−4 with the part below the x-axis flipped up. It touches the x-axis at x=±2 and has a local maximum of 4 at x=0. A horizontal line strictly between 0 and 4 crosses it four times. At k=4 it meets the graph three times, at k=0 twice, and above 4 twice. C includes the endpoints, where the count drops."},"m2059":{"id":"m2059","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Sum of squared roots fixes a radical coefficient","tier":3,"traps":["setup"],"answer":"3","accept":["3"],"stem":"The solutions to x2−2a​x−9=0, where a is a positive constant, are p and q. If p2+q2=30, what is the value of a?","explain":"The sum of the roots is p+q=2a​ and the product is pq=−9. Then p2+q2=(p+q)2−2pq=4a+18. Setting 4a+18=30 gives a=3. Trap: using (p+q)2+2pq, which gives a=12."},"m2060":{"id":"m2060","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Biquadratic with a ratio between zeros","tier":4,"traps":["setup"],"answer":"20","accept":["20"],"stem":"The function f(x)=x4−kx2+36, where k is a constant, has four distinct real zeros. The greatest zero is 3 times the least positive zero. What is the value of k?","explain":"The zeros come in pairs ±r and ±s with 0<r<s, where r2 and s2 are the roots of u2−ku+36=0. So r2s2=36 and r2+s2=k. With s=3r: r2(9r2)=36, so r2=2 and s2=18, and k=2+18=20. Trap: using the least zero, −s, instead of the least positive zero."},"m2061":{"id":"m2061","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Vertex form from symmetry and a minimum value","tier":2,"traps":["setup"],"answer":"19","accept":["19"],"stem":"The function f(x)=a(x−h)2+k, where a, h, and k are constants, has a minimum value of −8, and f(1)=f(5)=4. What is the value of f(0)?","explain":"Since f(1)=f(5), the axis of symmetry is halfway between, at x=3, so h=3; the minimum value gives k=−8. Then f(5)=a(2)2−8=4, so a=3, and f(0)=3(−3)2−8=19."},"m2062":{"id":"m2062","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"The form that shows the zeros","tier":1,"traps":["concept"],"answer":"B","stem":"The function f is defined by f(x)=2x2−12x+10. Which of the following equivalent forms displays the x-intercepts of the graph of y=f(x) as constants or coefficients?","choices":["f(x)=2(x−3)2−8","f(x)=2(x−1)(x−5)","f(x)=2x(x−6)+10","f(x)=2(x2−6x+5)"],"explain":"Factored form a(x−r)(x−s) shows the zeros r and s directly: 1 and 5. A is vertex form, which shows the vertex (3,−8). C and D are equivalent but show neither the zeros nor the vertex."},"m2063":{"id":"m2063","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Interpreting an exponent with a divisor","tier":2,"traps":["concept"],"answer":"A","stem":"The function V(t)=1,200(0.85)t/3 gives the value, in dollars, of a piece of equipment t years after it was purchased. Which statement is true?","choices":["The value decreases by 15% every 3 years.","The value decreases by 15% every year.","The value decreases by 5% every year.","The value decreases by 45% every 3 years."],"explain":"The exponent 3t​ increases by 1 every 3 years, so every 3 years the value is multiplied by 0.85, a 15% decrease. C divides 15% by 3, but compounding doesn't work that way (the yearly factor is 0.851/3≈0.947, a decrease of about 5.3%)."},"m2064":{"id":"m2064","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Exponential model with an offset start","tier":3,"traps":["setup"],"answer":"289","accept":["289"],"stem":"A bacteria culture had 500 cells at 9:00 a.m. and grows by 20% each hour. The number of cells is modeled by N(t)=a(1.2)t, where t is the number of hours after 6:00 a.m. To the nearest whole number, what is the value of a?","explain":"9:00 a.m. is t=3, so a(1.2)3=500 and a=1.728500​≈289.35, which rounds to 289. Trap: a=500 assumes the clock starts at 9:00 a.m."},"m2065":{"id":"m2065","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Horizontal stretch moves the vertex","tier":2,"traps":["concept"],"answer":"3/2","accept":["3/2","1.5"],"stem":"The function f is defined by f(x)=(x−3)2+k, where k is a constant, and g(x)=f(2x). For what value of x does g reach its minimum?","explain":"g(x)=(2x−3)2+k, which is smallest when 2x−3=0, at x=23​. Replacing x with 2x compresses the graph horizontally by a factor of 2, so the vertex moves from x=3 to x=1.5. Trap: x=6 stretches instead of compresses."},"m2066":{"id":"m2066","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Symmetry gives the axis","tier":2,"traps":["setup"],"answer":"11","accept":["11"],"stem":"The function f(x)=x2+bx+c, where b and c are constants, satisfies f(1)=f(7) and f(4)=−5. What is the value of f(0)?","explain":"Equal outputs at x=1 and x=7 put the axis of symmetry halfway, at x=4, so −2b​=4 and b=−8. Then f(4)=16−32+c=−5, so c=11=f(0)."},"m2067":{"id":"m2067","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Matching a cubic to its graph","tier":1,"traps":["sign"],"figure":"
\"The
","answer":"D","stem":"The graph of the polynomial function y=f(x) is shown. Which of the following could define f?","choices":["f(x)=(x−3)(x+1)(x+4)","f(x)=−(x+3)(x−1)(x−4)","f(x)=(x+3)(x−1)2(x−4)","f(x)=(x+3)(x−1)(x−4)"],"explain":"The graph crosses the x-axis at −3, 1, and 4, so the factors are (x+3), (x−1), and (x−4). It rises to the right, so the leading coefficient is positive (B would fall to the right). C has a squared factor, which would make the graph touch, not cross, at x=1. A has the signs of the zeros reversed."},"m2068":{"id":"m2068","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Touching versus crossing","tier":2,"traps":["concept"],"answer":"C","stem":"The function f is defined by f(x)=(x−2)(x+5)2. Which statement about the graph of y=f(x) is true?","choices":["It crosses the x-axis at x=−2 and touches it without crossing at x=5.","It touches the x-axis without crossing at x=2 and crosses it at x=−5.","It crosses the x-axis at x=2 and touches it without crossing at x=−5.","It crosses the x-axis at three different points."],"explain":"The zeros are 2 and −5. The factor (x+5) is squared, so f doesn't change sign there: the graph touches the axis at x=−5. The single factor (x−2) makes the graph cross at x=2."},"m2069":{"id":"m2069","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Exponential from a ratio and a difference","tier":3,"traps":["setup"],"answer":"2","accept":["2"],"stem":"For the exponential function f(x)=abx, where a and b are positive constants, f(x+3)=27f(x) for all values of x, and f(2)−f(1)=12. What is the value of f(0)?","explain":"Since f(x+3)=abx+3=b3⋅abx, b3=27 and b=3. Then f(2)−f(1)=9a−3a=6a=12, so a=2, and f(0)=a=2."},"m2070":{"id":"m2070","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Asymptote from a value and a difference","tier":3,"traps":["concept"],"answer":"2","accept":["2"],"stem":"The function g is defined by g(x)=a⋅2x+k, where a and k are constants. If g(0)=5 and g(2)−g(1)=6, what value does g(x) approach as x decreases without bound?","explain":"The difference cancels k: g(2)−g(1)=4a−2a=2a=6, so a=3. Then g(0)=3+k=5, so k=2. As x decreases without bound, 3⋅2x approaches 0, so g(x) approaches k=2. Trap: answering 5, the y-intercept."},"m2071":{"id":"m2071","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Launch speed from the maximum height","tier":3,"traps":["setup"],"answer":"1","accept":["1"],"stem":"An object is launched upward, and its height, in feet, t seconds after launch is h(t)=−16t2+vt+5, where v is a positive constant. The object's maximum height is 69 feet. How many seconds after reaching its maximum height is the object 53 feet above the ground?","explain":"The vertex is at t=32v​, and the maximum height is 5+64v2​. Setting this equal to 69 gives v2=4,096, so v=64 and the maximum occurs at t=2. Solving −16t2+64t+5=53 gives t2−4t+3=0, so t=1 or t=3. On the way down, the object is at 53 feet 1 second after its maximum. Trap: answering 3, the time since launch."},"m2072":{"id":"m2072","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Vertex form from symmetry and a relative point","tier":3,"traps":["setup"],"answer":"17","accept":["17"],"stem":"The graph of y=a(x−h)2+k, where a, h, and k are constants, has the line x=3 as its axis of symmetry. The graph passes through the points (1,5) and (h+2,k−12). What is the maximum value of y?","explain":"The axis of symmetry gives h=3. The point (h+2,k−12) says that 2 units from the vertex, y is 12 less than k: a(2)2=−12, so a=−3. The point (1,5) is also 2 units from the vertex, so 5=k−12 and k=17. Since a<0, the maximum value of y is k=17."},"m2073":{"id":"m2073","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Removable gap in a rational function","tier":2,"traps":["extraneous"],"answer":"5","accept":["5"],"stem":"The function f is defined by f(x)=x−kx2−9​, where k is a constant, and f is undefined only when x=−3. For what value of x does f(x)=2?","explain":"Undefined only at x=−3 means k=−3. Then f(x)=x+3(x−3)(x+3)​=x−3 for x=−3. Setting x−3=2 gives x=5, which is allowed."},"m2074":{"id":"m2074","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Vertex from a factored form with a constant","tier":1,"traps":["partial"],"answer":"-4","accept":["-4"],"stem":"The function p(x)=(x−a)(x−6), where a is a constant, has a graph whose vertex has an x-coordinate of 4. What is the y-coordinate of the vertex?","explain":"The vertex is halfway between the zeros a and 6: 2a+6​=4, so a=2. Then p(4)=(2)(−2)=−4."},"m2075":{"id":"m2075","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Converting a yearly rate to a monthly rate","tier":3,"traps":["percent"],"answer":"B","stem":"The value of a car is modeled by V(t)=25,000(0.84)t, where t is the number of years since it was purchased. Which of the following is closest to the percent decrease in the car's value each month?","choices":["1.3%","1.4%","1.5%","16%"],"explain":"The monthly factor m satisfies m12=0.84, so m=0.841/12≈0.9856, a decrease of about 1.44% per month. A divides 16% by 12 (about 1.33%), which ignores compounding: twelve monthly decreases of 1.33% would leave about 85.1% of the value, not 84%. D is the yearly decrease."},"m2076":{"id":"m2076","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Composition with a domain restriction","tier":2,"traps":["setup"],"answer":"2","accept":["2"],"stem":"The functions f and g are defined by f(x)=2x2−3 and g(x)=x+7​. If f(g(k))=15, what is the value of k?","explain":"f(g(k))=2(k+7​)2−3=2(k+7)−3=2k+11. Setting 2k+11=15 gives k=2, which is in the domain of g (k≥−7)."},"m2077":{"id":"m2077","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Quadratic table with an unknown constant","tier":3,"traps":["setup"],"figure":"
x0123
f(x)cc−5c−6c−3
","answer":"159/2","accept":["159/2","79.5"],"stem":"The table shows four values of the quadratic function f, where c is a constant. If f(5)=3f(0), what is the value of f(8)?","explain":"Write f(x)=ax2+bx+c; the table gives f(0)=c. From f(1)=c−5: a+b=−5. From f(2)=c−6: 4a+2b=−6, so 2a+b=−3. Then a=2 and b=−7 (check: f(3)=18−21+c=c−3). Now f(5)=50−35+c=15+c, and 15+c=3c gives c=7.5. So f(8)=128−56+7.5=79.5."},"m2078":{"id":"m2078","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Exponential from a product and a quotient","tier":3,"traps":["setup"],"answer":"2/3","accept":["2/3",".6666",".6667","0.666","0.667"],"stem":"For the exponential function f(x)=abx, where a and b are positive constants, f(1)⋅f(3)=324 and f(1)f(3)​=9. What is the value of f(−1)?","explain":"The quotient gives b2=9, so b=3. The product gives (ab)(ab3)=a2b4=324, so 81a2=324, a2=4, and a=2. Then f(−1)=2⋅3−1=32​. Enter 2/3, .6666, or .6667."},"m2079":{"id":"m2079","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Signs of coefficients from a graph's features","tier":3,"traps":["concept"],"answer":"B","stem":"The graph of y=ax2+bx+c, where a, b, and c are constants, is a parabola that opens downward and has its vertex in Quadrant IV. Which of the following must be true?","choices":["a<0, b<0, and c<0","a<0, b>0, and c<0","a<0, b>0, and c>0","a>0, b>0, and c<0"],"explain":"Opening downward means a<0. The vertex's x-coordinate −2ab​ is positive (Quadrant IV), and with a<0 that forces b>0. The vertex is the highest point and its y-value is negative, so every point is below the x-axis, including the y-intercept: c<0."},"m2080":{"id":"m2080","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Factors from a table of values","tier":1,"traps":["sign"],"figure":"
xp(x)
−20
15
30
","answer":"A","stem":"For the polynomial function p, the table shows selected values. Which of the following must be a factor of p(x)?","choices":["x2−x−6","x2+x−6","x−1","x2−5x+6"],"explain":"p(−2)=0 and p(3)=0 mean (x+2) and (x−3) are factors, so their product x2−x−6 is a factor. B corresponds to zeros at 2 and −3 (signs flipped). C is wrong because p(1)=5, not 0."},"m2081":{"id":"m2081","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Number of intersections with a horizontal line","tier":2,"traps":["setup"],"answer":"D","stem":"The function f is defined by f(x)=∣x−3∣−4. For which values of c does the graph of y=c intersect the graph of y=f(x) at exactly two points?","choices":["c≥−4","c>3","c<−4","c>−4"],"explain":"The graph is a V with its vertex, the lowest point, at (3,−4). A horizontal line above the vertex meets both arms: two points. At c=−4 it touches only the vertex (one point), and below it misses entirely. So c>−4."},"m2082":{"id":"m2082","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Shifted input in a composition","tier":2,"traps":["setup"],"answer":"5","accept":["5"],"stem":"The function h is defined by h(x)=x2+4x. For what positive value of a is h(a−2)=21?","explain":"h(a−2)=(a−2)2+4(a−2)=a2−4a+4+4a−8=a2−4. Setting a2−4=21 gives a=5 (positive)."},"m2083":{"id":"m2083","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Doubling time from a model","tier":2,"traps":["setup"],"answer":"60","accept":["60"],"stem":"The number of cells in a culture is modeled by N(t)=1,500(2)t/20, where t is time in minutes. How many minutes does it take for the number of cells to grow from 3,000 to 24,000?","explain":"The exponent 20t​ means the count doubles every 20 minutes. From 3,000 to 24,000 is a factor of 8, or three doublings: 3×20=60 minutes. Trap: measuring from t=0 (when N=1,500) gives 80."},"m2084":{"id":"m2084","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Horizontal shift of zeros","tier":2,"traps":["sign"],"answer":"-3","accept":["-3"],"stem":"The function f is defined by f(x)=(x−4)(x+2). The function g is defined by g(x)=f(x+k), where k is a constant, and the zeros of g are 1 and 7. What is the value of k?","explain":"g(x)=0 when x+k=4 or x+k=−2, so the zeros of g are 4−k and −2−k. Matching {1,7}: 4−k=7 gives k=−3, and then −2−(−3)=1. The graph shifted right 3, which means k=−3. Trap: k=3, from thinking f(x+3) shifts right."},"m2085":{"id":"m2085","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Reading a + b + c from a graph","tier":3,"traps":["concept"],"figure":"
\"A
","answer":"0","accept":["0"],"stem":"The graph of y=ax2+bx+c, where a, b, and c are constants, is shown. What is the value of a+b+c?","explain":"a+b+c is the value of y when x=1, since a(1)2+b(1)+c=a+b+c. The graph passes through (1,0), so a+b+c=0. (Expanding −(x+1)2+4 gives a=−1, b=−2, c=3.)"},"m2086":{"id":"m2086","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Parameter that sets the minimum","tier":2,"traps":["partial"],"answer":"3","accept":["3"],"stem":"The function f is defined by f(x)=x2−2kx+k2+4k−12, where k is a constant. If the minimum value of f is 0, what is the value of k?","explain":"x2−2kx+k2=(x−k)2, so f(x)=(x−k)2+4k−12. The minimum is 4k−12, at x=k. Setting 4k−12=0 gives k=3."},"m2087":{"id":"m2087","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Vertex fixes two coefficients","tier":2,"traps":["sign"],"answer":"6","accept":["6"],"stem":"The graph of f(x)=ax2+4x+c, where a and c are constants, has its vertex at (−2,1). What is the value of a+c?","explain":"The vertex's x-coordinate is −2a4​=−2, so a=1. Then f(−2)=4−8+c=1, so c=5, and a+c=6."},"m2088":{"id":"m2088","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Maximum revenue with an unknown demand constant","tier":3,"traps":["setup"],"answer":"12960","accept":["12960"],"stem":"A theater estimates that if it charges p dollars per ticket, it will sell N−40p tickets, where N is a constant. The theater's ticket revenue is greatest when the price is $18. What is the greatest possible ticket revenue, in dollars?","explain":"Revenue is R(p)=p(N−40p), a parabola opening downward with zeros at p=0 and p=40N​. Its maximum is halfway between the zeros, at p=80N​, so 80N​=18 and N=1,440. At $18 the theater sells 1,440−720=720 tickets, for revenue of 18×720=12,960 dollars."},"m2089":{"id":"m2089","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Exponential model rewritten in months","tier":2,"traps":["units"],"figure":"
t024
P5,0007,20010,368
","answer":"C","stem":"The table shows the population P of a town t years after 2020. The population grows by the same percent each year. Which function models the population m months after 2020?","choices":["P=5,000(1.2)12m","P=5,000(1.44)m/12","P=5,000(1.2)m/12","P=5,000(1.0167)m"],"explain":"Every 2 years the population is multiplied by 5,0007,200​=1.44, so each year it is multiplied by 1.44​=1.2, and P=5,000(1.2)t. Since t=12m​, P=5,000(1.2)m/12. B treats the 2-year factor as yearly, A multiplies by 12 instead of dividing, and D spreads the 20% evenly over the months; the true monthly factor is 1.21/12≈1.0153."},"m2090":{"id":"m2090","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Shifting a parabola to a new vertex","tier":2,"traps":["sign"],"answer":"1","accept":["1"],"stem":"The function f is defined by f(x)=x2−4x+7. The function g is defined by g(x)=f(x−k)+m, where k and m are constants. The vertex of the graph of y=g(x) is (5,1). What is the value of k+m?","explain":"f(x)=(x−2)2+3, so its vertex is (2,3). Replacing x with x−k moves the graph k units right, and adding m moves it m units up, so the vertex of g is (2+k,3+m). Then k=3 and m=−2, so k+m=1. Trap: thinking f(x−k) moves the graph left, which gives k=−3."},"m3001":{"id":"m3001","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Converting a compound unit","tier":2,"traps":["units"],"answer":"14.5","accept":["14.5"],"stem":"A car's fuel efficiency is 34 miles per gallon. If 1 mile is approximately 1.609 kilometers and 1 gallon is approximately 3.785 liters, what is the car's fuel efficiency in kilometers per liter, rounded to the nearest tenth?","explain":"Convert the distance up and the volume down: 1 gal34 mi​⋅1 mi1.609 km​⋅3.785 L1 gal​=3.78554.706​≈14.45, which rounds to 14.5. Trap: multiplying by 3.785 gives about 207."},"m3002":{"id":"m3002","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Scale applied to area","tier":2,"traps":["units"],"answer":"36","accept":["36"],"stem":"On a map, 1 centimeter represents 2.5 kilometers. A rectangular park is shown on the map as 3.2 centimeters by 1.8 centimeters. What is the actual area of the park, in square kilometers?","explain":"Scale each side first: 3.2×2.5=8 km and 1.8×2.5=4.5 km. The area is 8×4.5=36 km². Trap: multiplying the map area (5.76 cm²) by 2.5 once gives 14.4; areas scale by 2.52."},"m3003":{"id":"m3003","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Combined rate with an unknown difference","tier":3,"traps":["setup"],"answer":"6","accept":["6"],"stem":"Working together, pumps A and B fill a tank in 4 hours. Working alone, pump A fills the tank in 6 fewer hours than pump B does alone. How many hours does pump A take to fill the tank alone?","explain":"If A takes a hours, B takes a+6. Rates add: a1​+a+61​=41​. Multiply by 4a(a+6): 4(2a+6)=a2+6a, so a2−2a−24=0 and a=6 (rejecting −4). B takes 12 hours. Trap: averaging times or adding them."},"m3004":{"id":"m3004","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Worker-days","tier":2,"traps":["setup"],"answer":"5","accept":["5"],"stem":"Six painters, working at the same rate, can paint 4 identical houses in 10 days. At that rate, how many days would it take 9 painters to paint 3 of these houses?","explain":"Painting 4 houses takes 6×10=60 painter-days, or 15 painter-days per house. Three houses need 45 painter-days, and 9 painters supply 9 painter-days per day: 945​=5 days."},"m3005":{"id":"m3005","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Ratio after changes to both groups","tier":2,"traps":["partial"],"answer":"25","accept":["25"],"stem":"The ratio of boys to girls in a club is 3:5. After 6 boys join the club and 4 girls leave it, the numbers of boys and girls are equal. How many girls were in the club originally?","explain":"Start with 3k boys and 5k girls. Then 3k+6=5k−4, so k=5: originally 15 boys and 25 girls. Trap: answering 21, the number of girls after 4 left."},"m3006":{"id":"m3006","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Square units","tier":2,"traps":["units"],"answer":"969","accept":["969"],"stem":"Carpet costs $4.50 per square foot. To the nearest dollar, how much would it cost to carpet a rectangular room that measures 4 meters by 5 meters? (1 foot = 0.3048 meter)","explain":"The room is 20 m². One square foot is 0.30482≈0.0929 m², so the room is 0.3048220​≈215.28 ft². The cost is 215.28×4.50≈$968.75, or $969. Trap: dividing by 0.3048 only once gives about $295."},"m3007":{"id":"m3007","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Average speed for a round trip","tier":2,"traps":["setup"],"answer":"24","accept":["24"],"stem":"A cyclist rides 30 kilometers to a lake at an average speed of 20 kilometers per hour and returns along the same route at an average speed of 30 kilometers per hour. What is the cyclist's average speed for the whole round trip, in kilometers per hour?","explain":"Average speed is total distance over total time: 1.5+160​=24. Trap: averaging the two speeds gives 25, but the cyclist spends more time at the slower speed."},"m3008":{"id":"m3008","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Density across two shapes","tier":2,"traps":["units"],"answer":"261","accept":["261"],"stem":"A solid metal cube with an edge length of 3 centimeters has a mass of 210.6 grams. To the nearest gram, what is the mass of a solid sphere of the same metal with a radius of 2 centimeters?","explain":"Density: 27210.6​=7.8 g/cm³. The sphere's volume is 34​π(2)3≈33.51 cm³, so its mass is 7.8×33.51≈261.4 grams, or 261."},"m3009":{"id":"m3009","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Equal percent increase and decrease","tier":2,"traps":["percent"],"answer":"30","accept":["30"],"stem":"A price is increased by p%, and then the new price is decreased by p%. The final price is 9% less than the original price. What is the value of p?","explain":"The multipliers are (1+100p​)(1−100p​)=1−10,000p2​. Setting this equal to 0.91 gives 10,000p2​=0.09, so p=30. Trap: thinking equal percent changes cancel."},"m3010":{"id":"m3010","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Percent of an overlapping group","tier":2,"traps":["percent"],"answer":"100/3","accept":["100/3","33.33"],"stem":"At a school, 40% of the students are in the band, and 25% of the band members are also in the choir. The choir has 120 members, which is 30% of the students at the school. What percent of the choir members are also in the band?","explain":"The school has 0.30120​=400 students. The band has 0.40×400=160 members, and 0.25×160=40 of them are in the choir. So 12040​≈33.33% of the choir is in the band. Trap: answering 25%, which is a percent of the band, not of the choir."},"m3011":{"id":"m3011","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Undoing a discount and a tax","tier":1,"traps":["percent"],"answer":"100","accept":["100"],"stem":"A jacket is discounted 15%, and then 8% sales tax is added to the discounted price. The final cost is $91.80. What was the original price of the jacket, in dollars?","explain":"x(0.85)(1.08)=91.80, so 0.918x=91.80 and x=100. Trap: adding back 15% and subtracting 8% from $91.80 doesn't undo the multipliers."},"m3012":{"id":"m3012","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Percent change of a percentage","tier":2,"traps":["percent"],"answer":"25","accept":["25"],"stem":"A town's unemployment rate rose from 4% to 5%. By what percent did the unemployment rate increase?","explain":"Percent change compares the change to the starting value: 45−4​=0.25, a 25% increase. Trap: 1 is the change in percentage points, not the percent increase."},"m3013":{"id":"m3013","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Reversing the base","tier":2,"traps":["percent"],"answer":"50/3","accept":["50/3","16.66","16.67"],"stem":"The population of City A is 20% greater than the population of City B. The population of City B is what percent less than the population of City A?","explain":"If B is 100, then A is 120. B is 20 less than A, and relative to A that's 12020​≈16.67%. Trap: 20% uses B as the base, but the question compares to A."},"m3014":{"id":"m3014","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Concentration by evaporation","tier":2,"traps":["setup"],"answer":"25/2","accept":["25/2","12.5"],"stem":"A 50-liter salt solution is 30% salt. How many liters of water must evaporate for the solution to become 40% salt?","explain":"The salt stays at 0.30×50=15 liters. For 15 liters to be 40% of the solution, the solution must be 0.4015​=37.5 liters. So 50−37.5=12.5 liters must evaporate. Trap: thinking a 10-point change requires removing 10% of the volume."},"m3015":{"id":"m3015","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"\"Percent of\" versus \"percent less than\"","tier":2,"traps":["percent"],"answer":"90","accept":["90"],"stem":"The value of p is 150% of q, and q is 40% less than r. The value of p is what percent of r?","explain":"\"40% less than r\" means q=0.60r, not 0.40r. Then p=1.5(0.60r)=0.90r, so p is 90% of r. Trap: using q=0.40r gives 60%."},"m3016":{"id":"m3016","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Unknown second change from a net change","tier":2,"traps":["percent"],"answer":"28","accept":["28"],"stem":"A club's membership increased by 25% from 2022 to 2023 and then decreased by x% from 2023 to 2024. The membership in 2024 was 10% less than in 2022. What is the value of x?","explain":"1.25(1−100x​)=0.90, so 1−100x​=0.72 and x=28. Trap: 25+10=35 treats the percent changes as if they add."},"m3017":{"id":"m3017","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Removing two values with a relationship","tier":2,"traps":["partial"],"answer":"165/2","accept":["165/2","82.5"],"stem":"The mean of 12 numbers is 45. Two of the numbers are removed, and the mean of the remaining 10 numbers is 43. One of the removed numbers is 3 times the other. What is the greater of the two removed numbers?","explain":"The original total is 12×45=540; the remaining total is 10×43=430. The removed numbers sum to 110. If they are n and 3n, then 4n=110, so n=27.5 and the greater is 82.5."},"m3018":{"id":"m3018","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Linear transformation of data","tier":2,"traps":["concept"],"answer":"A","stem":"A data set has mean m and standard deviation s. Each value in the data set is multiplied by 3, and then 5 is added to each result. What are the mean and standard deviation of the new data set?","choices":["Mean 3m+5, standard deviation 3s","Mean 3m+5, standard deviation 3s+5","Mean 3m, standard deviation 3s","Mean 3m+5, standard deviation s"],"explain":"Multiplying every value by 3 multiplies both the mean and the spread by 3. Adding 5 shifts every value, which moves the mean up 5 but doesn't change how spread out the values are. So the mean is 3m+5 and the standard deviation is 3s."},"m3019":{"id":"m3019","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Removing an outlier","tier":2,"traps":["concept"],"figure":"
\"A
","answer":"B","stem":"The dot plot shows the number of books each of 16 students read over the summer. If the student who read 14 books is removed from the data, which of the following is true?","choices":["The mean decreases, and the median decreases.","The mean decreases, and the median stays the same.","The mean stays the same, and the median decreases.","Both the mean and the median stay the same."],"explain":"With 16 values, the median is the average of the 8th and 9th values, both 4. With 15 values, the median is the 8th value, still 4. The mean drops from 1671​≈4.44 to 1557​=3.8 because the largest value is gone."},"m3020":{"id":"m3020","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Weighted combined mean","tier":2,"traps":["setup"],"answer":"30","accept":["30"],"stem":"Class A has 20 students with a mean test score of 78. Class B has a mean test score of 84. The mean score of all the students in both classes combined is 81.6. How many students are in class B?","explain":"20+b20(78)+84b​=81.6, so 1,560+84b=1,632+81.6b, which gives 2.4b=72 and b=30. (The combined mean is closer to 84, so class B must be larger.)"},"m3021":{"id":"m3021","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Median from a frequency table after adding data","tier":2,"traps":["setup"],"figure":"
Rating12345
Frequency47964
","answer":"1/2","accept":["1/2","0.5",".5"],"stem":"The frequency table summarizes a data set of 30 survey ratings. If 10 more ratings of 5 are added to the data set, by how much does the median increase?","explain":"Originally, the 15th and 16th values are both 3 (cumulative counts 4, 11, 20), so the median is 3. With 40 values, the median averages the 20th and 21st values: the 20th is 3 and the 21st is 4, so the median is 3.5. It increases by 0.5."},"m3022":{"id":"m3022","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Comparing spread from frequencies","tier":2,"traps":["concept"],"figure":"
Frequencies
Value1020304050
Data set X61116
Data set Y13731
","answer":"D","stem":"The tables show the frequency of each value in data sets X and Y. Which statement correctly compares the standard deviations of the two data sets?","choices":["The standard deviation of Y is greater.","The standard deviations are equal, because the data sets have the same range.","The standard deviations are equal, because the data sets have the same mean.","The standard deviation of X is greater."],"explain":"Both data sets have mean 30 and range 40, but X has most of its values at the extremes (10 and 50), far from the mean, while Y clusters at 30. Values farther from the mean make the standard deviation larger, so X's is greater. Same mean or range doesn't imply the same spread."},"m3023":{"id":"m3023","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Mean, median, and unique mode together","tier":3,"traps":["setup"],"answer":"4","accept":["4"],"stem":"A list of five positive integers has a mean of 10, a median of 12, and a unique mode of 14. What is the greatest possible value of the smallest integer in the list?","explain":"The sum is 50. In order: a≤b≤12≤d≤e. The unique mode 14 must appear at least twice, and it can only be d and e, so a+b=50−12−28=10. If a=b=5, then 5 would also appear twice, and 14 would no longer be the unique mode. So a<b, and the greatest a is 4 (with b=6)."},"m3024":{"id":"m3024","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Adding a value equal to the mean","tier":3,"traps":["concept"],"answer":"C","stem":"A data set of 25 values has a mean of 18 and a standard deviation of 4. A new value of 18 is added to the data set. Which of the following describes the new data set?","choices":["The mean is 18, and the standard deviation is 4.","The mean is 18, and the standard deviation is greater than 4.","The mean is 18, and the standard deviation is less than 4.","The mean is greater than 18, and the standard deviation is less than 4."],"explain":"Adding a value equal to the mean keeps the mean at 18. That value is 0 away from the mean, so it adds nothing to the total squared distance while increasing the count. The typical distance from the mean shrinks a little, so the standard deviation decreases."},"m3025":{"id":"m3025","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Two-variable data: models and scatterplots","sub":"Residual from a line of best fit","tier":1,"traps":["sign"],"figure":"
\"A
","answer":"4","accept":["4"],"stem":"The scatterplot shows the relationship between hours studied, x, and test score, y, for 10 students, along with the line of best fit y=1.8x+12. For the student who studied 20 hours and scored 52, how much greater is the actual score than the score predicted by the line of best fit?","explain":"Predicted: 1.8(20)+12=48. Actual minus predicted: 52−48=4. Trap: subtracting in the other order gives −4."},"m3026":{"id":"m3026","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Two-variable data: models and scatterplots","sub":"Interpreting a slope with units","tier":1,"traps":["units"],"answer":"B","stem":"A researcher models the relationship between the elevation E, in meters, of a weather station and the average July temperature T, in degrees Celsius, with T=31.4−0.0062E. Which statement is the best interpretation of 0.0062 in this context?","choices":["For each increase of 1°C, the predicted elevation decreases by 0.0062 meter.","For each increase of 1 meter in elevation, the predicted average July temperature decreases by 0.0062°C.","At an elevation of 0 meters, the predicted average July temperature is 0.0062°C.","For each increase of 100 meters in elevation, the predicted temperature decreases by 0.0062°C."],"explain":"The slope multiplies E, so it is the change in T per 1-meter change in E; the minus sign means a decrease. D's rate is off by a factor of 100 (that would be 0.62°C per 100 meters). A reverses the variables, and C describes the intercept, 31.4."},"m3027":{"id":"m3027","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Two-variable data: models and scatterplots","sub":"Exponential model fitted to data","tier":1,"traps":["percent"],"answer":"A","stem":"A biologist models the number of fish in a pond x years after 2020 with y=300(0.92)x. According to the model, which statement is true?","choices":["The number of fish decreases by 8% each year.","The number of fish decreases by 92% each year.","The number of fish decreases by 24 each year.","The number of fish decreases by 0.92% each year."],"explain":"Each year the count is multiplied by 0.92, which keeps 92% and loses 8%. C would be a linear model; the drop of 24 happens only in the first year."},"m3028":{"id":"m3028","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Two-variable data: models and scatterplots","sub":"Same model, different time unit","tier":3,"traps":["units","concept"],"answer":"0.7225","accept":[".7225"],"stem":"A biologist models the number of bacteria in a sample with y=200(0.85)x, where x is the number of days since the sample was taken. A second biologist rewrites the same model as y=200dw, where w is the number of 2-day periods since the sample was taken. What is the value of d?","explain":"Since w counts 2-day periods, x=2w. Then y=200(0.85)2w=200(0.852)w=200(0.7225)w. So d=0.7225: the count keeps 72.25% of its value every 2 days. Enter .7225. Trap: doubling the 15% decrease to get d=0.70."},"m3029":{"id":"m3029","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Two-variable data: models and scatterplots","sub":"Where a model overestimates","tier":2,"traps":["setup"],"figure":"
x246810
y815182528
","answer":"3","accept":["3"],"stem":"A linear model for the data in the table is y=2.5x+4. For how many of the five data points does the model's predicted value exceed the actual value of y?","explain":"Predicted values: 9, 14, 19, 24, 29. Compare with actual 8, 15, 18, 25, 28. The prediction is greater at x=2, 6, and 10: three points."},"m3030":{"id":"m3030","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Two-variable data: models and scatterplots","sub":"Line of best fit read from a graph","tier":2,"traps":["setup"],"figure":"
\"A
","answer":"2","accept":["2"],"stem":"The scatterplot shows the number of visitors to a museum, in hundreds, and the daily high temperature for 12 days. The line of best fit passes through the points (10,12) and (30,4). According to the line, what is the predicted number of visitors, in hundreds, on a day with a high temperature of 35°C?","explain":"Slope: 30−104−12​=−0.4. From (30,4), five more degrees lowers the prediction by 0.4×5=2, giving 2 (hundred visitors)."},"m3031":{"id":"m3031","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Conditional probability from a two-way table","tier":1,"traps":["setup"],"figure":"
LibraryHomeCaféTotal
Sophomore18421575
Junior27362285
Senior35302590
Total8010862250
","answer":"27/80","accept":["27/80",".3375"],"stem":"The table shows the results of a survey of students about their preferred study location. If a student who prefers the library is selected at random, what is the probability that the student is a junior?","explain":"The condition limits us to the 80 students who prefer the library. Of them, 27 are juniors: 8027​=0.3375. Trap: 8527​ is the probability that a junior prefers the library, which reverses the condition."},"m3032":{"id":"m3032","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Conditional probability with complements","tier":2,"traps":["setup"],"answer":"5/16","accept":["5/16",".3125"],"stem":"In a group of 200 people, 120 have a library card, and 45 of the people with a library card also have a museum pass. A total of 70 people in the group have a museum pass. If a person who does NOT have a library card is selected at random, what is the probability that the person has a museum pass?","explain":"People without a library card: 200−120=80. Museum-pass holders without a library card: 70−45=25. So the probability is 8025​=165​=0.3125."},"m3033":{"id":"m3033","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Inclusion and exclusion","tier":2,"traps":["setup"],"answer":"1/3","accept":["1/3",".3333","0.333"],"stem":"An integer from 1 to 60, inclusive, is chosen at random. What is the probability that the integer is divisible by 4 or by 6?","explain":"Multiples of 4: 15. Multiples of 6: 10. Numbers counted twice are multiples of 12: 5. So 15+10−5=20 numbers, and the probability is 6020​=31​. Trap: 6025​ double counts 12, 24, 36, 48, and 60."},"m3034":{"id":"m3034","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Probability after adding items","tier":2,"traps":["setup"],"answer":"32","accept":["32"],"stem":"A bag contains only red and blue marbles, in the ratio 3:5. After 8 red marbles are added, the probability of drawing a red marble at random is 21​. How many marbles were in the bag before the red marbles were added?","explain":"Start with 3k red and 5k blue. After adding 8 red, red equals blue: 3k+8=5k, so k=4. The bag held 8k=32 marbles."},"m3035":{"id":"m3035","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Completing a two-way table","tier":2,"traps":["setup"],"figure":"
FloweredDid not flowerTotal
Fertilizer A132180
Fertilizer B84
Total268400
