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PRACTICE TEST 11. A
You don’t need to find the value of a to answer this question. It’s more direct to recognize that the denominator ofthe expression you’re solving for is exactly twice the left side of the given equation:
2. CTo find the average of four expressions, add them up and divide by 4. The sum is 8x + 4. When you divide that by 4,you get 2x + 1.
3. ATo solve an equation with the unknown in an exponent, rewrite the equation so that both sides have the same base:
4. BThe easiest way to find the answer to this question is to check the answer choices, starting with the smallest, untilyou find one that meets the given conditions. A multiple of 2 and 5 is a multiple of 10, so already you know that theanswer is (A), (B), or (C). Starting with (A), divide each by 7 until you find one that leaves a remainder of 2. (A) 20divided by 7 is 2 with a remainder of 6—that’s not it. (B) 30 divided by 7 is 4 with a remainder of 2—that’s it.
5. EThe two rectangles have the same height, so if CDEF has twice the area of ABCF, then the base of CDEF is twice thebase of ABCF. The longer base is given as 2x + 2, so the shorter base is half that, or x + 1. You’re looking for AE, whichis the sum of the lengths of the bases:
6. BThe second equation expresses y2 in terms of x, so you can substitute that expression for y2 in the first equation andsolve for x:
7. EA rate of x pages per minute is the same as 100 pages per how many minutes? If y is the number of minutes it takesto print 100 pages, then you can set up the proportion:
8. CThe given values of x are evenly spaced, and ƒ(x) is a linear (i.e., straight-line) function, so the values of ƒ(x) will beevenly spaced. The unknown k is halfway between 2 and 8, so k = 5.
9. BAfter spending 30% on rent, Jackie has 70% of her earnings left. Half of that 70% is 35%, which goes for food andtransportation. So the ratio of rent to food and transportation is 30:35, or 6:7. Now you can set up a proportion andsolve for the rent:
10. D
Mark up the figure. Because the triangles are congruent, ∠C also measures 60°. And because they’re both righttriangles, ∠BDA and ∠CAD both measure 30°:
Now you have the other two angles inside a triangle with x, so you can set their sum equal to 180 and solve for x:
11. CTo solve an equation with fractions, first multiply both sides by whatever you have to to clear all denominators. Inthis case, multiply both sides by 4:
12. CPlug a = 10 and b = 3 into the definition:
13. CIf you add the equations as presented, the y’s drop out:
The only answer choice that works is (C). To confirm, plug x = 7 back into one of the original equations to find y :
14. DTo solve an equation with absolute value, consider the two possibilities.
If |2x − 3| = 13, then what’s inside the absolute value signs equals either 13 or –13:
OR:
15. EExpand the factorials, cancel factors common to the numerator and denominator, and then calculate:
16. BMark up the figure. CD = 5 and AC is 3 times that, so = 15:
B is the midpoint of , so AB = BC = 7.5:
Now you can see that BD = BC + CD = 7.5 + 5 = 12.5.
17. AParallel lines have the same slope, so you’re looking for a line with a slope of –2. Only (A) and (B) have that slope,and of those, only (A) has y-intercept +4.
18. EAdd to the figure. Diagonal BD will divide the quadrilateral into two familiar triangles:
ABD is a right triangle with legs 3 and 4, so BD = 5, and therefore BCD is a 5-12-13 right triangle.
The area of the 3-4-5 triangle is
And the area of the 5-12-13 right triangle is
The quadrilateral’s area, then, is 6 + 30 = 36
19. CApply the inside function first:
Then apply the outside function to the result:
20. EPut much more simply, what the question is asking is: At what point does cost equal revenue? In algebraic terms, forwhat value of x does C equal R?
21. AMark up the figure. The squares are identical, so AD = DG and triangle ADG is isosceles. That makes the measure of∠DGA 75 degrees, and the measure of ∠ADG is therefore 180° – 75° – 75° = 30 degrees. Now you know every angle inthe figure:
The measure of ∠ADE is 30° + 90° = 120 degrees.
22. D
If is the midpoint of and C(x, y), then 6 is the average of and x, and is theaverage of 4 and y :
and:
23. CSimplify the equation:
Don’t jump to the conclusion that if 9 equals x2, then x must be 3. x could just as well be –3:
24. CTo satisfy the Triangle Inequality Theorem, adding the two shorter sides must give you more than the longest side.In (B), (D), and (E), the two shorter sides add up to less than the longest side, and in (A), they add up to the same asthe longest side. Only in (C) do they add up to more.
25. AIf sin x is 0.5, then the side opposite x is one-half the hypotenuse. In other words, this is a 30-60-90 triangle withfamiliar side ratios:
Tangent is opposite over adjacent, so
26. E
When x and y have different signs, (that is, when the point is in the second or fourth quadrant), is negative, so for
no point in those quadrants can be greater than 1. That eliminates (A) and (C). When x and y are both positive, will be greater than 1 if x is greater than y. Point B is above the line x = y, so for that point y > x. When x and y are both
negative, will be greater than 1 if x is less than y. Point D is below the line x = y, so for that point y < x. Point E,
however, is above the line x = y, so for that point y > x, and > 1.