","answer":"7/11","accept":["7/11",".6363",".6364","0.636"],"stem":"A study classified 400 seedlings by fertilizer type and by whether they flowered. Some entries in the table are missing. If a seedling that did not flower is chosen at random, what is the probability that it received fertilizer B?","explain":"Fertilizer A seedlings that did not flower: 180−132=48. All seedlings that did not flower: 48+84=132 (check: 400−268=132). So P(B∣did not flower)=13284​=117​≈0.636. Trap: dividing by B's row total (220) answers a different question."},"m3036":{"id":"m3036","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Conditioning on \"at least one\"","tier":3,"traps":["setup"],"answer":"2/11","accept":["2/11",".1818","0.181","0.182"],"stem":"Two fair six-sided dice are rolled. Given that at least one of the dice shows a 5, what is the probability that the sum of the two dice is 8?","explain":"Outcomes with at least one 5: 6 with the first die showing 5, plus 6 with the second, minus the one counted twice, (5,5): 11 outcomes. Of these, the sum is 8 for (5,3) and (3,5). So the probability is 112​≈0.1818. Trap: using 12 outcomes double counts (5,5)."},"m3037":{"id":"m3037","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Inference from sample statistics and margin of error","sub":"Interpreting a margin of error","tier":1,"traps":["concept"],"answer":"D","stem":"A random sample of likely voters in a city found that 62% support a new park, with a margin of error of 4 percentage points. Which conclusion is most appropriate?","choices":["Exactly 62% of all likely voters in the city support the new park.","Between 58% and 66% of the people in the sample support the new park.","The proportion of all city residents who support the park is certainly greater than 58%.","It is plausible that between 58% and 66% of all likely voters in the city support the new park."],"explain":"A margin of error gives a range of plausible values for the population, here likely voters in the city: 62%±4%. B is wrong because the sample's value is known exactly (62%). A and C claim certainty, and C also widens the population to all residents."},"m3038":{"id":"m3038","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Inference from sample statistics and margin of error","sub":"Sample size and margin of error","tier":2,"traps":["concept"],"answer":"C","stem":"Two researchers each select a random sample of students from the same large university and ask how many hours they slept the previous night. Researcher A surveys 400 students and Researcher B surveys 1,600 students. Both use the same method to compute a margin of error. Which of the following is most likely true?","choices":["Researcher B's margin of error is larger than Researcher A's.","The margins of error are about the same, because the samples come from the same population.","Researcher B's margin of error is smaller than Researcher A's.","Researcher B's sample mean is closer to 8 hours than Researcher A's."],"explain":"Larger random samples give more precise estimates, so the margin of error shrinks as the sample grows (quadrupling the sample roughly halves it). D assumes something about the data that the question never gives."},"m3039":{"id":"m3039","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Inference from sample statistics and margin of error","sub":"Plausible count from a margin of error","tier":2,"traps":["setup"],"answer":"B","stem":"A school has 12,000 students. In a random sample of 300 of them, 45 said they bike to school. The margin of error for the sample proportion is 4 percentage points. Based on these results, which of the following is a plausible number of students at the school who bike to school?","choices":["1,000","1,500","2,400","2,600"],"explain":"The sample proportion is 30045​=15%, so plausible proportions run from 11% to 19%. For 12,000 students, that is 1,320 to 2,280. Only 1,500 falls in that range."},"m3040":{"id":"m3040","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Inference from sample statistics and margin of error","sub":"Comparing two estimates with margins","tier":2,"traps":["concept"],"answer":"C","stem":"Based on random samples, researchers estimate that the mean daily screen time of teenagers is 5.2 hours in County A, with a margin of error of 0.6 hour, and 4.1 hours in County B, with a margin of error of 0.4 hour. Which conclusion is best supported?","choices":["The mean daily screen time of teenagers in County A is exactly 1.1 hours greater than in County B.","The two counties plausibly have the same mean daily screen time.","It is plausible that the mean daily screen time is greater for teenagers in County A than for teenagers in County B.","Every teenager in County A spends more time on screens than every teenager in County B."],"explain":"County A's plausible range is 4.6 to 5.8 hours; County B's is 3.7 to 4.5 hours. The ranges don't overlap, so it is plausible that County A's mean is greater. B fails because no single value lies in both ranges. A claims a precision the estimates don't have, and D confuses means with individuals."},"m3041":{"id":"m3041","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Inference from sample statistics and margin of error","sub":"The population a result extends to","tier":2,"traps":["scope"],"answer":"D","stem":"A researcher selected a random sample of 500 high school seniors in Ohio and found that 71% planned to attend college. To which population can the result most appropriately be generalized?","choices":["All high school students in Ohio","All high school seniors in the United States","Only the 500 seniors who were surveyed","All high school seniors in Ohio"],"explain":"A random sample represents the population it was drawn from: Ohio high school seniors. B includes other grades and C includes other states, neither of which was sampled. C is too narrow; random sampling is exactly what allows a generalization beyond the sample."},"m3042":{"id":"m3042","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Evaluating statistical claims: observational studies and experiments","sub":"Correlation without random assignment","tier":1,"traps":["concept"],"answer":"B","stem":"A survey of 2,000 randomly selected adults found that those who drank at least three cups of coffee a day reported sleeping fewer hours, on average, than those who drank less coffee. Which conclusion is best supported?","choices":["Drinking at least three cups of coffee a day causes adults to sleep less.","Among adults, drinking more coffee is associated with sleeping fewer hours.","Sleeping fewer hours causes adults to drink more coffee.","The results apply only to the 2,000 adults surveyed."],"explain":"Random selection lets the result extend to adults generally, but this is an observational study: no one was assigned to drink coffee. That supports an association, not a cause in either direction (A and C)."},"m3043":{"id":"m3043","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Evaluating statistical claims: observational studies and experiments","sub":"Random assignment without random sampling","tier":2,"traps":["scope"],"answer":"A","stem":"A researcher recruited 120 volunteers from a running club. Half were randomly assigned to follow a new stretching routine and half to follow their usual routine. After 8 weeks, the stretching group had significantly fewer injuries. Which conclusion is best supported?","choices":["The new routine reduces injuries for members of the running club like those who volunteered.","The new routine reduces injuries for all adults.","The new routine is associated with fewer injuries, but no causal conclusion can be drawn.","No conclusion can be drawn, because the volunteers were not selected at random."],"explain":"Random assignment supports a cause-and-effect conclusion. But the participants were volunteers from one club, not a random sample of a wider group, so the conclusion is limited to people like them. B overgeneralizes; C ignores the random assignment; D throws out a valid experiment."},"m3044":{"id":"m3044","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Evaluating statistical claims: observational studies and experiments","sub":"A lurking variable","tier":2,"traps":["concept"],"answer":"D","stem":"A city found that neighborhoods with more ice cream shops had higher rates of sunburn. A council member concluded that ice cream causes sunburn. Which of the following best explains why this conclusion is not supported?","choices":["The study should have measured ice cream sales instead of the number of shops.","Sunburn rates cannot be measured accurately.","The relationship would be causal only if every neighborhood had an ice cream shop.","A third factor, such as the amount of sunny weather and outdoor activity, could explain both measurements."],"explain":"The data are observational, and a lurking variable (sunny, outdoor-heavy neighborhoods) could raise both the number of ice cream shops and sunburn rates. A changes the measurement but not the design; it would still show only association."},"m3045":{"id":"m3045","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Evaluating statistical claims: observational studies and experiments","sub":"A sampling method that misses part of the population","tier":2,"traps":["scope"],"answer":"B","stem":"To estimate the average time residents of a city spend commuting to work, a researcher surveyed every tenth person boarding city buses between 7 a.m. and 9 a.m. on one Tuesday. Why might the estimate be biased?","choices":["Surveying every tenth person is not a form of sampling.","The sample includes only bus riders at rush hour, who may commute differently from residents who drive, walk, or work other hours.","The sample is too large to give a reliable estimate.","The researcher should have surveyed only people who drive to work."],"explain":"The method reaches only people boarding buses during the morning rush, so residents who commute by car, bike, or foot, or at other times, can't be selected. Their commutes may differ. Systematic selection (every tenth person) isn't the problem; the place and time are."},"m4001":{"id":"m4001","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Percent change in a cylinder's volume","tier":2,"traps":["percent"],"answer":"76/5","accept":["76/5","15.2"],"stem":"The radius of a right circular cylinder is increased by 20%, and its height is decreased by 20%. By what percent does the volume of the cylinder increase?","explain":"V=πr2h. The new volume is π(1.2r)2(0.8h)=1.44×0.8×πr2h=1.152V, an increase of 15.2%. Trap: thinking the changes cancel (0%), or squaring nothing (which gives a 4% decrease)."},"m4002":{"id":"m4002","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Cone and cylinder with a shared radius","tier":2,"traps":["setup"],"answer":"9/4","accept":["9/4","2.25"],"stem":"A right circular cone and a right circular cylinder have the same radius. The volume of the cone is 43​ the volume of the cylinder. What is the ratio of the height of the cone to the height of the cylinder?","explain":"31​πr2hcone​=43​πr2hcyl​, so hcone​=49​hcyl​. The ratio is 49​, or 2.25. Trap: forgetting the 31​ in the cone formula gives 43​."},"m4003":{"id":"m4003","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Sphere inscribed in a cube","tier":1,"traps":["partial"],"answer":"B","stem":"A sphere is inscribed in a cube, touching each face of the cube. The volume of the cube is 216 cubic inches. What is the volume of the sphere, in cubic inches?","choices":["72π","36π","288π","9π"],"explain":"The cube's edge is 3216​=6, which equals the sphere's diameter, so r=3. V=34​π(3)3=36π. C uses r=6."},"m4004":{"id":"m4004","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Overlap of two circles","tier":3,"traps":["setup"],"answer":"C","stem":"Two circles each have radius 6, and each circle passes through the center of the other. What is the area of the region inside both circles?","choices":["12π−93​","24π−363​","24π−183​","12π"],"explain":"The centers and either intersection point form an equilateral triangle with side 6, so each circle's arc bounding the overlap spans 120°. The overlap is two circular segments. Each segment is a 120° sector minus a triangle: 31​π(36)−21​(6)(6)sin120°=12π−93​. Two segments: 24π−183​≈44.2. A counts only one segment."},"m4005":{"id":"m4005","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Doubling volume, not surface area","tier":3,"traps":["concept"],"answer":"A","stem":"A cube has a surface area of 150 square centimeters. A second cube has twice the volume of the first. What is the surface area of the second cube, in square centimeters?","choices":["15034​","300","1502​","600"],"explain":"The first cube's edge is 5 (since 6s2=150), so its volume is 125. Doubling the volume multiplies the edge by 32​, and surface area scales with the square of the edge: 150⋅(32​)2=15034​≈238. B assumes area doubles with volume; D doubles the edge."},"m4006":{"id":"m4006","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Similar solids","tier":2,"traps":["concept"],"answer":"D","stem":"Two right circular cylinders are similar. Their surface areas are 48π and 108π square units, and the volume of the smaller cylinder is 64π cubic units. What is the volume of the larger cylinder?","choices":["96π","144π","324π","216π"],"explain":"The area ratio is 48108​=49​, so the length ratio is 23​ and the volume ratio is (23​)3=827​. The larger volume is 64π⋅827​=216π. B uses the area ratio for volume; A uses the length ratio."},"m4007":{"id":"m4007","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"A tilted square from one diagonal","tier":3,"traps":["setup"],"answer":"7","accept":["7"],"stem":"In the xy-plane, A(1,2) and C(7,6) are opposite vertices of square ABCD. What is the greatest y-coordinate of any vertex of the square?","explain":"The diagonals of a square have equal length, bisect each other, and are perpendicular. The center is the midpoint of AC, (4,4). From the center to C is a move of (3,2); a perpendicular move of the same length is (−2,3) or (2,−3). So the other two vertices are (4−2,4+3)=(2,7) and (4+2,4−3)=(6,1). The greatest y-coordinate is 7. Trap: assuming the sides are horizontal and vertical, which puts vertices at (1,6) and (7,2); those make a 6-by-4 rectangle, not a square."},"m4008":{"id":"m4008","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Water displacement","tier":2,"traps":["setup"],"answer":"36/25","accept":["36/25","1.44"],"stem":"A cylindrical container with an inside radius of 5 centimeters holds some water. A solid metal sphere with a radius of 3 centimeters is dropped in and is completely submerged, and no water spills. By how many centimeters does the water level rise?","explain":"The rise in water fills a cylinder whose volume equals the sphere's: π(5)2h=34​π(3)3=36π. So 25h=36 and h=1.44."},"m4009":{"id":"m4009","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Regular hexagon","tier":1,"traps":["concept"],"answer":"A","stem":"What is the area of a regular hexagon with a side length of 4?","choices":["243​","123​","483​","96"],"explain":"A regular hexagon splits into 6 equilateral triangles with side 4. Each has area 43​​(4)2=43​, so the hexagon's area is 243​."},"m4010":{"id":"m4010","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Pyramid from a slant height","tier":2,"traps":["partial"],"answer":"400","accept":["400"],"stem":"A right pyramid has a square base with an edge length of 10 inches. The slant height of each triangular face, measured from the midpoint of a base edge to the apex, is 13 inches. What is the volume of the pyramid, in cubic inches?","explain":"The height, half the base edge, and the slant height form a right triangle: h=132−52​=12. Volume: 31​(102)(12)=400. Trap: using 13 as the height gives about 433."},"m4011":{"id":"m4011","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Midsegment area ratio","tier":2,"traps":["partial"],"answer":"27","accept":["27"],"stem":"In triangle ABC, points D and E are the midpoints of AB and AC, respectively. The area of triangle ADE is 9. What is the area of quadrilateral DBCE?","explain":"Triangle ADE is similar to ABC with scale factor 21​, so its area is 41​ of ABC's. ABC has area 36, and the quadrilateral is 36−9=27. Trap: answering 36 (the whole triangle) or 9 (thinking halves)."},"m4012":{"id":"m4012","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Area from perimeter and diagonal, twice","tier":3,"traps":["setup"],"answer":"32","accept":["32"],"stem":"Rectangle R has a diagonal of length 13 and an area of 60. Rectangle S has the same perimeter as rectangle R and a diagonal of length 15. What is the area of rectangle S?","explain":"For a rectangle with sides ℓ and w, (ℓ+w)2=ℓ2+w2+2ℓw=(diagonal)2+2(area). For R: (ℓ+w)2=169+120=289, so ℓ+w=17. Rectangle S has the same perimeter, so its sides also sum to 17: 289=225+2A, which gives A=32. (The sides of S never need to be found.)"},"m4013":{"id":"m4013","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Volume equals surface area for a cylinder","tier":2,"traps":["setup"],"answer":"3","accept":["3"],"stem":"A right circular cylinder has a height equal to twice its radius. Its volume, in cubic units, is numerically equal to its total surface area, in square units, including both bases. What is the radius of the cylinder?","explain":"With h=2r: volume πr2(2r)=2πr3, and surface area 2πr2+2πr(2r)=6πr2. Setting them equal: 2πr3=6πr2, so r=3. Trap: leaving out the two bases gives r=2."},"m4014":{"id":"m4014","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Parallel lines with algebraic angles","tier":1,"traps":["setup"],"figure":"
\"Two
","answer":"110","accept":["110"],"stem":"In the figure, lines ℓ and m are parallel and are intersected by a transversal. The two marked angles have measures (3x+10)° and (5x−30)°, as shown. What is the measure, in degrees, of the angle marked y°?","explain":"The two labeled angles are alternate interior angles, so they are equal: 3x+10=5x−30, which gives x=20, and each measures 70°. The angle y° sits on line m next to the 70° angle (they form a straight angle), so y=180−70=110. Trap: entering 70 or x=20."},"m4015":{"id":"m4015","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Exterior angle with a ratio","tier":2,"traps":["partial"],"answer":"50","accept":["50"],"stem":"An exterior angle of a triangle measures 130°. The two interior angles not adjacent to it have measures in the ratio 2:3. What is the measure, in degrees, of the smallest interior angle of the triangle?","explain":"The exterior angle equals the sum of the two remote interior angles: 2k+3k=130, so k=26, giving 52° and 78°. The third interior angle is adjacent to the exterior angle: 180−130=50°. The smallest is 50. Trap: 52 overlooks the third angle."},"m4016":{"id":"m4016","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Parallel segment creates similar triangles","tier":1,"traps":["setup"],"answer":"15","accept":["15"],"stem":"In triangle ABC, point D is on AB and point E is on AC so that DE is parallel to BC. If AD=6, DB=4, and DE=9, what is the length of BC?","explain":"Triangles ADE and ABC are similar, with ADAB​=610​. So BC=9⋅610​=15. Trap: using ADDB​ gives 6."},"m4017":{"id":"m4017","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Triangle inequality count","tier":2,"traps":["setup"],"answer":"13","accept":["13"],"stem":"Two sides of a triangle have lengths 7 and 12. The third side has an integer length. How many different lengths are possible for the third side?","explain":"The third side must be greater than 12−7=5 and less than 12+7=19. The integers 6 through 18 work: 13 values. Trap: including 5 or 19, which give a degenerate \"triangle.\""},"m4018":{"id":"m4018","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Isosceles cases","tier":2,"traps":["concept"],"answer":"D","stem":"One angle of an isosceles triangle measures 40°. Which of the following could NOT be the measure of another angle of the triangle?","choices":["40°","70°","100°","80°"],"explain":"If 40° is a base angle, the angles are 40, 40, 100. If 40° is the vertex angle, the base angles are each 70°. So 40, 70, and 100 are all possible, but 80 is not."},"m4019":{"id":"m4019","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Interior angle as a multiple of the exterior angle","tier":2,"traps":["concept"],"answer":"12","accept":["12"],"stem":"Each interior angle of a regular polygon is 5 times as large as each of its exterior angles. How many sides does the polygon have?","explain":"An interior angle and its exterior angle add to 180°: 5e+e=180, so e=30°. Exterior angles of any polygon sum to 360°, so there are 30360​=12 sides."},"m4020":{"id":"m4020","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Altitude to the hypotenuse","tier":2,"traps":["concept"],"answer":"6","accept":["6"],"stem":"In right triangle ABC, the right angle is at C, and CD is the altitude to hypotenuse AB. If AD=4 and DB=9, what is the length of CD?","explain":"The altitude to the hypotenuse creates triangles ADC and CDB that are similar, so CDAD​=DBCD​, which means CD2=4×9=36 and CD=6."},"m4021":{"id":"m4021","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Angle between two parallels","tier":2,"traps":["setup"],"figure":"
\"Two
","answer":"85","accept":["85"],"stem":"In the figure, lines ℓ and m are parallel. What is the value of x?","explain":"Draw a line through the bend parallel to ℓ and m. It splits the bend into two angles that are alternate interior angles with 38° and 47°. So x=38+47=85."},"m4022":{"id":"m4022","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Area of a 13-14-15 triangle","tier":3,"traps":["setup"],"answer":"84","accept":["84"],"stem":"A triangle has side lengths 13, 14, and 15. What is the area of the triangle?","explain":"Draw the altitude h to the side of length 14, splitting it into pieces x and 14−x. Then 132−x2=152−(14−x)2, so 169−x2=225−196+28x−x2, giving x=5 and h=12. The area is 21​(14)(12)=84."},"m4023":{"id":"m4023","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Areas of similar triangles","tier":1,"traps":["concept"],"answer":"225/2","accept":["225/2","112.5"],"stem":"Triangle ABC is similar to triangle DEF, with A, B, and C corresponding to D, E, and F. If AB=4, DE=10, and the area of triangle ABC is 18, what is the area of triangle DEF?","explain":"The scale factor is 410​=2.5, and areas scale by its square: 18×6.25=112.5. Trap: multiplying by 2.5 gives 45."},"m4024":{"id":"m4024","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Isosceles triangle with an exterior angle","tier":3,"traps":["concept"],"answer":"60","accept":["60"],"stem":"In triangle ABC, AB=AC. The exterior angle at vertex C measures (3x)°, and angle A measures (x+20)°. What is the measure, in degrees, of angle B?","explain":"Since AB=AC, angles B and C are equal. The exterior angle at C is 180−C=3x, so C=180−3x. Angle sum: (x+20)+2(180−3x)=180, so −5x=−200 and x=40. Then C=60°, so B=60° (and A=60°: the triangle turns out to be equilateral)."},"m4025":{"id":"m4025","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Point dividing a segment in a ratio","tier":2,"traps":["setup"],"answer":"6","accept":["6"],"stem":"In the xy-plane, point P lies on the segment from A(−3,2) to B(9,11) so that AP:PB=1:2. What is the sum of the coordinates of P?","explain":"P is 31​ of the way from A to B: x=−3+31​(12)=1 and y=2+31​(9)=5. The sum is 6. Trap: going 21​ or 32​ of the way."},"m4026":{"id":"m4026","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"From sine of one angle to tangent of the other","tier":1,"traps":["concept"],"answer":"5/12","accept":["5/12",".4166",".4167","0.416","0.417"],"stem":"In right triangle ABC, the right angle is at C and sinA=1312​. What is the value of tanB?","explain":"With sinA=1312​, the side opposite A is 12 and the hypotenuse is 13, so the third side is 5. For angle B, the opposite side is 5 and the adjacent side is 12: tanB=125​≈0.4167. Trap: 512​ is tanA."},"m4027":{"id":"m4027","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Complementary angles","tier":1,"traps":["concept"],"answer":"13","accept":["13"],"stem":"For the acute angles (3x+10)° and (2x+15)°, sin(3x+10)°=cos(2x+15)°. What is the value of x?","explain":"The sine of an angle equals the cosine of its complement, so the angles add to 90°: 5x+25=90 and x=13. (The angles are 49° and 41°.)"},"m4028":{"id":"m4028","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Hidden right angle","tier":3,"traps":["concept"],"answer":"1","accept":["1"],"stem":"In triangle PQR, the measure of angle P is (4a)°, the measure of angle Q is (5a)°, and the measure of angle R is (9a)°. What is the value of cosQsinP​?","explain":"The angles sum to 18a=180, so a=10: P=40°, Q=50°, R=90°. The triangle is a right triangle even though no right angle was given, and P and Q are complementary. So sin40°=cos50° and the ratio is 1. Trap: deciding there isn't enough information because the triangle isn't labeled as right."},"m4029":{"id":"m4029","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Radians to degrees, then a supplement","tier":2,"traps":["partial"],"answer":"75","accept":["75"],"stem":"An angle measures 127π​ radians. What is the measure, in degrees, of its supplement?","explain":"127π​⋅π180°​=105°. Its supplement is 180−105=75. Trap: entering 105."},"m4030":{"id":"m4030","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Equilateral triangle in a circle","tier":2,"traps":["setup"],"answer":"C","stem":"An equilateral triangle is inscribed in a circle of radius 8. What is the perimeter of the triangle?","choices":["24","48","243​","123​"],"explain":"The center splits the triangle into three congruent triangles with two sides of 8 and a 120° angle between them. Half of one such triangle is a 30-60-90 triangle with hypotenuse 8, so half a side is 43​ and a side is 83​. The perimeter is 243​. A takes the side to equal the radius."},"m4031":{"id":"m4031","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Sign of sine from the quadrant","tier":2,"traps":["sign"],"answer":"B","stem":"If cosθ=−53​ and π<θ<23π​, what is the value of sinθ?","choices":["54​","−54​","−43​","34​"],"explain":"sin2θ=1−259​=2516​, so sinθ=±54​. Between π and 23π​ (Quadrant III), sine is negative: −54​."},"m4032":{"id":"m4032","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"A sliding ladder","tier":2,"traps":["partial"],"answer":"4","accept":["4"],"stem":"A 25-foot ladder leans against a vertical wall with its top 24 feet above the level ground. The bottom of the ladder then slides 8 feet farther from the wall. How many feet does the top of the ladder slide down the wall?","explain":"At first the base is 252−242​=7 feet from the wall. After sliding it is 15 feet away, so the top is 625−225​=20 feet high. It slid 24−20=4 feet. Trap: assuming the top slides 8 feet too."},"m4033":{"id":"m4033","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Cosine of the smaller acute angle","tier":2,"traps":["setup"],"answer":"B","stem":"A right triangle has legs of lengths a and 2a, where a>0. What is the cosine of the smaller acute angle of the triangle?","choices":["55​​","525​​","21​","2"],"explain":"The hypotenuse is a2+4a2​=a5​. The smaller angle is opposite the shorter leg a, so its adjacent leg is 2a: cos=a5​2a​=5​2​=525​​. A is the cosine of the larger acute angle."},"m4034":{"id":"m4034","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Angle of elevation","tier":1,"traps":["concept"],"answer":"C","stem":"From a point on level ground 50 meters from the base of a tower, the angle of elevation to the top of the tower is 32°. Which expression gives the height of the tower, in meters?","choices":["50sin32°","tan32°50​","50tan32°","50cos32°"],"explain":"The height is opposite the 32° angle and the 50 meters is adjacent: tan32°=50h​, so h=50tan32°. A would be right only if 50 were the line of sight (the hypotenuse)."},"m4035":{"id":"m4035","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Area from perimeter and hypotenuse","tier":3,"traps":["setup"],"answer":"60","accept":["60"],"stem":"A right triangle has a perimeter of 40 and a hypotenuse of length 17. What is the area of the triangle?","explain":"Let the legs be a and b. Then a+b=40−17=23 and a2+b2=289. Since (a+b)2=a2+b2+2ab, 529=289+2ab, so ab=120 and the area is 21​ab=60. (The legs are 8 and 15.)"},"m4036":{"id":"m4036","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Side from a cosine","tier":1,"traps":["setup"],"answer":"12","accept":["12"],"stem":"In right triangle ABC, the right angle is at B, cosA=0.6, and AB=9. What is the length of BC?","explain":"cosA=hypotenuseadjacent​=ACAB​, so AC=0.69​=15. Then BC=152−92​=12. Trap: treating 9 as the hypotenuse."},"m4037":{"id":"m4037","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Square inscribed in a circle","tier":2,"traps":["concept"],"answer":"D","stem":"A square is inscribed in a circle of radius r. What is the ratio of the area of the square to the area of the circle?","choices":["π4​","π1​","2π​","π2​"],"explain":"The square's diagonal is a diameter, 2r, so its side is 2​2r​=r2​ and its area is 2r2. Dividing by πr2 gives π2​. A uses a side of 2r, which is the square circumscribed around the circle."},"m4038":{"id":"m4038","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Two special triangles sharing an altitude","tier":3,"traps":["setup"],"figure":"
\"Triangle
","answer":"A","stem":"In triangle ABC, angle B measures 30° and angle C measures 45°. The altitude from A to BC has length 6. What is the length of BC?","choices":["6+63​","12+62​","63​","12"],"explain":"The altitude meets BC at D. In right triangle ABD with a 30° angle at B, the leg opposite 30° is 6, so BD=63​. In right triangle ADC with a 45° angle, DC=6. So BC=6+63​≈16.4. B uses the hypotenuses AB=12 and AC=62​ instead of the legs along BC."},"m4039":{"id":"m4039","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Radius given, a coefficient unknown","tier":2,"traps":["setup","sign"],"answer":"-5","accept":["-5"],"stem":"In the xy-plane, the graph of x2+y2−6x+ky=2, where k is a positive constant, is a circle with radius 6. What is the y-coordinate of the center of the circle?","explain":"Complete the square: (x−3)2+(y+2k​)2=2+9+4k2​. The radius is 6, so 11+4k2​=36, k2=100, and k=10. The center is (3,−2k​)=(3,−5). Trap: taking the center's y-coordinate as +2k​."},"m4040":{"id":"m4040","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Constant that sets the radius","tier":2,"traps":["sign"],"answer":"29","accept":["29"],"stem":"In the xy-plane, the graph of x2+8x+y2−4y=k, where k is a constant, is a circle with radius 7. What is the value of k?","explain":"Completing the squares gives (x+4)2+(y−2)2=k+16+4=k+20. The radius squared is 49, so k+20=49 and k=29."},"m4041":{"id":"m4041","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Arc length from a sector area","tier":2,"traps":["setup"],"answer":"A","stem":"A sector of a circle with radius 12 has an area of 30π. What is the length of the arc of the sector?","choices":["5π","25π​","10π","215π​"],"explain":"The sector is 144π30π​=245​ of the circle. The arc is the same fraction of the circumference: 245​⋅24π=5π. (Shortcut: sector area =21​rs, so 30π=6s and s=5π.) C doubles the arc."},"m4042":{"id":"m4042","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Central and inscribed angles on the same arc","tier":2,"traps":["concept"],"answer":"70","accept":["70"],"stem":"Points A, B, and C lie on a circle with center O, and angle ACB and angle AOB intercept the same arc. Angle AOB measures (3x−10)° and angle ACB measures (x+20)°. What is the measure, in degrees, of angle ACB?","explain":"A central angle is twice an inscribed angle on the same arc: 3x−10=2(x+20), so x=50. Angle ACB measures 50+20=70° (and angle AOB is 140°). Trap: setting the angles equal, or entering x=50."},"m4043":{"id":"m4043","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Slope of a tangent line","tier":2,"traps":["sign"],"answer":"-3/4","accept":["-3/4","-0.75","-.75"],"stem":"A circle in the xy-plane has center (2,−1), and the point (5,3) lies on the circle. What is the slope of the line tangent to the circle at (5,3)?","explain":"The radius to (5,3) has slope 5−23−(−1)​=34​. A tangent line is perpendicular to the radius, so its slope is −43​, or −0.75. Trap: 34​ is the radius's slope."},"m4044":{"id":"m4044","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Lattice points on a line inside a circle","tier":3,"traps":["partial"],"answer":"6","accept":["6"],"stem":"For how many integer values of k does the point (k,k) lie inside the circle (x−1)2+(y+2)2=25?","explain":"A point is inside the circle when its distance from the center (1,−2) is less than the radius 5: (k−1)2+(k+2)2<25. Expanding: 2k2+2k+5<25, so k2+k−10<0. The roots of k2+k−10=0 are 2−1±41​​, about −3.70 and 2.70, so the integers −3 through 2 work: 6 values."},"m4045":{"id":"m4045","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Circle tangent to both axes through a point","tier":3,"traps":["partial"],"answer":"6","accept":["6"],"stem":"A circle in the first quadrant is tangent to both the x-axis and the y-axis and passes through the point (2,1). What is the sum of all possible values of the circle's radius?","explain":"Tangent to both axes in Quadrant I means the center is (r,r) and the equation is (x−r)2+(y−r)2=r2. Substituting (2,1): (2−r)2+(1−r)2=r2, so r2−6r+5=0 and r=1 or r=5. Both circles work, and the sum is 6. Trap: finding only one of the two circles."},"m4046":{"id":"m4046","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Two parallel chords","tier":2,"traps":["setup"],"answer":"14","accept":["14"],"stem":"Two parallel chords of a circle with radius 10 have lengths 16 and 12, and they lie on opposite sides of the center. What is the distance between the two chords?","explain":"The distance from the center to a chord is r2−(half-chord)2​: 100−64​=6 for the 16-chord and 100−36​=8 for the 12-chord. On opposite sides, the distances add: 6+8=14. Trap: subtracting (2), which applies when both chords are on the same side."},"m4047":{"id":"m4047","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Angle from arc length and sector area","tier":2,"traps":["setup"],"answer":"D","stem":"A sector of a circle has an arc length of 6π and an area of 27π. What is the measure, in radians, of the sector's central angle?","choices":["3π​","94π​","23π​","32π​"],"explain":"A sector's area equals 21​rs, where s is the arc length: 27π=21​r(6π), so r=9. The angle is θ=rs​=96π​=32π​."},"m4048":{"id":"m4048","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Shift and scale by area","tier":2,"traps":["concept"],"answer":"B","stem":"In the xy-plane, circle A has equation (x+2)2+y2=9. Circle B is the image of circle A shifted down 6 units and resized so that its area is twice the area of circle A. Which equation represents circle B?","choices":["(x+2)2+(y+6)2=36","(x+2)2+(y+6)2=18","(x+2)2+(y−6)2=18","(x−4)2+y2=18"],"explain":"Shifting down 6 moves the center from (−2,0) to (−2,−6), giving (y+6)2. Twice the area means πrB2​=2π(9), so rB2​=18. A doubles the radius, which quadruples the area. C shifts up; D shifts right."},"m4049":{"id":"m4049","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Tangent segments and a kite","tier":2,"traps":["concept"],"answer":"34","accept":["34"],"stem":"From a point P outside a circle with center O and radius 5, two segments are drawn tangent to the circle at points A and B. If OP=13, what is the perimeter of quadrilateral OAPB?","explain":"A radius meets a tangent at a right angle, so triangle OAP is a right triangle with legs 5 and PA and hypotenuse 13: PA=12. By symmetry PB=12. The perimeter is 5+5+12+12=34."},"m4050":{"id":"m4050","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Chord through the origin of a known length","tier":4,"traps":["setup"],"answer":"3/4","accept":["3/4","0.75",".75"],"stem":"In the xy-plane, the line y=mx, where m is a positive constant, intersects the circle x2+y2−10x=0 at the origin and at a point P. The distance from the origin to P is 8. What is the value of m?","explain":"The circle is (x−5)2+y2=25, so its diameter runs from (0,0) to (10,0). Any point P on the circle forms a right angle with the ends of that diameter, so the origin, P, and (10,0) make a right triangle with hypotenuse 10 and leg 8. The other leg is 6, and the slope of the line from the origin to P is the tangent of the angle at the origin: m=86​=43​. Algebraically: substituting y=mx gives x=1+m210​ at P, so OP=1+m2​10​=8 and m2=169​."},"m4051":{"id":"m4051","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Sector area with a radian angle","tier":1,"traps":["concept"],"answer":"C","stem":"A circle has an area of 64π. What is the area of a sector of this circle with a central angle of 43π​ radians?","choices":["48π","12π","24π","6π"],"explain":"The sector is 2π3π/4​=83​ of the circle: 83​⋅64π=24π. A uses 43​ of the circle, treating the angle as a fraction of π rather than 2π."},"m4101":{"id":"m4101","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Partly filled cone poured into a cylinder","tier":3,"traps":["concept","setup"],"answer":"3","accept":["3"],"stem":"A cone-shaped cup, with its vertex pointing down, has a height of 12 centimeters and a top radius of 4 centimeters. The cup is filled with water to a depth of 9 centimeters. All of the water is then poured into an empty cylinder with a radius of 3 centimeters. What is the depth of the water in the cylinder, in centimeters?","explain":"The water in the cup is itself a cone of height 9. It is similar to the full cup, so its radius is 4⋅129​=3. Its volume is 31​π(3)2(9)=27π. In the cylinder, π(3)2h=27π, so h=3. Trap: using the cup's full radius of 4 for the water, which gives 48π and a depth of 316​."},"m4102":{"id":"m4102","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Sphere inscribed in a cone","tier":4,"traps":["setup","concept"],"answer":"C","stem":"A sphere is inscribed in a right circular cone with base radius 6 and height 8, so that the sphere touches the cone's base and its curved surface. What is the volume of the sphere?","choices":["3256π​","332π​","36π","48π"],"explain":"Slice the cone through its axis. The cross-section is an isosceles triangle with base 12 and height 8, so each slanted side is 62+82​=10, and the sphere becomes the circle inscribed in this triangle. Let the radius be r; the center is r above the base, so it is 8−r from the apex. The radius drawn to a slanted side is perpendicular to it, forming a right triangle similar to the half-triangle with legs 6 and 8 and hypotenuse 10: 8−rr​=106​. So 10r=48−6r and r=3. The volume is 34​π(3)3=36π. A uses half the height as the radius."},"m4103":{"id":"m4103","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Cylinder inscribed in a sphere","tier":3,"traps":["setup"],"answer":"D","stem":"A right circular cylinder with height 8 is inscribed in a sphere of radius 5, so that the circular edges of both bases lie on the sphere. What is the volume of the cylinder?","choices":["200π","24π","128π","72π"],"explain":"The sphere's center is the midpoint of the cylinder's axis, 4 units from each base. A radius of the sphere drawn to a point on a base's edge is the hypotenuse of a right triangle with legs 4 (along the axis) and r (the cylinder's radius): r2+42=52, so r=3. The volume is π(3)2(8)=72π. A uses the sphere's radius, and C uses half the height as the radius."},"m4104":{"id":"m4104","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Volume from three face areas","tier":3,"traps":["concept"],"answer":"192","accept":["192"],"stem":"A rectangular box has faces with areas of 24, 32, and 48 square inches. What is the volume of the box, in cubic inches?","explain":"If the edges are a, b, and c, the face areas are ab, bc, and ca. Multiplying them gives (ab)(bc)(ca)=(abc)2, so (abc)2=24×32×48=36,864 and abc=192. (The edges are 4, 6, and 8.) Trap: multiplying the three areas without taking the square root."},"m4105":{"id":"m4105","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Frustum as a fraction of a cone","tier":4,"traps":["concept","partial"],"answer":"19/27","accept":["19/27",".7037","0.703","0.704"],"stem":"A right circular cone has a height of 12. A plane parallel to the base cuts the cone 4 units above the base, dividing it into a smaller cone and a frustum (the piece between the plane and the base). The volume of the frustum is what fraction of the volume of the original cone?","explain":"The smaller cone on top has height 12−4=8 and is similar to the original with scale factor 128​=32​. Volumes scale by the cube of the scale factor, so the small cone has (32​)3=278​ of the original volume. The frustum has the rest: 1−278​=2719​. Enter 19/27, .7037, or 0.703. Trap: using 124​ as the scale factor for the small cone; the cut is 4 units above the base, so the small cone is 8 tall."},"m4106":{"id":"m4106","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Recasting a sphere into cubes","tier":3,"traps":["partial"],"answer":"113","accept":["113"],"stem":"A solid metal sphere with a radius of 6 centimeters is melted down and recast into solid cubes with an edge length of 2 centimeters. If no metal is lost, what is the greatest number of complete cubes that can be made?","explain":"The sphere's volume is 34​π(6)3=288π≈904.8 cubic centimeters. Each cube has volume 23=8. Then 8904.8​≈113.1, so 113 complete cubes can be made. Trap: using the surface area 4πr2 instead of the volume, or rounding up; only complete cubes count."},"m4107":{"id":"m4107","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Isosceles trapezoid from its diagonals","tier":4,"traps":["setup"],"answer":"B","stem":"An isosceles trapezoid has bases of lengths 10 and 24, and each of its diagonals has length 25. What is the area of the trapezoid?","choices":["408","6821​","13621​","3421​"],"explain":"Drop perpendiculars from the ends of the shorter base to the longer base; they cut off 224−10​=7 at each end. A diagonal runs from a top vertex to the far bottom vertex, covering a horizontal distance of 10+7=17. So h2+172=252, h2=336, and h=421​. The area is 21​(10+24)(421​)=6821​. A treats 25 as the length of a leg instead of a diagonal; C forgets the 21​."},"m4108":{"id":"m4108","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Inscribed and circumscribed hexagons","tier":4,"traps":["concept"],"answer":"3/4","accept":["3/4","0.75",".75"],"stem":"Regular hexagon P is inscribed in a circle, so all of its vertices lie on the circle. Regular hexagon Q is circumscribed about the same circle, so each of its sides is tangent to the circle. What is the ratio of the area of hexagon P to the area of hexagon Q?","explain":"Let the circle's radius be r. A regular hexagon splits into 6 equilateral triangles from its center. For P, the radius r is the distance from the center to a vertex. For Q, r is the distance from the center to the midpoint of a side, which is the height of one of its equilateral triangles, so Q's side is 3​2r​. The hexagons are similar with side ratio 2r/3​r​=23​​, so the area ratio is (23​​)2=43​. Enter 3/4 or .75."},"m4109":{"id":"m4109","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Overlapping semicircles inside a square","tier":4,"traps":["setup","concept"],"figure":"
\"Square
","answer":"A","stem":"In the figure, each side of square ABCD has length 8. Four semicircles are drawn inside the square, each with one side of the square as its diameter. What is the total area of the four shaded regions, where two semicircles overlap?","choices":["32π−64","16π−32","64−16π","8π"],"explain":"Each semicircle has radius 4 and area 21​π(4)2=8π, so the four together have area 32π. Every point of the square lies in at least one semicircle, and the shaded petals are exactly the points covered twice. Adding the four semicircle areas counts the square once and the petals a second time: 32π=64+(petals). So the petals have total area 32π−64≈36.5. B is the area of only two of the petals. C is the area of the square outside a circle inscribed in it, a different figure."},"m4110":{"id":"m4110","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Square tucked into a right angle","tier":3,"traps":["setup"],"answer":"24/7","accept":["24/7","3.428","3.429"],"stem":"A right triangle has legs of lengths 6 and 8. A square is drawn inside the triangle so that two of its sides lie along the legs and the vertex opposite the right angle lies on the hypotenuse. What is the side length of the square?","explain":"Place the right angle at the origin with the legs along the axes, so the hypotenuse is the line 8x​+6y​=1. The square's far vertex is (s,s), so 8s​+6s​=1, which gives 247s​=1 and s=724​. Enter 24/7, 3.428, or 3.429. (Similar triangles give the same result: the small triangle above the square has legs s and 6−s in the ratio 8:6.)"},"m4111":{"id":"m4111","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Rhombus from a diagonal ratio and perimeter","tier":3,"traps":["setup"],"answer":"96","accept":["96"],"stem":"The diagonals of a rhombus have lengths in the ratio 3:4, and the perimeter of the rhombus is 40. What is the area of the rhombus?","explain":"Each side is 440​=10. The diagonals of a rhombus bisect each other at right angles, so half-diagonals 3k and 4k are the legs of a right triangle with hypotenuse 10: 5k=10 and k=2. The diagonals are 12 and 16, and the area is 21​(12)(16)=96. Trap: using 3 and 4 times 10 as the diagonals themselves."},"m4112":{"id":"m4112","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Two triangles from any interior point","tier":3,"traps":["concept"],"answer":"30","accept":["30"],"stem":"Point P lies inside rectangle ABCD, where AB=10 and BC=6. What is the sum of the areas of triangles PAB and PCD?","explain":"Triangles PAB and PCD have bases AB and CD, each of length 10. Their heights are the distances from P to these two opposite sides, which add up to the rectangle's width, 6. So the total area is 21​(10)h1​+21​(10)h2​=5(h1​+h2​)=5(6)=30, half the rectangle, wherever P is."},"m4113":{"id":"m4113","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Cube, sphere, cube","tier":4,"traps":["concept"],"answer":"A","stem":"A cube is inscribed in a sphere, so the cube's vertices lie on the sphere. The sphere is inscribed in a larger cube, so it touches each face of the larger cube. What is the ratio of the volume of the larger cube to the volume of the smaller cube?","choices":["33​","3","9","22​"],"explain":"If the small cube has edge s, its space diagonal is s3​, which is the sphere's diameter. The large cube's edge equals the sphere's diameter, s3​. The edges are in the ratio 3​:1, so the volumes are in the ratio (3​)3=33​. B uses the ratio of the squares of the edges, and D uses a face diagonal instead of the space diagonal."},"m4114":{"id":"m4114","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"A cube that sticks out of the water","tier":4,"traps":["concept","setup"],"answer":"15/4","accept":["15/4","3.75"],"stem":"A rectangular tank is 50 centimeters long and 40 centimeters wide and holds water to a depth of 15 centimeters. A solid cube with edges of 20 centimeters is placed in the tank and rests on the bottom. By how many centimeters does the water level rise?","explain":"The cube is taller than the water, so it is not fully submerged and doesn't displace its full volume. Instead, the water now fills the tank's base minus the cube's footprint: 50×40−20×20=1,600 square centimeters. The water's volume, 50×40×15=30,000, is unchanged, so the new depth is 1,60030,000​=18.75 centimeters. That is less than 20, so the cube's top is indeed above the water. The level rises 18.75−15=3.75 centimeters. Trap: dividing the cube's whole volume by the tank's base, 2,0008,000​=4; a depth of 19 would leave the cube sticking out, so it couldn't have displaced all of its volume."},"m4115":{"id":"m4115","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Alternating squares and circles","tier":3,"traps":["concept"],"answer":"C","stem":"A square is inscribed in a circle. That circle is inscribed in a larger square, and the larger square is inscribed in a larger circle. What is the ratio of the area of the largest circle to the area of the smallest square?","choices":["2π","2π​","π","4"],"explain":"Let the smallest square have side s. Its diagonal, s2​, is the diameter of the small circle, so that circle's radius is 2s2​​. The larger square's side equals that diameter, s2​, and the largest circle's diameter is the larger square's diagonal, s2​⋅2​=2s. So the largest circle has radius s and area πs2, and the ratio to the smallest square's area s2 is π. B compares the smallest circle, whose area is 2πs2​, with the smallest square."},"m4116":{"id":"m4116","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Prism height from total surface area","tier":3,"traps":["setup"],"answer":"D","stem":"A right prism has bases that are equilateral triangles with side length 6. The total surface area of the prism, including both bases, is 183​+180. What is the volume of the prism?","choices":["1803​","303​","453​","903​"],"explain":"Each base has area 43​​(6)2=93​, so the two bases account for 183​. The three rectangular faces account for the remaining 180: 3×6×h=180, so h=10. The volume is the base area times the height: 93​×10=903​. C counts the base area as 21​ of 93​; A doubles the base area."},"m4117":{"id":"m4117","sec":"math","domain":"Geometry and Trigonometry","skill":"Area and volume","sub":"Slicing a corner off a cube","tier":3,"traps":["concept"],"answer":"180","accept":["180"],"stem":"A cube has edges of length 6. A plane passes through three vertices of the cube that are each adjacent to the same vertex V, cutting off a small pyramid that contains V. What is the volume of the remaining solid?","explain":"The piece cut off is a pyramid with vertex V whose three edges at V are mutually perpendicular edges of the cube, each of length 6. Use one right-triangle face as the base: its area is 21​(6)(6)=18, and the third edge, 6, is the height. The pyramid's volume is 31​(18)(6)=36. The remaining solid has volume 216−36=180. Trap: forgetting the 31​, which gives 216−108=108."},"m4121":{"id":"m4121","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Angles in an arithmetic sequence","tier":4,"traps":["concept","partial"],"answer":"9","accept":["9"],"stem":"The interior angles of a convex polygon, listed in increasing order, form an arithmetic sequence. The smallest angle measures 120°, and each angle is 5° greater than the one before it. How many sides does the polygon have?","explain":"With n sides, the angles are 120,125,…,120+5(n−1), and they sum to 120n+25n(n−1)​. Interior angles of an n-gon sum to 180(n−2). Setting these equal and simplifying gives n2−25n+144=0, so n=9 or n=16. With 16 sides the largest angle would be 120+75=195°, impossible in a convex polygon. So n=9, whose largest angle is 160°. Trap: answering 16 without checking convexity."},"m4122":{"id":"m4122","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Two isosceles triangles sharing a side","tier":3,"traps":["setup"],"answer":"47","accept":["47"],"stem":"In triangle ABC, AB=AC. Point D lies on BC so that AD=BD. If angle DAC measures 39°, what is the measure, in degrees, of angle ABC?","explain":"Let angle ABC=x. Since AB=AC, angle ACB=x too. Since AD=BD, triangle ABD is isosceles, so angle BAD=x. Then angle BAC=x+39, and the angles of triangle ABC give (x+39)+x+x=180, so x=47. Check: angle ADB=180−2(47)=86°, so angle ADC=94°, and in triangle ADC, 39+47+94=180."},"m4123":{"id":"m4123","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"A chain of three equal segments","tier":3,"traps":["setup"],"figure":"
\"Triangle
","answer":"105","accept":["105"],"stem":"In the figure, point D lies on AC, and AB=BD=DC. If angle C measures 25°, what is the measure, in degrees, of angle ABC? (Note: Figure not drawn to scale.)","explain":"Since BD=DC, triangle BDC is isosceles, so angle DBC=25°. Angle BDA is an exterior angle of triangle BDC, so it equals 25+25=50°. Since AB=BD, angle A = angle BDA=50°, and angle ABD=180−50−50=80°. So angle ABC=80+25=105°. Check: 50+105+25=180."},"m4124":{"id":"m4124","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Parallel cut with an area split","tier":3,"traps":["concept"],"answer":"2","accept":["2"],"stem":"In triangle ABC, point D lies on AB and point E lies on AC so that DE is parallel to BC. The area of triangle ADE is 16, and the area of quadrilateral DBCE is 20. What is the value of DBAD​?","explain":"Triangle ADE is similar to triangle ABC, whose area is 16+20=36. Areas scale by the square of the scale factor, so ABAD​=3616​​=32​. Then AD=32​AB and DB=31​AB, so DBAD​=2. Trap: comparing 16 with 20 directly, or forgetting the square root."},"m4125":{"id":"m4125","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Area of one piece cut by the altitude","tier":3,"traps":["setup"],"answer":"54","accept":["54"],"stem":"In right triangle ABC, the right angle is at C, AC=15, and BC=20. Point D lies on AB so that CD is perpendicular to AB. What is the area of triangle ACD?","explain":"The hypotenuse is AB=152+202​=25. The area of triangle ABC is 21​(15)(20)=150, and also 21​(25)(CD), so CD=12. In right triangle ACD, AD=152−122​=9. The area of triangle ACD is 21​(9)(12)=54. (Triangle ACD is similar to triangle ABC with scale factor 2515​, so its area is (53​)2×150=54.)"},"m4126":{"id":"m4126","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Triangle inequality with a variable side","tier":3,"traps":["partial"],"answer":"9","accept":["9"],"stem":"A triangle has side lengths x, 2x+1, and 15, where x is an integer. How many different values of x are possible?","explain":"Each side must be shorter than the sum of the other two. From x+(2x+1)>15: x>314​, so x≥5. From x+15>2x+1: x<14, so x≤13. The third condition, (2x+1)+15>x, holds for every positive x. The integers 5 through 13 work: 9 values."},"m4127":{"id":"m4127","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Counting obtuse triangles","tier":4,"traps":["partial","concept"],"answer":"7","accept":["7"],"stem":"A triangle has side lengths 7, 9, and n, where n is an integer. For how many values of n is the triangle obtuse?","explain":"First, the triangle inequality gives 2<n<16, so n is between 3 and 15. A triangle is obtuse when the square of its longest side is greater than the sum of the squares of the other two. If n is the longest side (n≥10), obtuse means n2>49+81=130, so n=12,13,14,15. If 9 is the longest side (n≤9), obtuse means 81>49+n2, so n2<32 and n=3,4,5. That's 4+3=7 values. Trap: checking only the case where n is the longest side."},"m4128":{"id":"m4128","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"A zigzag between parallel lines","tier":3,"traps":["concept"],"figure":"
\"Two
","answer":"60","accept":["60"],"stem":"In the figure, lines ℓ and m are parallel. What is the value of x? (Note: Figure not drawn to scale.)","explain":"Draw a line through each bend, parallel to ℓ and m. At the first bend, the angle x splits into a part equal to the 40° angle at ℓ (alternate interior angles) and a part equal to the angle the middle segment makes with the horizontal. At the second bend, the 70° angle splits into that same middle-segment angle and a part equal to the 50° angle at m. So the middle-segment angle is 70−50=20°, and x=40+20=60. (A shortcut: angles at bends pointing one way add up to the angles pointing the other way: x+50=40+70.)"},"m4129":{"id":"m4129","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Reflecting a point across a slanted line","tier":4,"traps":["setup"],"answer":"8","accept":["8"],"stem":"In the xy-plane, the point P(3,5) is reflected across the line y=x+1 to the point P′. What is the sum of the coordinates of P′?","explain":"The segment PP′ is perpendicular to y=x+1, so it has slope −1, and its midpoint lies on the line. Points on the line through P with slope −1 have the form (3+t,5−t), and their coordinates always sum to 8. In particular, P′ is on this line, so the sum of its coordinates is 8. (To find P′ itself: the midpoint (3+2t​,5−2t​) must satisfy y=x+1, so t=1 and P′=(4,4).)"},"m4130":{"id":"m4130","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Distance from a point to a line","tier":4,"traps":["setup"],"answer":"4","accept":["4"],"stem":"In the xy-plane, what is the shortest distance from the point (1,2) to the line 3x+4y=31?","explain":"The shortest path is along the perpendicular. The line has slope −43​, so the perpendicular through (1,2) has slope 34​: points (1+3t,2+4t). Substituting into the line: 3(1+3t)+4(2+4t)=31, so 11+25t=31 and t=0.8. The foot of the perpendicular is (3.4,5.2), and the distance is 2.42+3.22​=16​=4. (Each step of t moves 5 units, since 32+42​=5, so the distance is 5(0.8)=4.)"},"m4131":{"id":"m4131","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Rebuilding a triangle from its midpoints","tier":3,"traps":["concept"],"answer":"32","accept":["32"],"stem":"The midpoints of the three sides of a triangle are (1,2), (4,6), and (7,2). What is the perimeter of the triangle?","explain":"Each segment joining two midpoints is parallel to the third side and half as long. The triangle formed by the midpoints has sides 32+42​=5, 32+42​=5, and 6, for a perimeter of 16. The original triangle's sides are twice as long, so its perimeter is 32. (Its vertices are (4,−2), (−2,6), and (10,6).)"},"m4132":{"id":"m4132","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Two shadows from one lamppost","tier":3,"traps":["setup"],"answer":"6","accept":["6"],"stem":"A woman who is 6 feet tall stands 12 feet from the base of a lamppost, and her shadow is 4 feet long. She walks directly away from the lamppost until her shadow is 6 feet long. How many feet did she walk?","explain":"The lamppost, the woman, and the tip of her shadow form similar right triangles. At first, the big triangle's base is 12+4=16 and her shadow is 4, so the lamppost is 6×416​=24 feet tall. When her shadow is s and she stands d feet away, s6​=d+s24​, so d+s=4s and d=3s. With s=6, d=18. She walked 18−12=6 feet. Trap: answering 18, her new distance from the post."},"m4133":{"id":"m4133","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Crossing segments between parallel sides","tier":3,"traps":["setup"],"figure":"
\"Segment
","answer":"12","accept":["12"],"stem":"In the figure, AE and BD intersect at C, and AB is parallel to DE. If AB=9, DE=6, and AE=20, what is the length of AC? (Note: Figure not drawn to scale.)","explain":"Since AB∥DE, angle BAC = angle DEC (alternate interior angles), and the vertical angles at C are equal, so triangle ABC is similar to triangle EDC with ratio AB:ED=9:6=3:2. So AC:CE=3:2, and AC=53​×20=12. Trap: matching A with D instead of E, or splitting 20 in the ratio 2:3."},"m4134":{"id":"m4134","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"An angle formed by a diagonal of a regular octagon","tier":3,"traps":["concept"],"answer":"45/2","accept":["45/2","22.5"],"stem":"In regular octagon ABCDEFGH, what is the measure, in degrees, of angle ACB?","explain":"The interior angles of an octagon sum to 180(8−2)=1,080°, so each angle of a regular octagon is 135°. Triangle ABC has AB=BC and angle B=135°, so its other two angles are each 2180−135​=22.5°. Enter 45/2 or 22.5."},"m4135":{"id":"m4135","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"Two regular polygons, one with twice the sides","tier":3,"traps":["setup"],"answer":"12","accept":["12"],"stem":"A regular polygon has n sides, and a second regular polygon has 2n sides. Each interior angle of the second polygon is 15° greater than each interior angle of the first. What is the value of n?","explain":"Work with exterior angles, which are 180° minus the interior angles. The exterior angles are n360​ and 2n360​, and a 15° larger interior angle means a 15° smaller exterior angle: n360​−n180​=15. So n180​=15 and n=12. (Check: a 12-gon has 150° angles and a 24-gon has 165° angles.)"},"m4136":{"id":"m4136","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"A vertex on a line fixed by an area","tier":3,"traps":["setup"],"answer":"B","stem":"In the xy-plane, a triangle has vertices (0,0), (8,0), and (k,2k), where k is a positive constant. The area of the triangle is 40. What is the distance from (0,0) to (k,2k)?","choices":["10","55​","53​","105​"],"explain":"The base from (0,0) to (8,0) has length 8, and the height is the y-coordinate of the third vertex, 2k. So 21​(8)(2k)=8k=40 and k=5. The third vertex is (5,10), at distance 25+100​=125​=55​ from the origin. A is the height alone, and D uses k=10."},"m4137":{"id":"m4137","sec":"math","domain":"Geometry and Trigonometry","skill":"Lines, angles, and triangles","sub":"A median equal to half the side","tier":3,"traps":["concept"],"answer":"55","accept":["55"],"stem":"In triangle ABC, M is the midpoint of BC, and AM=BM. If angle B measures 35°, what is the measure, in degrees, of angle C?","explain":"Since M is the midpoint, BM=CM, so AM=BM=CM. Triangle ABM is isosceles, so angle BAM=35°. Triangle ACM is isosceles too, so angle CAM = angle C. The angles of triangle ABC then give 35+(35+C)+C=180, so C=55°. (In fact angle A=90°: a point that is equally far from B, C, and the midpoint of BC always makes a right angle with B and C.)"},"m4141":{"id":"m4141","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Isosceles triangle from a base-angle cosine","tier":3,"traps":["setup"],"answer":"48","accept":["48"],"stem":"In triangle ABC, AB=AC=10 and cosB=0.6. What is the area of triangle ABC?","explain":"Draw the altitude from A to BC; in an isosceles triangle it meets BC at its midpoint M. In right triangle ABM, BM=ABcosB=10(0.6)=6 and AM=102−62​=8. So BC=12 and the area is 21​(12)(8)=48. Trap: using BM=6 as the whole base, which gives 24."},"m4142":{"id":"m4142","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"A tangent and a perimeter fix the triangle","tier":3,"traps":["setup"],"answer":"120","accept":["120"],"stem":"In right triangle ABC, angle C is the right angle and tanA=125​. The perimeter of the triangle is 60. What is the area of the triangle?","explain":"tanA=ACBC​=125​, so the legs are 5k and 12k for some k>0, and the hypotenuse is (5k)2+(12k)2​=13k. The perimeter is 30k=60, so k=2: the legs are 10 and 24, and the area is 21​(10)(24)=120. Trap: treating 5 and 12 as the actual leg lengths."},"m4143":{"id":"m4143","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Height from two angles of elevation","tier":4,"traps":["setup"],"figure":"
\"A
","answer":"D","stem":"From point A on level ground, the angle of elevation to the top of a tower is 30°. From point B, which is 20 meters closer to the tower along the same straight line, the angle of elevation is 45°. What is the height of the tower, in meters?","choices":["103​","20(3​−1)","203​","10(3​+1)"],"explain":"Let the height be h. From B, the 45° angle makes an isosceles right triangle, so B is h meters from the base. From A, the distance is h+20, and tan30°=h+20h​=3​1​. So h3​=h+20, h(3​−1)=20, and h=3​−120​=220(3​+1)​=10(3​+1)≈27.3. B multiplies by 3​−1 instead of dividing."},"m4144":{"id":"m4144","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Third side of a 60-degree triangle with an altitude","tier":4,"traps":["setup"],"answer":"7","accept":["7"],"stem":"In triangle ABC, AB=8, AC=5, and angle A measures 60°. What is the length of BC?","explain":"Drop an altitude from C to AB at point H. Triangle AHC is a 30-60-90 triangle with hypotenuse 5, so AH=25​ and CH=253​​. Then HB=8−25​=211​, and in right triangle CHB, BC2=(211​)2+(253​​)2=4121​+475​=49. So BC=7."},"m4145":{"id":"m4145","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Inradius from tangent lengths","tier":4,"traps":["concept"],"answer":"3","accept":["3"],"stem":"A circle is inscribed in a right triangle with legs of lengths 9 and 12, so the circle is tangent to all three sides. What is the radius of the circle?","explain":"The hypotenuse is 15. The two tangent segments from any vertex to the circle are equal. At the right angle, the radius drawn to each leg forms a square, so both tangent segments from that vertex have length r. The legs are then split into r+(9−r) and r+(12−r), and the hypotenuse is made of the other two tangent segments: (9−r)+(12−r)=15. So 21−2r=15 and r=3. (Also: area =54=r⋅29+12+15​=18r.)"},"m4146":{"id":"m4146","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Distances to the sides of an equilateral triangle","tier":4,"traps":["concept"],"answer":"B","stem":"Point P lies inside an equilateral triangle with side length 12. What is the sum of the perpendicular distances from P to the three sides of the triangle?","choices":["12","63​","18","43​"],"explain":"Connect P to the three vertices, splitting the triangle into three triangles whose bases are the sides (each 12) and whose heights are the distances d1​, d2​, d3​. Their areas add up to the whole: 21​(12)(d1​+d2​+d3​)=43​​(12)2=363​. So d1​+d2​+d3​=63​, the height of the triangle, wherever P is. D is twice the distance from the center to one side."},"m4147":{"id":"m4147","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Tangent from a cosine in the fourth quadrant","tier":3,"traps":["sign"],"answer":"C","stem":"If cosθ=53​ and 23π​<θ<2π, what is the value of tanθ?","choices":["34​","−43​","−34​","43​"],"explain":"On the unit circle, cosθ=53​ means the point is (53​,±54​). For 23π​<θ<2π the point is in Quadrant IV, where y is negative, so sinθ=−54​. Then tanθ=cosθsinθ​=−34​. A ignores the quadrant, and B inverts the ratio."},"m4148":{"id":"m4148","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Altitude to the hypotenuse with a sine","tier":4,"traps":["setup"],"answer":"15","accept":["15"],"stem":"In right triangle ABC, the right angle is at C, and CD is the altitude to hypotenuse AB. If sinA=53​ and AD=16, what is the length of BC?","explain":"Triangle ADC has a right angle at D and shares angle A with triangle ABC. Since sinA=53​, cosA=54​. In triangle ADC, cosA=ACAD​, so AC=4/516​=20. In triangle ABC, tanA=ACBC​=43​, so BC=15. (Check: AB=25 and AD=ABAC2​=25400​=16.) Trap: applying sinA to triangle ADC as if AD were opposite angle A."},"m4149":{"id":"m4149","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Rectangle from a diagonal and an angle","tier":3,"traps":["setup"],"answer":"A","stem":"The diagonal of a rectangle has length 10 and makes a 30° angle with the longer side of the rectangle. What is the area of the rectangle?","choices":["253​","503​","50","25"],"explain":"The diagonal is the hypotenuse of a 30-60-90 right triangle whose legs are the sides of the rectangle. The side opposite the 30° angle is 10×21​=5, and the longer side is 53​. The area is 5×53​=253​≈43.3. B doubles the area of the rectangle; D uses only the short side."},"m4150":{"id":"m4150","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"A ladder that slides from one angle to another","tier":3,"traps":["partial"],"answer":"A","stem":"A ladder leans against a vertical wall, making a 60° angle with the level ground, and its top is 63​ feet above the ground. The bottom of the ladder slides directly away from the wall until the ladder makes a 45° angle with the ground. How many feet did the bottom of the ladder move?","choices":["62​−6","63​−62​","63​−6","6"],"explain":"At 60°, the ladder is the hypotenuse of a 30-60-90 triangle whose long leg is 63​, so the ladder is 12 feet long and its bottom is 6 feet from the wall. At 45°, the bottom is 12cos45°=62​ feet from the wall. It moved 62​−6≈2.49 feet. B is how far the top slid down the wall."},"m4151":{"id":"m4151","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Two special triangles on one diagonal","tier":3,"traps":["setup"],"answer":"B","stem":"In quadrilateral ABCD, diagonal AC has length 12. Triangle ABC is an isosceles right triangle with its right angle at B. Triangle ACD is a right triangle with its right angle at D, and angle CAD measures 30°. Points B and D are on opposite sides of AC. What is the area of quadrilateral ABCD?","choices":["72+363​","36+183​","36+363​","18+183​"],"explain":"In triangle ABC, the hypotenuse is 12, so each leg is 2​12​=62​ and the area is 21​(62​)2=36. In triangle ACD, the hypotenuse is 12, so CD=6 (opposite the 30° angle) and AD=63​, for an area of 21​(6)(63​)=183​. The quadrilateral is the two triangles together: 36+183​. C forgets the 21​ for triangle ACD."},"m4152":{"id":"m4152","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Area from a product of sine and cosine","tier":4,"traps":["concept"],"answer":"12","accept":["12"],"stem":"A right triangle has a hypotenuse of length 10, and one of its acute angles, θ, satisfies sinθcosθ=0.24. What is the area of the triangle?","explain":"The legs are 10sinθ (opposite θ) and 10cosθ (adjacent to θ). The area is 21​(10sinθ)(10cosθ)=50sinθcosθ=50(0.24)=12. There's no need to find θ or the legs themselves."},"m4153":{"id":"m4153","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"A sine twice a cosine","tier":3,"traps":["concept"],"answer":"1/2","accept":["1/2","0.5",".5"],"stem":"In right triangle ABC, angle C is the right angle and sinA=2cosA. What is the value of tanB?","explain":"Dividing by cosA gives tanA=cosAsinA​=2. Angles A and B are complementary, and the leg opposite A is adjacent to B, so tanB=tanA1​=21​. Enter 1/2 or .5. Trap: answering 2, which is tanA."},"m4154":{"id":"m4154","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"Two angles of depression from one height","tier":3,"traps":["setup"],"answer":"C","stem":"From the top of a lighthouse 30 meters above sea level, the angles of depression to two boats that are directly in line with the lighthouse are 45° and 30°. How far apart are the two boats, in meters?","choices":["303​","30−103​","303​−30","103​"],"explain":"An angle of depression from the top equals the angle of elevation from the boat. For the 45° boat, the horizontal distance is 30. For the 30° boat, tan30°=d30​, so d=1/3​30​=303​≈52.0. The boats are 303​−30≈22.0 meters apart. B and D multiply by tan30° instead of dividing."},"m4155":{"id":"m4155","sec":"math","domain":"Geometry and Trigonometry","skill":"Right triangles and trigonometry","sub":"A 60-degree line inside a square","tier":3,"traps":["setup"],"answer":"D","stem":"Square ABCD has side length 8. Point E lies on CD so that angle EAB measures 60°. What is the area of triangle ADE?","choices":["323​","3163​​","3643​​","3323​​"],"explain":"Since angle DAB is 90°, angle DAE=90−60=30°. Triangle ADE has a right angle at D, with AD=8 adjacent to the 30° angle, so DE=8tan30°=3​8​=383​​. The area is 21​(8)(383​​)=3323​​≈18.5. A uses DE=83​, which would put E past C; C forgets the 21​."},"m4161":{"id":"m4161","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Center on a line, tangent to an axis, through a point","tier":4,"traps":["partial","setup"],"answer":"28","accept":["28"],"stem":"A circle in the xy-plane is tangent to the x-axis, its center lies on the line y=2x, and it passes through the point (3,2). What is the sum of all possible values of the circle's radius?","explain":"The center is (a,2a), and because the circle is tangent to the x-axis, the radius is the center's distance from that axis, ∣2a∣. The point (3,2) is on the circle: (3−a)2+(2−2a)2=4a2. Expanding: 9−6a+a2+4−8a+4a2=4a2, so a2−14a+13=0 and a=1 or a=13. The radii are 2 and 26, with a sum of 28. Trap: stopping after one solution."},"m4162":{"id":"m4162","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Radius from an inscribed angle and its arc","tier":3,"traps":["concept"],"answer":"18","accept":["18"],"stem":"In a circle, an inscribed angle measuring 35° intercepts an arc of length 7π. What is the radius of the circle?","explain":"An inscribed angle is half of the central angle on the same arc, so the arc measures 70°. Its length is 36070​⋅2πr=187πr​. Setting this equal to 7π gives r=18. Trap: using 35° as the arc's measure, which gives r=36."},"m4163":{"id":"m4163","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Sector from its perimeter and area","tier":4,"traps":["partial"],"answer":"10","accept":["10"],"stem":"A sector of a circle has a perimeter of 20 (the two radii plus the arc) and an area of 24. What is the sum of all possible values of the radius of the circle?","explain":"With radius r and central angle θ in radians, the arc is rθ and the area is 21​r2θ. The perimeter gives rθ=20−2r, so the area is 21​r(20−2r)=10r−r2=24. Then r2−10r+24=0, so r=4 or r=6. Both work: r=4 gives θ=3 and r=6 gives θ=34​, and each is less than 2π. The sum is 10."},"m4164":{"id":"m4164","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Tangent segment from a point on an axis","tier":3,"traps":["concept"],"answer":"10","accept":["10"],"stem":"In the xy-plane, a line through the point (a,0), where a>6, is tangent to the circle x2+y2=36 at the point T. If the distance from (a,0) to T is 8, what is the value of a?","explain":"A tangent line is perpendicular to the radius at the point of tangency, so the origin, T, and (a,0) form a right triangle with the right angle at T. Its legs are the radius 6 and the tangent segment 8, and its hypotenuse is a: a=36+64​=10."},"m4165":{"id":"m4165","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Circle through three points with a right angle","tier":3,"traps":["concept"],"answer":"5","accept":["5"],"stem":"A circle in the xy-plane passes through the points (0,0), (8,0), and (0,6). What is the radius of the circle?","explain":"The segments from (0,0) to the other two points lie along the axes, so they meet at a right angle. A right angle inscribed in a circle intercepts a semicircle, so the segment from (8,0) to (0,6) is a diameter, with length 64+36​=10. The radius is 5, and the center is the midpoint (4,3)."},"m4166":{"id":"m4166","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Completing the square after dividing out a coefficient","tier":3,"traps":["setup"],"answer":"4","accept":["4"],"stem":"In the xy-plane, the graph of 2x2+2y2−12x+8y=6 is a circle. What is the radius of the circle?","explain":"Divide every term by 2 first: x2+y2−6x+4y=3. Completing the square: (x−3)2+(y+2)2=3+9+4=16, so the radius is 4. Trap: completing the square without dividing by 2, which gives 6+9+4=19 and a radius of 19​."},"m4167":{"id":"m4167","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"A chord of one circle tangent to another","tier":4,"traps":["concept"],"figure":"
\"Two
","answer":"B","stem":"Two circles have the same center. A chord of the larger circle is tangent to the smaller circle, and the chord has length 16. What is the area of the region between the two circles?","choices":["256π","64π","16π","32π"],"explain":"Let the radii be R and r. The radius of the small circle to the point of tangency is perpendicular to the chord and bisects it, so it splits the chord into halves of length 8. The center, the point of tangency, and an end of the chord form a right triangle: r2+82=R2. The area between the circles is πR2−πr2=π(R2−r2)=64π, even though neither radius is known. A uses the whole chord, 16, as the leg."},"m4168":{"id":"m4168","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Opposite angles of an inscribed quadrilateral","tier":3,"traps":["concept"],"answer":"100","accept":["100"],"stem":"Quadrilateral ABCD is inscribed in a circle. Angle A measures (3x+10)°, angle B measures (x+50)°, and angle C measures (2x+20)°. What is the measure, in degrees, of angle D?","explain":"Angles A and C are inscribed angles that intercept arcs making up the whole circle, so A+C=21​(360°)=180°. Then 5x+30=180, so x=30. Angle B=30+50=80°, and since B and D are also opposite angles, D=180−80=100°. Trap: setting A+B=180; the supplementary pairs are the opposite angles."},"m4169":{"id":"m4169","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Common tangent of two touching circles","tier":4,"traps":["setup"],"answer":"12","accept":["12"],"stem":"Two circles with radii 4 and 9 are tangent to each other externally. A line is tangent to both circles, at points P and Q, and both circles lie on the same side of the line. What is the length of PQ​?","explain":"The centers are 4+9=13 apart. The radii to P and Q are both perpendicular to the tangent line, so they are parallel, and the centers with P and Q form a trapezoid with two right angles. Drawing a segment from the smaller circle's center parallel to PQ​ creates a right triangle with hypotenuse 13 and one leg 9−4=5. The other leg equals PQ: 132−52​=12. Trap: using the sum of the radii, 13, as the leg instead of their difference."},"m4170":{"id":"m4170","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Square inscribed in a semicircle","tier":4,"traps":["setup"],"figure":"
\"A
","answer":"20","accept":["20"],"stem":"A square is inscribed in a semicircle of radius 5, with one side of the square on the diameter and its other two vertices on the arc. What is the area of the square?","explain":"By symmetry the square is centered on the diameter. If its side is s, a top corner is 2s​ from the center horizontally and s above the diameter. That corner is on the arc, so (2s​)2+s2=52, giving 45s2​=25 and s2=20. The area is 20. Trap: placing a corner at horizontal distance s instead of 2s​, which gives s2=12.5."},"m4171":{"id":"m4171","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Region between two tangents and an arc","tier":4,"traps":["setup","partial"],"figure":"
\"A
","answer":"A","stem":"From point P outside a circle with center O and radius 6, two lines are drawn tangent to the circle at points A and B. If OP=12, what is the area of the shaded region bounded by PA, PB, and the minor arc AB?","choices":["363​−12π","363​−24π","183​−6π","723​−12π"],"explain":"The radii OA and OB are perpendicular to the tangents. In right triangle OAP, the hypotenuse is 12 and the leg OA is 6, so angle AOP=60° and PA=63​. The kite OAPB is two such triangles, with area 2⋅21​(6)(63​)=363​. Angle AOB=120°, so the sector OAB has area 360120​π(6)2=12π. The shaded region is the kite minus the sector: 363​−12π≈24.7. B subtracts the major sector instead."},"m4172":{"id":"m4172","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Slope of a tangent line through the origin","tier":4,"traps":["concept"],"answer":"C","stem":"In the xy-plane, the line y=mx, where m is a positive constant, is tangent to the circle (x−4)2+y2=4. What is the value of m?","choices":["3​","21​","33​​","22​​"],"explain":"Let T be the point of tangency and C=(4,0) the center. The radius CT is perpendicular to the tangent line, so triangle OTC has a right angle at T, hypotenuse OC=4, and leg CT=2. The angle at the origin therefore has sin=42​=21​, so it measures 30°, and the slope is tan30°=33​​. B uses the sine of the angle as the slope; A uses the 60° angle at C."},"m4173":{"id":"m4173","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Arc length from a chord","tier":3,"traps":["partial"],"answer":"D","stem":"A chord of length 63​ is drawn in a circle with radius 6. What is the length of the minor arc cut off by the chord?","choices":["2π","8π","6π","4π"],"explain":"The perpendicular from the center to the chord bisects it, making a right triangle with hypotenuse 6 and half-chord 33​. Then sin(half-angle)=633​​=23​​, so the half-angle is 60° and the central angle is 120°. The minor arc is 360120​(2π⋅6)=4π. B is the major arc, and A uses the 60° half-angle as the central angle."},"m4174":{"id":"m4174","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Two equal circles whose intersections are a set distance apart","tier":3,"traps":["setup"],"answer":"6","accept":["6"],"stem":"In the xy-plane, the circles x2+y2=25 and (x−d)2+y2=25, where d is a positive constant, intersect at two points that are 8 units apart. What is the value of d?","explain":"Subtracting the equations: x2−(x−d)2=0, so 2dx−d2=0 and x=2d​. The intersection points are (2d​,±y), which are 2y apart, so y=4. Then (2d​)2+16=25, 2d​=3, and d=6."},"m4175":{"id":"m4175","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Triangle on a diameter","tier":3,"traps":["concept"],"answer":"120","accept":["120"],"stem":"Segment AB is a diameter of a circle with radius 13. Point C lies on the circle, and AC=10. What is the area of triangle ABC?","explain":"An angle inscribed in a semicircle is a right angle, so angle ACB=90°. The hypotenuse is the diameter, 26, so BC=262−102​=24. The area is 21​(10)(24)=120. Trap: using the radius, 13, as the hypotenuse."},"m4176":{"id":"m4176","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Circle through two points and tangent to an axis","tier":4,"traps":["setup"],"answer":"5","accept":["5"],"stem":"A circle in the xy-plane passes through the points (1,1) and (7,1) and is tangent to the x-axis. Its center lies above the x-axis. What is the radius of the circle?","explain":"The center is equally far from (1,1) and (7,1), so it lies on the perpendicular bisector x=4: the center is (4,k). Tangency to the x-axis makes the radius k. Then (1−4)2+(1−k)2=k2, so 9+1−2k=0 and k=5. The radius is 5."},"m4177":{"id":"m4177","sec":"math","domain":"Geometry and Trigonometry","skill":"Circles","sub":"Area between a chord and its arc","tier":3,"traps":["partial"],"answer":"A","stem":"Points A and B lie on a circle with center O and radius 6, and angle AOB measures 60°. What is the area of the region between chord AB and the minor arc AB?","choices":["6π−93​","6π−183​","12π−93​","6π−9"],"explain":"The sector OAB has area 36060​π(6)2=6π. Triangle OAB has two sides of length 6 with a 60° angle between them, so it is equilateral, with area 43​​(6)2=93​. The region is the sector minus the triangle: 6π−93​≈3.26. B forgets the 21​ in the triangle's area; C uses a 120° sector."},"m1101":{"id":"m1101","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Infinitely many solutions, with a tempting second value","tier":3,"traps":["concept","partial"],"answer":"6","accept":["6"],"stem":"In the system below, k is a constant.