27. DIf the average of the 5 original scores was 83, then the sum of those 5 scores was 5 × 83 = 415. When each score isincreased by 2, the sum goes up to 425, one-fifth of which is 85. So the new average is 85 and statement I is true.Statement II, however, is not true because if both the lowest and highest scores go up by the same amount, thedifference between those scores remains the same. So I is true and II is not, and the answer is either (A) or (D). Whatabout III? To find the lowest score that one of the 5 students can get, imagine that after the increase the other 4students all get perfect 100s. Four 100s add up to 400, which leaves 425 – 400 = 25 for the fifth and lowest possiblescore for the 5 students. III is true.
28. CYou could do this one by picking numbers, but for any set of numbers you might pick, more than one of the answerchoices will be true. You will therefore have to pick at least two sets of numbers to find the correct answer. Thestatement that must be true is the one that holds for all a, b, c, and d that fit the given conditions. (A) is tempting:You might think that the product of the larger a and c will be greater than the product of the smaller b and d, and it isif all you pick are positive integers. (A) doesn’t hold when you consider negatives, however. But (C) holds for allpossible values of a, b, c, and d. The sum of two larger quantities is greater than the sum of two smaller quantities.
29. BLook for “(sin2 + cos2)” in the expression:
30. ASketch a diagram:
The net change is south 20 and west 60:
So you’re looking for the hypotenuse of a right triangle with legs of 20 and 60:
31. BThis time there’s no inside and outside functions. What you’re looking for here is the product of the functions:
32. BThe volume of the tank is 100 cm × 20 cm × 40 cm = 80,000 cubic centimeters. The volume of the bucket is πr2h ≈(3.14) (92) (20) = 5,086.8. To figure out how many buckets fit into the tank, divide:
80,000 ÷ 5,086.8 ≈ 15.7
The nearest choice is (B), 16.
33. DSarah can fill her first-period slot with any of the 4 subjects, leaving 3 for the second, 2 for the third, and 1 for thefourth:
4 × 3 × 2 × 1 = 24
34. BThat and are parallel tells you that the outside triangle ACE and the inside triangle BCD are similar.Corresponding sides are proportional:
35. EFor what values of x can you take the square root of x2 – 4? Only those values for which x2 – 4 is nonnegative:
If you use the short side from (1, 1) to (3, 1) as the base, then the base is 2 and the height is 6:
Plug b = 2 and h = 6 into the formula for the area of a triangle:
37. ASimplify:
Notice that the inequality sign flipped when both sides were divided by –2.
38. BWhen you raise i to successive integer exponents, a pattern develops:
Every fourth power the pattern repeats. A similar pattern develops with –i:
Whenever the exponent is a multiple of 4, i n and (–i)n are both equal to 1. (B) is the only multiple of 4 among thechoices. When n = 24, in + (–i)n = 1 + 1 = 2, which is a positive integer.
39. DBetween 0 and 2π, cos x ranges from –1 to 1. The maximum value of f(x) = 1 − cos x will be when cos x is at its least;that is, when cos x = –1:
40. EThe triangle is isosceles; since AB = BC, x = z. Since y > 60, that leaves less than 120 degrees to split between x and z,so x and z are both less than 60. Angle a is an exterior angle, so it’s equal to the sum of the remote interior angles:
a = x + y
Now you can see that II is true, because y > 60 and z < 60, and you can see that III is true, because a = x + y and x = z. Iis not true: y is greater than z and a + z = 180, so a + y is greater than 180.
41. D
Imagine (or use your graphing calculator) the graph of between 0 and 1. At x = 1, f(x) = 1. As x getssmaller, f(x) gets bigger:
As x approaches 0, f(x) gets infinitely large. The range is all real numbers greater than 1.
42. EThis one’s hard to sketch because it’s 3-D. Here’s a cross section:
Drop an altitude, and you make two right triangles:
That altitude is the radius of the circle formed by the cross section of the spheres. (Can you picture it?)
43. C
In algebraic terms, the remainder is –k + 14. If that’s equal to 4, then
44. DThe total portfolio is 50,000 + 75,000 = 125,000 dollars. Eighty percent of that is (0.80)(125,000) = 100,000 dollars.Thus, she needs to convert 100,000 – 75,000 = 25,000 dollars.
45. D
A sphere of radius 5 has volume If that’s also the volume of a cube, then an edge of that cube
is the cube root of that. Use your calculator:
46. CThe equation x2 = y2 is true whenever x = ± y. Choice (A) shows x = y. Choice (B) shows x = –y. Choice (C) shows both.
47. A is a radius of the circle, and
You’re looking for the choice for which x2 + y2 is greater than 34. Use your calculator. In (A) x2 + y2 = 37. So it looks likethe answer is (A). If you have lots of time, you can check the other answer choices, just to be sure. In (B) x2 + y2 = 32.5.In (C) x2 + y2 = 26.5. In (D) x2 + y2 = 32. And in (E) x2 + y2 = 34. Only (A) is greater than 34.
48. BYou don’t actually need to find the inverse function to answer this question if you remember that the slopes of
inverse functions are reciprocals. The slope of the given f(x) = 3x – 1 is 3, so the slope of f−1(x) is The slope of a
line perpendicular to f(x) will have a slope that’s the negative reciprocal of 3, or So and and therefore
49. AThe question is asking, in other words, for how many positive integer values of x does y turn out to be a positiveinteger?
The expression will be positive whenever 4x is greater than
The expression will be an integer whenever is an integer.
Half of x2 is an integer whenever x is even, so y will be a positive integer if x is an even number greater than 0 and lessthan 8—in other words, if x is 2, 4, or 6. That’s 3 points.
50. BMark up the figure:
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