kx+3y=6 12x+ky=12

For what value of k does the system have infinitely many solutions?","explain":"Infinitely many solutions means the second equation is a multiple of the first, so all three ratios match: 12k​=k3​=126​. From k3​=21​, k=6, and then 12k​=21​ too. Trap: solving only 12k​=k3​, which gives k=±6. With k=−6 the x- and y-ratios are −21​ but the constants' ratio is 21​, so the lines are parallel and there is no solution."},"m1102":{"id":"m1102","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Undoing a self-composition","tier":3,"traps":["partial"],"answer":"7","accept":["7"],"stem":"The function f is linear and increasing, and f(f(x))=4x+9 for all values of x. What is the value of f(2)?","explain":"Write f(x)=ax+b. Then f(f(x))=a(ax+b)+b=a2x+ab+b. Matching: a2=4 and b(a+1)=9. Since f is increasing, a=2, so 3b=9 and b=3. Then f(2)=2(2)+3=7. Trap: a=−2 also gives a2=4 (with b=−9), but that function is decreasing."},"m1103":{"id":"m1103","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"Two linear functions that commute","tier":4,"traps":["partial","setup"],"answer":"1","accept":["1"],"stem":"The functions f and g are defined by f(x)=ax+6 and g(x)=2x+a, where a is a constant. If f(g(x))=g(f(x)) for all values of x, what is the sum of all possible values of a?","explain":"f(g(x))=a(2x+a)+6=2ax+a2+6 and g(f(x))=2(ax+6)+a=2ax+12+a. The x-terms already match, so the constants must too: a2+6=12+a, or a2−a−6=0. Then (a−3)(a+2)=0, so a=3 or a=−2, with a sum of 1."},"m1104":{"id":"m1104","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Intercepts in a ratio, including the case through the origin","tier":4,"traps":["partial","concept"],"answer":"7/6","accept":["7/6","1.166","1.167"],"stem":"A line in the xy-plane passes through the point (2,3), and its x-intercept is 3 times its y-intercept. What is the sum of all possible slopes of the line?","explain":"If the intercepts are not zero, the line is 3bx​+by​=1, where b is the y-intercept. Substituting (2,3): 3b2​+b3​=1, so 3b11​=1 and b=311​. This line has slope −3bb​=−31​. There is a second line: the one through the origin, whose intercepts are both 0, and 0=3×0. It passes through (2,3), so its slope is 23​. The sum is −31​+23​=67​. Enter 7/6, 1.166, or 1.167. Trap: missing the line through the origin."},"m1105":{"id":"m1105","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Two equations, three unknowns","tier":3,"traps":["setup"],"answer":"6","accept":["6"],"stem":"If 2a+3b+c=20 and a+2b+c=14, what is the value of a+b?","explain":"Subtract the second equation from the first: (2a−a)+(3b−2b)+(c−c)=20−14, so a+b=6. The individual values of a, b, and c can't be found from two equations, and they aren't needed."},"m1106":{"id":"m1106","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Counting lattice points in a region","tier":4,"traps":["partial"],"answer":"16","accept":["16"],"stem":"How many points (x,y), where x and y are both integers, satisfy both y>∣x∣−3 and y<2?","explain":"For each integer x, the integer y-values run from ∣x∣−2 up to 1, which is 1−(∣x∣−2)+1=4−∣x∣ values, as long as that is positive. For x=0: 4 points. For x=±1: 3 each. For x=±2: 2 each. For x=±3: 1 each (only y=1). For ∣x∣≥4: none. The total is 4+6+4+2=16. Trap: counting y=2 or y=∣x∣−3, which the strict inequalities exclude."},"m1107":{"id":"m1107","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"An inequality true for every nonnegative x","tier":4,"traps":["concept"],"answer":"D","stem":"The inequality ax+5≤3x+b, where a and b are constants, is true for every value of x that is greater than or equal to 0. Which of the following must be true?","choices":["a≥3 and b≥5","a≤3 and b≤5","a=3 and b=5","a≤3 and b≥5"],"explain":"Rewrite the inequality as (a−3)x≤b−5. At x=0 it says 0≤b−5, so b≥5. If a−3 were positive, the left side would grow without bound as x increases and eventually exceed b−5, so a−3≤0, that is, a≤3. Both conditions must hold. C is one case that works, but it doesn't have to be the case: a=0, b=6 also works."},"m1108":{"id":"m1108","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"A later start, a faster pace, a lead","tier":3,"traps":["setup","partial"],"answer":"170","accept":["170"],"stem":"Car A leaves a town at 9:00 a.m. and travels at a constant 50 miles per hour. Car B leaves the same town at 9:30 a.m. and travels the same route at a constant 65 miles per hour. How many minutes after 9:00 a.m. will car B be 10 miles ahead of car A?","explain":"Let t be the number of hours after 9:00 a.m. Car A has gone 50t miles and car B, which has driven for t−0.5 hours, has gone 65(t−0.5) miles. Car B is 10 miles ahead when 65(t−0.5)−50t=10: 15t−32.5=10, so t=1542.5​=617​ hours, which is 170 minutes. Trap: finding when car B catches up (130 minutes) and stopping there."},"m1109":{"id":"m1109","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Drain and replace to raise a concentration","tier":3,"traps":["setup"],"answer":"8","accept":["8"],"stem":"A radiator holds 40 liters of a mixture that is 25% antifreeze. Some of the mixture is drained and replaced with the same amount of pure antifreeze, so the radiator again holds 40 liters, now 40% antifreeze. How many liters were drained?","explain":"Let x be the number of liters drained. The radiator starts with 0.25(40)=10 liters of antifreeze. Draining removes 0.25x liters of antifreeze, and adding pure antifreeze puts back x liters. The target is 0.40(40)=16 liters: 10−0.25x+x=16, so 0.75x=6 and x=8. Trap: ignoring the antifreeze that leaves with the drained mixture, which gives 6."},"m1110":{"id":"m1110","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"A rate raised by a percent, fee unchanged","tier":3,"traps":["setup"],"answer":"75","accept":["75"],"stem":"A plumber charges a fixed fee plus an hourly rate. A 3-hour job costs $255. After the plumber raises the hourly rate by 20% and keeps the fixed fee the same, a 3-hour job costs $291. What is the fixed fee, in dollars?","explain":"The fee is the same both times, so the $36 increase comes entirely from the raise on 3 hours of work: 3(0.2r)=36, so r=60. Then F+3(60)=255 gives F=75. Trap: applying the 20% raise to the whole bill."},"m1111":{"id":"m1111","sec":"math","domain":"Algebra","skill":"Systems of two linear equations in two variables","sub":"Infinitely many solutions with a constant in two places","tier":4,"traps":["concept","partial"],"answer":"11","accept":["11"],"stem":"In the system below, a and b are constants and a>0.

ax+4y=8 3x+(a+1)y=b

If the system has infinitely many solutions, what is the value of a+b?","explain":"The equations must be multiples of each other: 3a​=a+14​=b8​. From the first two, a(a+1)=12, so a2+a−12=0 and a=3 or a=−4. Since a>0, a=3, and the common ratio is 33​=1. Then b8​=1, so b=8, and a+b=11."},"m2101":{"id":"m2101","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"A composition that returns its input","tier":4,"traps":["setup"],"answer":"23","accept":["23"],"stem":"The function f is defined by f(x)=x2−6x+c, where c is a constant. If f(f(3))=3, what is the sum of all possible values of c?","explain":"First, f(3)=9−18+c=c−9. Then f(c−9)=(c−9)2−6(c−9)+c=c2−18c+81−6c+54+c=c2−23c+135. Setting this equal to 3: c2−23c+132=0, so (c−11)(c−12)=0. The values are 11 and 12, with a sum of 23."},"m2102":{"id":"m2102","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"a + b + c as a function value","tier":3,"traps":["concept"],"answer":"34/9","accept":["34/9","3.777","3.778"],"stem":"The graph of y=ax2+bx+c, where a, b, and c are constants, has its vertex at (3,2) and a y-intercept of 6. What is the value of a+b+c?","explain":"The sum a+b+c is the value of y when x=1. In vertex form, y=a(x−3)2+2. The y-intercept gives 9a+2=6, so a=94​. At x=1: y=94​(4)+2=916​+918​=934​. Enter 34/9, 3.777, or 3.778."},"m2103":{"id":"m2103","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Yearly rate from a tripling time","tier":3,"traps":["concept"],"answer":"B","stem":"A population triples every 8 years. Which of the following is closest to the percent increase in the population each year?","choices":["3.0%","14.7%","25%","37.5%"],"explain":"If the yearly growth factor is g, then 8 years of growth multiply the population by g8=3, so g=31/8≈1.147, an increase of about 14.7% each year. C divides 200% by 8 as if the growth added the same amount each year, and D divides 300% by 8."},"m2104":{"id":"m2104","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Remainder after dividing by a quadratic","tier":4,"traps":["concept"],"answer":"A","stem":"When the polynomial p(x) is divided by x−2, the remainder is 5. When p(x) is divided by x+1, the remainder is −4. What is the remainder when p(x) is divided by (x−2)(x+1)?","choices":["3x−1","3x+1","x+3","9"],"explain":"Dividing by a quadratic leaves a remainder of degree less than 2, so p(x)=(x−2)(x+1)q(x)+ax+b for some polynomial q. The remainder theorem gives p(2)=5 and p(−1)=−4. At those inputs the first term is 0, so 2a+b=5 and −a+b=−4. Subtracting, 3a=9, so a=3 and b=−1. The remainder is 3x−1."},"m2105":{"id":"m2105","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Distance between zeros from the vertex","tier":3,"traps":["concept"],"answer":"6","accept":["6"],"stem":"The function f is defined by f(x)=(x−a)(x−b), where a and b are constants. The vertex of the graph of y=f(x) is (5,−9). What is the value of ∣a−b∣?","explain":"The zeros a and b are symmetric about the vertex's x-coordinate, so they are 5−d and 5+d for some d≥0. Then f(5)=(5−(5−d))(5−(5+d))=(d)(−d)=−d2=−9, so d=3. The zeros are 2 and 8, and ∣a−b∣=6."},"m2106":{"id":"m2106","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"One root four times the other","tier":3,"traps":["partial"],"answer":"10","accept":["10"],"stem":"The equation x2−kx+16=0, where k is a positive constant, has two real solutions, one of which is 4 times the other. What is the value of k?","explain":"Let the solutions be r and 4r. Their product is 4r2=16, so r=2 or r=−2. Their sum is 5r=k, and k is positive, so r=2 and k=10. (The solutions are 2 and 8.)"},"m2107":{"id":"m2107","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Counting integer factorizations","tier":3,"traps":["partial"],"answer":"8","accept":["8"],"stem":"For how many integer values of b can x2+bx+24 be written as (x+m)(x+n), where m and n are integers?","explain":"(x+m)(x+n)=x2+(m+n)x+mn, so mn=24 and b=m+n. The positive factor pairs of 24 are 1×24, 2×12, 3×8, and 4×6, giving b=25,14,11,10. The negative pairs give b=−25,−14,−11,−10. That's 8 values. Trap: forgetting the negative factor pairs."},"m2108":{"id":"m2108","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"A line through the origin tangent to a parabola","tier":3,"traps":["concept"],"answer":"B","stem":"In the xy-plane, the line y=mx, where m is a positive constant, intersects the graph of y=x2+2 at exactly one point. What is the value of m?","choices":["2​","22​","8","4"],"explain":"Setting x2+2=mx gives x2−mx+2=0. Exactly one intersection means the discriminant is 0: m2−4(1)(2)=0, so m2=8 and m=22​. C is m2, not m."},"m2109":{"id":"m2109","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"A circle and a parabola meeting three times","tier":4,"traps":["concept"],"answer":"-5","accept":["-5"],"stem":"In the xy-plane, the graphs of x2+y2=25 and y=x2+k, where k is a constant, intersect at exactly three points. What is the value of k?","explain":"Both graphs are symmetric about the y-axis, so intersection points come in pairs (±x,y) unless x=0. An odd number of points means (0,k), the parabola's vertex, is on the circle: k2=25, so k=5 or k=−5. Check each. Substituting x2=y−k into the circle gives y2+y−(25+k)=0. For k=5: y=5 or y=−6; y=−6 is below the vertex, so the only point is (0,5). For k=−5: y=−5 (the vertex) or y=4, which gives x2=9 and two more points. So k=−5."},"m2110":{"id":"m2110","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Projecting an exponential pattern forward","tier":3,"traps":["concept"],"answer":"2030","accept":["2030"],"stem":"A town's population grows exponentially. It was 2,000 in 2010 and 4,500 in 2020. In what year will the population reach 10,125?","explain":"In 10 years the population was multiplied by 2,0004,500​=2.25. From 4,500 to 10,125 is another factor of 4,50010,125​=2.25, which takes another 10 years. So the population reaches 10,125 in 2030. Trap: treating the growth as linear (adding 2,500 per decade), which puts it in 2042 or 2043."},"m2111":{"id":"m2111","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Rewriting two powers in terms of one","tier":3,"traps":["setup"],"answer":"50","accept":["50"],"stem":"If 32x=5, what is the value of 9x+1+272x/3?","explain":"Rewrite both terms with 32x. First, 9x+1=9⋅9x=9⋅32x=9(5)=45. Second, 272x/3=(33)2x/3=32x=5. The sum is 50. Trap: writing 9x+1 as 9x+9."},"m2112":{"id":"m2112","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"A constant that makes the fraction linear","tier":3,"traps":["concept"],"answer":"7","accept":["7"],"stem":"The expression x+2x2+kx+6​, where k is a constant, is equivalent to a linear expression for all x=−2. What is the value of that linear expression when x=4?","explain":"For the fraction to simplify to a linear expression, x+2 must be a factor of the numerator, so the numerator is 0 at x=−2: 4−2k+6=0, giving k=5. Then x2+5x+6=(x+2)(x+3), and the expression is x+3, which equals 7 at x=4. Trap: substituting x=4 before finding k."},"m2113":{"id":"m2113","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Factored form with a known maximum","tier":3,"traps":["setup"],"answer":"24","accept":["24"],"stem":"The function f is defined by f(x)=a(x−2)(x−8), where a is a constant. The maximum value of f is 27. What is the value of f(4)?","explain":"The vertex is halfway between the zeros, at x=5, so f(5)=a(3)(−3)=−9a=27 and a=−3. Then f(4)=−3(2)(−4)=24."},"m2114":{"id":"m2114","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Half-life at a non-multiple of the period","tier":3,"traps":["concept"],"answer":"B","stem":"A radioactive substance loses half of its mass every 6 hours. Which of the following is closest to the percent of the original mass that remains after 15 hours?","choices":["12.5%","17.7%","18.75%","25%"],"explain":"After t hours the fraction remaining is (21​)t/6. For t=15: (21​)2.5=42​1​≈0.177, or about 17.7%. As a check, 15 hours is between 2 half-lives (25%) and 3 half-lives (12.5%), which rules out A and D. C draws a straight line from 25% at 12 hours to 12.5% at 18 hours, but the decay curve bends below that line."},"m2115":{"id":"m2115","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Sum and product hidden in exponents","tier":3,"traps":["setup"],"answer":"80","accept":["80"],"stem":"For x>1, xa⋅xb=x12 and (xa)b=x32, where a and b are constants. What is the value of a2+b2?","explain":"Multiplying powers adds exponents and raising a power to a power multiplies them, so a+b=12 and ab=32. Then a2+b2=(a+b)2−2ab=144−64=80. (The values are 4 and 8.)"},"m2116":{"id":"m2116","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"Zeros after a horizontal compression","tier":3,"traps":["concept"],"answer":"1","accept":["1"],"stem":"The zeros of the function f are −4 and 6, and f has no other zeros. The function g is defined by g(x)=f(2x)+1. What is the sum of all values of x for which g(x)=1?","explain":"g(x)=1 exactly when f(2x)=0, which happens when 2x=−4 or 2x=6. So x=−2 or x=3, with a sum of 1. Trap: doubling the zeros instead of halving them, which gives 4."},"m3101":{"id":"m3101","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Dilute, then strengthen","tier":4,"traps":["setup"],"answer":"36","accept":["36"],"stem":"A chemist has 30 liters of a solution that is 40% acid. She adds pure water until the solution is 25% acid. She then adds a solution that is 60% acid until the mixture is 40% acid again. How many liters of the 60% solution does she add?","explain":"The original solution contains 0.40(30)=12 liters of acid. Adding water doesn't change the acid, so at 25% the total volume is 0.2512​=48 liters. Adding y liters of the 60% solution: 48+y12+0.6y​=0.40, so 12+0.6y=19.2+0.4y, 0.2y=7.2, and y=36. Trap: forgetting that the water raised the volume to 48 liters; with 30 liters the equation gives y=0."},"m3102":{"id":"m3102","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Work rates with a late start","tier":3,"traps":["setup"],"answer":"12","accept":["12"],"stem":"Working together, Ana and Ben can paint a fence in 6 hours. One day, Ana paints alone for 4 hours, and then Ben joins her and they finish the fence together in 4 more hours. How many hours would it take Ana to paint the fence alone?","explain":"Let Ana paint a of the fence per hour and Ben b. Together, a+b=61​. On the second day, Ana works 8 hours and Ben 4 hours: 8a+4b=1. Since 4a+4b=64​=32​, the extra 4a is 1−32​=31​, so a=121​. Ana alone takes 12 hours."},"m3103":{"id":"m3103","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"The missing decrease in three changes","tier":3,"traps":["concept"],"answer":"27.27","accept":["27.27"],"stem":"A price is increased by 25%, then decreased by x%, and then increased by 10%. The final price equals the original price. What is the value of x, to the nearest hundredth?","explain":"Each change multiplies the price: 1.25(1−100x​)(1.10)=1. So 1−100x​=1.3751​≈0.72727, and x≈27.27. Trap: adding the percents (25+10=35), which ignores that each change applies to a different amount."},"m3104":{"id":"m3104","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Comparing two markup-then-discount prices","tier":3,"traps":["concept"],"answer":"5","accept":["5"],"stem":"Two stores buy the same jacket at the same wholesale price. Store A raises the wholesale price by 40% and then offers 20% off that price. Store B raises the wholesale price by 30% and then offers 10% off that price. The final price at store B is greater than the final price at store A by what percent of the wholesale price?","explain":"Store A's price is 1.40×0.80=1.12 times the wholesale price, and store B's is 1.30×0.90=1.17 times. The difference is 0.05 of the wholesale price, or 5%. Trap: computing the difference as a percent of store A's price, 1.120.05​≈4.5%."},"m3105":{"id":"m3105","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Group sizes from a combined mean","tier":3,"traps":["setup"],"answer":"2","accept":["2"],"stem":"The mean age of a group of doctors and lawyers is 40. The mean age of the doctors is 35, and the mean age of the lawyers is 50. What is the ratio of the number of doctors to the number of lawyers?","explain":"Let there be d doctors and ℓ lawyers. The total age is 35d+50ℓ=40(d+ℓ), so 10ℓ=5d and ℓd​=2. (The combined mean is twice as close to the doctors' mean as to the lawyers', so there are twice as many doctors.) Trap: inverting the ratio and answering 21​."},"m3106":{"id":"m3106","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Which factory made the defective phone","tier":4,"traps":["concept"],"answer":"5/8","accept":["5/8","0.625",".625"],"stem":"A company makes 40% of its phones at factory X and 60% at factory Y. Of the phones made at factory X, 5% are defective; of those made at factory Y, 2% are defective. If a phone chosen at random is defective, what is the probability that it was made at factory X?","explain":"Picture 1,000 phones: 400 from X, of which 0.05(400)=20 are defective, and 600 from Y, of which 0.02(600)=12 are defective. Of the 32 defective phones, 20 came from X, so the probability is 3220​=85​. Enter 5/8 or .625. Trap: answering 0.05, the chance that a phone from X is defective, which reverses the condition."},"m3107":{"id":"m3107","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Two draws of the same color","tier":3,"traps":["concept"],"answer":"13/28","accept":["13/28",".4642",".4643","0.464"],"stem":"A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random, one after the other, without replacement. What is the probability that both marbles are the same color?","explain":"Both red: 85​⋅74​=5620​. Both blue: 83​⋅72​=566​. Together: 5626​=2813​. Enter 13/28, .4642, or .4643. Trap: drawing with replacement, 6425​+649​=3217​."},"m3108":{"id":"m3108","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Drops to liters over a week","tier":3,"traps":["units"],"answer":"10.08","accept":["10.08"],"stem":"A leaky faucet drips 1 drop every 3 seconds, and 20 drops make 1 milliliter. How many liters of water does the faucet drip in one week?","explain":"A week has 7×24×60×60=604,800 seconds, so the faucet drips 3604,800​=201,600 drops. That is 20201,600​=10,080 milliliters, or 10.08 liters. Trap: stopping at milliliters."},"m3109":{"id":"m3109","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Average speed over three equal distances","tier":3,"traps":["concept"],"answer":"40","accept":["40"],"stem":"A driver covers the first third of a trip at 30 miles per hour, the second third at 40 miles per hour, and the last third at 60 miles per hour. What is the driver's average speed for the whole trip, in miles per hour?","explain":"Average speed is total distance divided by total time. Say the trip is 120 miles (any length works). Each third is 40 miles, taking 3040​+4040​+6040​=34​+1+32​=3 hours. The average speed is 3120​=40 miles per hour. Trap: averaging the three speeds, 330+40+60​≈43.3; the driver spends more time at the slower speeds."},"m3110":{"id":"m3110","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Reversing a percent across two groups","tier":3,"traps":["concept"],"answer":"48","accept":["48"],"stem":"In a class, 60% of the students are girls. Of the girls, 25% walk to school; of the boys, 40% walk to school. To the nearest whole percent, what percent of the students who walk to school are girls?","explain":"Take 100 students: 60 girls, of whom 15 walk, and 40 boys, of whom 16 walk. Of the 31 walkers, 15 are girls: 3115​≈48.4%, which rounds to 48. Trap: answering 25, the percent of girls who walk; the question asks what percent of walkers are girls."},"m3111":{"id":"m3111","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Equal yearly percent from a two-year total","tier":3,"traps":["concept"],"answer":"25","accept":["25"],"stem":"A town's population grew from 40,000 to 62,500 over two years, increasing by the same percent each year. What was the percent increase each year?","explain":"If the yearly growth factor is g, then 40,000g2=62,500, so g2=1.5625 and g=1.25: a 25% increase each year. Trap: splitting the 56.25% total increase evenly, which gives 28.125%."},"m3112":{"id":"m3112","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Percent change of a ratio","tier":3,"traps":["concept"],"answer":"25","accept":["25"],"stem":"A city's land area increased by 20% after it annexed nearby land, and over the same period its population increased by 50%. By what percent did the city's population density, in people per square mile, increase?","explain":"Density is population divided by area. The new density is 1.20A1.50P​=1.25⋅AP​, a 25% increase. Trap: subtracting the percents, 50−20=30."},"m3113":{"id":"m3113","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Largest value under mean, median, and mode constraints","tier":4,"traps":["concept","partial"],"answer":"31","accept":["31"],"stem":"A list of 7 positive integers has a mean of 10, a median of 9, and a unique mode of 12. What is the greatest possible value of the largest integer in the list?","explain":"The sum is 7×10=70. In order, the fourth value is the median, 9. The mode 12 must appear at least twice, and since it is greater than 9 it must fill the fifth and sixth positions. To make the largest value as big as possible, make the others as small as possible. The three smallest must be less than 9 and all different (if any value appeared twice, 12 would no longer be the unique mode), so the least they can be is 1, 2, and 3. Then 1+2+3+9+12+12+L=70 gives L=31. Trap: using 1, 1, 1 for the smallest values, which creates a second mode."},"m3114":{"id":"m3114","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Adding values exactly one standard deviation away","tier":4,"traps":["concept"],"answer":"A","stem":"A data set of 10 values has a mean of 50 and a standard deviation of 5. Two values, 45 and 55, are added to the data set. Which statement about the standard deviation of the new data set of 12 values is true?","choices":["It is equal to 5.","It is less than 5.","It is greater than 5.","It cannot be determined from the information given."],"explain":"The two new values balance each other, so the mean stays 50. The standard deviation is based on the average of the squared distances from the mean, and for the original data that average squared distance is 52=25. Each new value is exactly 5 from the mean, so it adds a squared distance of exactly 25, the same as the existing average. Adding values that match the average doesn't change it, so the standard deviation is still 5. (This holds whether the standard deviation is computed by dividing by n or by n−1.) B would be right if the new values were closer to the mean than 5, and C if they were farther."},"m3115":{"id":"m3115","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"At least one, by the complement","tier":3,"traps":["concept"],"answer":"7/19","accept":["7/19",".3684","0.368"],"stem":"A box contains 20 light bulbs, 4 of which are defective. Two bulbs are chosen at random without replacement. What is the probability that at least one of the two bulbs is defective?","explain":"The complement of \"at least one defective\" is \"neither defective.\" The probability that both are good is 2016​⋅1915​=380240​=1912​. So the probability of at least one defective bulb is 1−1912​=197​. Enter 7/19, .3684, or 0.368. Trap: adding 204​+204​, which double counts and ignores the change after the first draw."},"m3116":{"id":"m3116","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Ink cost for an hour of printing","tier":3,"traps":["units"],"answer":"17.28","accept":["17.28"],"stem":"A printer prints 12 pages per minute. Each ink cartridge costs $36 and prints 1,500 pages. What is the cost of ink, in dollars, for one hour of continuous printing?","explain":"In one hour the printer prints 12×60=720 pages. Ink costs 1,50036​=0.024 dollars per page, so 720 pages cost 720×0.024=17.28 dollars."},"m3117":{"id":"m3117","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Adding one ingredient to change a ratio","tier":3,"traps":["setup"],"answer":"3","accept":["3"],"stem":"A paint mixture is 3 parts blue paint to 5 parts white paint. How many liters of blue paint must be added to 12 liters of this mixture to make a mixture that is half blue and half white?","explain":"The 12 liters contain 83​(12)=4.5 liters of blue and 85​(12)=7.5 liters of white. Adding only blue leaves the white at 7.5 liters, so the blue must also reach 7.5 liters: add 3 liters. Trap: working with \"parts\" as liters, which gives 2."},"m3118":{"id":"m3118","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Round trip with a rest","tier":3,"traps":["setup"],"answer":"5.4","accept":["27/5","5.4"],"stem":"A hiker walks up a trail at 2 miles per hour, rests at the top for 30 minutes, and walks back down the same trail at 3 miles per hour. The whole trip, including the rest, takes 5 hours. How many miles long is the trail, one way?","explain":"Walking time is 5−0.5=4.5 hours. If the trail is d miles, 2d​+3d​=4.5, so 65d​=4.5 and d=5.4. Trap: forgetting to remove the rest, which gives 6."},"m3119":{"id":"m3119","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Probability and conditional probability","sub":"Building the table from percents","tier":4,"traps":["setup","concept"],"answer":"15/37","accept":["15/37",".4054","0.405"],"stem":"In a survey of 200 students, 60% have a part-time job. Of the students with a job, 25% also play a sport. Of the students without a job, 55% play a sport. If a student who plays a sport is chosen at random, what is the probability that the student has a part-time job?","explain":"Build the counts. With a job: 0.60(200)=120 students, of whom 0.25(120)=30 play a sport. Without a job: 80 students, of whom 0.55(80)=44 play a sport. Of the 30+44=74 students who play a sport, 30 have a job, so the probability is 7430​=3715​. Enter 15/37, .4054, or 0.405. Trap: answering 0.25, the probability of playing a sport given a job."},"m1112":{"id":"m1112","sec":"math","domain":"Algebra","skill":"Linear equations in two variables","sub":"Perpendicular lines sharing a constant","tier":3,"traps":["setup","sign"],"answer":"-2","accept":["-2"],"stem":"In the xy-plane, line ℓ passes through the points (−3,2) and (5,k), and line m passes through the points (0,0) and (k,−4), where k is a nonzero constant. If line ℓ is perpendicular to line m, what is the value of k?","explain":"The slope of ℓ is 8k−2​ and the slope of m is k−4​. Perpendicular slopes multiply to −1: 8k−2​⋅k−4​=−1, so −4(k−2)=−8k, which gives 4k=−8 and k=−2. Check: the slopes are −21​ and 2."},"m1113":{"id":"m1113","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Area of a region bounded by an absolute value","tier":3,"traps":["setup"],"answer":"36","accept":["36"],"stem":"In the xy-plane, what is the area of the region of points that satisfy both y≥∣x−2∣ and y≤6?","explain":"The graph of y=∣x−2∣ is a V with its vertex at (2,0). It meets the line y=6 where ∣x−2∣=6, at x=−4 and x=8. The region is the triangle with vertices (2,0), (−4,6), and (8,6): base 12 along y=6 and height 6. Its area is 21​(12)(6)=36."},"m1114":{"id":"m1114","sec":"math","domain":"Algebra","skill":"Linear inequalities in one or two variables","sub":"Area of a diamond-shaped region","tier":3,"traps":["concept"],"answer":"32","accept":["32"],"stem":"In the xy-plane, what is the area of the region of points (x,y) that satisfy ∣x∣+∣y∣≤4?","explain":"In each quadrant the inequality is a linear one; in Quadrant I it is x+y≤4 with x,y≥0, a right triangle with legs 4 and area 8. By symmetry the region is four such triangles, a square standing on a corner with vertices (±4,0) and (0,±4). Its area is 4×8=32 (or half the product of its diagonals, 21​(8)(8))."},"m1115":{"id":"m1115","sec":"math","domain":"Algebra","skill":"Linear functions","sub":"A percent relationship between two outputs","tier":3,"traps":["setup"],"answer":"24","accept":["24"],"stem":"For the linear function f, f(0)=12, and f(10) is 20% greater than f(5). What is the value of f(20)?","explain":"Write f(x)=ax+12. Then 10a+12=1.2(5a+12)=6a+14.4, so 4a=2.4 and a=0.6. So f(20)=0.6(20)+12=24. Trap: assuming the 20% applies to the slope alone."},"m1116":{"id":"m1116","sec":"math","domain":"Algebra","skill":"Linear equations in one variable","sub":"Exactly one solution to an absolute value equation","tier":4,"traps":["concept"],"answer":"-5/2","accept":["-5/2","-2.5"],"stem":"The equation ∣2x−5∣=x+k, where k is a constant, has exactly one solution. What is the value of k?","explain":"Split into cases. If x≥25​: 2x−5=x+k, so x=k+5, valid when k+5≥25​, that is, k≥−25​. If x<25​: 5−2x=x+k, so x=35−k​, valid when 35−k​<25​, that is, k>−25​. For k>−25​ both cases give a solution (two in all); for k<−25​ neither does. At k=−25​ only the first case works, giving the single solution x=25​. Graphically, the line y=x+k passes through the vertex of the V. Enter -5/2 or -2.5."},"m2117":{"id":"m2117","sec":"math","domain":"Advanced Math","skill":"Nonlinear equations in one variable and systems of equations in two variables","sub":"Two quadratics whose zeros differ by 1","tier":4,"traps":["setup"],"answer":"-4","accept":["-4"],"stem":"The zeros of f(x)=x2+bx+c are r and s, where b and c are constants. The zeros of g(x)=x2+cx+b are r+1 and s+1. What is the value of b+c?","explain":"For f: r+s=−b and rs=c. For g: (r+1)+(s+1)=−c and (r+1)(s+1)=b. The first gives −b+2=−c, so c=b−2. The second gives rs+(r+s)+1=b, so c−b+1=b and c=2b−1. Then b−2=2b−1, so b=−1, c=−3, and b+c=−4. (Check: x2−x−3 has zeros 21±13​​, and x2−3x−1 has zeros 23±13​​, each 1 more.)"},"m2118":{"id":"m2118","sec":"math","domain":"Advanced Math","skill":"Equivalent expressions","sub":"Reducing a cubic with the defining equation","tier":4,"traps":["concept"],"answer":"3","accept":["3"],"stem":"If x2−3x−1=0, what is the value of x3−10x?","explain":"The equation says x2=3x+1. Multiply by x: x3=3x2+x=3(3x+1)+x=10x+3. So x3−10x=3, for either solution of the equation."},"m2119":{"id":"m2119","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"A function defined by its step-to-step change","tier":4,"traps":["concept"],"answer":"103","accept":["103"],"stem":"The function f satisfies f(0)=3 and f(x+1)=f(x)+2x+1 for all values of x. What is the value of f(10)?","explain":"Each step from x to x+1 adds 2x+1, so f(10)=f(0)+(1+3+5+⋯+19). The sum of the first 10 odd numbers is 102=100, so f(10)=103. In fact f(x)=x2+3 fits both conditions, since (x+1)2+3=x2+3+2x+1."},"m2120":{"id":"m2120","sec":"math","domain":"Advanced Math","skill":"Nonlinear functions","sub":"A cubic's third zero from its y-intercept","tier":3,"traps":["sign"],"answer":"6","accept":["6"],"stem":"The polynomial p(x)=x3+ax2+bx+c, where a, b, and c are constants, has zeros at x=1, x=2, and x=k. If p(0)=−12, what is the value of k?","explain":"A cubic with leading coefficient 1 and zeros 1, 2, and k is p(x)=(x−1)(x−2)(x−k). Then p(0)=(−1)(−2)(−k)=−2k=−12, so k=6. Trap: a sign slip giving −6."},"m3120":{"id":"m3120","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Cheapest way to buy an exact amount","tier":3,"traps":["setup"],"answer":"37.7","accept":["37.7"],"stem":"A market sells apples in bags of 6 for $4.50 or individually for $0.85 each. What is the least amount, in dollars, that it costs to buy exactly 50 apples?","explain":"Bagged apples cost 64.50​=0.75 dollars each, cheaper than single apples, so use as many bags as possible without going over 50: 8 bags hold 48 apples. The other 2 are bought singly. The cost is 8(4.50)+2(0.85)=36+1.70=37.70 dollars. Enter 37.7. Trap: buying 9 bags gives 54 apples, not exactly 50."},"m3121":{"id":"m3121","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Discount and tax in either order","tier":3,"traps":["concept"],"answer":"86.4","accept":["432/5","86.4"],"stem":"A coat is discounted 20%, and then 8% sales tax is added to the discounted price, for a total of $86.40. If instead the 8% tax had been added to the original price and the 20% discount had then been applied to that total, what would the coat cost, in dollars?","explain":"Both orders multiply the original price by the same two factors, 0.80 and 1.08, and the order of multiplication doesn't matter. So the cost is $86.40 either way. (The original price is 0.86486.40​=100 dollars.) Enter 86.4. Trap: assuming the order changes the result."},"m3122":{"id":"m3122","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Percentages","sub":"Overlap from two percents and \"neither\"","tier":3,"traps":["setup"],"answer":"40","accept":["40"],"stem":"In a survey, 70% of the people said they like tea, 60% said they like coffee, and 10% said they like neither. What percent of the people said they like both?","explain":"Since 10% like neither, 90% like at least one. Adding the tea and coffee groups counts the people who like both twice: 70+60−(both)=90, so both =40%. Trap: answering 30, from 70+60−100, which ignores the 10% who like neither."},"m3123":{"id":"m3123","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"Ratios, rates, proportional relationships, and units","sub":"Filling and draining at once from half full","tier":4,"traps":["setup"],"answer":"9","accept":["9"],"stem":"Pipe A alone can fill a tank in 6 hours. Pipe B alone can empty the full tank in 9 hours. The tank is half full when both pipes are opened at the same time. How many hours will it take to fill the tank?","explain":"Per hour, pipe A fills 61​ of the tank and pipe B empties 91​, so the net rate is 61​−91​=181​ of the tank per hour. Half a tank takes 1/181/2​=9 hours. Trap: adding the rates, or filling the whole tank (18 hours)."},"m3124":{"id":"m3124","sec":"math","domain":"Problem-Solving and Data Analysis","skill":"One-variable data: distributions and measures of center and spread","sub":"Two removals tracked through the means","tier":3,"traps":["setup"],"answer":"26","accept":["26"],"stem":"The mean of 6 numbers is 15. When one number is removed, the mean of the remaining numbers is 14. When a second number is removed, the mean of the remaining numbers is 16. What is the sum of the two numbers that were removed?","explain":"Work with totals. The 6 numbers sum to 6×15=90. After the first removal, 5 numbers sum to 5×14=70, so the first number removed was 20. After the second removal, 4 numbers sum to 4×16=64, so the second was 6. Their sum is 26. (Or directly: 90−64=26.)"},"r1051":{"id":"r1051","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Agrees with the data, disputes the cause","tier":3,"traps":["half"],"answer":"B","stem":"Based on the texts, how would the author of Text 2 most likely respond to Okafor's argument in Text 1?","passage":"Text 1
Over the past forty years, the average body size of many North American songbirds has shrunk, while their wings have grown slightly longer. Ornithologist Ben Okafor attributes the change to warming temperatures: smaller bodies shed heat more easily, he argues, so warmer summers favor smaller birds.

Text 2
The measurements behind the shrinking-songbird studies are impressive, and I don't doubt that the birds are getting smaller. But body size also depends on how well chicks are fed. Insect populations have declined sharply over the same forty years, and undernourished chicks grow into smaller adults. Until the two explanations are tested against each other, warming should be treated as one possible cause rather than the established one.","choices":["By arguing that the measurements of songbird body size are too imprecise to show any real change over forty years","By accepting that songbirds have become smaller while questioning whether warming is the reason","By agreeing that warming explains the shrinking bodies but arguing that it cannot account for the longer wings","By claiming that warming caused the insect declines that left chicks undernourished"],"explain":"The author of Text 2 accepts the data (\"I don't doubt that the birds are getting smaller\") but offers a competing cause, food shortage, and treats warming as \"one possible cause.\" A contradicts that acceptance. C has the author agreeing about warming, which the text never does. D links insect decline to warming, a claim neither text makes."},"r1052":{"id":"r1052","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"What both authors would accept","tier":2,"traps":["half"],"answer":"A","stem":"Based on the texts, both authors would most likely agree with which statement?","passage":"Text 1
Urban planners in many cities now favor \"road diets,\" which reduce a street from four car lanes to two and add bike lanes and a center turn lane. Supporters point to studies showing fewer crashes on converted streets. Critics worry about traffic jams, but on streets that carry fewer than about 20,000 vehicles a day, travel times usually change very little.

Text 2
Road diets are not a cure-all. On busy commercial corridors, removing lanes can push traffic onto nearby residential streets, creating new dangers there. Planners should study traffic volumes carefully before converting a street, because a design that works well on a quiet avenue may fail on a crowded one.","choices":["Whether a road diet works well on a given street depends in part on how much traffic that street carries.","Road diets usually make residential streets more dangerous by pushing traffic onto them from busy corridors.","Road diets should be avoided on most city streets until planners can study their effects on crash rates.","Adding bike lanes is the main reason that streets converted by road diets have fewer crashes."],"explain":"Text 1 limits its claim to streets with fewer than about 20,000 vehicles a day, and Text 2 warns that a design for a \"quiet avenue may fail on a crowded one.\" Both tie success to traffic volume. B overstates Text 2's warning about busy corridors, and Text 1 never mentions it. C goes beyond Text 2, which urges study, not avoidance, and contradicts Text 1. Neither text says why crashes fall (D)."},"r1053":{"id":"r1053","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Text 2 extends Text 1","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice best describes how Text 2 relates to Text 1?","passage":"Text 1
Honeybees perform a \"waggle dance\" to tell nestmates where food is. The angle of the dance relative to vertical shows the direction of the food relative to the sun, and the length of the waggle shows the distance. For decades, researchers assumed this ability was entirely instinctive.

Text 2
A study led by biologist Shu Lin raised colonies in which young bees could not watch older bees dance before their first flights. These bees danced with larger errors in angle and distance than bees that had observed experienced dancers. Over time their dances improved, but they never fully matched those of bees that had learned by watching.","choices":["Text 2 confirms the assumption in Text 1 by showing that bees dance correctly even without watching other bees.","Text 2 gives a more detailed account of how the angle of the dance indicates the direction of food.","Text 2 argues that the dance's distance signal is learned while its direction signal is entirely instinctive.","Text 2 presents experimental evidence that complicates the assumption described at the end of Text 1."],"explain":"Text 1 ends with the assumption that the dance is \"entirely instinctive.\" Text 2 shows that bees who couldn't watch others danced worse, so learning also plays a role. That complicates the assumption rather than confirming it (A). C splits the two signals, but the untrained bees made errors in both angle and distance. The angle-direction mechanism is explained in Text 1, not Text 2 (B)."},"r1054":{"id":"r1054","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"A response to a recommendation","tier":3,"traps":["half"],"answer":"C","stem":"Based on the texts, how would the author of Text 2 most likely respond to Durand's argument in Text 1?","passage":"Text 1
Museums that hold artifacts taken during colonial rule have begun returning some of them to the communities they came from. Curator Elise Durand argues that every such object should be returned, since its presence in a foreign museum reflects a history of unequal power.

Text 2
Repatriation is often right, but it requires more than shipping an object home. Many communities want a say in how objects are cared for and displayed, and some have asked museums to keep items temporarily while local facilities are built. Treating return as a single act, rather than as a partnership, risks repeating the old pattern of decisions made by museums alone.","choices":["By rejecting the idea that colonial-era artifacts should be returned, since many communities lack facilities to care for them","By arguing that museums, which already have the necessary facilities, should decide how returned objects are displayed","By agreeing that objects should often be returned but insisting that how they are returned also matters","By insisting that every object be returned immediately, since any delay repeats the old pattern of unequal power"],"explain":"Text 2 opens with agreement (\"Repatriation is often right\") and then argues for partnership, including some communities' requests for delayed return. A contradicts the first sentence. D ignores the requests to keep items temporarily. B attributes to the author a view the text criticizes: decisions \"made by museums alone.\""},"r1055":{"id":"r1055","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Different standards of evidence","tier":3,"traps":["scope"],"answer":"D","stem":"Based on the texts, how would the author of Text 2 most likely view the news outlets' reaction described in Text 1?","passage":"Text 1
The Voynich manuscript, a fifteenth-century book written in an unknown script, has resisted every attempt at decoding. In 2019 a researcher announced that it was written in an early Romance language, citing a few dozen words that seemed to match. Several news outlets quickly reported the claim as a breakthrough.

Text 2
Proposed solutions to the Voynich manuscript appear every few years, and nearly all follow the same pattern: a handful of words are matched to a chosen language, and the matches are presented as proof. A real decipherment would need to produce readable, grammatical sentences across many pages, and none has done so. Until one does, enthusiasm should be held in reserve.","choices":["As justified, because the 2019 proposal, unlike earlier ones, produced readable sentences across many pages","As mistaken, because the manuscript's script has been shown to come from a much later period than the fifteenth century","As reasonable, because a Romance language is a likely source for a fifteenth-century European manuscript","As premature, because matching a few words falls short of the evidence a genuine decipherment would provide"],"explain":"Text 2 describes a pattern of matching \"a handful of words\" and says a real solution must produce readable sentences across many pages, so the excitement over a few dozen matching words would seem premature. A contradicts the text (\"none has done so\"). Text 2 never questions the date (B), and a plausible guess about the language isn't the evidence it demands (C)."},"r1056":{"id":"r1056","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Qualified agreement about a technology","tier":2,"traps":["half"],"answer":"C","stem":"Based on the texts, how would the author of Text 2 most likely respond to Fenn's proposal?","passage":"Text 1
Green roofs, which are covered with soil and plants, can lower building temperatures, absorb rainwater, and provide habitat for insects. Architect Lydia Fenn argues that cities should require them on all new commercial buildings, since their benefits clearly outweigh their costs.

Text 2
A green roof's benefits depend on what grows on it. Studies of roofs planted with a single species of sedum, the cheapest option, found that they absorbed less water and supported far fewer insects than roofs with diverse native plants. A requirement that ignores planting choices could produce many roofs that deliver only a fraction of the promised benefits.","choices":["By arguing that green roofs cost more to build and maintain than they save in energy and stormwater costs","By agreeing that because green roofs lower temperatures and absorb rain, their benefits always outweigh their costs","By suggesting that the proposal's success would depend largely on the kinds of plants grown on the roofs","By recommending that the requirement specify sedum, since it is the least expensive planting option"],"explain":"Text 2 doesn't reject green roofs; it warns that benefits \"depend on what grows\" on them and that a requirement ignoring planting choices could disappoint. B misses that warning. D reverses the text's finding about sedum."},"r1057":{"id":"r1057","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"A shared assumption","tier":3,"traps":["irrelevant"],"answer":"A","stem":"Based on the texts, both authors would most likely agree with which statement?","passage":"Text 1
Economist Rahul Mehta argues that a four-day workweek raises productivity: workers who know they have a long weekend focus better and waste less time. In trials at several firms, output per hour rose by as much as 20%.

Text 2
The four-day week may work in offices, where output is hard to measure and focus matters most. But a hospital or a bus company cannot simply compress its hours; patients and passengers need service five or more days a week. For such employers, a shorter week means hiring more workers, which raises costs regardless of how productive each worker becomes.","choices":["Workers on a four-day schedule may well get more done in each hour that they spend working.","A four-day workweek reduces costs for most employers because their workers become more productive.","Hospitals and bus companies should adopt a four-day workweek once their staff become more productive.","Output per hour is easy to measure in offices, which is why the office trials showed such clear gains."],"explain":"Text 1 reports gains in output per hour. Text 2 doesn't dispute them; it argues that costs rise \"regardless of how productive each worker becomes,\" granting that productivity may improve. B conflicts with Text 2's point about hiring costs, and D contradicts Text 2's remark that office output is \"hard to measure.\""},"r1058":{"id":"r1058","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Response to a generalization","tier":2,"traps":["scope"],"answer":"B","stem":"Based on the texts, how would Kowal (Text 2) most likely respond to the historians' view described in Text 1?","passage":"Text 1
For much of the twentieth century, historians described the medieval period in Europe as a time of stagnation, when scientific inquiry nearly vanished between the fall of Rome and the Renaissance.

Text 2
Medieval scholars translated and commented on Greek and Arabic works on optics, astronomy, and medicine; they built mechanical clocks; and they founded the first universities. Historian Agnes Kowal argues that the Renaissance would be hard to imagine without this groundwork.","choices":["She would agree that scientific inquiry nearly vanished, while noting that medieval scholars preserved older works.","She would argue that it overlooks the medieval contributions on which later scientific work came to depend.","She would argue that the Renaissance was less important to science than the medieval period that preceded it.","She would claim that the Greek and Arabic works translated by medieval scholars contributed little to later science."],"explain":"Text 2 lists medieval achievements and reports Kowal's view that the Renaissance depended on this \"groundwork,\" which contradicts the picture of stagnation. A keeps the stagnation claim that her list of achievements disputes. She never ranks the periods (C), and Text 2 credits the Greek and Arabic works rather than dismissing them (D)."},"r1059":{"id":"r1059","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Methodological objection","tier":3,"traps":["half"],"answer":"D","stem":"Based on the texts, how would the author of Text 2 most likely respond to Reyes's conclusion?","passage":"Text 1
In a survey of 3,000 adults, people who reported reading fiction at least weekly scored higher on a test of empathy than people who rarely read fiction. Psychologist Tomas Reyes concludes that reading fiction builds empathy by giving readers practice imagining other people's minds.

Text 2
Correlations between fiction reading and empathy are well documented, but they cannot tell us which way the influence runs. People who are already curious about others may simply be more drawn to novels. Experiments that assign some participants to read fiction and others to read nonfiction have produced mixed results, with effects that are small or that disappear in repeated studies.","choices":["By arguing that the survey found no real relationship between fiction reading and empathy among the 3,000 adults","By agreeing that controlled experiments have now clearly confirmed that reading fiction builds empathy","By claiming that reading nonfiction builds empathy more effectively than reading fiction does","By noting that the survey data are consistent with empathy leading to fiction reading rather than the reverse"],"explain":"Text 2 grants the correlation but says it can't show \"which way the influence runs\": empathetic people may choose novels. That is an alternative explanation for Reyes's data. A denies the correlation Text 2 accepts; B contradicts the \"mixed results\"; C isn't claimed."},"r1060":{"id":"r1060","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Different emphases on one artist","tier":2,"traps":["half"],"answer":"A","stem":"Which choice best describes a difference in how the authors of Text 1 and Text 2 approach the Gee's Bend quilts?","passage":"Text 1
The quilts made by the women of Gee's Bend, Alabama, are often praised for their bold geometry, and critics have compared them to abstract paintings of the mid-twentieth century. Seen in a gallery, their improvised patterns of color and shape hold their own beside any modernist canvas.

Text 2
Comparing the Gee's Bend quilts to modernist painting can be flattering, but it risks hiding what the quilts are. Many were pieced from worn work clothes and feed sacks and were made to keep families warm. Their designs grew out of a local tradition passed between generations, not from conversations in art schools.","choices":["Text 1 compares the quilts to modernist art, while Text 2 stresses the materials, uses, and tradition behind them.","Text 1 criticizes the quilts for imitating modernist painting, while Text 2 defends them as part of a local tradition.","Text 1 traces the history of the quilts' materials, while Text 2 focuses mainly on their colors and shapes.","Text 1 argues that the quilts belong in galleries, while Text 2 argues that they should be kept out of museums."],"explain":"Text 1 values the quilts as art comparable to modernist painting. Text 2 cautions that the comparison \"risks hiding\" their origins in reused fabric, practical use, and local tradition. Text 1 praises the quilts rather than criticizing them (B), C reverses which text discusses history, and Text 2 never says the quilts should be kept out of museums (D)."},"r1061":{"id":"r1061","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Reconciling two findings","tier":3,"traps":["half"],"answer":"C","stem":"Based on the texts, how would Vanterpool (Text 2) most likely respond to Haas's explanation in Text 1?","passage":"Text 1
Field studies of wolves in Yellowstone National Park found that after wolves were reintroduced in 1995, elk spent less time browsing in open river valleys, and willows and aspens along the rivers began to recover. Ecologist Martin Haas describes this as a \"landscape of fear\": the mere risk of wolf attack changed where elk fed.

Text 2
Tracking collars placed on elk show that they did not, in fact, avoid areas where wolves were most common; elk and wolves are active at different times of day. Ecologist Rhea Vanterpool argues that the recovery of riverside trees is better explained by a drop in elk numbers, which fell sharply after wolves returned, along with hunting by people and other predators.","choices":["She would argue that the riverside trees recovered only in areas where wolves were rarely found.","She would agree that fear of wolves kept elk away from the river valleys where the trees grew.","She would accept that riverside trees recovered but credit fewer elk, not changed elk behavior.","She would claim that wolves had little effect on elk, since elk and wolves are active at different times of day."],"explain":"Both texts accept that the trees recovered. Vanterpool rejects the behavioral explanation (elk didn't avoid wolf areas) in favor of a numbers explanation: fewer elk. D twists the timing detail; she says elk numbers fell sharply after wolves returned, partly because of them. B is Haas's view, and A isn't supported."},"r1062":{"id":"r1062","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Shared concern, different solutions","tier":2,"traps":["half"],"answer":"B","stem":"Which choice best describes how the approaches in the two texts are related?","passage":"Text 1
Rising sea levels threaten coastal wetlands, which may drown if waters rise faster than sediment can build them up. Hydrologist Peter Lund argues for \"thin-layer placement\": spraying sediment dredged from shipping channels onto marshes to raise them.

Text 2
Building marshes up in place will help for a while, but the most durable solution is to give them somewhere to go. Where land behind a marsh is low and undeveloped, removing levees lets the wetland migrate inland as the sea rises. Ecologist Ana Morales notes that many marshes are blocked by roads and seawalls, and she urges planners to protect the open land behind them.","choices":["Text 1 says rising seas threaten wetlands, while Text 2 says the real threat comes from roads and seawalls.","Both respond to rising seas, but Text 1 raises marshes in place while Text 2 lets them move inland.","Both texts argue that dredged sediment should be used to raise marshes before they are allowed to migrate inland.","Text 2 rejects the method in Text 1 because sediment from shipping channels would harm the marshes."],"explain":"Both texts aim to save wetlands from rising seas. Text 1 proposes adding sediment to raise marshes; Text 2 grants this \"will help for a while\" but prefers letting marshes migrate inland. So Text 2 doesn't reject the method (D). In Text 2, roads and seawalls block migration; the threat is still the rising sea (A). Text 1 never discusses migration (C)."},"r1063":{"id":"r1063","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Text 2 offers a counterexample","tier":2,"traps":["scope"],"answer":"A","stem":"Based on the texts, how would the author of Text 2 most likely respond to the view described in Text 1?","passage":"Text 1
Many linguists hold that every human language uses recursion: the ability to embed one phrase inside another of the same type, as in \"the dog that chased the cat that ate the mouse.\" On this view, recursion is the core feature that separates human language from animal communication.

Text 2
Linguist Daniel Everett reported that Pirahã, a language spoken in the Amazon, does not appear to embed clauses within clauses. Speakers express the same ideas in strings of separate, simple sentences. Other linguists have challenged his analysis, but the debate itself shows that the universality of recursion is a claim to be tested, not an assumption to be made.","choices":["By suggesting that the view treats as settled a claim about all languages that remains open to question","By agreeing that recursion appears in every human language, including Pirahã, once it is analyzed correctly","By arguing that some animal communication systems use recursion and so are not fundamentally different","By claiming that Everett's analysis has shown that recursion is not a feature of human language at all"],"explain":"Text 2 ends by saying universality \"is a claim to be tested, not an assumption to be made,\" which is a response to Text 1's view. D goes too far: Everett's analysis concerns one language and is still debated. Text 2 doesn't side with the universal view (B) or discuss animal communication (C)."},"r1064":{"id":"r1064","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Same evidence, different lessons","tier":3,"traps":["half"],"answer":"C","stem":"Based on the texts, how would the author of Text 2 most likely respond to Fairbanks's claim in Text 1?","passage":"Text 1
After Lake Erie's walleye population crashed in the mid-twentieth century, fisheries managers imposed strict catch limits. Within about fifteen years the population had rebounded to record levels, which manager Lois Fairbanks cites as proof that catch limits work.

Text 2
The walleye recovery coincided with sharp reductions in phosphorus pollution under a new water-quality agreement, which ended the algae blooms that had starved walleye eggs of oxygen. Catch limits likely helped. But a recovery with two simultaneous causes cannot prove the power of either one alone.","choices":["By denying that the walleye population recovered, since record levels were reached only briefly","By claiming that catch limits slowed the recovery by keeping the fishing industry from removing invasive species","By arguing that the recovery can't show how much catch limits mattered, since pollution fell at the same time","By agreeing that the recovery proves that catch limits, not pollution controls, were the main cause"],"explain":"Text 2 grants that catch limits \"likely helped\" but points to a second cause occurring at the same time, so the recovery \"cannot prove the power of either one alone.\" That disputes \"proof,\" not the limits themselves. B and A contradict the text; D reverses its conclusion."},"r1065":{"id":"r1065","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Literary interpretation versus author's statement","tier":3,"traps":["scope"],"answer":"B","stem":"Based on the texts, how would Weil (Text 2) most likely respond to the interpretation described in Text 1?","passage":"Text 1
Readers have long interpreted the empty chair that appears in every chapter of Tove Lindqvist's novel as a symbol of the narrator's dead brother. The chair is always described in the same words, and the narrator never sits in it.

Text 2
In a 1998 interview, Lindqvist said she had included the chair only because her own kitchen had one that no one used, and she was surprised by readers' interpretations. Critic Jonah Weil argues that her comment settles nothing: a symbol can work in a novel whether or not its author planned it, and the evidence of the text itself, such as the repeated wording, supports the readers' view.","choices":["He would reject it, since Lindqvist said the chair was based on one in her own kitchen and was not meant as a symbol.","He would consider it well supported by evidence in the novel itself, whatever Lindqvist said about the chair.","He would argue that the chair represents Lindqvist's own kitchen and therefore her life rather than the brother.","He would claim that the repeated wording shows that the chair stands for the narrator rather than the brother."],"explain":"Weil says the author's comment \"settles nothing\" and that textual evidence like the repeated wording \"supports the readers' view.\" A is the conclusion Weil argues against. C confuses the chair's origin with its meaning in the novel, and D uses his evidence for a reading he never proposes."},"r1066":{"id":"r1066","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"A limitation acknowledged in both","tier":2,"traps":["irrelevant"],"answer":"D","stem":"Based on the texts, both authors would most likely agree with which statement?","passage":"Text 1
Satellite images of nighttime lights have become a popular way to estimate economic activity in countries with unreliable statistics: brighter regions tend to be wealthier. Economist Farah Nasser cautions, however, that lights capture cities far better than farms, where much economic activity happens in daylight and leaves no glow.

Text 2
Nighttime light data have transformed research on regions where official figures are missing or doubtful. Still, light measures energy use as much as income; a region with cheap electricity may glow more brightly than an equally wealthy region where power is expensive.","choices":["Nighttime light data are more reliable than official statistics, which are often missing or doubtful in poorer regions.","Nighttime light data overstate economic activity in farming regions, where much of the work happens in daylight.","Regions with cheap electricity tend to be wealthier, which is why they glow more brightly at night.","Nighttime light data are useful for estimating economic activity, but they can be misleading in some regions."],"explain":"Both texts value the data (a \"popular way\" to estimate activity; it has \"transformed research\") and both name a distortion: farms leave no glow (Text 1), and cheap power inflates brightness (Text 2). B reverses Text 1: farm activity is understated, not overstated. C reverses Text 2, which says equally wealthy regions can differ in brightness. Neither text ranks the data above official statistics everywhere (A)."},"r1067":{"id":"r1067","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"How Text 1's author would view Text 2's finding","tier":3,"traps":["half"],"answer":"A","stem":"Based on the texts, how would Ruiz (Text 1) most likely respond to the results in Text 2?","passage":"Text 1
Psychologist Carla Ruiz argues that multitasking is a myth: the brain cannot attend to two demanding tasks at once and instead switches rapidly between them, losing time and accuracy with every switch.

Text 2
In a recent experiment, participants who listened to familiar instrumental music while solving arithmetic problems performed as well as participants who worked in silence. But when the music had lyrics, accuracy dropped by nearly 15%. The researchers suggest that tasks drawing on the same mental resources, such as two that both involve language, interfere with each other far more than tasks that do not.","choices":["She would see the lyrics result as support for her view and the instrumental result as consistent with it.","She would say that the results show multitasking improves performance when one of the tasks involves familiar music.","She would reject the results, since doing arithmetic while hearing music doesn't count as performing two tasks.","She would conclude that the results disprove her view, since participants handled music and arithmetic at once."],"explain":"Ruiz's claim concerns \"two demanding tasks.\" Text 2's lyrics condition shows the cost she predicts, while familiar instrumental music, which may not be demanding, caused no loss. Both results fit her view. Nothing in Text 2 shows improvement (B), nothing suggests she would throw the results out (C), and the no-cost result involves a task that may not be demanding, so it doesn't disprove her (D)."},"r1068":{"id":"r1068","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Different scales of explanation","tier":2,"traps":["scope"],"answer":"D","stem":"Based on the texts, how would Brandt (Text 2) most likely respond to the explanation in Text 1?","passage":"Text 1
The decline of the medieval Norse settlements in Greenland is often blamed on the Little Ice Age, a period of cooling that began in the fourteenth century and shortened the growing season for the hay the settlers' cattle depended on.

Text 2
Archaeologist Niels Brandt points out that by the fourteenth century the Greenland Norse had already shifted much of their diet to seals, which were plentiful despite the cold. What they lost, he argues, was trade: the walrus ivory they exported to Europe fell out of fashion, and ships from Norway stopped coming. Without trade, the settlements could no longer obtain iron, timber, or new settlers.","choices":["By denying that the climate cooled, since seals remained plentiful in Greenland throughout the fourteenth century","By agreeing that the settlements failed because cattle starved, though he would add that trade made the losses worse","By claiming that the settlers abandoned seal hunting once the colder climate made it too dangerous to go to sea","By arguing that it overstates failing hay harvests, given the settlers' reliance on seals, and ignores lost trade"],"explain":"Brandt notes that the settlers relied on seals, which weakens the case that failing hay doomed them, and offers lost trade as the cause. He never disputes the cooling itself (A). C contradicts his statement that seals stayed plentiful."},"r1069":{"id":"r1069","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Text 2 tests Text 1's prediction","tier":3,"traps":["reverse"],"answer":"B","stem":"Which choice best describes how Text 2 relates to the prediction in Text 1?","passage":"Text 1
Some economists predicted that raising the minimum wage in a region would reduce employment in low-wage jobs, since businesses facing higher labor costs would hire fewer workers. Fast-food restaurants, with many low-wage employees and thin profit margins, seemed especially likely to cut jobs.

Text 2
When one state raised its minimum wage while a neighboring state did not, economists compared fast-food restaurants on both sides of the border. Employment at restaurants in the state that raised its wage did not fall relative to employment at restaurants across the border; if anything, it rose slightly. One possible explanation is that higher pay reduced costly turnover, offsetting part of the wage increase.","choices":["Text 2 describes a study confirming that fast-food restaurants cut jobs, though fewer than economists had predicted.","Text 2 describes a study whose results do not support the prediction, and it offers a possible reason.","Text 2 argues that the prediction holds only for restaurants located near the border between two states.","Text 2 explains why restaurants with thin profit margins are especially likely to reduce employment."],"explain":"The prediction was job losses. The study found no relative drop, perhaps a slight rise, and suggests lower turnover as a reason. That contradicts the prediction rather than confirming it (A). The border is just the study's design, not a limit on the prediction (C)."},"r1070":{"id":"r1070","sec":"rw","domain":"Craft and Structure","skill":"Cross-Text Connections","sub":"Common ground between rival views","tier":3,"traps":["half"],"answer":"C","stem":"Based on the texts, Obi and Henning would most likely agree with which statement?","passage":"Text 1
Historian Grace Obi argues that the steam engine drove Britain's industrial revolution: by freeing factories from rivers, which had powered water mills, it allowed industry to concentrate in cities near coal and labor.

Text 2
Water power remained cheaper than steam in Britain until the 1840s, and many mills kept using waterwheels for decades. Historian Lars Henning argues that the decisive change came earlier, in how work was organized: factories that gathered workers under one roof and divided tasks among them raised output long before steam became widespread.","choices":["Steam power was cheaper than water power by the time Britain's industrial revolution began to accelerate.","The way work was organized inside factories was the decisive cause of Britain's industrial revolution.","Bringing workers and production together in concentrated places contributed to Britain's industrial growth.","Factories powered by water produced more goods than factories powered by steam during the industrial revolution."],"explain":"Obi says steam let industry concentrate in cities near coal and labor; Henning credits factories that gathered workers under one roof. Both explanations involve bringing production together. B is Henning's view only. A contradicts Text 2, and neither text compares output by power source (D)."},"r1031":{"id":"r1031","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"A concession before a claim","tier":2,"traps":["scope"],"answer":"A","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"Seismologist Rafael Ortega has found a way to turn the fiber-optic cables that carry internet traffic into earthquake detectors. When a seismic wave stretches a buried cable, it changes how light scatters inside the glass fiber, so Ortega's team can record ground motion every few meters along hundreds of kilometers of cable. The method cannot yet reliably distinguish ground motion caused by earthquakes from vibrations caused by passing trucks and trains. Even so, Ortega argues, it could give regions that lack seismometers their first detailed records of earthquakes.","choices":["It acknowledges a shortcoming of the method before the text returns to what the method could still achieve.","It explains why Ortega's team decided to use existing internet cables instead of installing new seismometers.","It identifies the main reason the method has not yet been tested outside of a laboratory setting.","It presents evidence that challenges Ortega's account of how stretching a cable changes the light inside it."],"explain":"The underlined sentence names a limitation, and the next sentence begins \"Even so,\" returning to the method's potential. That is a concession followed by a renewed claim. Nothing says the method hasn't left the lab, and the sentence doesn't question how the method works, only what it can tell apart."},"r1032":{"id":"r1032","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"A shift in a character's perception","tier":3,"traps":["scope"],"answer":"C","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"The following text is from a short story. Inés has returned to her family's orchard after ten years away.

The trees were smaller than she remembered, their trunks gray and split, and the ladder her father had built still leaned against the wall of the shed. Inés walked the rows twice, counting. Somewhere in the second circuit she stopped counting what was missing and began to notice the young trees, waist-high and staked, planted in the gaps where the old ones had been. Someone, she realized, had been tending this place all along.","choices":["It explains why Inés believes that her father, rather than someone else, built the ladder by the shed.","It suggests that Inés regrets having stayed away from the orchard and the people who cared for it.","It marks a shift in what Inés notices, and that shift leads to the realization described in the final sentence.","It describes the orchard as Inés remembers it from before she left, when the trees were still young."],"explain":"Before the underlined sentence, Inés notices decline (\"smaller,\" \"gray and split\") and counts losses. The underlined sentence turns her attention to new growth, which produces the final realization that someone has been tending the orchard. Regret (B) may be plausible, but the text never states it, and the sentence describes the present, not the past (D)."},"r1033":{"id":"r1033","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Overall structure of an argument","tier":2,"traps":["half"],"answer":"B","stem":"Which choice best describes the overall structure of the text?","passage":"For decades, ornithologists assumed that the songs of young zebra finches were simply imperfect copies of their fathers' songs, improving with practice until they matched. Recordings analyzed by biologist Hana Sato complicate that picture. Sato found that young finches regularly insert notes that appear in no adult's song, and that these novel notes are more likely to survive into the adult song when a female responds to them. Learning, Sato suggests, is shaped not only by imitation but also by the reactions of listeners.","choices":["It describes a method for recording birdsong, explains the method's limitations, and proposes a better one.","It presents an assumption, describes findings that complicate it, and offers a revised explanation.","It presents two competing explanations of how finches learn songs and argues that both are equally supported.","It describes an unexpected finding about finch songs and then explains why most ornithologists have rejected it."],"explain":"The text moves from what ornithologists \"assumed,\" to Sato's recordings that \"complicate that picture,\" to her suggestion that listeners also shape learning. That is assumption, complicating evidence, revised explanation. No method is critiqued, and nothing says others rejected Sato's finding."},"r1034":{"id":"r1034","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Main purpose of an informational text","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice best states the main purpose of the text?","passage":"Most maps of the ocean floor are surprisingly coarse. Satellites can estimate seafloor depth from tiny bumps in the sea surface, but only at a resolution of several kilometers, too blurry to show many underwater volcanoes and canyons. Ship-based sonar produces far sharper images, yet in 2023 only about a quarter of the seafloor had been mapped that way. An international project now aims to finish a high-resolution map by 2030, in part by asking fishing and cargo ships to record sonar data as they travel.","choices":["To argue that satellites will soon replace ships as the main tool for mapping the seafloor","To compare the costs of satellite mapping and sonar mapping","To describe the underwater volcanoes and canyons discovered by sonar","To explain why much of the seafloor is poorly mapped and describe an effort to change that"],"explain":"The text explains the problem (satellite maps are coarse; little has been mapped by sonar) and then introduces a project to fix it. It favors sonar, not satellites, and it says nothing about costs or specific discoveries."},"r1035":{"id":"r1035","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Function of an example","tier":3,"traps":["half"],"answer":"D","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"Economists often assume that people value a gain and an equal loss about equally. Behavioral research suggests otherwise. In one classic design, participants who are given a coffee mug typically refuse to sell it for less than about twice what other participants, who were not given a mug, will pay for an identical one. Owning the mug for just a few minutes seems to make giving it up feel like a loss. Such findings have led some economists to revise their models of how prices are set in markets.","choices":["It describes a second experiment, conducted with different objects, that confirms the coffee mug result.","It explains why economists were slow to revise their models even after behavioral research appeared.","It qualifies the claim, made earlier in the text, that people value gains and equal losses about equally.","It offers a possible interpretation of the participants' behavior described in the previous sentence."],"explain":"The previous sentence reports what participants did; the underlined sentence explains why they might have done it (\"seems to make giving it up feel like a loss\"). It is an interpretation, not a new experiment. Trap: C sounds close, but the text rejects the equal-value assumption rather than qualifying it, and that rejection happens in \"Behavioral research suggests otherwise,\" not in the underlined sentence."},"r1036":{"id":"r1036","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Purpose of a poem excerpt","tier":3,"traps":["scope"],"answer":"B","stem":"Which choice best describes the overall purpose of the text?","passage":"The following text is from a poem. The speaker addresses a river near her childhood home.

You were the first road I knew,
the one that went somewhere without asking.
I would set a leaf on you
and watch it leave as if it had been called.
Now I come back with my questions
and you answer the way you always did:
by going.","choices":["To describe the dangers the river posed to the speaker as a child playing alone on its banks","To contrast the speaker's hopes for an answer from the river with the river's unchanging indifference","To explain how the river's course has gradually changed since the speaker first set leaves on it","To celebrate the river as a dependable source of answers to the questions the speaker brings back"],"explain":"The returning speaker brings \"questions,\" and the river \"answers\" only \"by going,\" the same way it always has. The poem sets her hopes for a response against the river's unchanging motion. Trap: D takes \"answer\" literally; the joke of the last line is that going is no answer at all. Nothing describes danger or a changed course."},"r1037":{"id":"r1037","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Function of a quotation","tier":2,"traps":["scope"],"answer":"A","stem":"Which choice best describes the function of the underlined quotation in the text as a whole?","passage":"When the city of Medellín built cable cars to connect hillside neighborhoods to its metro system, planners expected the main benefit to be shorter commutes. They were right about the commutes, which shrank dramatically for many residents. But surveys conducted by urban researcher Paula Restrepo found something planners had not measured. \"People told us they felt, for the first time, that they were part of the city,\" Restrepo said. Several studies have since found that neighborhoods near the new stations saw declines in violent crime.","choices":["It provides a specific example of the unexpected benefit that is introduced in the previous sentence.","It explains why commutes became shorter for residents of the hillside neighborhoods after the cable cars were built.","It suggests that residents disagreed with the planners about whether shorter commutes were the most important goal.","It presents a claim that the final sentence, about declines in violent crime, calls into question."],"explain":"The previous sentence says Restrepo found \"something planners had not measured.\" The quotation shows what it was: a new sense of belonging. The final sentence adds a further benefit rather than questioning the quotation."},"r1038":{"id":"r1038","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"How a paragraph is organized","tier":2,"traps":["half"],"answer":"C","stem":"Which choice best describes the overall structure of the text?","passage":"The Antarctic midge is the continent's largest land animal, yet it is barely six millimeters long. It survives temperatures far below freezing by losing up to 70% of its body water, which keeps ice crystals from forming in its cells. It also produces sugars that act as natural antifreeze. Even its life cycle is adapted to the cold: larvae take two years to mature, feeding slowly during the brief summers, while the wingless adults live only about ten days.","choices":["It compares the Antarctic midge with other insects that survive freezing temperatures in polar regions.","It presents a debate among scientists about which of the midge's adaptations matters most for its survival.","It introduces an insect and then details several of the ways in which it is suited to a harsh environment.","It traces the history of scientists' efforts to explain how the Antarctic midge survives losing body water."],"explain":"After introducing the midge, each sentence adds another adaptation: dehydration, antifreeze sugars, and a cold-adapted life cycle. There is no comparison with other insects, no debate, and no history of research."},"r1039":{"id":"r1039","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"A sentence that anticipates an objection","tier":3,"traps":["scope"],"answer":"A","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"Some historians argue that the printing press spread literacy in sixteenth-century Europe mainly by making books cheaper. Historian Ruth Abara points to a less obvious factor: printed books used standardized letterforms, which were easier for new readers to learn than the varied handwriting in manuscripts. Abara acknowledges that lower prices mattered, since a reader who cannot afford a book has no use for clear type. But she argues that cheaper books alone would not explain why literacy rose fastest among readers who had little earlier exposure to manuscripts.","choices":["It grants a point made by other historians while setting up Abara's case for an additional factor.","It provides the main evidence for Abara's claim that standardized letterforms made reading easier to learn.","It explains why manuscripts were more expensive to produce than printed books in sixteenth-century Europe.","It reveals that Abara ultimately agrees with the historians described at the beginning of the text."],"explain":"The underlined sentence grants the other historians' point (price mattered), and the next sentence begins \"But,\" arguing that price alone isn't enough. That is a concession inside a larger argument for another factor. Trap: D overstates the concession; Abara agrees that price mattered but not that it was the main factor."},"r1040":{"id":"r1040","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Main purpose of a character sketch","tier":2,"traps":["scope"],"answer":"C","stem":"Which choice best states the main purpose of the text?","passage":"The following text is from a novel. Mr. Abernathy runs the village post office.

Mr. Abernathy weighed every parcel twice, once on the scale and once in his hands, as though the scale might be lying. He read postcards only when they were addressed to people he disliked, and then only to confirm that his dislike was justified. When the new postmistress from the county office suggested that he might retire, he thanked her warmly, offered her tea, and posted a letter of complaint about her that same afternoon.","choices":["To explain how Mr. Abernathy came to run the village post office and why he refuses to retire from it","To show that Mr. Abernathy is more careful and efficient than the new postmistress from the county office","To portray Mr. Abernathy as a suspicious and quietly calculating man through a series of his small actions","To describe the procedures used in rural post offices before the county office took control of them"],"explain":"Each sentence adds a habit that shows distrust (weighing parcels twice, reading postcards to confirm a dislike) or a hidden agenda (warm thanks, then a complaint). The text builds a character portrait. It gives no history, no comparison of efficiency, and no general account of procedures."},"r1041":{"id":"r1041","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Function of a final sentence","tier":3,"traps":["half"],"answer":"B","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"When a population of guppies is moved from a stream full of predators to one without them, the fish change quickly. Within about thirty generations, males grow brighter, since conspicuous colors no longer get them eaten, and females begin producing fewer, larger offspring. Evolutionary biologists once assumed changes like these required thousands of years. The guppies suggest that, under strong enough pressure, evolution can be watched within a single researcher's career.","choices":["It introduces a new question about evolution that the guppy findings, by themselves, cannot answer.","It draws a broader conclusion from the guppy findings that contrasts with an earlier assumption.","It explains why male guppies grow brighter colors after they are moved to streams without predators.","It describes the method biologists used to measure how quickly the guppies changed across generations."],"explain":"The sentence before it states the old assumption (thousands of years). The underlined sentence generalizes from the guppies: evolution can be seen within a career. That is a broader conclusion that contradicts the assumption. The explanation for brighter colors (C) comes earlier."},"r1042":{"id":"r1042","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Comparing what two examples do","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice best describes the function of the examples of \"pea\" and \"cherry\" in the text?","passage":"Many words enter English through mistakes. \"Pea\" exists because speakers heard the older word \"pease,\" which referred to one pea or many, as a plural and invented a singular to match. \"Cherry\" arose the same way from \"cherise.\" In both cases, speakers reshaped a word to fit a grammatical pattern they expected, and the new form eventually replaced the original.","choices":["They show that English has borrowed many common words from French, often changing their spelling.","They suggest that most English plurals were once singular nouns that speakers came to misunderstand.","They demonstrate that older word forms are usually replaced within a generation.","They illustrate a process by which speakers can form new words from a misunderstanding of old ones."],"explain":"Both examples show speakers mistaking a word for a plural and creating a new singular, the \"mistakes\" named in the first sentence. The text says nothing about French borrowing, most plurals, or how quickly forms change."},"r1043":{"id":"r1043","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Purpose of contrasting details","tier":3,"traps":["half"],"answer":"D","stem":"Which choice best describes the purpose of the details about the dollhouse repair?","passage":"The following text is from a memoir. The narrator describes her grandfather, a carpenter.

My grandfather's hands were scarred across every knuckle, and two of his fingernails had grown back crooked after a hammer missed. He could split a board along the grain with a single blow of a hatchet. Yet when he repaired my dollhouse, he worked with tweezers and a jeweler's lens for an entire evening, setting a door hinge no larger than a grain of rice.","choices":["To suggest that the grandfather preferred building small toys to the full-sized carpentry he did for a living","To show that the grandfather's injuries had made heavy carpentry difficult, so he turned to smaller projects","To explain why the narrator decided to learn carpentry herself after watching her grandfather at work","To reveal a delicacy in the grandfather's work that contrasts with the roughness described earlier"],"explain":"\"Yet\" signals a contrast: after scarred hands and a hatchet split, the grandfather uses tweezers and a jeweler's lens. The details reveal a careful delicacy alongside his strength. The text doesn't say he preferred toys, that his injuries limited him, or anything about the narrator's career."},"r1044":{"id":"r1044","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"A sentence that limits a claim","tier":3,"traps":["scope"],"answer":"C","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"A study of records from three hospitals found that patients whose rooms faced a garden recovered from surgery slightly faster, on average, than patients whose rooms faced a brick wall. The study's authors proposed that the view reduces stress. But in those hospitals, garden-view rooms were also more likely to be in newer wings, where staffing levels tended to be higher. A trial that randomly assigns room views would be needed to separate these effects.","choices":["It provides further evidence that garden views reduce patients' stress after surgery.","It explains why hospitals have begun placing rooms with garden views in their newer wings.","It points out another factor that could explain the reported difference in recovery.","It shows that patients in older wings recovered faster than patients in newer wings did."],"explain":"The underlined sentence introduces a confound: garden-view rooms were also in better-staffed wings, so staffing, not the view, might explain the faster recovery. The final sentence follows directly: only random assignment could separate the effects."},"r1045":{"id":"r1045","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"The role of a definition","tier":2,"traps":["scope"],"answer":"A","stem":"Which choice best describes the function of the first sentence in the text as a whole?","passage":"Astronomers describe some planets as \"tidally locked\": the same side always faces the star, just as the same side of the Moon always faces Earth. On such a planet, one hemisphere would bake in permanent daylight while the other froze in permanent night. For years, this seemed to rule out life on tidally locked planets orbiting small, dim stars. Recent climate models, however, suggest that thick atmospheres could carry enough heat to the night side to keep a band of the planet temperate.","choices":["It defines a term that the rest of the text uses to discuss whether certain planets could support life.","It presents a claim about tidally locked planets that the climate models described later disprove.","It explains why one hemisphere of the Moon is always in daylight while the other is always in darkness.","It compares the climates of Earth and the Moon to show why the Moon could not support life."],"explain":"The first sentence explains what \"tidally locked\" means, and the rest of the text applies that idea to the question of life on such planets. The climate models revise an assumption about habitability, not the definition."},"r1046":{"id":"r1046","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Main purpose of a review","tier":2,"traps":["half"],"answer":"B","stem":"Which choice best states the main purpose of the text?","passage":"Director Yuki Tanaka's documentary about the last lighthouse keepers of Hokkaido is patient to the point of stillness: long shots of fog, of a kettle heating, of a man polishing a lens he will polish again tomorrow. Some viewers will find it slow. But the film's refusal to hurry is its argument. By the end, the keepers' repetitive routines seem less like tedium than like a form of attention the rest of us have lost.","choices":["To explain how the last lighthouse keepers of Hokkaido maintained the lenses of their lighthouses","To argue that a quality some viewers may dislike in a documentary is central to its meaning","To criticize a documentary for moving so slowly that many viewers will lose interest before the end","To compare a documentary about lighthouse keepers with other recent films about remote places"],"explain":"The reviewer concedes the film is slow (\"Some viewers will find it slow\") and then claims the slowness is \"its argument.\" The text defends the quality rather than criticizing it (C). The lens-polishing is a detail, not the subject (A)."},"r1047":{"id":"r1047","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Function of a question in the text","tier":3,"traps":["scope"],"answer":"D","stem":"Which choice best describes the function of the underlined question in the text as a whole?","passage":"A team of engineers recently built a small footbridge entirely from 3D-printed concrete. The printer laid down the concrete in thin layers, adding steel cable between them for strength, and the finished bridge used far less concrete than a conventional one would have. But would it last? Sensors embedded in the structure have since recorded how it responds to temperature changes and foot traffic, and so far its performance matches the engineers' models.","choices":["It suggests that the engineers doubted their own design and expected the bridge to fail.","It introduces a criticism of 3D-printed concrete that the rest of the text ultimately endorses.","It shifts the focus of the text from the bridge itself to the printer that produced the concrete.","It raises a natural concern about the bridge that the following sentence begins to address."],"explain":"The question asks whether the bridge is durable; the next sentence reports sensor data showing that, so far, it performs as predicted. Nothing says the engineers doubted the design, and the text doesn't endorse a criticism."},"r1048":{"id":"r1048","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Structure of a narrative","tier":2,"traps":["half"],"answer":"A","stem":"Which choice best describes the overall structure of the text?","passage":"The following text is from a short story.

The storm took the roof off the schoolhouse on a Tuesday. By Wednesday, the children were holding lessons in the church basement, where the hymnals had to be cleared from the tables each morning. By Friday, Mrs. Odom had taught them how to measure the basement in hymnal-lengths, and the oldest boy had calculated, to everyone's satisfaction, that the room was exactly 212 hymnals long.","choices":["It describes a disruption and then shows how the characters adapt to it, even finding some fun in it.","It presents a conflict between Mrs. Odom and her students over the move and then shows how it is resolved.","It describes the damage caused by a storm in detail and then explains what caused the storm.","It compares the old schoolhouse with the church basement to show which is better suited to teaching."],"explain":"The storm disrupts school (Tuesday), the class moves (Wednesday), and by Friday a lesson grows out of the new setting \"to everyone's satisfaction.\" That is disruption followed by adaptation. There's no conflict with Mrs. Odom, no account of causes, and no comparison of buildings."},"r1049":{"id":"r1049","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Distinguishing evidence from interpretation","tier":3,"traps":["scope"],"answer":"B","stem":"Which choice best describes the function of the underlined sentence in the text as a whole?","passage":"In the Atacama Desert, archaeologists have found mummies that are about two thousand years older than the oldest Egyptian mummies. The Chinchorro people who made them removed the skin, reinforced the skeleton with sticks, and then rebuilt the body with clay, reeds, and the original skin. Unlike Egyptian mummification, which was largely reserved for the elite, Chinchorro mummification was given to people of every age and status, including infants. Some researchers interpret this as evidence of a belief that every member of the community deserved to be remembered.","choices":["It explains how the Chinchorro learned their method of mummification from contact with Egyptians.","It presents an observation about Chinchorro burial practices that the final sentence then interprets.","It provides evidence that Chinchorro mummies are about two thousand years older than Egyptian mummies.","It challenges the interpretation that some researchers offer in the final sentence of the text."],"explain":"The underlined sentence states a fact (mummification was given to everyone), and the final sentence interprets it (\"Some researchers interpret this as…\"). A contradicts the timeline: the Chinchorro practice is older. The age comparison (C) comes in the first sentence."},"r1050":{"id":"r1050","sec":"rw","domain":"Craft and Structure","skill":"Text Structure and Purpose","sub":"Main purpose of a scientific explanation","tier":2,"traps":["scope"],"answer":"C","stem":"Which choice best states the main purpose of the text?","passage":"When a woodpecker strikes a tree, its head decelerates at forces that would cause a concussion in a human. For years, researchers believed the bird's skull worked like a shock absorber, cushioning the brain from each blow. High-speed video analyzed by biomechanist Sam Van Wassenbergh suggests otherwise: the skull barely flexes at all. Instead, the woodpecker's small brain appears to be what protects it, since smaller brains experience much less force from the same deceleration.","choices":["To describe the high-speed video methods researchers use to study how birds move their heads","To argue that helmets for people should be designed to work like the skulls of woodpeckers","To present research that challenges a common explanation of how woodpeckers avoid brain injury","To explain why woodpeckers strike trees and how often they do so in a single day"],"explain":"The text states the old belief (the skull absorbs shock) and then reports video evidence that \"suggests otherwise,\" with an alternative explanation (small brain size). Filming methods are only a detail, and helmets and the reasons woodpeckers peck aren't discussed."},"r1001":{"id":"r1001","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Contrast with the evidence that follows","tier":3,"traps":["echo","reverse"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Critics have long described the composer's late string quartets as , as though each surprising turn had simply occurred to her at the keyboard. Her sketchbooks tell a different story: nearly every passage that sounds spontaneous was drafted and redrafted, some more than twenty times.","choices":["meticulous","derivative","improvisatory","austere"],"explain":"The critics' view is the one the sketchbooks contradict: the music sounds spontaneous, \"as though each surprising turn had simply occurred to her.\" That is improvisatory. Trap: meticulous describes what the sketchbooks reveal, which is the opposite of the critics' description."},"r1002":{"id":"r1002","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Balance, not completeness","tier":2,"traps":["half"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Historian Lena Marsh's account of the 1912 textile strike is admirably . She gives the mill owners' correspondence as much attention as the strikers' testimony, and where the surviving records are silent about who acted first, she declines to assign blame.","choices":["evenhanded","exhaustive","polemical","speculative"],"explain":"Both details describe fairness: equal attention to both sides and a refusal to blame without evidence. That is evenhanded. Polemical (aggressively one-sided) and speculative are the opposite. Trap: exhaustive sounds like praise for a careful history, but nothing here says she covers everything; the point is balance."},"r1003":{"id":"r1003","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Direction of a bias","tier":2,"traps":["reverse"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Because the region's bird surveys were conducted only from roadsides, ecologist Talia Brandt cautions that population estimates based on them may be : species that avoid roads are almost certainly undercounted, while those drawn to open roadside habitat may be overrepresented.","choices":["inflated","skewed","redundant","provisional"],"explain":"The colon explains the problem: some species are undercounted and others overrepresented. The estimates are biased in different directions, or skewed. Trap: inflated fits only the overrepresented species and contradicts the undercounting. Provisional means temporary, which the text doesn't address."},"r1004":{"id":"r1004","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Precise opposite of a recipe","tier":2,"traps":["echo"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"In her memoir, chef Rosa Delgado recalls that her grandmother's recipes were less instructions than : \"a pinch of this,\" \"enough\" of that, and a reminder to taste and adjust. They assumed the cook already knew what the finished dish should taste like.","choices":["prescriptions","inventories","transcriptions","suggestions"],"explain":"\"Less instructions than \" needs something looser than instructions, and the examples (\"a pinch,\" \"enough,\" \"taste and adjust\") are vague guidance. Suggestions fits. Prescriptions would be even more exact than instructions, and inventories (lists of ingredients) misses the guidance about method."},"r1005":{"id":"r1005","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Temporal precision","tier":3,"traps":["half"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Economist Ama Owusu argues that the cash-transfer program's benefits were real but . Payments reached nearly every eligible household in the first year, yet within three years most families' incomes had fallen back to where they were before the program began.","choices":["short-lived","pervasive","superficial","negligible"],"explain":"The evidence is about time: benefits were felt widely at first but faded within three years. Short-lived captures that. Trap: pervasive matches the first year only. Negligible contradicts \"real,\" and superficial (shallow) isn't what the time-based evidence shows."},"r1006":{"id":"r1006","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Contrast with \"deliberate\"","tier":2,"traps":["half"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Some readers regard the unresolved ending of Mira Castellanos's novel as , but Castellanos insists that leaving the central mystery unsolved was intentional: in her view, a tidy resolution would have betrayed the uncertainty her characters live with.","choices":["an homage","a revelation","a compromise","an oversight"],"explain":"\"But Castellanos insists that leaving the mystery unsolved was intentional\" answers a charge that it was unintentional, so readers see it as an oversight. A compromise implies a deliberate trade-off, which is closer to the author's view than to her critics'."},"r1007":{"id":"r1007","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Reuse, not imitation","tier":2,"traps":["echo"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Architect Jun Park's design for the new public library is strikingly . Rather than ordering new materials, it reuses the brick, the timber beams, and even the steel window frames of the shoe factory that once stood on the site.","choices":["ostentatious","resourceful","derivative","provisional"],"explain":"The design makes clever use of what was already there, which is resourceful. Trap: derivative means imitating someone else's work; reusing a building's materials isn't copying a design. Ostentatious (showy) clashes with the thrift described."},"r1008":{"id":"r1008","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Mixed sources","tier":2,"traps":["irrelevant"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Linguist Omar Haddad notes that the coastal dialect's vocabulary is remarkably : words borrowed from Arabic, Portuguese, Hindi, and Swahili sit side by side, and speakers often draw on three or four of these sources in a single sentence.","choices":["archaic","insular","eclectic","esoteric"],"explain":"Words drawn from many sources, used together, make the vocabulary eclectic. Insular (closed off from outside influence) is the opposite. Nothing says the words are old (archaic) or understood only by specialists (esoteric)."},"r1009":{"id":"r1009","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"\"Passing\" versus four decades","tier":2,"traps":["reverse"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Early reviewers dismissed the painter's interest in industrial scenes as a passing , but she returned to factories, cranes, and rail yards again and again over the next four decades.","choices":["obsession","conviction","commission","whim"],"explain":"The reviewers expected the interest to be brief (\"passing\") and unserious (\"dismissed\"): a whim. The \"but\" clause shows they were wrong. Trap: obsession describes what the interest turned out to be, and \"a passing obsession\" contradicts itself."},"r1010":{"id":"r1010","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Neither yes nor no","tier":2,"traps":["half"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The senator's statement on the trade agreement was carefully : she praised its labor protections and criticized its environmental provisions but declined to say how she would vote.","choices":["conciliatory","unequivocal","noncommittal","dismissive"],"explain":"Praise for one part, criticism of another, and no stated vote: she avoided taking a position, so the statement was noncommittal. Unequivocal means the opposite. Trap: conciliatory (meant to make peace) captures the mixed tone but not the refusal to commit."},"r1011":{"id":"r1011","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Uncommon sense of a common word","tier":3,"traps":["echo"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The park ranger's praise for the Copper Ridge trail was : she called it the most beautiful route in the park but warned that its loose rock made it unsuitable for anyone without climbing experience.","choices":["unreserved","qualified","grudging","effusive"],"explain":"Here qualified means limited by a condition: the praise comes with a serious warning attached. Unreserved and effusive describe praise without limits. Trap: grudging means given reluctantly, but the ranger isn't reluctant; she calls it the most beautiful route."},"r1012":{"id":"r1012","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Reveal without meaning to","tier":3,"traps":["reverse"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"In his wartime letters home, the general's confident tone occasionally his private doubts: a reassurance repeated a little too often, a plan described in suspiciously exhaustive detail.","choices":["betrays","conceals","resolves","exaggerates"],"explain":"The examples after the colon are clues that leak through the confident surface, so the tone sometimes betrays (unintentionally reveals) his doubts. Trap: a confident tone usually conceals doubt, but the colon lists signs of doubt showing through, the opposite of concealment."},"r1013":{"id":"r1013","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"What a statistic leaves out","tier":2,"traps":["reverse"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Sociologist Imani Clarke cautions that the importance of rural libraries is easily by circulation statistics, which count the books residents borrow but not the hours they spend using library computers, attending classes, or getting help with tax forms.","choices":["inflated","understated","documented","replaced"],"explain":"Circulation figures miss much of what libraries do, so they make libraries look less important than they are: the importance is understated. Trap: inflated reverses the direction of the error."},"r1014":{"id":"r1014","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Narrowing a theory","tier":3,"traps":["half"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The new observations do not overturn the theory of planet formation so much as it: the theory's central claim survives, but its predictions now appear to hold only for stars within a narrower range of masses than astronomers had assumed.","choices":["circumscribe","corroborate","supplant","popularize"],"explain":"The theory survives but now applies to fewer stars: its scope is limited, or circumscribed. Supplant means replace, which \"do not overturn\" rules out. Trap: corroborate (confirm) fits \"survives\" but ignores the narrowing."},"r1015":{"id":"r1015","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Minimal composition","tier":1,"traps":["irrelevant"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Unlike her contemporaries, who filled their canvases with crowded markets and elaborate interiors, the painter favored compositions: a single tree, a strip of horizon, and broad fields of unbroken color.","choices":["ornate","vivid","cluttered","spare"],"explain":"A single tree, a strip of horizon, and unbroken color describe minimal compositions, which are spare, in contrast to her contemporaries' crowded scenes. Ornate and cluttered describe the contemporaries. Vivid (bright) isn't supported."},"r1016":{"id":"r1016","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"A suspicion that turns out wrong","tier":2,"traps":["reverse"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Marine chemist Leilani Kahale initially suspected that her temperature readings had been by a faulty sensor. But recalibrating the instrument changed none of the readings, and she concluded that the unusually warm water was real.","choices":["confirmed","supplemented","distorted","anticipated"],"explain":"She suspected the sensor had made the readings inaccurate (distorted) and then found it hadn't, so the warm water was real. Trap: confirmed is what she eventually concluded about the water, not what she suspected the sensor did."},"r1017":{"id":"r1017","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"\"Deceptively\" reverses the surface","tier":3,"traps":["half"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The playwright's dialogue is deceptively . Her characters speak in short, ordinary sentences, yet nearly every line carries a second meaning that becomes clear only when the play is seen a second time.","choices":["ornate","opaque","verbose","simple"],"explain":"\"Deceptively\" means appearing one way while being another. The dialogue looks plain (short, ordinary sentences) but holds hidden meanings, so it is deceptively simple. Ornate and verbose contradict \"short, ordinary sentences.\" Trap: opaque describes the hidden meanings, but \"deceptively opaque\" would mean the dialogue only seems hard to understand."},"r1018":{"id":"r1018","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Old view versus new records","tier":2,"traps":["reverse"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Historians once regarded the mountain trade route as to the region's economy, a minor path used by a handful of traveling merchants. Newly translated tax records, however, show that it carried more goods than any other route in the region.","choices":["peripheral","integral","detrimental","antecedent"],"explain":"The old view is glossed right after the blank: \"a minor path.\" That is peripheral, of little importance. Trap: integral is what the new records show, the view the \"however\" sentence contrasts with."},"r1019":{"id":"r1019","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Unequal distribution","tier":1,"traps":["irrelevant"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Because the vaccine must stay frozen until shortly before use, its distribution in remote areas has been : clinics without reliable freezers have received far fewer doses than urban hospitals.","choices":["accelerated","uneven","uniform","unprecedented"],"explain":"Some places get far fewer doses than others, so distribution is uneven. Uniform is the opposite, and nothing indicates speed or novelty."},"r1020":{"id":"r1020","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Small improvements","tier":2,"traps":["echo"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Physicist Dara Okonkwo describes her lab's new particle detector as rather than revolutionary: it uses the same basic design as earlier instruments, but nearly every component has been refined to reduce background noise.","choices":["derivative","experimental","incremental","obsolete"],"explain":"Same design, many small refinements: progress by steps, which is incremental. Trap: derivative suggests unoriginal copying, a criticism; the text describes improvement, and Okonkwo is describing her own lab's work favorably."},"r1021":{"id":"r1021","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Mixed feelings","tier":2,"traps":["half"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"In her essay, Noor Siddiqui's tone toward her childhood town is : she writes fondly of its harvest festivals and its neighbors' kindness, yet she also recounts, without softening it, the narrowness of outlook that eventually drove her to leave.","choices":["ambivalent","nostalgic","scathing","indifferent"],"explain":"Fondness and criticism together make the tone ambivalent (mixed). Nostalgic covers only the fond half and scathing only the critical half. Indifferent (not caring) contradicts both."},"r1022":{"id":"r1022","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Degree supported by evidence","tier":2,"traps":["reverse"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Anthropologist Kenji Mori found that the island's fishing rules, though never written down, were observed: during two years of fieldwork, he recorded only three violations.","choices":["sporadically","nominally","scrupulously","reluctantly"],"explain":"Only three violations in two years means the rules were followed carefully, or scrupulously. Sporadically (occasionally) and nominally (in name only) reverse the evidence. Nothing indicates reluctance."},"r1023":{"id":"r1023","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Intended versus actual","tier":2,"traps":["reverse"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Although the museum's founders intended its collection to be , acquiring objects from every continent and historical period, budget shortfalls meant that by 1950 the collection consisted mostly of pottery from nearby towns.","choices":["provincial","encyclopedic","esoteric","ephemeral"],"explain":"The founders' intention is glossed after the comma: objects from everywhere and every period. That is encyclopedic. Trap: provincial describes what the collection actually became, the outcome the \"although\" sentence contrasts with the intention."},"r1024":{"id":"r1024","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Uncommon sense: to restrain","tier":2,"traps":["irrelevant"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Ecologists introduced barn owls to the island in hopes that the birds would the growth of its rat population. Instead, the owls preyed mostly on native seabirds, and the number of rats kept climbing.","choices":["mirror","document","accelerate","check"],"explain":"The ecologists hoped the owls would hold back the rats; here check means restrain. \"Instead … the number of rats kept climbing\" shows the hope failed. Accelerate reverses the goal; mirror and document aren't things predators do to prey."},"r1025":{"id":"r1025","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Evidence that weakens an assumption","tier":1,"traps":["reverse"],"answer":"D","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The newly digitized census data the long-standing assumption that the town's population peaked in 1920: the figures show that it continued to grow for another two decades.","choices":["reinforce","explain","predate","undercut"],"explain":"Data showing growth after 1920 contradict the assumption of a 1920 peak, so they undercut it. Reinforce is the reverse."},"r1026":{"id":"r1026","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Character trait from behavior","tier":2,"traps":["reverse"],"answer":"A","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The diplomat was known for her : in negotiations, she rarely revealed her own position until the other side had fully stated theirs, and she seldom spoke to the press at all.","choices":["reticence","candor","volatility","magnanimity"],"explain":"Withholding her position and avoiding the press show reticence, a reluctance to reveal thoughts. Candor (frankness) is the opposite. Nothing suggests mood swings (volatility) or generosity (magnanimity)."},"r1027":{"id":"r1027","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Expected opposition","tier":2,"traps":["reverse"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Rather than the earlier study's findings, as some critics had expected, the new paper accepts them and builds on them, returning to the same field sites to ask a question the original authors never considered.","choices":["extending","citing","challenging","applying"],"explain":"\"Rather than … the new paper accepts them\" sets up a contrast with accepting, so the blank must mean rejecting or disputing: challenging. Trap: extending describes what the paper actually does."},"r1028":{"id":"r1028","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"A charge the evidence rebuts","tier":2,"traps":["reverse"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"The poet's late work is often called , but her notebooks suggest that its compressed, fragmentary lines were not a sign of declining powers but a deliberate experiment with silence and omission.","choices":["experimental","faltering","prolific","derivative"],"explain":"The notebooks rebut a charge of \"declining powers,\" so critics call the work faltering. Trap: experimental is the poet's own view, which the notebooks support, not the critics' label."},"r1029":{"id":"r1029","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Significant but small","tier":2,"traps":["half"],"answer":"C","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Pharmacologist Sunita Rao warns that the drug's benefits, though statistically significant, are : patients who took it recovered, on average, only half a day sooner than those who received a placebo.","choices":["spurious","substantial","modest","inconsistent"],"explain":"The colon shows the benefit is small: half a day. Modest fits. Substantial reverses it. Trap: spurious (false) contradicts \"statistically significant\": the benefit is real, just small."},"r1030":{"id":"r1030","sec":"rw","domain":"Craft and Structure","skill":"Words in Context","sub":"Not a literal depiction","tier":2,"traps":["reverse"],"answer":"B","stem":"Which choice completes the text with the most logical and precise word or phrase?","passage":"Medieval European world maps were not representations of geography. Many placed Jerusalem at the center and illustrated scenes from scripture, reflecting the mapmakers' beliefs as much as the shape of the land.","choices":["symbolic","literal","decorative","ambitious"],"explain":"The maps reflected beliefs as much as geography, so they were not straightforward, literal depictions. Trap: symbolic describes what the maps were, so \"not symbolic\" contradicts the evidence. The illustrations make decorative tempting, but \"not decorative\" is also contradicted."},"r2001":{"id":"r2001","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea of a literary passage","tier":3,"traps":["scope"],"answer":"A","stem":"Which choice best states the main idea of the text?","passage":"The following text is from a novel. Teodora has just been promoted to head chef of the restaurant where she has worked for nine years.

Teodora had imagined the moment many times, and in every version she felt triumphant. Instead, standing in the empty kitchen after the others had gone home, she noticed only the small things she would now be responsible for: the hum of the walk-in freezer, the knife with the loose handle, the schedule taped crookedly to the wall. The promotion had not lifted a weight from her; it had handed her a new one, and she was surprised to find that she did not mind its heft.","choices":["Teodora's promotion brings responsibilities she hadn't expected, and she finds she is willing to carry them.","Teodora regrets accepting a promotion that she now realizes will keep her in the kitchen long after the others leave.","Teodora feels just as triumphant about her promotion as she had always imagined she would.","Teodora worries that the restaurant's equipment, including the freezer and the knives, is in poor condition."],"explain":"Her imagined triumph gives way to a sense of new responsibility (\"handed her a new one\"), but she \"did not mind its heft.\" B ignores that final acceptance; C contradicts \"Instead.\" The freezer and the loose knife are examples of responsibility, not the main point (D)."},"r2002":{"id":"r2002","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea of a dense informational text","tier":2,"traps":["scope"],"answer":"C","stem":"Which choice best states the main idea of the text?","passage":"Researchers once believed that trees in a forest compete with one another for every resource. Studies by forest ecologist Mei Takahashi suggest a more complicated picture. Underground networks of fungi connect the roots of many trees, and through them, carbon and nutrients can pass from one tree to another. In Takahashi's experiments, seedlings shaded by taller trees received carbon through these networks, which helped them survive until a gap opened in the canopy above.","choices":["Fungi that connect the roots of different trees harm the trees by drawing carbon and nutrients from them.","Seedlings shaded by taller trees grow faster than seedlings that receive direct sunlight.","Research suggests that trees can share resources through fungal networks, not only compete for them.","Forest ecologists now agree that trees in a forest never compete with one another for resources."],"explain":"The text revises the old view of pure competition by describing resource sharing through fungal networks. D overstates the revision: \"more complicated\" isn't \"never compete.\" B misreads the seedling detail; the carbon helped them survive, not grow faster than seedlings in the sun."},"r2003":{"id":"r2003","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"A detail about a character's motive","tier":3,"traps":["half"],"answer":"B","stem":"According to the text, why does Mr. Ferris begin to involve the boy's mother in the lessons?","passage":"The following text is from a short story. Mr. Ferris, a retired schoolteacher, has started tutoring his neighbor's son.

The boy was quick, quicker than he let on, and Mr. Ferris soon understood that the wrong answers were not mistakes. Each time the boy erred, his mother, listening from the next room, would come in to sit beside him. Mr. Ferris began, without saying anything, to leave the door open and to pause, now and then, to ask the mother what she thought.","choices":["He believes that the mother understands the subject better than he does and can explain it more clearly.","He realizes that the boy gives wrong answers on purpose so that his mother will join him.","He wants the mother to see for herself how poorly her son is doing in his lessons.","He finds that the boy's questions are often too difficult for him to answer without help."],"explain":"Mr. Ferris sees that \"the wrong answers were not mistakes\": each error brings the mother into the room. Leaving the door open and asking her opinion gives the boy the company he was seeking. C assumes the boy is struggling, which the text contradicts (\"quick, quicker than he let on\")."},"r2004":{"id":"r2004","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea with a concession","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice best states the main idea of the text?","passage":"Lab-grown diamonds are chemically identical to mined diamonds and can now be produced for a fraction of the cost. Jewelers once predicted that consumers would never accept them, but sales have risen steadily, especially among younger buyers. Economist Rosa Lind argues that this shift may permanently lower the prices of mined diamonds, whose value has depended largely on the belief that they are rare.","choices":["Lab-grown diamonds are chemically different from mined diamonds and are therefore less valuable to most buyers.","Younger consumers prefer mined diamonds to lab-grown diamonds because they believe mined diamonds are rare.","Jewelers have successfully persuaded consumers to reject lab-grown diamonds despite their lower cost.","Wider acceptance of lab-grown diamonds may lower the value of mined diamonds, which depends on scarcity."],"explain":"The text describes rising acceptance of lab-grown diamonds and Lind's argument about its effect on mined-diamond prices. A contradicts \"chemically identical.\" B and C reverse the sales trend."},"r2005":{"id":"r2005","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"What a speaker reveals indirectly","tier":3,"traps":["scope"],"answer":"D","stem":"Which choice best describes how the speaker's father communicates with her, according to the text?","passage":"The following text is from a poem. The speaker describes her father's workshop.

Every tool had a painted outline on the pegboard,
so you could see at a glance what was missing.
He never told me to put things back.
He only stood, some evenings, looking at the wall,
at the white shape of a saw I had borrowed,
until I brought it home.","choices":["He scolds her whenever she forgets to return one of his tools to its place on the pegboard.","He refuses to let her borrow his tools until she promises to put each one back.","He writes notes on the pegboard beside each outline reminding her what she needs to return.","He conveys his expectations silently, through the outlines and his attention to them."],"explain":"\"He never told me to put things back\"; instead he stood looking at the empty outline until she returned the saw. He communicates without words. A contradicts \"never told me.\" She had borrowed the saw, so B is wrong too."},"r2006":{"id":"r2006","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea of a historical account","tier":2,"traps":["scope"],"answer":"B","stem":"Which choice best states the main idea of the text?","passage":"In the 1850s, London's River Thames carried the city's untreated sewage, and during the hot summer of 1858 the smell became so overwhelming that Parliament, which met beside the river, considered relocating. The crisis, known as the Great Stink, finally pushed lawmakers to fund engineer Joseph Bazalgette's plan for a vast network of sewers that carried waste away from the city center. Completed over the following decades, the system sharply reduced outbreaks of cholera, though most officials at the time still believed the disease spread through foul air rather than contaminated water.","choices":["Parliament moved away from the Thames after the Great Stink of 1858 made its meeting place unbearable.","A stench in 1858 led to sewers that improved public health, though officials misunderstood cholera.","Joseph Bazalgette's sewer system proved to officials that cholera was spread through contaminated water.","London's sewers were completed within a single year after the Great Stink, ending cholera outbreaks."],"explain":"The text links the Great Stink to funding for the sewers, notes the drop in cholera, and adds that officials held the wrong theory of disease. A confuses \"considered relocating\" with relocating. C isn't stated, and D contradicts \"over the following decades.\""},"r2007":{"id":"r2007","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"A character's perspective on a change","tier":2,"traps":["scope"],"answer":"A","stem":"Which choice best describes Abuela Carmen's attitude toward the change described in the text?","passage":"The following text is from a novel. Abuela Carmen has lived in the same apartment for fifty years.

When the building's new owners replaced the old iron mailboxes with gleaming steel ones, the other tenants were delighted. Abuela Carmen said nothing, but every morning she went down to check her new box at exactly the time the mail had always come, though the carrier now arrived in the afternoon. \"The box is new,\" she told her grandson. \"I am not.\"","choices":["She keeps to her long-standing habits even though her surroundings have changed.","She is privately as delighted by the new mailboxes as the other tenants are but keeps it to herself.","She resents the new owners for replacing the mailboxes and refuses to use her new box.","She is confused about when the mail now arrives because the carrier's schedule has changed."],"explain":"She keeps checking at the old time, and her line \"The box is new. I am not\" shows she knowingly holds to her routine. The text says she \"said nothing,\" so C fails. D misreads deliberate habit as confusion."},"r2008":{"id":"r2008","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea with a surprising result","tier":3,"traps":["scope"],"answer":"C","stem":"Which choice best states the main idea of the text?","passage":"When a city in the Netherlands removed traffic lights, signs, and curbs from a busy intersection, planners expected chaos. Instead, accidents fell. Traffic engineer Hans Monderman, who designed the change, argued that drivers who are unsure who has the right of way slow down and make eye contact with pedestrians and cyclists, while clear rules invite drivers to stop paying attention. Critics note, however, that the approach may be harder for people with visual impairments, who rely on curbs and signals to navigate.","choices":["Traffic lights and signs are the main cause of accidents at busy intersections in Dutch cities.","Drivers in the Netherlands pay closer attention to pedestrians and cyclists than drivers in other countries do.","Removing traffic controls from one intersection reduced accidents, though the approach has drawbacks.","Pedestrians with visual impairments prefer intersections without curbs because they are easier to cross."],"explain":"The text reports the surprising drop in accidents, Monderman's explanation, and the critics' concern. A overgeneralizes from one intersection; D reverses the critics' point."},"r2009":{"id":"r2009","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea of a two-part claim","tier":2,"traps":["half"],"answer":"C","stem":"Which choice best states the main idea of the text?","passage":"Octopuses can change color in a fraction of a second, yet most species have only one type of light receptor in their eyes, which should make them colorblind. Biologists have proposed that octopuses may detect color in another way: their unusual U-shaped pupils could separate light by wavelength, letting the animals sense color by how blurry different parts of an image appear. The hypothesis has not yet been confirmed experimentally.","choices":["Octopuses are colorblind, which explains why their unusual pupils are shaped like the letter U.","Experiments have confirmed that the U-shaped pupils of octopuses separate light by its wavelength.","Biologists have proposed an unconfirmed explanation of how octopuses might perceive color.","Octopuses have more types of light receptors in their eyes than most other animals that change color."],"explain":"The text presents a puzzle (color change but one receptor type) and a proposed solution that \"has not yet been confirmed.\" B contradicts that last sentence; D contradicts \"only one type.\""},"r2010":{"id":"r2010","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Purpose of a character's action","tier":3,"traps":["half"],"answer":"B","stem":"Based on the text, what is the most likely reason Grandpa Ike pours a second cup of coffee each Sunday?","passage":"The following text is from a short story.

Every Sunday, Grandpa Ike set two cups on the porch table, filled both with coffee, and drank from one. When my cousin asked who the other cup was for, he said, \"Whoever comes.\" In all the summers I visited, no one ever came, and every Sunday at noon he poured the second cup, cold and untouched, over the roses, which were the best on the street.","choices":["He is expecting a particular friend who once promised to visit him on a Sunday.","He wants to show that he is ready to welcome any visitor, even though none arrives.","He uses the second cup of coffee to fertilize his roses, which are the best on the street.","He forgets each week that he has already poured a cup of coffee for himself."],"explain":"His answer, \"Whoever comes,\" shows the cup is for no one in particular; it signals openness to a visitor. A contradicts \"Whoever.\" The roses get the leftover coffee, but that's a consequence, not the reason he pours it (C)."},"r2011":{"id":"r2011","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Detail that supports the main point","tier":2,"traps":["scope"],"answer":"D","stem":"According to the text, how did Petra's builders make it possible for a large population to live in a dry place?","passage":"The ancient city of Petra, carved into sandstone cliffs in what is now Jordan, received only about 15 centimeters of rain a year. Yet it supported as many as 20,000 people. The Nabataeans who built it captured almost every drop: channels cut into the rock carried runoff from the surrounding hills to underground cisterns, and clay pipes delivered water throughout the city. Archaeologists estimate that the system could store millions of liters.","choices":["They used clay pipes to import water from rivers in the hills surrounding the city.","They kept the city's population small enough that the limited rainfall could support it.","They chose a location in the hills where rain fell far more often than in the surrounding desert.","They built a system that collected rainwater and runoff and stored it for later use."],"explain":"The text describes channels carrying runoff to cisterns and pipes distributing the stored water. The pipes moved water within the city, not from distant rivers (A). C contradicts the low rainfall."},"r2012":{"id":"r2012","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea of a reflective passage","tier":3,"traps":["scope"],"answer":"A","stem":"Which choice best states the main idea of the text?","passage":"The following text is from an essay by a translator.

People often ask which of my translations is most faithful, as if faithfulness were a single thing. But a sentence can be faithful to the original's meaning and unfaithful to its music, or faithful to its rhythm while losing a pun on which the whole paragraph turns. Every translation chooses among loyalties. The best I can do is choose deliberately, and know what I have given up.","choices":["Translation means choosing among kinds of faithfulness, and the translator should choose knowingly.","A good translation preserves every feature of the original, including its meaning, rhythm, and wordplay.","Faithfulness to an original text's meaning is less important than faithfulness to its sound and rhythm.","Readers who care about faithfulness should avoid translations and learn to read works in the original."],"explain":"The writer argues that faithfulness has several forms that conflict (\"Every translation chooses among loyalties\") and that the translator should \"choose deliberately.\" B is the view the essay rejects. C ranks one loyalty above another, which the writer never does."},"r2013":{"id":"r2013","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Main idea that reverses an assumption","tier":2,"traps":["scope"],"answer":"B","stem":"Which choice best states the main idea of the text?","passage":"For years, physicians advised parents to keep peanuts away from infants to prevent peanut allergies. A large clinical trial changed that advice. Infants at high risk of allergy who were regularly fed small amounts of peanut protein starting before their first birthday were far less likely to develop a peanut allergy by age five than similar infants who avoided peanuts. Guidelines in many countries now recommend introducing peanut foods early for most infants.","choices":["Infants at high risk of allergy should not be given peanut foods until after their fifth birthday.","A trial found that early exposure to peanuts lowers allergy risk, reversing earlier advice.","Peanut allergies that develop in early childhood usually disappear by the time children turn five.","Physicians have stopped testing infants for allergies because early exposure prevents them."],"explain":"The text contrasts old advice (avoid peanuts) with a trial showing early exposure reduced allergies, and notes that guidelines changed. A is the old advice the trial overturned. The trial measured who developed an allergy, not whether allergies disappear (C)."},"r2014":{"id":"r2014","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"A character's realization","tier":3,"traps":["scope"],"answer":"D","stem":"Which choice best describes what Rafael realizes at the end of the text?","passage":"The following text is from a novel. Rafael, a young violinist, has just played for a famous teacher.

Rafael finished the sonata and waited for the verdict. The old man only asked him to play the opening again, slower. Then again, slower still, until the first phrase stretched so long that Rafael could hear, inside each note, the small unevenness of his bow that speed had always hidden. When the teacher finally nodded, Rafael understood that he had not been judged at all. He had been shown.","choices":["The teacher was disappointed by his playing and had decided not to accept him as a student.","He should always play the opening of the sonata as slowly as possible when performing it.","The teacher had not listened carefully to his first performance of the sonata.","The teacher's requests were meant to show him a flaw rather than to judge him."],"explain":"Playing slower exposed \"the small unevenness of his bow,\" and Rafael concludes he \"had not been judged\" but \"shown.\" The slow playing was a teaching method, not a rule for performing (B). Nothing indicates disappointment (A)."},"r2015":{"id":"r2015","sec":"rw","domain":"Information and Ideas","skill":"Central Ideas and Details","sub":"Detail inside a technical passage","tier":2,"traps":["half"],"answer":"C","stem":"According to the text, what is the purpose of expansion joints?","passage":"Most bridges expand in summer and contract in winter. To keep the structure from cracking, engineers include expansion joints: gaps, often bridged by interlocking metal teeth, that let sections of the deck slide slightly as their length changes. On a long steel bridge, the total movement across a joint can exceed half a meter between the hottest and coldest days of the year.","choices":["To strengthen a bridge so that it can support heavy traffic in both summer and winter","To prevent the sections of a bridge from expanding on the hottest days of the year","To let parts of a bridge change length with temperature without cracking","To connect sections of a bridge made of steel to sections made of concrete"],"explain":"Joints \"let sections of the deck slide slightly as their length changes,\" which keeps the structure \"from cracking.\" They allow expansion rather than prevent it (B). Traffic and mixed materials aren't mentioned."},"r2034":{"id":"r2034","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Comparing a gap across conditions","tier":3,"traps":["half"],"figure":"
Mean Grain Yield (tons per hectare)
VarietyNormal wateringDrought
K5.04.0
M6.03.6
","answer":"B","stem":"Which choice most effectively uses data from the table to support Petrov's conclusion?","passage":"Plant physiologist Nadia Petrov grew two varieties of wheat under normal watering and under drought conditions and measured grain yield. She concluded that variety K is more drought tolerant than variety M: although K yields less than M when water is plentiful, it loses a smaller share of its yield when water is scarce.","choices":["Under normal watering, variety M yielded 1.0 ton per hectare more than variety K, a difference of 20%.","Under drought, variety K's yield fell by 20% from its normal level, while variety M's yield fell by 40%.","Under drought, variety K yielded 4.0 tons per hectare, while variety M yielded 3.6 tons per hectare.","Both varieties yielded less under drought, and variety K yielded more than variety M under drought."],"explain":"The conclusion is about the share of yield lost. K: 5.05.0−4.0​=20%. M: 6.06.0−3.6​=40%. Only B compares the shares. A supports the first half of the sentence (\"K yields less…\"), not the conclusion. C gives one number with no comparison."},"r2035":{"id":"r2035","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Completing a claim with a trend","tier":2,"traps":["irrelevant"],"figure":"
\"A
","answer":"A","stem":"Which choice most effectively uses data from the graph to complete the statement?","passage":"Ornithologists counted migrating hawks passing over a ridge each autumn. The graph shows the counts for two species. Counts of broad-winged hawks varied widely from year to year, but counts of red-tailed hawks ","choices":["fell steadily, declining from each year to the next from 2019 through 2024.","were lower than counts of broad-winged hawks in every year from 2019 through 2024.","were highest in 2019, when about 1,500 red-tailed hawks were counted.","never exceeded 2,000 in any year that the ornithologists counted."],"explain":"The sentence contrasts the wild swings of broad-winged hawks with red-tailed hawks, so the completion should describe a steady pattern: a decline every year. B, C, and D are all true, but none provides the contrast with \"varied widely.\""},"r2036":{"id":"r2036","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Data that weaken a claim","tier":3,"traps":["reverse"],"figure":"
Average Cars Entering Downtown per Weekday (thousands)
City20202022
City with bike share8476
Nearby city without bike share7063
","answer":"C","stem":"Which choice most effectively uses data from the table to support the researcher's point?","passage":"A city introduced a bike-share program in 2021. A local official claims that the program reduced car traffic downtown, citing a drop in the average number of cars entering downtown each weekday after 2021. A transportation researcher notes that the official's comparison leaves out a nearby city that did not introduce bike sharing.","choices":["The city with bike share had more downtown car traffic than the nearby city in both years.","Car traffic in the city with bike share fell from 84,000 to 76,000 cars per weekday.","Car traffic fell by about 10% in both cities, even though only one introduced bike sharing.","In 2022, the nearby city had 63,000 cars entering downtown per weekday."],"explain":"The researcher's point is that the comparison needs a city without bike sharing. Traffic fell about 9.5% (84 to 76) in one city and 10% (70 to 63) in the other, so the drop happened without bike sharing too, which undercuts the official's causal claim. B is the official's own evidence."},"r2037":{"id":"r2037","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Reading a two-variable table carefully","tier":3,"traps":["half"],"figure":"
How Visitors First Heard About the Park (percent of visitors)
SourcePark XPark YPark Z
Friends or family412835
Social media223930
Travel guides251821
Other121514
","answer":"D","stem":"Which choice most effectively uses data from the table to support the researchers' claim?","passage":"Researchers surveyed visitors to three national parks about how they first heard about the park. The researchers claim that social media was a more important source of information for visitors to Park Y than for visitors to Park X.","choices":["Social media was cited by 30% of Park Z visitors, more than any other source except friends or family.","Friends or family were cited by 41% of Park X visitors, more than any other source for that park.","Travel guides were cited by fewer Park Y visitors (18%) than Park X visitors (25%).","Social media was cited by 39% of Park Y visitors, the most of any source, but by just 22% of Park X visitors."],"explain":"The claim compares social media's importance for Park Y and Park X, so the evidence must give social media's figure for both parks: 39% versus 22% (and for Park Y it is the top source). A is about Park Z. B and C describe other sources, not social media."},"r2038":{"id":"r2038","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Graph showing a crossover","tier":2,"traps":["irrelevant"],"figure":"
\"A
","answer":"C","stem":"Which choice most effectively uses data from the graph to support the analyst's claim?","passage":"An energy analyst tracked the share of a region's electricity generated from coal and from wind between 2016 and 2024. The analyst claims that by the end of the period, wind had overtaken coal as a source of electricity in the region.","choices":["Coal's share of electricity fell in every year shown.","Wind's share of electricity rose from 9% in 2016 to 20% in 2020.","In 2024, wind generated 33% of the region's electricity, while coal generated 17%.","In 2016, coal generated more than five times as much of the region's electricity as wind did."],"explain":"\"Overtaken by the end of the period\" requires comparing the two sources at the end: in 2024, wind (33%) exceeds coal (17%). A and B each describe only one source, and D describes the beginning of the period."},"r2039":{"id":"r2039","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Choosing the right comparison group","tier":3,"traps":["scope"],"figure":"
Average Gain in Reading Score (points) After One Year
GroupUsual lessonsNew program
Struggling readers615
Strong readers58
","answer":"B","stem":"Which choice most effectively uses data from the table to support Aydin's claim?","passage":"In a study of a new reading program, students were randomly assigned either to the program or to their school's usual lessons. Education researcher Kemal Aydin claims that the program helped struggling readers more than it helped strong readers.","choices":["Struggling readers in the program gained 15 points, the most of any group in the study.","The program raised gains by 9 points for struggling readers but by only 3 points for strong readers.","Strong readers in the program gained 8 points, while strong readers in usual lessons gained 5 points.","Struggling readers gained more than strong readers in both the program and usual lessons."],"explain":"\"Helped more\" means a bigger improvement compared with usual lessons for each group: 15−6=9 versus 8−5=3. A ignores the baseline. D is true but shows only that struggling readers gain more in general, not that the program helped them more."},"r2040":{"id":"r2040","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Bar graph and a qualified claim","tier":2,"traps":["half"],"figure":"
\"A
","answer":"D","stem":"Which choice most effectively uses data from the graph to support the researcher's claim?","passage":"A sleep researcher measured how long it took participants to fall asleep after using a phone, reading a printed book, or doing neither during the hour before bed. She claims that reading a printed book did not delay sleep, while using a phone did.","choices":["Participants took 26 minutes to fall asleep after using a phone, nearly twice as long as after doing neither.","Participants fell asleep fastest after doing neither, taking an average of only 14 minutes.","Participants took one minute longer to fall asleep after reading a printed book than after doing neither.","Sleep came about as fast after reading (15 minutes) as after doing neither (14), but took 26 minutes after phone use."],"explain":"The claim has two parts: reading did not delay sleep (compared with doing neither) and phones did. D gives the evidence for both. A covers only the phone, C only the book, and B mentions neither."},"r2041":{"id":"r2041","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Percent change versus absolute change","tier":3,"traps":["reverse"],"figure":"
Number of Adult Frogs Counted
Species20152025
Species A400560
Species B80200
","answer":"A","stem":"Which choice most effectively uses data from the table to support the group's claim?","passage":"A conservation group reported population counts for two endangered frog species at a wildlife refuge. The group claims that, relative to their starting populations, species B recovered more strongly than species A between 2015 and 2025.","choices":["Species B's population increased by 150%, while species A's increased by 40%.","Species A's population increased by 160 frogs, while species B's increased by 120 frogs.","In 2025, there were more adult frogs of species A than of species B.","Both species had more adult frogs in 2025 than in 2015."],"explain":"\"Relative to their starting populations\" calls for percent change: B grew 80120​=150% and A grew 400160​=40%. B uses absolute changes, which actually favor species A, the opposite of the claim."},"r2042":{"id":"r2042","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Completing an explanation with data","tier":2,"traps":["irrelevant"],"figure":"
Retreat of Glacier Front, 2000–2020 (meters)
Glacier2000–20102010–2020Total
R (ends on land)210260470
S (ends in ocean)8901,1202,010
","answer":"B","stem":"Which choice most effectively uses data from the table to complete the statement?","passage":"Glaciologists measured how far the front edge of two glaciers retreated between 2000 and 2020. Glacier R ends on land, while Glacier S ends in the ocean, where warm seawater can melt it from below. Consistent with the idea that ocean contact speeds retreat, the data show that ","choices":["Glacier R retreated farther in 2010–2020 (260 meters) than in 2000–2010 (210 meters).","Glacier S retreated more than four times as far as Glacier R over the twenty years.","Glacier S retreated 890 meters in 2000–2010 and 1,120 meters in 2010–2020.","both glaciers retreated faster in the second decade than in the first."],"explain":"The idea is that ocean contact speeds retreat, so the evidence must compare the ocean glacier with the land glacier: 2,010 meters versus 470 is more than four times as far. A and D concern changes over time, not ocean contact, and C gives one number with no comparison."},"r2043":{"id":"r2043","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Data that contradict a prediction","tier":3,"traps":["reverse"],"figure":"
Bumblebee Visits per Hour by Wind Speed
Wind speed (km/h)0–56–1011–1516–20
Visits per hour48413335
","answer":"C","stem":"Which choice most effectively uses data from the table to complete the statement?","passage":"Ecologist Juno Abara predicted that bumblebees would visit flowers less often on windy days, since wind makes landing difficult. She recorded visits per hour to a meadow's flowers at different wind speeds. The data do not support her prediction for the highest wind speeds tested, however: ","choices":["visits were most frequent at wind speeds of 0–5 km/h.","visits fell from 48 per hour to 41 per hour as wind speed rose from 0–5 km/h to 6–10 km/h.","visits were slightly more frequent at 16–20 km/h than at 11–15 km/h.","there were fewer visits at 16–20 km/h than at 0–5 km/h."],"explain":"The sentence says the data don't fit the prediction at the highest wind speeds. Visits rose slightly (33 to 35) from 11–15 to 16–20 km/h, which is not what the prediction expects. A, B, and D all fit the prediction."},"r2044":{"id":"r2044","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Combining two columns","tier":3,"traps":["half"],"figure":"
Museum Attendance by Type of Visit
Total visitorsFirst-time visitors
Regular weekday (average)1,200240
Free Thursday evening (average)1,500600
","answer":"D","stem":"Which choice most effectively uses data from the table to support the director's claim?","passage":"A museum offered free admission on Thursday evenings to attract visitors who rarely visit museums. The museum's director claims that the program succeeded: on free evenings, a larger share of visitors were first-time visitors than on regular days.","choices":["The museum averaged 300 more visitors on free Thursday evenings than on regular weekdays.","The museum averaged 600 first-time visitors on free Thursday evenings, compared with 240 on regular weekdays.","Most visitors on free Thursday evenings were not first-time visitors.","First-time visitors made up 40% of visitors on free Thursday evenings but only 20% on regular weekdays."],"explain":"The claim is about the share of first-time visitors: 1,500600​=40% versus 1,200240​=20%. A compares total attendance. B compares counts, but total attendance also differed, so counts alone can't show a larger share. C is true but works against the director's emphasis."},"r2045":{"id":"r2045","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Scatter of data and a correlation claim","tier":2,"traps":["scope"],"figure":"
\"A
","answer":"A","stem":"Which choice most effectively uses data from the scatterplot to support the team's claim?","passage":"A team of agronomists measured soil organic carbon and average crop yield on 10 farms. The team claims that farms with more soil organic carbon tended to have higher yields.","choices":["Farms above 3% organic carbon yielded over 5 tons per hectare; farms below 1.5% yielded under 4 tons.","One farm had soil organic carbon of 2.5% and a yield of 4.2 tons per hectare, lower than a farm with 2.2%.","No farm had a yield greater than 6 tons per hectare, regardless of its level of soil organic carbon.","Soil organic carbon ranged from about 1% to about 4% across the 10 farms in the study."],"explain":"A \"tended to\" claim is supported by comparing high-carbon farms with low-carbon farms: the highest-carbon farms yield above 5 tons, the lowest below 4. B is a single point, and C and D describe ranges, not the relationship."},"r2046":{"id":"r2046","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Distinguishing a total from a rate","tier":3,"traps":["half"],"figure":"
Reported Cases in 2024
CountyPopulationCases
F500,0001,000
G120,000360
","answer":"C","stem":"Which choice most effectively uses data from the table to support the researcher's argument?","passage":"A public health researcher compared two counties' rates of a disease. She argues that although County F reported more cases, the disease was actually more common in County G once population size is taken into account.","choices":["County F had 640 more reported cases than County G in 2024, even though it is the larger county.","County F's population of 500,000 was more than four times County G's population of 120,000.","County G had 3 cases per 1,000 residents, while County F had 2 cases per 1,000 residents.","County G had 360 reported cases in 2024 out of a population of 120,000."],"explain":"\"Taken population size into account\" means comparing rates: 120,000360​=3 per 1,000 and 500,0001,000​=2 per 1,000. A supports only the \"more cases\" half of the sentence, and B gives population without cases."},"r2047":{"id":"r2047","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Graph of change over time with a turning point","tier":2,"traps":["irrelevant"],"figure":"
\"A
","answer":"A","stem":"Which choice most effectively uses data from the graph to support the historian's claim?","passage":"A historian studying a nineteenth-century port city claims that the city's population grew rapidly after a railroad connected it to inland farms in 1850, after decades of little change.","choices":["The population grew by about 3,000 from 1820 to 1850 but by about 39,000 from 1850 to 1870.","The population was about 63,000 in 1870, three times what it had been in 1820.","The population was larger in 1860 than in 1850, and it was larger still in 1870, two decades after the railroad.","The population was about 21,000 in 1820 and about 24,000 in 1850."],"explain":"The claim contrasts slow growth before 1850 with rapid growth after. A gives both: about 3,000 over thirty years, then about 39,000 over twenty. D covers only the years before 1850, C only the years after, and B merges the two periods into one overall change that hides the contrast."},"r2048":{"id":"r2048","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Matching a claim to the right row","tier":3,"traps":["half"],"figure":"
Time for a Tablet to Dissolve
Water temperatureTrial 1 (seconds)Trial 2 (seconds)Average (seconds)
20°C96104100
40°C586260
60°C434142
","answer":"D","stem":"Which choice most effectively uses data from the table to support the student's conclusion?","passage":"Students in a chemistry class tested how temperature affects the rate at which a tablet dissolves. One student concluded that raising the water temperature from 20°C to 40°C shortened dissolving time more than raising it from 40°C to 60°C did.","choices":["At 60°C, the average dissolving time was 42 seconds, less than half the average time at 20°C.","In trial 1, the tablet dissolved in 96 seconds at 20°C and in 43 seconds at 60°C.","The average dissolving time decreased each time the water temperature was raised by 20 degrees.","The average time fell 40 seconds from 20°C to 40°C but only 18 seconds from 40°C to 60°C."],"explain":"The conclusion compares two changes: 100−60=40 seconds and 60−42=18 seconds. C is true but doesn't compare the two intervals, and B skips 40°C entirely."},"r2049":{"id":"r2049","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Survey data and a negative claim","tier":2,"traps":["reverse"],"figure":"
Average Daily Library Visits
Time of dayBefore extensionAfter extension
Morning and afternoon820835
Evening0310
","answer":"B","stem":"Which choice most effectively uses data from the table to support the council member's claim?","passage":"A city surveyed residents before and after it extended library hours into the evening. A council member claims that the change did not reduce daytime library use.","choices":["After the extension, the library averaged 310 evening visits per day, up from none before the change.","Average daytime visits were about the same after the extension (835 per day) as before it (820 per day).","Total daily visits rose from 820 before the extension to 1,145 after it, an increase of about 40%.","Before the extension, the library had no evening visits because it closed in the late afternoon."],"explain":"The claim is specifically about daytime use not falling. Daytime visits went from 820 to 835, so they didn't drop. A and C show the extension attracted visitors but say nothing about whether daytime use fell."},"r2050":{"id":"r2050","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Quantitative)","sub":"Two-condition bar graph","tier":3,"traps":["half"],"figure":"
\"A
","answer":"B","stem":"Which choice most effectively uses data from the graph to support the researchers' conclusion?","passage":"Psychologists tested whether background music affects memory. Participants memorized word lists in silence, with instrumental music, or with music that had lyrics, and were tested immediately and again one week later. The researchers concluded that music with lyrics hurt long-term recall more than it hurt immediate recall.","choices":["Participants who heard lyrics recalled 38% of words one week later, the lowest of any condition at either time.","Compared with silence, lyrics cut recall by 11 points on the immediate test but by 17 points a week later.","Recall was lower one week later than on the immediate test in every condition, including silence.","Instrumental music lowered immediate recall by only 2 percentage points compared with silence."],"explain":"\"Hurt long-term recall more than immediate recall\" requires comparing the lyrics penalty at both times: 72−61=11 points immediately and 55−38=17 points a week later. C describes forgetting in general, and D is about instrumental music."},"r2016":{"id":"r2016","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Finding that would weaken a hypothesis","tier":3,"traps":["reverse"],"answer":"A","stem":"Which finding, if true, would most directly weaken Brandt's hypothesis?","passage":"Many songbirds sing more loudly in cities than in forests. Biologist Ilse Brandt hypothesizes that the change is a direct response to traffic noise: birds raise their volume only when background noise is high, much as people raise their voices in a noisy room.","choices":["City birds sing just as loudly on quiet Sunday mornings, when traffic is light, as during weekday rush hours.","City birds sing more loudly in parks next to highways than in parks far from any major road, even at the same time of day.","Forest birds that hear recordings of traffic noise played through speakers begin singing more loudly within a few days.","City birds sing at noticeably higher pitches than forest birds of the same species, which helps their songs carry over noise."],"explain":"Brandt says birds raise volume only when noise is high. If they sing just as loudly when traffic is light, the volume doesn't track noise, which weakens the hypothesis. B and C support it. D concerns pitch, not volume."},"r2017":{"id":"r2017","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Quotation matching a two-part claim","tier":3,"traps":["half"],"answer":"C","stem":"Which quotation from the novel most effectively illustrates this claim?","passage":"In the novel The Lantern Keeper, the narrator describes her aunt Maud as someone whose generosity is real but always comes with a reminder of what it cost her.","choices":["\"Maud kept her savings in a tin box under the floorboards and counted every coin of it each Friday night after supper.\"","\"When the roof leaked that spring, Maud paid for all of the repairs herself, without telling any of us what it cost.\"","\"Maud bought me a good wool coat that winter and mentioned at every meal that week how she would now go without boots.\"","\"Maud was the first in the family to finish school, and she reminded us of it at every holiday dinner for twenty years.\""],"explain":"The claim has two parts: real generosity and a reminder of its cost. The coat is generous, and the repeated mention of going without boots is the reminder. B shows generosity without any reminder. D shows reminding, but about schooling, not generosity."},"r2018":{"id":"r2018","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Evidence that supports a causal claim","tier":2,"traps":["irrelevant"],"answer":"B","stem":"Which finding, if true, would most directly support Asante's claim?","passage":"Coral reefs that bleach during marine heat waves sometimes recover. Marine ecologist Kofi Asante claims that reefs with abundant plant-eating fish recover faster, because the fish graze on algae that would otherwise smother young corals before they can grow.","choices":["Plant-eating fish are more common on reefs in warm tropical waters than on reefs in cooler waters farther from the equator.","After a heat wave, reefs closed to fishing for plant-eating fish had fewer algae and more young corals than fished reefs.","Corals bleach when heat causes them to expel the tiny algae that live inside their tissues and supply most of their food.","Some coral species bleach at lower water temperatures than others, and those species tend to recover more slowly."],"explain":"The claim links plant-eating fish to faster recovery through algae grazing. Protected reefs with more of these fish having fewer algae and more young corals supports that chain directly. C is about the algae inside coral tissue, a different thing from the algae that smother young corals."},"r2019":{"id":"r2019","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Quotation that illustrates a poem's claim","tier":2,"traps":["half"],"answer":"D","stem":"Which quotation from the poem most effectively illustrates this claim?","passage":"In the poem \"Harbor Town in Winter,\" the speaker suggests that the town, though quiet in winter, is still full of life.","choices":["\"The summer crowds are gone, their laughter gone, / the boardwalk bare beneath a pale gray sky\"","\"The gulls have taken the pier for their own / and quarrel there from morning until dark\"","\"I walked the empty beach from end to end / and met no one but wind and the long tide\"","\"The shutters closed, the boats pulled up and tarred, / yet every kitchen window burns till late\""],"explain":"The claim has two parts: quiet and still alive. Closed shutters and boats pulled up show the quiet; windows lit late show life continuing. A and C show only emptiness. B shows gulls' activity, but not the town's life."},"r2020":{"id":"r2020","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Weakening a proposed explanation","tier":3,"traps":["reverse"],"answer":"D","stem":"Which finding, if true, would most directly weaken the claim that water was poured to reduce friction?","passage":"Some historians argue that ancient Egyptian builders moved the enormous stone blocks of the pyramids on wooden sledges dragged over sand, and that workers poured water in front of the sledges to reduce friction. A wall painting from a tomb shows a worker pouring liquid in front of a sledge carrying a statue.","choices":["Other tomb paintings show workers carrying water jars at construction sites where large stone blocks were being moved.","Laboratory tests show that sand with the right amount of water roughly halves the force needed to drag a sledge across it.","Remains of wooden sledges strong enough to carry heavy stone blocks have been found buried near several of the pyramid sites.","Laboratory tests show that wetting sand increases, rather than decreases, the force needed to drag a heavy sledge across it."],"explain":"The explanation depends on water reducing friction. Tests showing that wetting sand makes dragging harder would undercut it. B would support the claim, and A and C are consistent with it."},"r2021":{"id":"r2021","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Which quotation shows a contrast","tier":3,"traps":["half"],"answer":"C","stem":"Which quotation from the memoir most effectively illustrates this claim?","passage":"In her memoir, the scientist Lucinda Ford describes her relationship to her field notebooks as both practical and deeply personal.","choices":["\"My notebooks are waterproof and have numbered pages, so that none of my data can ever be lost, torn out, or altered.\"","\"When I reread my old notebooks from graduate school, I sometimes find mistakes I made as a student and wince.\"","\"I record the wind and time of every reading, and I've kept each notebook since I was nineteen, as others keep letters.\"","\"I prefer a pencil to a pen in the field, because ink smears in the rain and pencil marks last for decades.\""],"explain":"Recording wind and time is practical; keeping every notebook \"as others keep letters\" is personal. A and D are only practical, and B doesn't show a personal attachment."},"r2022":{"id":"r2022","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Support for a generalization from a study","tier":2,"traps":["scope"],"answer":"A","stem":"Which finding, if true, would most strongly support Rahman's conclusion?","passage":"Economist Ayesha Rahman studied a program that gave small grants to rural women in Bangladesh to start businesses. She concludes that the grants had lasting effects: women who received them were more likely to be running profitable businesses years after the program ended.","choices":["Seven years later, recipients were twice as likely as similar women without grants to own a profitable business.","In the first month after the program began, most recipients used their grants to buy livestock or tools for a business.","Many recipients said in interviews years later that they still enjoyed running their own businesses and planned to continue.","The program gave grants to more women in its second year than in its first, and nearly all of them started businesses."],"explain":"\"Lasting effects\" requires evidence long after the program, and \"profitable businesses\" requires profit. A gives both, with a comparison group. B is too early to show lasting effects, and C shows enjoyment, not profitability."},"r2023":{"id":"r2023","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Evidence for a negative claim","tier":3,"traps":["reverse"],"answer":"B","stem":"Which finding, if true, would most directly support Castro's argument?","passage":"A popular theory holds that the dinosaurs died out gradually over millions of years as volcanic eruptions slowly changed Earth's climate, and that the asteroid impact 66 million years ago merely finished off species that were already in steep decline. Paleontologist Rubén Castro disputes this, arguing that dinosaurs were thriving until the impact.","choices":["Volcanic eruptions released large amounts of gas both before and after the asteroid struck Earth 66 million years ago.","Fossil records from the last million years before the impact show no decline in the number of dinosaur species.","The asteroid impact created a crater more than 150 kilometers wide and threw enormous amounts of debris into the atmosphere.","Some mammal species survived the asteroid impact and spread into many of the habitats that dinosaurs had occupied."],"explain":"Castro claims dinosaurs were thriving right up to the impact. Stable species counts in the final million years directly contradict the \"steep decline\" in the popular theory. A and C don't address whether dinosaurs were declining, and D is about mammals."},"r2024":{"id":"r2024","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Illustrating a character's contradiction","tier":2,"traps":["half"],"answer":"B","stem":"Which quotation from the play most effectively illustrates this claim?","passage":"In the play Second Harvest, the character Oren insists that he cares nothing for other people's opinions, but his actions suggest otherwise.","choices":["OREN: \"I have never asked anyone in this town for anything, and I never will.\" [He sits down heavily and folds his arms.]","OREN: \"Let them talk. I need no one's approval.\" [He glances at the window to see whether the neighbors are watching.]","OREN: \"The harvest will be late this year. Everyone knows it, and everyone is pretending not to worry.\"","OREN: \"My father built this house with his own hands, and I will not sell it.\" [He runs a hand along the doorframe.]"],"explain":"The claim pairs a stated indifference with behavior that contradicts it. In B, Oren says he needs no one's approval while checking whether the neighbors are watching. The actions in A and D fit his words rather than undercutting them, and C has no such contrast."},"r2025":{"id":"r2025","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Ruling out an alternative explanation","tier":3,"traps":["irrelevant"],"answer":"C","stem":"Which finding, if true, would most directly address the skeptic's concern?","passage":"Students at a high school who took an optional course in personal finance had higher credit scores ten years later than students who did not take the course. A school board member argues that the course caused the higher scores. A skeptic replies that students who chose the course may already have had families who discussed money at home.","choices":["Students who took the course reported years later that it was one of the most useful classes they took in high school.","The course covered budgeting, saving, and the use of credit cards, topics that are closely related to credit scores.","When a similar school made the course required, classes that took it later had higher credit scores than earlier classes.","Students who took the course were more likely than other students to attend college and to earn higher incomes afterward."],"explain":"The skeptic worries that students selected themselves into the course. Making the course required removes self-selection, so improved scores afterward point to the course itself. D could even support the skeptic, since college attendance may reflect family background."},"r2026":{"id":"r2026","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Quotation supporting a reading of a novel","tier":3,"traps":["half"],"answer":"A","stem":"Which quotation from the novel most effectively illustrates this claim?","passage":"In Elena Moreau's novel The Salt Road, the narrator's grandmother is presented as someone who expresses love through practical acts rather than words.","choices":["\"She never once said she was proud of me, but on the morning of my exam, new pencils lay sharpened on my desk.\"","\"She told me every night, before she turned off the light, that I was the light of her life and her greatest joy.\"","\"She was the most practical person I have ever known, and she had no patience for nonsense of any kind.\"","\"She would sit by the window for hours in the evening, saying nothing at all, while I did my homework beside her.\""],"explain":"The claim has two parts: love shown through acts, not words. Never saying she was proud (no words) while leaving sharpened pencils (a practical act of care) fits both. B expresses love in words, and C describes practicality without love."},"r2027":{"id":"r2027","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Undermining an assumption in a proposal","tier":2,"traps":["reverse"],"answer":"D","stem":"Which finding, if true, would most directly undermine the council's assumption?","passage":"A city council proposes planting thousands of trees in a low-income neighborhood to reduce summer heat, assuming that residents there lack shade because there is little space for trees along the streets.","choices":["Neighborhoods with more street trees have lower summer temperatures, especially on the hottest afternoons.","Most residents of the neighborhood say in surveys that they support the council's tree-planting plan.","Newly planted trees take several years to grow large enough to provide significant shade for a street.","The neighborhood's streets have as many open planting spaces as streets in wealthier neighborhoods do."],"explain":"The assumption is that there's little space for trees. Abundant open planting spaces contradict it. A supports the plan's goal, not the assumption, and C raises a separate concern about timing."},"r2028":{"id":"r2028","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Support for an interpretation of an ancient object","tier":3,"traps":["scope"],"answer":"C","stem":"Which finding, if true, would most strongly support the researchers' argument?","passage":"The Antikythera mechanism, a bronze device recovered from a shipwreck dated to about 70 BCE, contains dozens of interlocking gears. Researchers have long argued that it was a calculating device used to predict the positions of the Sun, the Moon, and possibly the planets.","choices":["The same shipwreck also contained bronze statues and coins from several Greek cities, suggesting a wealthy cargo.","The device's gears were cut from a single sheet of bronze with remarkable precision for the period.","Inscriptions on the device name the planets and describe the cycles of lunar and solar eclipses.","Other shipwrecks from the same period have yielded scientific instruments such as simple sundials."],"explain":"Inscriptions naming planets and describing eclipse cycles tie the device's purpose to astronomical prediction. The other choices describe the setting or materials without bearing on what the device calculated."},"r2029":{"id":"r2029","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Which data would test a claim of preference","tier":3,"traps":["irrelevant"],"answer":"B","stem":"Which finding, if true, would most directly support Nwosu's claim?","passage":"Some bumblebees are more likely to visit flowers with stripes than plain flowers of the same color. Entomologist Grace Nwosu claims that the bees are not born with this preference but learn it, because striped flowers in their region tend to hold more nectar.","choices":["Bees in the region visit striped flowers more often than plain flowers of the same color and size.","Bees raised in a lab where plain flowers held more nectar than striped ones came to prefer plain flowers.","Striped flowers in the region hold more nectar, on average, than plain flowers of the same color.","Bees can see stripes in ultraviolet light that are invisible to humans, and many flowers have such stripes."],"explain":"To show the preference is learned, the evidence must show it changing with experience. Bees that learned to prefer plain flowers when plain flowers paid off show exactly that. A and C restate facts from the passage, which are consistent with either a learned or an inborn preference."},"r2030":{"id":"r2030","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Weakening a claim about a cause","tier":2,"traps":["reverse"],"answer":"A","stem":"Which finding, if true, would most directly weaken the company's claim?","passage":"A study found that people who own dogs live longer, on average, than people who do not. A pet-food company claims in an advertisement that owning a dog increases life expectancy.","choices":["People who are already healthy and physically active are more likely than others to choose to get a dog.","Dog owners walk more each day, on average, than people without dogs, and regular walking is linked to longer lives.","The study included thousands of participants from several countries and followed them for more than a decade.","Owning a cat is associated with a smaller increase in life expectancy than owning a dog."],"explain":"If healthier people are more likely to get dogs, their longer lives might reflect their health, not the dogs. B suggests a way dogs could help, which supports the claim. C strengthens the study's credibility."},"r2031":{"id":"r2031","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Quotation illustrating a speaker's change","tier":3,"traps":["half"],"answer":"D","stem":"Which quotation from the poem most effectively illustrates this claim?","passage":"In the poem \"Field Notes,\" the speaker describes how her view of a marsh changes from boredom to fascination.","choices":["\"The marsh is wide and flat and smells of salt, / and the tide comes in twice daily, as it always has\"","\"I came because my mother asked me to, / and I stayed because the summer had nowhere else to go\"","\"By August, the heat had driven everyone indoors, / and the marsh was mine alone from dawn to dusk\"","\"At first I saw only reeds, and more reeds; / by August I could name the forty birds that hid in them.\""],"explain":"The claim describes a change from boredom to fascination. \"Only reeds, and more reeds\" conveys boredom, and learning the names of forty birds shows growing fascination. C marks time passing but shows no change in the speaker's view."},"r2032":{"id":"r2032","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Strengthening a claim with a natural experiment","tier":3,"traps":["scope"],"answer":"D","stem":"Which finding, if true, would most directly support Oyelaran's claim?","passage":"Linguist Marta Oyelaran claims that children learn new words faster when adults label objects the children are already looking at, rather than redirecting the children's attention to a new object.","choices":["Children whose parents talk to them more often tend to have much larger vocabularies by the time they start kindergarten.","Toddlers pay more attention to brightly colored toys than to plain ones, and they hold the colored toys longer.","Older children learn new words faster than younger children do, whether or not adults label objects for them.","Toddlers whose parents named toys the toddlers were already holding learned more words than toddlers shown other toys."],"explain":"The claim compares labeling what the child is already attending to with redirecting attention. D tests exactly that comparison. A concerns the amount of talk, not its timing, and B and C are unrelated to labeling."},"r2033":{"id":"r2033","sec":"rw","domain":"Information and Ideas","skill":"Command of Evidence (Textual)","sub":"Evidence against a dating claim","tier":3,"traps":["reverse"],"answer":"C","stem":"Which finding, if true, would most directly support Verbeek's argument?","passage":"A painting long attributed to the seventeenth-century artist Judith Leyster was recently reexamined. Art historian Paul Verbeek argues that it is a nineteenth-century imitation rather than Leyster's work.","choices":["The painting's subject, a young musician holding a lute, appears in several of Leyster's confirmed works from the 1630s.","The painting was owned by a single family for most of the twentieth century before it was sold at auction.","Chemical analysis shows that the painting contains a blue pigment that was not manufactured until the 1820s.","The painting's canvas is a type of linen commonly used by Dutch artists, including Leyster, in the 1600s."],"explain":"A pigment that didn't exist until the 1820s couldn't appear in a seventeenth-century painting, so the work must be later. A and D point toward the opposite conclusion, and B says nothing about when it was painted."},"r2051":{"id":"r2051","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Three facts that must all fit","tier":3,"traps":["half"],"answer":"C","stem":"Which choice most logically completes the text?","passage":"Archaeologists excavating a Bronze Age village in Scotland found bronze tools but no traces of copper or tin ore, the two metals needed to make bronze, within 200 kilometers of the site. They did find molds for casting tools and drips of solidified bronze on the workshop floor. Because bronze can be melted down and recast without losing quality, the archaeologists concluded that the village's metalworkers most likely ","choices":["mined small deposits of copper and tin near the village that the chemical analysis failed to detect.","imported all of their tools already cast and never worked bronze themselves.","obtained finished bronze objects through trade and recast them into new tools.","made most of their tools from a local metal and used imported bronze only for decoration."],"explain":"Three facts must fit together: no ore nearby, casting evidence on site, and bronze can be recast without losing quality. Only recasting traded bronze satisfies all three. A contradicts the absence of ore, B ignores the molds and drips, and D contradicts the bronze tools that were found."},"r2052":{"id":"r2052","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"What a control rules out","tier":3,"traps":["reverse"],"answer":"A","stem":"Which choice most logically completes the text?","passage":"Ecologists noticed that a species of lizard on a small island had longer legs than the same species on the mainland. To test whether the difference was inherited or caused by the environment, they raised hatchlings from both populations in identical cages with identical food and perches. If the island lizards raised in captivity also developed longer legs than the mainland lizards raised in captivity, that result would suggest that ","choices":["the difference in leg length is at least partly inherited.","the difference in leg length is caused entirely by differences in the lizards' environments.","island lizards grow faster than mainland lizards in the wild.","longer legs make lizards better at climbing."],"explain":"Raising both groups in the same conditions removes environmental differences. If island lizards still grow longer legs, something other than the environment, likely heredity, is responsible. B reverses the logic of the control. C and D go beyond what the experiment tests."},"r2053":{"id":"r2053","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Explaining a mismatch in records","tier":3,"traps":["scope"],"answer":"D","stem":"Which choice most logically completes the text?","passage":"In 1906 an earthquake struck San Francisco. Official reports at the time listed about 375 deaths, but historians now estimate that more than 3,000 people died. The official count included few residents of Chinatown, whose deaths were often not recorded, and city leaders were eager to reassure investors that the city was safe to rebuild. These circumstances suggest that the official death toll was low because ","choices":["the earthquake itself was less destructive than historians now believe, since most deaths came later.","most residents of Chinatown had left the city in the weeks before the earthquake struck.","historians have overestimated the population of San Francisco in 1906 by relying on investors' reports.","many deaths in Chinatown went unrecorded and city leaders wanted to reassure investors."],"explain":"The text gives two circumstances: many deaths weren't recorded, and officials wanted to reassure investors. The completion must reflect both. A contradicts the historians' estimate; B and C are unsupported."},"r2054":{"id":"r2054","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"An unexpected result in an experiment","tier":3,"traps":["half"],"answer":"B","stem":"Which choice most logically completes the text?","passage":"A biologist expected that adding nitrogen fertilizer to a grassland plot would increase the number of plant species growing there, since nitrogen is often scarce in grasslands. Instead, after five years, fertilized plots had fewer species than unfertilized plots, although total plant mass was higher. A few tall, fast-growing grasses dominated the fertilized plots and shaded the ground. This pattern suggests that ","choices":["nitrogen is not actually scarce in grasslands, so adding more of it had little effect on most species.","the fertilizer helped a few fast-growing species grow so large that they crowded out the others.","the fertilizer was toxic to most grassland plants, killing all but a few tolerant species.","the unfertilized plots produced more total plant mass because they supported a greater variety of species."],"explain":"More total mass but fewer species, with tall grasses shading the ground, points to competition: the fertilizer favored a few fast growers that crowded out others. C would not explain the higher plant mass, and D contradicts it. Nothing supports A."},"r2055":{"id":"r2055","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Conclusion from a pattern across sites","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice most logically completes the text?","passage":"Linguists studying a group of related languages in the Pacific found that the word for \"sweet potato\" is nearly identical across languages spoken on islands thousands of kilometers apart, and closely resembles the word used in Quechua, a language of South America, where the sweet potato was first domesticated. Since most researchers doubt that the plant could have crossed the ocean on its own, the linguistic evidence suggests that ","choices":["the sweet potato was first domesticated in the Pacific islands and later carried to South America.","Quechua is the ancestor of the Pacific languages, which explains the many words they share with it.","the Pacific languages borrowed the word from one another rather than from any language outside the region.","people carried the sweet potato, and its name, from South America to the Pacific islands."],"explain":"A shared word resembling the Quechua name, plus a plant unlikely to have crossed the ocean on its own, points to people carrying both the plant and its name. A contradicts the text. B overreaches from a single shared word to the ancestry of whole languages, and C contradicts the borrowing the text implies."},"r2056":{"id":"r2056","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Ruling out the obvious cause","tier":3,"traps":["reverse"],"answer":"B","stem":"Which choice most logically completes the text?","passage":"A town's water utility found that household water use fell 12% after it installed new meters that let residents check their usage on a phone app. But the decline was the same in households that never downloaded the app as in households that used it daily. The new meters also replaced old ones that had been undercounting water use, and bills were based on the new readings. These findings suggest that the drop in water use was most likely caused by ","choices":["the app, which showed residents how much water they used and encouraged them to conserve it.","residents' responses to their higher and more accurate water bills rather than by the app itself.","a mistake in the new meters, which undercounted water use in households that never used the app.","households that used the app daily and shared conservation tips with neighbors who did not."],"explain":"The decline didn't depend on using the app, so the app isn't the cause (A). The new meters fixed undercounting, so bills likely went up, a reason everyone would cut back. C reverses which meters were inaccurate."},"r2057":{"id":"r2057","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"What a date range implies","tier":2,"traps":["scope"],"answer":"C","stem":"Which choice most logically completes the text?","passage":"A poem attributed to the medieval poet Aldric of Wensley survives in a single manuscript. The manuscript's paper contains a watermark that was used only between 1440 and 1460, and Aldric is known to have died in 1398. Since manuscripts are usually copies, the poem itself may be older than the paper. The watermark therefore shows only that ","choices":["Aldric could not have written the poem, since he died more than forty years before the paper was made.","the poem was composed between 1440 and 1460 by a writer who borrowed Aldric's name.","this particular copy of the poem was made no earlier than 1440, decades after Aldric's death.","Aldric wrote the poem shortly before his death and someone copied it immediately afterward."],"explain":"The watermark dates the paper, so this copy was made in 1440 or later. Because the text stresses that manuscripts are usually copies, the poem itself could be older, so A and B conclude too much. Nothing supports D."},"r2058":{"id":"r2058","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Distinguishing correlation from mechanism","tier":3,"traps":["half"],"answer":"A","stem":"Which choice most logically completes the text?","passage":"Students who eat breakfast tend to earn higher grades than students who skip it. To test whether breakfast itself improves performance, researchers provided free breakfast at school to a randomly chosen half of the students in a district. A year later, the grades of students offered breakfast had improved slightly more than those of other students, but the improvement was much smaller than the original grade gap between breakfast eaters and skippers. This result suggests that ","choices":["breakfast helps a little, but other differences between the groups likely explain most of the gap.","breakfast has no effect on students' grades, since the improvement was smaller than the original gap.","the original grade gap was caused entirely by breakfast, since grades improved when breakfast was provided.","students who skip breakfast would earn higher grades if their schools served lunch earlier in the day."],"explain":"The random assignment produced a small real effect, so breakfast matters somewhat, ruling out B. But the effect was much smaller than the original gap, so other factors must account for most of it, ruling out C."},"r2059":{"id":"r2059","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"A prediction from a mechanism","tier":3,"traps":["reverse"],"answer":"A","stem":"Which choice most logically completes the text?","passage":"Some desert beetles collect drinking water from fog: tiny bumps on their backs attract water droplets, which then roll down grooves into the beetles' mouths. Engineers designing fog-harvesting nets have copied this structure. If the bumps work by attracting water while the surrounding surface repels it, then a net coated with a uniformly water-attracting material, with no repelling areas, would most likely ","choices":["collect less water, since droplets would spread into a film instead of gathering and rolling off.","collect more water, since every part of the net, not just the bumps, would draw droplets out of the fog.","collect water exactly as effectively as the beetles do, since it would copy their bumps precisely.","stop attracting water altogether, since the water-attracting material would repel the fog."],"explain":"The mechanism depends on contrast: attracting spots gather droplets and repelling surroundings let them roll away. Remove the repelling areas and water would spread into a film rather than gathering into droplets that roll down. B assumes more attraction is always better, which the mechanism contradicts."},"r2060":{"id":"r2060","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Inference about an author's audience","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice most logically completes the text?","passage":"A seventeenth-century cookbook written in England gives measurements in cups and spoons rather than by weight, assumes readers know how to judge when an oven is hot by holding a hand inside it, and includes many recipes for preserving fruit but few for elaborate dishes requiring imported spices. These features suggest that the book was most likely written for ","choices":["readers who had never cooked before and needed simple measurements they could follow exactly.","professional cooks in wealthy homes who prepared fruit preserves for large households.","readers living outside England who could not obtain the imported spices used in English cooking.","experienced cooks in ordinary households rather than professional cooks in wealthy homes."],"explain":"Household measures and few imported spices point away from wealthy professional kitchens (B); assuming readers can judge oven heat by hand points away from beginners (A). Experienced household cooks fit all three features."},"r2061":{"id":"r2061","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Inference from a missing result","tier":3,"traps":["reverse"],"answer":"B","stem":"Which choice most logically completes the text?","passage":"Astronomers searched the light from 2,000 nearby stars for periodic dips that would indicate an orbiting planet passing in front of its star. A planet is detected this way only if its orbit is lined up edge-on as seen from Earth, which happens for only a small fraction of planetary systems. The astronomers found such dips for only 40 of the stars. Given these facts, it would be unreasonable to conclude that ","choices":["at least 40 of the 2,000 stars have one or more orbiting planets.","only 40 of the 2,000 stars that the astronomers studied have orbiting planets.","many of the 2,000 stars may have planets that this method could not detect.","planets whose orbits are not lined up edge-on cannot be found by this method."],"explain":"The question asks for the unreasonable conclusion. Because only edge-on orbits produce dips, many planets go undetected, so finding dips at 40 stars can't mean only 40 have planets. A, C, and D are all reasonable given the text."},"r2062":{"id":"r2062","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Combining a rule with an observation","tier":3,"traps":["half"],"answer":"C","stem":"Which choice most logically completes the text?","passage":"In many bird species, the sex of a chick that hatches first has an advantage, because it is larger than its later-hatching siblings. In one species of hawk, females are larger than males. Researchers found that hawk mothers in good years, when food was plentiful, tended to lay male eggs first, giving the smaller sex a head start. In poor years, they laid female eggs first. This suggests that in poor years, mothers ","choices":["lay more eggs than in good years so that at least some chicks survive the shortage of food.","are unable to control the order in which they lay male and female eggs when food is scarce.","may give the head start to the larger sex, whose bigger size helps it survive scarce years.","prefer to raise male chicks, since males need less food than females do in poor years."],"explain":"In good years mothers help the smaller sex catch up; in poor years they give the head start to females, the larger sex, concentrating advantage where survival is likeliest. B contradicts the pattern the researchers found. D reverses the poor-year behavior."},"r2063":{"id":"r2063","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Policy inference with a constraint","tier":2,"traps":["scope"],"answer":"D","stem":"Which choice most logically completes the text?","passage":"A city wanted to reduce traffic congestion by charging drivers a fee to enter downtown during rush hour. Before the fee took effect, the city expanded bus service on routes into downtown. Planners reasoned that without the expanded service, many commuters who could not afford the fee would have no practical way to reach their jobs. The planners' reasoning suggests that they believed ","choices":["buses would cause more congestion downtown than the cars they replaced during rush hour.","most commuters could easily afford the fee and would keep driving downtown anyway.","the fee would be unnecessary once bus service was expanded, since commuters would switch on their own.","a congestion fee would be fairer if commuters had an affordable alternative to driving."],"explain":"The planners expanded buses because some commuters couldn't afford the fee and would otherwise have no way to work. That shows concern for providing an affordable alternative. B contradicts their worry, and C misreads the buses as a replacement for the fee rather than a complement."},"r2064":{"id":"r2064","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"What a sample can and cannot show","tier":3,"traps":["scope"],"answer":"B","stem":"Which choice most logically completes the text?","passage":"A researcher interviewed 40 novelists who had each published at least five books and found that 33 of them wrote every day, even if only for a few minutes. She was careful not to conclude that daily writing leads to a successful career, since her study included only novelists who had already succeeded. Her caution reflects the possibility that ","choices":["novelists who write every day tend to publish fewer books than those who write only when inspired.","many writers who wrote daily never published five books and so were left out of her study.","most of the novelists she interviewed did not actually write every day, despite what they reported.","daily writing becomes unnecessary once a novelist has already published several successful books."],"explain":"Studying only successful novelists can't show that daily writing leads to success, because unsuccessful daily writers were never counted. B names that gap. C contradicts her finding, and A and D aren't implied."},"r2065":{"id":"r2065","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Reading a historical motive","tier":2,"traps":["scope"],"answer":"A","stem":"Which choice most logically completes the text?","passage":"In 1869 the Suez Canal opened, linking the Mediterranean and Red Seas. Before its completion, ships traveling between Britain and India had to sail around the southern tip of Africa, a journey of about 19,000 kilometers. Through the canal, the trip was roughly 11,000 kilometers. British merchants, who at first had been skeptical of the project, became some of the canal's heaviest users within a decade, which suggests that ","choices":["the savings in distance outweighed whatever doubts the merchants had when the canal first opened.","the canal was built mainly by British engineers, who persuaded merchants to use it.","British merchants shifted their trade away from India after 1869 toward ports in the Red Sea.","the route around Africa became shorter after the canal opened, which reduced merchants' doubts."],"explain":"Initially skeptical merchants became heavy users once the route cut about 8,000 kilometers, which suggests the practical savings overcame their doubts. C contradicts heavy use, and B and D aren't supported."},"r2066":{"id":"r2066","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Logical consequence of two findings","tier":3,"traps":["reverse"],"answer":"C","stem":"Which choice most logically completes the text?","passage":"A study of a coastal town found that houses built on elevated foundations suffered far less flood damage in a major storm than nearby houses built at ground level. However, the elevated houses were also, on average, twenty years newer and built under stricter building codes requiring stronger walls and roofs. To determine whether elevation alone reduces flood damage, researchers would most need to compare ","choices":["elevated houses in this town with elevated houses in a nearby town that the storm did not reach.","the total cost of building elevated houses and ground-level houses under the stricter codes.","elevated and ground-level houses that were built at about the same time under the same codes.","flood damage to the elevated houses in this storm with their damage in earlier, weaker storms."],"explain":"Age and building codes are confounding factors. To isolate elevation, the comparison should hold those constant while varying only elevation. A varies nothing, and B and D don't separate elevation from the other factors."},"r2067":{"id":"r2067","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Inference about a literary character","tier":3,"traps":["scope"],"answer":"D","stem":"Which choice most logically completes the text?","passage":"The following text is from a novel. Hiroshi has received a letter from his estranged brother.

Hiroshi placed the letter, unopened, on the windowsill, where it stayed through a week of rain. Each morning he moved it a few centimeters to keep it out of the damp, and each evening he moved it back, so that it would catch the last of the light. By Sunday the envelope had been handled more than any letter he had ever answered. Hiroshi's behavior suggests that he ","choices":["has forgotten that the letter exists and moves it only because it is in his way.","has already decided never to read the letter and is waiting for a chance to throw it away.","is worried that the rain has damaged the letter so badly that it can no longer be read.","values the letter and protects it, even though he is not yet ready to read what it says."],"explain":"He protects the letter from damp and keeps it in the light every day, which shows care, yet he leaves it unopened. A contradicts his daily attention. B goes too far: nothing shows a final decision. C misreads his care as concern over damage that hasn't occurred."},"r2068":{"id":"r2068","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Narrowing possible explanations","tier":3,"traps":["half"],"answer":"B","stem":"Which choice most logically completes the text?","passage":"Several species of frogs in Central America declined sharply in the 1990s. Biologists considered three explanations: habitat loss, pollution, and a fungal disease. The declines occurred even in protected mountain forests far from roads, farms, and factories, and the dead frogs showed skin infections caused by a chytrid fungus. These observations most strongly support the conclusion that ","choices":["habitat loss was the main cause, since frogs in protected forests had nowhere else to go.","the chytrid fungus is a more likely cause of the declines than habitat loss or pollution.","pollution from distant factories spread into the protected forests and weakened the frogs.","none of the three explanations can account for declines in forests far from roads and farms."],"explain":"Declines in protected forests far from roads and factories weigh against habitat loss and pollution, while infected skin points to the fungus. C invents a pathway the text doesn't mention."},"r2069":{"id":"r2069","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Inference requiring a numerical comparison","tier":3,"traps":["reverse"],"answer":"A","stem":"Which choice most logically completes the text?","passage":"A survey of a country's workers found that the average commute lengthened from 26 minutes to 29 minutes over a decade. During the same period, the share of workers who worked entirely from home rose from 5% to 18%. Because the survey's average was calculated only from workers who commuted, the lengthening suggests that ","choices":["people who still commuted were, on average, traveling longer than commuters had a decade earlier.","working from home caused the commutes of the remaining commuters to become longer over the decade.","most workers spent more time commuting than before, including those who worked from home.","the workers who switched to working from home had previously had the shortest commutes in the country."],"explain":"The average covers only commuters, so it shows that commuters' trips grew longer on average. C wrongly includes home workers, who don't commute. D is one possible explanation, but the data can't confirm it, and B claims a cause the data can't establish."},"r2070":{"id":"r2070","sec":"rw","domain":"Information and Ideas","skill":"Inferences","sub":"Completing a scientific argument","tier":3,"traps":["half"],"answer":"C","stem":"Which choice most logically completes the text?","passage":"Some plants produce caffeine in their nectar. Caffeine is bitter, and at high doses it deters insects, but the amounts in nectar are too low for bees to taste. In experiments, bees that fed on nectar containing low doses of caffeine were three times as likely to remember the flower's scent a day later. Plants producing caffeinated nectar may therefore benefit because ","choices":["the caffeine discourages bees from visiting the plants too often and taking too much nectar.","bees can taste the small amount of caffeine and prefer nectar with a slightly bitter flavor.","bees that remember a plant's scent are more likely to return to that plant and carry its pollen.","the caffeine protects the nectar from spoiling, so bees find it fresh when they arrive."],"explain":"The experiment shows improved memory for the flower's scent; a remembered flower is more likely to be revisited and pollinated. A applies to high doses, not the low doses in nectar, and B contradicts \"too low for bees to taste.\""},"r3001":{"id":"r3001","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Semicolon before a transition","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The Atacama Desert receives almost no rain in most some of its weather stations have never recorded measurable rainfall at all.","choices":["years, in fact,","years in fact","years; in fact,","years, in fact"],"explain":"Both sides are independent clauses, and \"in fact\" is a transition, not a conjunction, so commas alone can't join the clauses. Use a semicolon before \"in fact\" and a comma after it. A and D are comma splices; B fuses the clauses."},"r3002":{"id":"r3002","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"No comma after a title before a name","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"In 1922, Danish physicist Niels won the Nobel Prize in Physics for his model of the structure of atoms.","choices":["Bohr,","Bohr","Bohr—","Bohr;"],"explain":"\"Danish physicist Niels Bohr\" is the subject, and nothing should separate a subject from its verb (\"won\"). A title such as \"Danish physicist\" placed directly before a name takes no comma either. A, C, and D each put punctuation between the subject and the verb."},"r3003":{"id":"r3003","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"No punctuation inside a long subject-verb unit","tier":2,"traps":["rule","distance"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The tiny pores on the underside of a leaf that open and close to control the exchange of gases called stomata.","choices":["are,","are:","are;","are"],"explain":"The long subject (\"The tiny pores … gases\") is followed directly by its verb phrase \"are called.\" Nothing should interrupt \"are called.\" A, B, and C each split the verb from its completion."},"r3004":{"id":"r3004","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Colon before an explaining clause","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Historian Mónica Ruiz exposed the forged letters by noticing one unremarkable the postage stamps on them had not been issued until two years after the dates written at the top.","choices":["detail:","detail,","detail","detail, which"],"explain":"Both sides are independent clauses, and the second explains what the detail was, so a colon fits. B is a comma splice, C fuses the clauses, and D creates a relative clause (\"which the postage stamps … had not been issued\") that doesn't make sense."},"r3005":{"id":"r3005","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Closing a dash interruption","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The Svalbard Global Seed Vault—a facility built into a mountain on a remote Norwegian stores duplicate samples of seeds from gene banks around the world.","choices":["island—","island,","island;","island"],"explain":"The interruption opens with a dash, so it must close with a dash. That returns the reader to the main clause: \"The Svalbard Global Seed Vault … stores duplicate samples.\" Mixing a dash and a comma (B) is not standard, and D leaves the interruption open."},"r3006":{"id":"r3006","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Splice hidden behind a relative clause","tier":3,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The first sound recordings were made on cylinders coated with tinfoil, which wore out after only a few by the late 1880s, wax cylinders that could be played dozens of times had replaced them.","choices":["playings,","playings","playings, which","playings;"],"explain":"\"The first sound recordings … a few playings\" is an independent clause, and so is \"by the late 1880s, wax cylinders … had replaced them.\" A semicolon joins them. The comma before \"which\" earlier in the sentence can make A look natural, but it creates a comma splice. B fuses the clauses, and C makes no grammatical sense."},"r3007":{"id":"r3007","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Semicolons in a list with internal commas","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The expedition set up its three base camps in Lhasa, Kathmandu, Nepal; and Darjeeling, India.","choices":["Tibet,","Tibet:","Tibet;","Tibet"],"explain":"Each list item contains a comma (\"Lhasa, Tibet\"), so the items are separated by semicolons, as the rest of the sentence already shows (\"Nepal; and\"). Only a semicolon keeps the list consistent."},"r3008":{"id":"r3008","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Direct question after a colon","tier":3,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The question that guided ornithologist Peter Laine's thirty years of fieldwork was deceptively simple: why do some songbirds keep learning new songs throughout their ","choices":["lives.","lives?","lives?.","lives;"],"explain":"After the colon, the sentence asks a direct question (\"why do some songbirds keep learning…\"), so it ends with a question mark. A would be right for an indirect question (\"asked why some songbirds keep learning\"), but this question is direct. C doubles the end punctuation."},"r3009":{"id":"r3009","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Comma after an introductory clause","tier":1,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Although the comet will not pass near Earth again for another 2,000 astronomers have already calculated its path with remarkable precision.","choices":["years;","years,","years.","years:"],"explain":"\"Although … 2,000 years\" is a dependent clause, so it can't stand alone or be followed by a semicolon, period, or colon. A comma connects it to the main clause."},"r3010":{"id":"r3010","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Two independent clauses, no conjunction","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The Great Barrier Reef is the largest structure built by living organisms it stretches more than 2,300 kilometers along the northeastern coast of Australia.","choices":["organisms,","organisms","organisms;","organisms, which"],"explain":"Each side is an independent clause, so a semicolon is needed. A is a comma splice and B fuses the clauses. D produces \"which it stretches,\" which repeats the subject."},"r3011":{"id":"r3011","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"An interrupting transition","tier":2,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Most visitors praised the museum's new wing for its dramatic views of the harbor. Its have drawn criticism from curators, who say the lighting is too dim for viewing paintings.","choices":["galleries; however,","galleries however","galleries, however;","galleries, however,"],"explain":"\"However\" interrupts a single clause (\"Its galleries have drawn criticism\"), so it is set off by a pair of commas. A and C put a semicolon inside the clause, splitting the subject from its verb. B leaves the interrupter unpunctuated."},"r3012":{"id":"r3012","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"No comma before a \"that\" clause","tier":1,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"In her 2019 study, sociologist Erin Park commuters who walked or biked to work reported higher job satisfaction than those who drove.","choices":["found that","found, that","found: that","found; that"],"explain":"\"That commuters … drove\" is the object of \"found,\" and nothing should separate a verb from its object. B, C, and D each insert punctuation between them."},"r3013":{"id":"r3013","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Comma before a conjunction joining clauses","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The city's first subway line opened in the trains were so crowded during the first week that the operator added extra cars within a month.","choices":["1904; and","1904 and,","1904, and","1904, and,"],"explain":"Two independent clauses joined by \"and\" take a comma before the conjunction. The SAT treats a semicolon before a coordinating conjunction (A) as an error. B and D misplace or add a comma after \"and.\""},"r3014":{"id":"r3014","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Closing a pair of commas","tier":2,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Some desert plants, such as the barrel cactus and the store water in their stems and can survive months without rain.","choices":["saguaro","saguaro;","saguaro—","saguaro,"],"explain":"The example phrase opens with a comma after \"plants,\" so it closes with a comma after \"saguaro,\" returning to \"Some desert plants … store water.\" C closes with a dash, which doesn't match, and A leaves the phrase open."},"r3015":{"id":"r3015","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Introductory participial phrase","tier":1,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Having spent a decade cataloging the island's the botanist was startled to find a species she had never seen before.","choices":["orchids,","orchids;","orchids.","orchids:"],"explain":"\"Having spent … orchids\" is an introductory phrase, not a clause, so it is joined to the main clause with a comma. A semicolon, period, or colon would leave the phrase stranded as a fragment."},"r3016":{"id":"r3016","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Semicolon before \"nevertheless\"","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Red dwarf stars are too faint to be seen from Earth without a they make up roughly three-quarters of the stars in the Milky Way.","choices":["telescope, nevertheless,","telescope; nevertheless,","telescope, nevertheless","telescope nevertheless,"],"explain":"Two independent clauses are joined with a transition, so the transition takes a semicolon before it and a comma after it. A and C are comma splices, and D fuses the clauses."},"r3017":{"id":"r3017","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Colon to introduce a list","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The kit contained everything a field researcher would a GPS unit, a soil probe, sample bags, and a waterproof notebook.","choices":["need:","need;","need","need, including:"],"explain":"The clause before the blank is complete, and a list follows, so a colon is correct. A semicolon (B) must be followed by an independent clause. D places a colon after \"including,\" which leaves an incomplete phrase before the colon."},"r3018":{"id":"r3018","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Semicolon between clauses with a conditional","tier":3,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The bridge's main cables are made of thousands of steel wires bundled one wire breaks, the others share its load.","choices":["together, if","together if","together; if","together, which if"],"explain":"\"The bridge's main cables … bundled together\" is a complete clause, and so is \"if one wire breaks, the others share its load.\" A semicolon joins them. A is a comma splice. B attaches \"if one wire breaks\" to \"bundled together,\" which changes the meaning and leaves \"the others share its load\" stranded."},"r3019":{"id":"r3019","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Closing an appositive","tier":1,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Ada Lovelace, the daughter of the poet Lord is often described as the first computer programmer.","choices":["Byron","Byron;","Byron—","Byron,"],"explain":"The appositive \"the daughter of the poet Lord Byron\" opens with a comma, so it must close with one before the verb \"is.\" C mismatches the punctuation, and A leaves the appositive open."},"r3020":{"id":"r3020","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Essential clause takes no commas","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Bees visit only one type of flower on each foraging trip are especially effective pollinators.","choices":[", which","that",", that","that,"],"explain":"The clause identifies which bees are effective pollinators, so it is essential and takes no commas: \"Bees that visit … are especially effective pollinators.\" A would make the clause nonessential and claim that all bees visit only one type of flower."},"r3021":{"id":"r3021","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Comma before \"yet\" joining clauses","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Scholars had studied the medieval manuscript for more than a no one noticed the faint drawings in its margins until it was photographed under ultraviolet light.","choices":["century; yet","century, yet","century yet","century, yet,"],"explain":"\"Yet\" is a coordinating conjunction joining two independent clauses, so it takes a comma before it. A semicolon before a conjunction (A) is treated as an error, and nothing should follow \"yet.\""},"r3022":{"id":"r3022","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Indirect question ends like a statement","tier":3,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Biologists have long debated how salmon find the streams where they some suspect that the fish follow the scent of their home waters.","choices":["hatched;","hatched?","hatched,","hatched"],"explain":"\"Biologists have long debated how salmon find the streams where they hatched\" is a statement that contains an indirect question, so it takes no question mark. It is an independent clause, as is \"some suspect…,\" so a semicolon joins them. C is a comma splice, and D fuses the clauses."},"r3023":{"id":"r3023","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Closing a long dash interruption","tier":2,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The aye-aye—a lemur that taps on tree bark and listens for the echoes of hidden insect is found only on Madagascar.","choices":["larvae,","larvae;","larvae","larvae—"],"explain":"The interruption begins with a dash after \"aye-aye,\" so it must end with a dash before the verb \"is.\" The length of the interruption makes it easy to lose track of how it began."},"r3024":{"id":"r3024","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Introductory phrase before a clause","tier":1,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Throughout the long winter months at the research station in the scientists kept a daily log of the sea ice's thickness.","choices":["Antarctica;","Antarctica:","Antarctica,","Antarctica."],"explain":"\"Throughout … in Antarctica\" is an introductory phrase, not a clause. It can't be followed by a semicolon, colon, or period, which would leave it as a fragment. A comma joins it to the main clause."},"r3025":{"id":"r3025","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Name as an essential appositive","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The ancient Greek mathematician calculated the circumference of Earth with surprising accuracy by comparing the angles of shadows in two cities.","choices":["Eratosthenes,","Eratosthenes",", Eratosthenes,","Eratosthenes—"],"explain":"The name identifies which mathematician is meant, so it is essential and takes no commas, and nothing should come between the subject and its verb \"calculated.\" C treats the name as optional, but the sentence needs it."},"r3026":{"id":"r3026","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Transition between two clauses","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The new vaccine must be stored at temperatures below hospitals without ultracold freezers cannot use it.","choices":["−70°C, therefore,","−70°C therefore,","−70°C; therefore,","−70°C, therefore"],"explain":"Two independent clauses joined by the transition \"therefore\" need a semicolon before the transition and a comma after it. A and D are comma splices; B fuses the clauses."},"r3027":{"id":"r3027","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Semicolons in a list of names with descriptions","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The festival's lineup featured Ana Tijoux, a Chilean rapper; Bomba Estéreo, a Colombian and Rodrigo y Gabriela, a Mexican guitar duo.","choices":["band;","band,","band:","band"],"explain":"Each item already contains a comma (\"Ana Tijoux, a Chilean rapper\"), so semicolons separate the items, as the sentence does after \"rapper.\" A comma (B) would blur where one item ends and the next begins."},"r3028":{"id":"r3028","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Transition after a full stop","tier":1,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The first trial of the drug produced no measurable effect. the second trial, which used a higher dose, reduced symptoms in most participants.","choices":["However","However;","However:","However,"],"explain":"An introductory transition is followed by a comma. A semicolon or colon after \"However\" would separate it from the clause it introduces, and no punctuation at all leaves the transition running into the clause."},"r3029":{"id":"r3029","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Colon before a quotation that explains","tier":3,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The sign at the trailhead gave hikers a single piece of “Stay on the marked path.”","choices":["advice;","advice:","advice that","advice, which"],"explain":"The clause before the blank is complete, and the quotation spells out what the advice was, so a colon introduces it. A semicolon (A) needs an independent clause after it. C and D turn the quotation into a broken relative clause."},"r3030":{"id":"r3030","sec":"rw","domain":"Standard English Conventions","skill":"Boundaries","sub":"Introductory appositive phrase","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"A former engineer who designed bridges for three Teresa Okoye now builds furniture from reclaimed timber.","choices":["decades","decades;","decades,","decades:"],"explain":"\"A former engineer … three decades\" is an introductory phrase describing Teresa Okoye, so it is joined to the main clause with a comma. Without the comma, the phrase runs into her name."},"r3031":{"id":"r3031","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Agreement with a distant singular subject","tier":2,"traps":["distance"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The collection of letters written by the explorer to her sisters during her three voyages to the Arctic now held at the national archive in Oslo.","choices":["are","were","is","have been"],"explain":"The subject is \"collection,\" which is singular; \"letters,\" \"sisters,\" and \"voyages\" are inside modifying phrases. \"Now\" calls for the present tense: \"is.\" A and D are plural, and B is plural and past."},"r3032":{"id":"r3032","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Inverted sentence with a compound subject","tier":3,"traps":["distance"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Among the artifacts recovered from the sixteenth-century shipwreck a bronze astrolabe and a set of brass navigational dividers.","choices":["were","was","is","has been"],"explain":"In an inverted sentence the subject comes after the verb: \"a bronze astrolabe and a set of … dividers,\" a compound (plural) subject. \"Recovered\" places the event in the past, so \"were\" is correct. B, C, and D are singular; \"artifacts\" is not the subject."},"r3033":{"id":"r3033","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Collective noun takes a singular pronoun","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"After two years of hearings, the city's transportation committee released final report, which recommended extending light-rail service to the airport.","choices":["their","its","it's","they're"],"explain":"\"Committee\" is a single body, so the possessive pronoun is \"its.\" A is plural; C means \"it is\" and D means \"they are.\""},"r3034":{"id":"r3034","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Plural possessive","tier":2,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"For decades, the three findings on lake sediments were scattered across a dozen journals; in 2021 they were finally collected in a single volume.","choices":["researcher's","researchers","researchers's","researchers'"],"explain":"The findings belong to three researchers, so the plural possessive \"researchers'\" is needed. A is singular possessive, B is plural with no possession, and C isn't a standard form."},"r3035":{"id":"r3035","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"A finite verb after a long subject","tier":3,"traps":["rule","distance"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Harriet Tubman, who escaped slavery in 1849 and later guided dozens of others to freedom along the network of safe houses known as the Underground Railroad, as a scout and spy for the Union Army during the Civil War.","choices":["serving","served","having served","to serve"],"explain":"Strip the clause between the commas and the sentence reads \"Harriet Tubman as a scout and spy.\" It still needs a main verb, and only \"served\" is one. A, C, and D are verb forms that can't serve as a sentence's main verb."},"r3036":{"id":"r3036","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Past tense in a past narrative","tier":1,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"In 1928, Alexander Fleming noticed that a mold had killed the bacteria growing in one of his laboratory dishes. He soon the substance produced by the mold \"penicillin.\"","choices":["names","will name","named","is naming"],"explain":"The passage narrates past events (\"In 1928,\" \"noticed,\" \"had killed\"), so the simple past \"named\" keeps the tense consistent."},"r3037":{"id":"r3037","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Avoiding a dangling modifier","tier":3,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Using a technique called radiocarbon dating, ","choices":["the bones were found by researchers to be about 9,000 years old.","the age of the bones was determined to be about 9,000 years.","it was determined that the bones were about 9,000 years old.","researchers determined that the bones were about 9,000 years old."],"explain":"The opening phrase describes someone using a technique, so the subject right after the comma must be the people who used it: \"researchers.\" In A and B, the phrase seems to describe the bones or their age using the technique, and in C the subject \"it\" names no one."},"r3038":{"id":"r3038","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"Each\" takes a singular verb","tier":2,"traps":["distance"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Each of the more than thirty paintings in Claude Monet's series of Rouen Cathedral the building's facade at a different time of day.","choices":["depicts","depict","are depicting","have depicted"],"explain":"The subject is \"Each,\" which is singular; \"paintings\" is part of the phrase \"of the more than thirty paintings.\" Only \"depicts\" is singular. The present tense fits a description of artworks."},"r3039":{"id":"r3039","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"Neither … nor\" agrees with the nearer subject","tier":3,"traps":["distance"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Neither the study's lead author nor her two coauthors able to explain why the bacteria stopped dividing after the third day.","choices":["were","was","is","has been"],"explain":"With \"neither … nor,\" the verb agrees with the nearer subject, \"her two coauthors,\" which is plural. The passage describes a completed study, so the past tense \"were\" fits. B, C, and D are singular."},"r3040":{"id":"r3040","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Parallel verbs in a series","tier":2,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The volunteer program trains residents to identify invasive plants, them from city parks, and replant the cleared areas with native species.","choices":["removing","removed","to be removing","remove"],"explain":"The series follows \"to\": \"to identify …, remove …, and replant ….\" The middle item must match the base forms \"identify\" and \"replant.\" The other choices break the parallel structure."},"r3041":{"id":"r3041","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"A gerund subject is singular","tier":2,"traps":["distance"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Studying the songs of humpback whales over several decades researchers trace how musical patterns spread from one population to another across the Pacific.","choices":["have helped","has helped","helping","to help"],"explain":"The subject is the gerund phrase \"Studying the songs …,\" which is singular even though it contains the plural nouns \"songs,\" \"whales,\" and \"decades.\" \"Has helped\" agrees and serves as the main verb. C and D aren't finite verbs."},"r3042":{"id":"r3042","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"One of\" takes a singular verb","tier":2,"traps":["distance"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"One of the oldest known maps of the night sky painted on the ceiling of an Egyptian tomb more than 3,000 years ago.","choices":["were","are","was","have been"],"explain":"The subject is \"One,\" not \"maps\" or \"sky.\" It's singular, and \"more than 3,000 years ago\" calls for the past tense: \"was.\""},"r3043":{"id":"r3043","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Verb in a relative clause matches its antecedent","tier":3,"traps":["distance"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The solar particles that the northern lights travel from the Sun to Earth in about two to three days.","choices":["causes","is causing","has caused","cause"],"explain":"\"That\" refers to \"particles,\" which is plural, so the verb in the relative clause is \"cause.\" The general fact calls for the present tense. A, B, and C are singular forms."},"r3044":{"id":"r3044","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Past perfect for an earlier past action","tier":2,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"By the time the rescue team reached the stranded climbers on the morning of June 14, the climbers on the ledge for nearly two days.","choices":["have been waiting","had been waiting","are waiting","will have waited"],"explain":"The waiting happened before another past event (the team's arrival), so the past perfect progressive \"had been waiting\" is correct. A is present perfect, which links to the present, not to a past moment."},"r3045":{"id":"r3045","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Plural pronoun for a plural noun","tier":1,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Hummingbirds can hover in place because they rotate wings in a figure-eight pattern, generating lift on both the forward and backward strokes.","choices":["its","it's","their","they're"],"explain":"The pronoun refers to \"Hummingbirds,\" which is plural, so \"their\" is correct. B and D are contractions."},"r3046":{"id":"r3046","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Irregular plural possessive","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The library's new reading room for young readers was designed with input from local students, and several of the suggestions, including a window seat and a reading nook, appear in the final plans.","choices":["children's","childrens'","childrens","children"],"explain":"\"Children\" is already plural, so its possessive adds an apostrophe and s: \"children's.\" \"Childrens\" isn't a word in standard English, so B and C fail, and D lacks possession."},"r3047":{"id":"r3047","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"As well as\" doesn't make a plural subject","tier":3,"traps":["distance"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The quartet's first violinist, as well as the two cellists who joined the group last year, at the national conservatory in Vienna.","choices":["were trained","have trained","was trained","are training"],"explain":"\"As well as the two cellists…\" is a phrase set off by commas; it doesn't make the subject plural. The subject is \"first violinist,\" so the singular \"was trained\" is correct. The other choices are plural."},"r3048":{"id":"r3048","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Plural subject with a long prepositional phrase","tier":2,"traps":["distance"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The effects of the three-year drought on the region's wheat harvest more severe than agricultural economists had predicted.","choices":["were","was","is","has been"],"explain":"The subject is \"effects,\" which is plural; \"drought\" and \"harvest\" are inside prepositional phrases. \"Had predicted\" places the comparison in the past, so \"were\" is correct."},"r3049":{"id":"r3049","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"Whose\" versus \"who's\"","tier":1,"traps":["rule"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The poet Ocean Vuong, first collection won the T.S. Eliot Prize, has also written a widely praised novel.","choices":["who's","whose","who","whom"],"explain":"The blank needs a possessive (the first collection belongs to the poet): \"whose.\" A means \"who is,\" which doesn't fit, and C and D aren't possessive."},"r3050":{"id":"r3050","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Relative clause verb after \"genes that\"","tier":2,"traps":["distance"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The study identified several genes that how well wheat plants tolerate drought, raising hopes for more resilient varieties.","choices":["influences","influencing","to influence","influence"],"explain":"\"That\" refers to \"genes,\" which is plural, and the relative clause needs a finite verb: \"influence.\" A is singular, and B and C aren't finite."},"r3051":{"id":"r3051","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Compound subject with \"and\"","tier":2,"traps":["distance"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The size and shape of a finch's beak closely to the kind of food the bird eats, a relationship Charles Darwin noticed in the Galápagos Islands.","choices":["corresponds","is corresponding","correspond","has corresponded"],"explain":"\"The size and shape\" is a compound subject joined by \"and,\" so it's plural: \"correspond.\" \"Beak\" is part of a prepositional phrase, which makes the singular forms tempting."},"r3052":{"id":"r3052","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"The number of\" takes a singular verb","tier":3,"traps":["distance"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"According to the park's annual surveys, the number of bird species recorded in the wetland nearly doubled since the restoration project began in 1998.","choices":["have","has","are","were"],"explain":"\"The number\" is singular; \"species\" belongs to the phrase \"of bird species.\" \"Since … 1998\" calls for the present perfect: \"has … doubled.\" A is plural; C and D don't form a correct verb with \"doubled\" here."},"r3053":{"id":"r3053","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"\"Its\" for a singular species","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The vaquita, a small porpoise found only in the northern Gulf of California, is critically endangered, and population has fallen to roughly ten animals.","choices":["its","their","it's","they're"],"explain":"The pronoun refers to \"The vaquita,\" which is singular, and the blank needs a possessive: \"its.\" C means \"it is.\" B is plural."},"r3054":{"id":"r3054","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"A series takes a singular verb","tier":3,"traps":["distance"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"A series of experiments conducted at three universities that the compound slows the growth of certain tumors in mice.","choices":["have shown","showing","are showing","has shown"],"explain":"The subject is \"series,\" which is singular; \"experiments\" and \"universities\" are inside modifying phrases. \"Has shown\" agrees. B isn't a finite verb."},"r3055":{"id":"r3055","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Modifier must touch what it modifies","tier":3,"traps":["rule"],"answer":"D","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Founded in 1636 as a school for training ministers, ","choices":["Harvard's original curriculum emphasized Latin, Greek, and theology.","Latin, Greek, and theology were emphasized in Harvard's original curriculum.","the emphasis of Harvard's original curriculum was on Latin, Greek, and theology.","Harvard originally emphasized Latin, Greek, and theology in its curriculum."],"explain":"The opening phrase describes something that was founded in 1636: Harvard. Only D puts \"Harvard\" right after the comma. A makes the curriculum the thing founded, and B and C make the subjects or the emphasis the thing founded."},"r3056":{"id":"r3056","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Present tense for a current state","tier":2,"traps":["rule"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The Rosetta Stone, which was discovered in Egypt in 1799, on display at the British Museum in London, where millions of visitors see it each year.","choices":["is","was","had been","will be"],"explain":"The main clause describes the stone's current state, as \"see it each year\" shows, so the present tense \"is\" is correct. The past tense belongs to the relative clause about its discovery."},"r3057":{"id":"r3057","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Subject-verb agreement after a quoted title","tier":3,"traps":["distance"],"answer":"B","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Hidden Figures, a book about the Black women mathematicians whose calculations helped launch the first American astronauts into orbit, adapted into a film in 2016.","choices":["were","was","have been","are"],"explain":"The subject is the book title, a single work, so the verb is singular even though the title and the appositive contain plural words (\"Figures,\" \"mathematicians,\" \"astronauts\"). \"In 2016\" calls for the past: \"was.\""},"r3058":{"id":"r3058","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Pronoun with a compound antecedent","tier":2,"traps":["rule"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"Marie Curie and her daughter Irène Joliot-Curie are the only mother and daughter to have each won a Nobel Prize, and research on radioactivity laid the groundwork for nuclear medicine.","choices":["her","its","their","one's"],"explain":"The pronoun refers to both women, a plural antecedent, so \"their\" is correct. A would refer to only one of them, and B and D don't fit people at all."},"r3059":{"id":"r3059","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Finite verb in a sentence with an appositive and a clause","tier":3,"traps":["rule","distance"],"answer":"A","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"The axolotl, a salamander that can regrow lost limbs, parts of its heart, and even sections of its brain, in the wild only in a few canals and lakes near Mexico City.","choices":["survives","surviving","having survived","to survive"],"explain":"After the long appositive, the sentence still needs a main verb for \"The axolotl.\" Only \"survives\" is a finite verb, and it agrees with the singular subject. The present tense fits a current fact."},"r3060":{"id":"r3060","sec":"rw","domain":"Standard English Conventions","skill":"Form, Structure, and Sense","sub":"Agreement with \"there\" constructions","tier":2,"traps":["distance"],"answer":"C","stem":"Which choice completes the text so that it conforms to the conventions of Standard English?","passage":"In the archive's oldest box of records, there a ship's log, three letters from the captain, and a hand-drawn map of the harbor.","choices":["is","was","are","has been"],"explain":"In a \"there\" sentence, the verb agrees with what follows: \"a ship's log, three letters …, and a hand-drawn map,\" a compound (plural) subject. The present tense matches a description of what the box contains now, so \"are\" is correct."},"r4001":{"id":"r4001","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A difference on one specific dimension","tier":3,"traps":["signal"],"answer":"A","stem":"The student wants to emphasize a difference between the atmospheres of Venus and Earth. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Whereas carbon dioxide makes up about 96% of Venus's atmosphere, it makes up only about 0.04% of Earth's.","Venus and Earth, unlike many other planets, are similar in size and mass, and both have volcanoes.","Although Venus and Earth both have volcanoes, Venus is also similar to Earth in size and in mass.","Like Earth's atmosphere, Venus's atmosphere contains carbon dioxide, and both planets have volcanoes."],"explain":"The goal names two things: a difference and atmospheres. A contrasts carbon dioxide levels in the two atmospheres. B uses \"unlike\" but describes a similarity between the two planets, C uses \"Although\" but lists only similarities, and D mentions the atmospheres only to say what they share."},"r4002":{"id":"r4002","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Aim and method of a study","tier":3,"traps":["half"],"answer":"D","stem":"The student wants to present the study's aim and its methodology. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Liang Wei found that urban foxes in London were more likely than rural foxes to approach and touch unfamiliar objects.","In 2022, ecologist Liang Wei studied urban foxes in London, finding that they were bolder than rural foxes.","Rural foxes often avoided food placed near unfamiliar objects, while urban foxes approached the objects, Liang Wei found.","To learn whether city foxes are bolder, Liang Wei put unfamiliar objects near food in 30 urban and 30 rural territories."],"explain":"The goal has two parts: the aim (whether city foxes are bolder) and the method (objects placed near food in 60 territories). D gives both. A, B, and C report results instead."},"r4003":{"id":"r4003","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Introducing a work to an unfamiliar audience","tier":2,"traps":["half"],"answer":"B","stem":"The student wants to introduce the novel to an audience unfamiliar with Achebe. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Chinua Achebe, who later wrote four more novels, published Things Fall Apart in 1958.","Things Fall Apart, a 1958 novel by Nigerian writer Chinua Achebe, follows an Igbo leader as colonizers arrive.","Translated into more than 50 languages, Things Fall Apart follows Okonkwo as British missionaries arrive.","Okonkwo, a leader in an Igbo village, is the main character of Achebe's 1958 novel Things Fall Apart."],"explain":"An audience unfamiliar with Achebe needs to know who he is and what the novel is about. B identifies Achebe as a Nigerian writer and summarizes the story. A names Achebe without saying who he is or what the novel is about, C never mentions Achebe, and D names him without identifying him."},"r4004":{"id":"r4004","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"An advantage relative to an alternative","tier":3,"traps":["half"],"answer":"C","stem":"The student wants to highlight an advantage of perovskite panels over silicon panels. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Perovskite panels currently degrade faster than silicon panels when exposed to moisture, researchers have found.","Like silicon panels, perovskite panels convert sunlight into electricity, and researchers are testing coatings for them.","Perovskite panels can be made at lower temperatures than silicon ones, which cuts manufacturing costs.","Researchers are testing coatings that could protect perovskite panels from the moisture that degrades them."],"explain":"The goal is an advantage over silicon. C compares manufacturing temperature and cost, favoring perovskite. A is a disadvantage, B is a similarity, and D describes ongoing research rather than an advantage."},"r4005":{"id":"r4005","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A generalization with an example","tier":3,"traps":["half"],"answer":"A","stem":"The student wants to make a generalization about animal navigation and support it with an example. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Many animals navigate by Earth's magnetic field; loggerhead turtles, for example, find home by its signature.","Loggerhead sea turtles return to the beach where they hatched, sometimes after spending decades at sea.","The mechanism that allows animals such as loggerhead turtles to sense magnetic fields is still debated.","European robins sense magnetic direction with eye proteins, and loggerhead turtles return to their home beaches."],"explain":"A states the generalization (\"Many animals navigate by Earth's magnetic field\") and then gives the turtles as an example of it. B gives the example without the generalization. C concerns an open question rather than a generalization, and D lists two facts without stating a general claim or connecting them."},"r4006":{"id":"r4006","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A similarity between two figures","tier":2,"traps":["signal"],"answer":"D","stem":"The student wants to emphasize a similarity between Anning and Ameghino. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Unlike Ameghino, who was Argentine and lived later, Anning was English and worked in the early 1800s.","Anning, who discovered the first correctly identified ichthyosaur, was excluded from the Geological Society.","Ameghino described hundreds of fossil species, while Anning discovered the first correctly identified ichthyosaur.","Anning and Ameghino both made major contributions to paleontology without the usual credentials of their fields."],"explain":"The goal is a similarity. Both made important discoveries while outside the usual credentials: Anning was excluded from the society, Ameghino lacked university training. A and C present differences, and B concerns only Anning."},"r4007":{"id":"r4007","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A result with its significance","tier":3,"traps":["half"],"answer":"B","stem":"The student wants to present the study's main finding and explain why it occurred. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Researchers compared 20 fields with wildflower strips and 20 without them to see how the strips affected crop pests.","Fields with wildflower strips had 40% fewer aphids, likely because the strips drew in aphid-eating insects.","Farmers who planted wildflower strips beside their fields used less pesticide than other farmers did.","Ladybugs and hoverflies, which eat aphids, were attracted to the strips of wildflowers planted beside farm fields."],"explain":"The goal pairs the finding (40% fewer aphids) with a reason (the strips drew aphid predators). B gives both. A describes the method, C a consequence, and D a fact without the finding."},"r4008":{"id":"r4008","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Specific result from a study with several results","tier":3,"traps":["scope"],"answer":"C","stem":"The student wants to emphasize how important library computers are to users without home internet. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["In a 2023 survey of 1,200 library users, 41% reported using the library's computers or internet access.","Most library users in a 2023 survey (64%) borrowed print books, while 41% used the library's computers.","In a 2023 survey, 78% of library users without home internet said they used library computers.","Nearly a quarter of library users attended a class or event, and 41% used the computers, a 2023 survey found."],"explain":"The goal concerns a specific group: users without home internet. Only C reports that group's figure. A gives the rate for all users, which is lower and doesn't address the group."},"r4009":{"id":"r4009","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Contrast of a claim with new evidence","tier":3,"traps":["half"],"answer":"A","stem":"The student wants to present the 2020 study's findings as a challenge to the long-standing theory. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["A 2020 study found no evidence for the long-held theory that Easter Island's population collapsed before 1722.","Easter Island's population collapsed before 1722, the theory holds, because islanders cut down all the trees.","Europeans arrived on Easter Island in 1722, bringing disease and enslavement, according to a 2020 study.","A 2020 study of radiocarbon dates and population records examined Easter Island's history before and after 1722."],"explain":"The goal requires both the old theory and the study's contrary finding. A names the theory and states that the study found no evidence for it. B restates the theory as fact, C reports what the study suggests without mentioning the theory, and D mentions the study without its finding."},"r4010":{"id":"r4010","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Explaining a term to a general audience","tier":2,"traps":["scope"],"answer":"D","stem":"The student wants to define biomimicry and give an example. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Biomimicry has been used in architecture, medicine, and robotics, as well as in the design of Velcro.","The nose of Japan's Shinkansen bullet train was redesigned to resemble a kingfisher's beak, reducing tunnel noise.","Both Velcro and the Shinkansen bullet train were inspired by nature, and both were redesigned to reduce noise.","Biomimicry, designing products by imitating nature, gave us Velcro, which was inspired by burrs."],"explain":"D defines the term and gives an example (Velcro). A lists uses without defining biomimicry, B offers an example without a definition, and C misstates the notes: only the train was redesigned to reduce noise."},"r4011":{"id":"r4011","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A cause and its effect over time","tier":2,"traps":["half"],"answer":"B","stem":"The student wants to explain how a single event in 1859 led to a large-scale environmental problem. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["To contain the rabbits that had damaged native plants, Australians built a fence 3,250 kilometers long.","Rabbits released in Australia in 1859, with few predators, grew to about 10 billion and damaged native plants.","By the 1920s, an estimated 10 billion rabbits lived in Australia, where they ate native plants and caused erosion.","Rabbits, which eat native plants and cause soil erosion, have few predators in Australia, where they were released."],"explain":"The goal links the 1859 release to a large environmental problem. B traces the chain: release, rapid growth without predators, vast numbers, damage. C gives the scale without the cause, and A describes a response."},"r4012":{"id":"r4012","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Emphasize a surprising scale","tier":2,"traps":["scope"],"answer":"C","stem":"The student wants to emphasize the surprising size of Pando as a single living thing. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Pando, a grove of about 47,000 aspen trees in Utah, covers about 43 hectares of land.","The trees in Pando are genetically identical and share one root system, so Pando is considered a single organism.","Though it looks like a forest, Pando is a single organism covering 43 hectares and weighing about 6,000 tons.","Pando's 47,000 trees are genetically identical, and together they weigh an estimated 6,000 metric tons."],"explain":"The goal combines \"single living thing\" with \"surprising size.\" C explains that the apparent forest is one organism and gives its area and weight. B and D explain the single-organism point without conveying size."},"r4013":{"id":"r4013","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Relative timing of two events","tier":3,"traps":["half"],"answer":"B","stem":"The student wants to emphasize that movable type existed in Asia long before Gutenberg's press. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Gutenberg used metal type and an oil-based ink, while Bi Sheng's earlier movable type was made of baked clay.","Movable type existed in China by 1040 and metal type in Korea by 1377, long before Gutenberg's press of 1440.","Bi Sheng's movable type, made of baked clay, was invented in China and later improved by Gutenberg.","Johannes Gutenberg developed the printing press in Europe around 1440, using metal type and an oil-based ink."],"explain":"The goal requires the Asian dates and the comparison with Gutenberg. B gives both the Chinese (1040) and Korean (1377) dates and places them \"long before\" 1440. C describes Bi Sheng's type without any timing, and A and D never say which came first."},"r4014":{"id":"r4014","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"An audience already familiar with the topic","tier":3,"traps":["scope"],"answer":"C","stem":"The student wants to tell an audience already familiar with CRISPR about a recent milestone. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["CRISPR, a tool adapted from a bacterial defense system, lets scientists edit DNA at precise locations.","CRISPR was adapted from a defense system that bacteria use against viruses and now edits human blood stem cells.","In 2023, a sickle cell treatment that edits patients' own blood stem cells became the first approved medicine to use CRISPR.","Scientists use CRISPR, a tool for editing DNA at precise locations, to treat diseases such as sickle cell disease."],"explain":"An audience familiar with CRISPR doesn't need it explained (A, B, D); what's new to them is the milestone. C reports the 2023 approval and why it is a first."},"r4015":{"id":"r4015","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A contrast within one group over time","tier":3,"traps":["signal"],"answer":"D","stem":"The student wants to emphasize that U.S. farms produced more even as the farm workforce shrank. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Unlike in 1900, when about 40% of U.S. workers were on farms, tractors were widely used on farms in 2000.","In 1900, about 40% of the U.S. workforce worked on farms, compared with less than 2% in 2000.","Tractors, fertilizers, and new crop varieties greatly increased the output of each U.S. farm worker after 1900.","The share of U.S. workers on farms fell from about 40% to under 2% from 1900 to 2000, yet output rose."],"explain":"The goal pairs two trends: the shrinking workforce and the rising output. D contrasts them directly. C explains why productivity rose but doesn't state either trend. A uses \"Unlike,\" but its contrast concerns tractors."},"r4016":{"id":"r4016","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A quote's context for the reader","tier":2,"traps":["half"],"answer":"A","stem":"The student wants to present Carson's quotation and explain her influence. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Carson, whose Silent Spring helped bring about the U.S. DDT ban, wrote, \"In nature nothing exists alone.\"","Rachel Carson, a marine biologist and writer, published Silent Spring in 1962, ten years before the DDT ban.","Silent Spring (1962) documented the harm caused by pesticides such as DDT and prompted public concern.","\"In nature nothing exists alone,\" wrote Rachel Carson, a marine biologist and writer, in her book Silent Spring."],"explain":"Only A includes the quotation and also explains Carson's influence (the book helped lead to the DDT ban)."},"r4017":{"id":"r4017","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Explaining a method's purpose","tier":3,"traps":["half"],"answer":"B","stem":"The student wants to explain why ground-penetrating radar is useful to archaeologists. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Ground-penetrating radar sends radio waves into the soil, where the waves reflect off buried walls and other objects.","Mapping buried structures without digging, ground-penetrating radar lets archaeologists study sites too large to excavate.","Archaeologists used ground-penetrating radar to map an entire Roman town in Italy in a single project.","By sending radio waves into the soil, ground-penetrating radar creates a detailed map of whatever lies underground."],"explain":"The goal is about usefulness. B connects what the method does (maps without digging) to an archaeological need (sites too large to excavate). A and D explain how it works but not why it helps, and C reports one use without saying what made the method valuable there."},"r4018":{"id":"r4018","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Comparing two quantities with a ratio","tier":2,"traps":["scope"],"answer":"D","stem":"The student wants to compare the size of the blue whale with the size of the largest known dinosaur. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["The blue whale, which can weigh about 150 metric tons, is the largest animal known to have lived.","Blue whales, which can weigh about 150 metric tons, feed mainly on tiny shrimp-like animals called krill.","Argentinosaurus, the largest known dinosaur, may have weighed about 70 metric tons.","A blue whale, at about 150 metric tons, can weigh more than twice as much as Argentinosaurus (about 70 tons)."],"explain":"A comparison needs both animals: D gives both weights and the relationship (\"more than twice as much\"). A implies a comparison without naming the dinosaur, B is about diet, and C gives only one side."},"r4019":{"id":"r4019","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Emphasize a limitation of a finding","tier":3,"traps":["half"],"answer":"A","stem":"The student wants to point out a limitation of the study's findings. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Its participants were volunteers eager to cut back, followed for just four weeks, so the results may not generalize.","Teenage participants who used the app reduced their screen time by 30 minutes a day on average over four weeks.","The study, which lasted four weeks, tested a new app designed to help teenagers reduce their screen time.","The researchers recruited volunteers who wanted to cut back on screen time and tested the app for four weeks."],"explain":"A limitation is a reason to be cautious about the results. A uses two notes (self-selected volunteers, a short study without follow-up) to explain why. B reports the result itself."},"r4020":{"id":"r4020","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Specify a place and a purpose","tier":2,"traps":["half"],"answer":"C","stem":"The student wants to describe where the Great Green Wall is and what it is meant to do. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Progress on the Great Green Wall has varied widely from one participating country to another.","The Great Green Wall aims to restore 100 million hectares of degraded land to support farming communities.","Spanning Africa's Sahel, the Great Green Wall aims to restore degraded land and slow desertification.","The Sahel, the region south of the Sahara, is the site of a project that has made uneven progress."],"explain":"The goal has two parts: location and purpose. C gives both. B gives a target without the location, and D gives a location without the project."},"r4021":{"id":"r4021","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"A comparison favoring one option for a stated reason","tier":3,"traps":["half"],"answer":"A","stem":"The student wants to argue that heat pumps use energy more efficiently than gas furnaces. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["A heat pump delivers about three units of heat per unit of electricity; a furnace, under one per unit of fuel.","Heat pumps cost more to install than gas furnaces, though both can heat homes effectively in most climates.","Both heat pumps and gas furnaces can heat homes, but heat pumps move heat while furnaces create it.","A heat pump moves heat from the outside air into a home, while a furnace burns fuel to create heat."],"explain":"Efficiency means heat delivered per unit of energy used. A compares exactly that. D explains how each works but makes no efficiency claim, and B works against the argument."},"r4022":{"id":"r4022","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Present a counterintuitive finding","tier":3,"traps":["scope"],"answer":"C","stem":"The student wants to present the experiment's results as evidence against a common assumption. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["The display with 24 varieties of jam attracted more visitors than the display with 6, as the assumption predicts.","A store tested the common assumption about choice by offering either 6 or 24 varieties of jam for tasting.","Despite the belief that more choice pleases shoppers, those offered 6 jams were ten times as likely to buy.","Shoppers offered 24 varieties of jam bought 3% of the time, while the larger display drew more visitors."],"explain":"The goal requires naming the assumption and showing results that cut against it. C does both: 30% versus 3% is ten times as likely. A reports a result that seems to fit the assumption, B describes only the method, and D gives figures from the notes without naming the assumption or making the comparison that undercuts it."},"r4023":{"id":"r4023","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Explain a name's origin","tier":2,"traps":["half"],"answer":"D","stem":"The student wants to explain why helium has its name. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Helium was first detected in sunlight in 1868 and was found on Earth in 1895, nearly thirty years later.","In 1868, astronomers saw an unexpected yellow line in sunlight that did not match any known element.","Helium, which was found on Earth in 1895, is named after Helios, a figure from Greek mythology.","Because helium was first detected in the Sun, it was named after Helios, the Greek god of the Sun."],"explain":"An explanation of the name needs the namesake (Helios) and the reason (first detected in the Sun). Only D links the two."},"r4024":{"id":"r4024","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Distinguish two similar terms","tier":3,"traps":["signal"],"answer":"B","stem":"The student wants to clarify the difference between weather and climate using an example. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Both weather and climate describe temperature and precipitation, whether over hours, days, or decades.","Weather covers hours or days, like one cold winter; climate covers decades, like a thirty-year warming.","A single cold winter and a thirty-year rise in winter temperatures are both related to Earth's climate.","Climate describes average conditions over decades, such as a thirty-year rise in average winter temperature."],"explain":"The goal is a difference, with an example. B contrasts the time scales and gives an example of each. A and C describe similarities or connections, and D defines only one term."},"r4025":{"id":"r4025","sec":"rw","domain":"Expression of Ideas","skill":"Rhetorical Synthesis","sub":"Emphasize the purpose of a design feature","tier":2,"traps":["half"],"answer":"D","stem":"The student wants to explain the purpose of the steel sphere in Taipei 101. Which choice most effectively uses relevant information from the notes to accomplish this goal?","passage":"While researching a topic, a student has taken the following notes:","choices":["Visitors to the 508-meter Taipei 101 can view its 660-metric-ton steel sphere from an observation deck.","Taipei 101, a 508-meter skyscraper in Taiwan, has a 660-metric-ton steel sphere hanging near its top.","The steel sphere hanging near the top of Taipei 101 is a 660-metric-ton device called a tuned mass damper.","The sphere in Taipei 101 sways opposite to the building, reducing its sway in strong winds and earthquakes."],"explain":"Purpose means what the sphere does for the building. D explains that it counteracts the building's sway. C names the device without saying what it does, and A describes it as a sight for visitors."},"r4026":{"id":"r4026","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Purpose, not contrast","tier":3,"traps":["signal"],"answer":"D","stem":"Which choice completes the text with the most logical transition?","passage":"Early in her career, the photographer Imogen Hart was told that portraits of factory workers would never sell. She resolved to prove her critics wrong. although she had little money, she spent three years photographing mill workers in their homes and at their machines, then exhibited the portraits in a gallery known for showing paintings by famous artists.","choices":["Nevertheless,","In contrast,","Even so,","To that end,"],"explain":"The sentence describes the steps Hart took to reach the goal stated just before (\"prove her critics wrong\"), so \"To that end\" fits. The contrast (\"although she had little money\") is already handled inside the sentence. Trap: three choices signal contrast, but the relationship to the previous sentence is purpose."},"r4027":{"id":"r4027","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Concession before a counterpoint","tier":3,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"Critics of electric cars point out that producing their batteries generates more carbon emissions than building a gasoline engine. studies that account for a car's full lifetime find that an electric car typically makes up for that difference within two years of driving, because charging it produces fewer emissions than burning gasoline.","choices":["Granted,","Still,","Similarly,","For example,"],"explain":"The first sentence states a critics' point; the second says that point doesn't settle the matter. \"Still\" signals that the conclusion holds despite what was just said. \"Granted\" (A) would introduce a concession, but the second sentence is the rebuttal, not the concession."},"r4028":{"id":"r4028","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Elaboration with \"in fact\"","tier":2,"traps":["signal"],"answer":"C","stem":"Which choice completes the text with the most logical transition?","passage":"The last known Tasmanian tiger, or thylacine, died in a zoo in 1936, but some residents of Tasmania insist that they have glimpsed the animal in the wild. more than a thousand reported sightings have been logged since the 1930s, although none has been confirmed with clear photographs or physical evidence.","choices":["However,","Instead,","In fact,","As a result,"],"explain":"The second sentence intensifies the first: not just \"some residents\" but more than a thousand reported sightings. \"In fact\" introduces a stronger version of the previous point. The \"although\" clause supplies its own contrast, so \"However\" (A) isn't needed."},"r4029":{"id":"r4029","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Consequence","tier":2,"traps":["signal"],"answer":"A","stem":"Which choice completes the text with the most logical transition?","passage":"Bamboo can grow as much as 90 centimeters in a single day, and many species reach full height in just a few months. bamboo forests that are harvested can regenerate far more quickly than forests of slow-growing hardwood trees.","choices":["Consequently,","Nevertheless,","Previously,","By contrast,"],"explain":"Fast regrowth is the cause, and quick regeneration after harvest is the result: \"Consequently.\" \"By contrast\" (D) mislabels the relationship; the comparison with hardwoods is inside the second sentence, not between the sentences."},"r4030":{"id":"r4030","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"A specific instance of a general claim","tier":2,"traps":["signal"],"answer":"A","stem":"Which choice completes the text with the most logical transition?","passage":"Many traditional building methods are surprisingly well suited to hot climates. the wind towers of Yazd, Iran, catch breezes high above the city and funnel them down into homes, cooling rooms without electricity.","choices":["For instance,","Moreover,","In conclusion,","Even so,"],"explain":"The wind towers are one example of a traditional method suited to heat. \"Moreover\" (B) would add a separate point, but this sentence illustrates the first one."},"r4031":{"id":"r4031","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Restating more precisely","tier":3,"traps":["signal"],"answer":"C","stem":"Which choice completes the text with the most logical transition?","passage":"The town's population has not so much declined as aged. the number of residents has barely changed since 1990, but the share of residents over sixty-five has doubled while the number of school-age children has fallen by half.","choices":["Moreover,","Nevertheless,","Specifically,","However,"],"explain":"The second sentence gives the precise details behind \"aged rather than declined\": steady numbers, a doubled share of older residents, half as many children. \"Specifically\" introduces exact details of the general claim. \"Moreover\" (A) would add a new point rather than spell out this one, and nothing is contrasted."},"r4032":{"id":"r4032","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Contrast across two sentences","tier":2,"traps":["signal"],"answer":"D","stem":"Which choice completes the text with the most logical transition?","passage":"Honeybees are often celebrated as the world's most important pollinators. many crops, including tomatoes and blueberries, are pollinated more effectively by wild bees such as bumblebees, which can vibrate flowers to release pollen.","choices":["Similarly,","Thus,","Specifically,","Yet"],"explain":"The second sentence qualifies the celebration of honeybees by pointing out crops that wild bees pollinate better. \"Yet\" signals that contrast. \"Thus\" (B) would wrongly make the second point follow from the first."},"r4033":{"id":"r4033","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Chronological sequence","tier":1,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"In 1912, the passenger ship Titanic sank after striking an iceberg in the North Atlantic. in 1914, thirteen nations signed the first International Convention for the Safety of Life at Sea, requiring enough lifeboats for everyone on board.","choices":["Previously,","Subsequently,","Meanwhile,","Likewise,"],"explain":"The convention came after the sinking (1914 follows 1912), so \"Subsequently\" fits. \"Previously\" (A) reverses the order, and \"Meanwhile\" (C) implies the events happened at the same time."},"r4034":{"id":"r4034","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Similar case in a different field","tier":2,"traps":["signal"],"answer":"D","stem":"Which choice completes the text with the most logical transition?","passage":"In medicine, checklists have sharply reduced errors during surgery by making sure that routine steps are never skipped. in aviation, pilots run through standardized checklists before every takeoff, a practice credited with preventing countless accidents.","choices":["However,","As a result,","Instead,","Similarly,"],"explain":"Both sentences describe checklists preventing errors, in surgery and in aviation. \"Similarly\" signals a parallel case. Aviation checklists aren't a result of surgical checklists (B)."},"r4035":{"id":"r4035","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Contrast with an expectation","tier":3,"traps":["signal"],"answer":"A","stem":"Which choice completes the text with the most logical transition?","passage":"Because sharks lack swim bladders, the gas-filled organs that keep many fish afloat, biologists long assumed that sharks must swim constantly to avoid sinking. many sharks rely on large, oil-filled livers that make them nearly buoyant, and some species rest motionless on the seafloor for hours.","choices":["In reality,","Moreover,","For instance,","Therefore,"],"explain":"The second sentence corrects the biologists' assumption, so \"In reality\" fits. \"Moreover\" (B) would add support, and \"Therefore\" (D) would present the correction as a consequence of the assumption."},"r4036":{"id":"r4036","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Additional point in the same direction","tier":2,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"Planting trees along city streets can lower summer temperatures by several degrees through shade and evaporation. street trees absorb rainwater that would otherwise flood storm drains, reducing the burden on a city's sewer system.","choices":["Even so,","In addition,","For example,","Instead,"],"explain":"The second sentence adds another benefit of street trees, so \"In addition\" fits. Stormwater absorption isn't an example of cooling (C), and nothing is contrasted (A, D)."},"r4037":{"id":"r4037","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Concession inside an argument","tier":3,"traps":["signal"],"answer":"C","stem":"Which choice completes the text with the most logical transition?","passage":"Some historians claim that the Library of Alexandria was destroyed in a single catastrophic fire. there was a fire during Julius Caesar's siege of the city in 48 BCE. But surviving sources suggest that the library declined gradually over several centuries, as funding dwindled and scholars left.","choices":["As a result,","Instead,","Granted,","Likewise,"],"explain":"The author concedes part of the historians' claim (there was a fire) before arguing, with \"But,\" that the decline was gradual. \"Granted\" introduces such a concession. \"Instead\" (B) would replace the claim rather than concede part of it."},"r4038":{"id":"r4038","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Result of a constraint","tier":2,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"The James Webb Space Telescope observes mainly infrared light, which is easily drowned out by heat. the telescope's instruments must be kept extremely cold, and a sunshield the size of a tennis court blocks heat from the Sun, Earth, and Moon.","choices":["Nevertheless,","Accordingly,","Similarly,","Meanwhile,"],"explain":"Because heat drowns out infrared light, the instruments must be kept cold; the second sentence describes measures that follow from the first. \"Accordingly\" signals that relationship."},"r4039":{"id":"r4039","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Contrast between two groups","tier":2,"traps":["signal"],"answer":"C","stem":"Which choice completes the text with the most logical transition?","passage":"In many bird species, males sing elaborate songs to attract mates and defend territories, while females rarely sing. in some tropical species, such as certain wrens, females sing as often as males, and pairs perform duets so precisely coordinated that they sound like a single bird.","choices":["Likewise,","Therefore,","By contrast,","For this reason,"],"explain":"The second sentence describes species that differ from the pattern in the first sentence, so \"By contrast\" fits. \"Likewise\" (A) would claim a similarity."},"r4040":{"id":"r4040","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Despite a drawback","tier":3,"traps":["signal"],"answer":"D","stem":"Which choice completes the text with the most logical transition?","passage":"Desalination plants, which remove salt from seawater, consume large amounts of energy and produce a salty brine that can harm marine life. several dry countries, including Israel and Saudi Arabia, now rely on desalination for much of their drinking water, since few other sources are available.","choices":["In other words,","Similarly,","For instance,","Even so,"],"explain":"The first sentence lists drawbacks; the second reports heavy reliance anyway. \"Even so\" signals a result that holds despite what came before. The countries aren't examples of the drawbacks (C)."},"r4041":{"id":"r4041","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Clarifying a technical term","tier":3,"traps":["signal"],"answer":"A","stem":"Which choice completes the text with the most logical transition?","passage":"Some frogs survive winter in a state called freeze tolerance. as much as two-thirds of the water in their bodies turns to ice, their hearts stop beating, and they stop breathing until spring thaws them out.","choices":["That is,","Afterward,","Nevertheless,","Similarly,"],"explain":"The second sentence explains what \"freeze tolerance\" means in practice. \"That is\" introduces an explanation of a term. Nothing is contrasted or compared, and the details describe the state itself, not what happens after it (B)."},"r4042":{"id":"r4042","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Alternative replacing what was rejected","tier":2,"traps":["signal"],"answer":"D","stem":"Which choice completes the text with the most logical transition?","passage":"The city council rejected a proposal to build a new parking garage downtown. it voted to expand bus service and add protected bike lanes on the main roads into the city center.","choices":["Likewise,","For example,","Similarly,","Instead,"],"explain":"The council chose a different course in place of the garage, so \"Instead\" fits. The bus and bike plan isn't like the garage proposal (A, C) or an example of it (B)."},"r4043":{"id":"r4043","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Strengthening a claim with a more extreme example","tier":3,"traps":["signal"],"answer":"A","stem":"Which choice completes the text with the most logical transition?","passage":"Tardigrades, microscopic animals often called water bears, can survive temperatures far below freezing. in laboratory tests, some have survived exposure to the vacuum and radiation of outer space.","choices":["Even more remarkably,","In contrast,","As a result,","Therefore,"],"explain":"Surviving outer space is an even more extreme feat than surviving cold, so \"Even more remarkably\" signals a stronger example of the same point. Surviving space doesn't follow from surviving cold, so \"As a result\" (C) and \"Therefore\" (D) don't fit."},"r4044":{"id":"r4044","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"A conclusion drawn from evidence","tier":2,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"Pollen grains found in sediment layers beneath the lake show a sharp rise in grass pollen and a sharp decline in tree pollen around 4,000 years ago. Charcoal from the same layers indicates frequent fires during that period. researchers conclude that early farmers likely cleared the region's forests with fire to create fields and pastures.","choices":["However,","Thus,","Meanwhile,","Likewise,"],"explain":"The first two sentences give evidence; the third draws a conclusion from it. \"Thus\" signals that inference."},"r4045":{"id":"r4045","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Simultaneous events","tier":2,"traps":["signal"],"answer":"C","stem":"Which choice completes the text with the most logical transition?","passage":"In 1969, as Apollo 11 astronauts walked on the Moon, millions of people around the world watched a live television broadcast of the landing. engineers at mission control in Houston monitored hundreds of data readouts, ready to call off the mission at the first sign of trouble.","choices":["Afterward,","For instance,","Meanwhile,","Therefore,"],"explain":"The engineers' monitoring happened at the same time as the moonwalk and the broadcast, so \"Meanwhile\" fits. \"Afterward\" (A) gets the timing wrong."},"r4046":{"id":"r4046","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Contrast inside a sentence already present","tier":3,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"For centuries, European astronomers believed that comets were disturbances in Earth's atmosphere. Tycho Brahe set out to test this belief. by comparing observations of the Great Comet of 1577 made from different locations, he showed that the comet was far beyond the Moon, even though most scholars of his day rejected the idea.","choices":["However,","Specifically,","Nevertheless,","In contrast,"],"explain":"The sentence describes exactly how Tycho tested the belief: comparing observations from different places. \"Specifically\" introduces those details. The contrast with other scholars is already handled by \"even though\" inside the sentence, so a contrast transition would be redundant."},"r4047":{"id":"r4047","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Explaining a cause after an effect","tier":3,"traps":["signal"],"answer":"D","stem":"Which choice completes the text with the most logical transition?","passage":"The salmon population in the river rebounded sharply after 2012. The reason was not a change in fishing rules, which stayed the same. the removal of two dams that year reopened more than 100 kilometers of spawning habitat that salmon had not been able to reach for nearly a century.","choices":["Similarly,","For instance,","In addition,","Rather,"],"explain":"The previous sentence rules out one explanation; this one gives the correct explanation in its place. \"Rather\" signals that correction. \"In addition\" (C) would add a second reason, but the fishing rules were ruled out, not counted."},"r4048":{"id":"r4048","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Example of a general difficulty","tier":2,"traps":["signal"],"answer":"C","stem":"Which choice completes the text with the most logical transition?","passage":"Translating poetry is especially difficult because sound and meaning are tightly linked. a rhyme or a pun that works in one language often disappears entirely in another, forcing the translator to choose between preserving the meaning and preserving the music.","choices":["Nevertheless,","In contrast,","For example,","Similarly,"],"explain":"A disappearing rhyme or pun illustrates how sound and meaning are linked, so \"For example\" fits. Nothing is contrasted or compared."},"r4049":{"id":"r4049","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Following from a definition","tier":3,"traps":["signal"],"answer":"A","stem":"Which choice completes the text with the most logical transition?","passage":"A keystone species is one whose influence on its ecosystem is far greater than its numbers would suggest. when sea otters disappeared from parts of the Pacific coast, the sea urchins they ate multiplied and devoured entire kelp forests, transforming the coastal ecosystem.","choices":["To illustrate,","For this reason,","Nonetheless,","Similarly,"],"explain":"The sea otters are an example that illustrates the definition of a keystone species. \"To illustrate\" signals that. \"For this reason\" (B) would make the otters' disappearance a result of the definition."},"r4050":{"id":"r4050","sec":"rw","domain":"Expression of Ideas","skill":"Transitions","sub":"Reversal of an earlier trend","tier":2,"traps":["signal"],"answer":"B","stem":"Which choice completes the text with the most logical transition?","passage":"For most of the twentieth century, the number of bald eagles in the continental United States fell steadily, in large part because the pesticide DDT thinned their eggshells. after DDT was banned in 1972, the population recovered so strongly that the species was removed from the endangered species list in 2007.","choices":["Similarly,","However,","For example,","As a result,"],"explain":"The second sentence reverses the trend described in the first (decline, then recovery), so \"However\" fits. The ban caused the recovery, but the recovery wasn't a result of the decline, so \"As a result\" (D) mislabels the link between the sentences."}}