2

MATHEMATICS LEVEL 2
For each of the following problems, decide which is the
BEST of the choices given. If the exact numerical value is not
one of the choices, select the choice that best approximates
this value. Then fill in the corresponding oval on the answer
sheet.
Notes: (1) A scientific or graphing calculator will be necessary for answering some (but not all) of the questions on this
test. For each question, you will have to decide whether or
not you should use a calculator.
(2) For some questions in this test you may have to decide
whether your calculator should be in the radian mode or the
degree mode.
(3) Figures that accompany problems on this test are intended to provide information useful in solving the problems.
They are drawn as accurately as possible EXCEPT when it
is stated in a specific problem that its figure is not drawn to
scale. All figures lie in a plane unless otherwise indicated.
(4) Unless otherwise specified, the domain of any function
f is assumed to be the set of all real numbers x for which f(x)
is a real number. The range of f is assumed to be the set of all
real numbers f(x), where x is in the domain of f.

1. If r s > r + s, then which of the following must

be true?

(5) Reference information that may be useful in answering

the questions on this test can be found below.

THE FOLLOWING INFORMATION IS FOR YOUR

REFERENCE IN ANSWERING SOME OF THE QUESTIONS ON THIS TEST.
Volume of a right circular cone with radius r and height h:

1 22
pr
r hh
3

V=

Lateral area of a right circular cone with circumference of

1
c
2
4 33
r
Volume of a sphere with radius r: V = pr
3
the base c and slant height : S =

Surface area of a sphere with radius r: S = 4r2

Volume of a pyramid with base area B and height h:
V=

1
Bh
3

USE THIS SPACE FOR SCRATCHWORK.

(A) r > s
(B) s < 0
(C) r < 0
(D) r < s
(E) s > 0

2. If f ( x ) = x +10, for which of the following

values of x does f(x) = f(x) ?
(A) 10 only
(B) 10 and 10 only
(C) All real x
(D) All real x except 10
(E) All real x except 10 and 10

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4 0 2 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

15!
3.
=
13! 2!

USE THIS SPACE FOR SCRATCHWORK.

(A)0 (B)0.58 (C)1 (D) 105 (E)210

y
B (13, 9)

A (1, 9)

C (13, 4)
x

O
Figure 1

4. In Figure 1, sin BAC =

5
(A)
13
5
(B)
12
12
(C)
13
12
(D)
5
13
(E)
5

5. Which of the following is the complete solution

set of the system:
A = {(x, y): x2 + y2 = 25} and
B = {(x, y): y = x + 1}?
(A) {(5, 5)}
(B) {(16, 9)}
(C) {(4, 3)}
(D) {(4, 3), (3, 4)}
(E) {(3, 4), (4, 3)}

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Practice Test 1 | 4 0 3

MATHEMATICS LEVEL 2Continued

6. If jk 0, then

jk

j
k
2
(B) j 2 _ j 2
k
(A) k 2 _

j
k

j
k =

USE THIS SPACE FOR SCRATCHWORK.

(C) jk 1
(D) j2 1
(E) k2 1

7. All of the following can be formed by the intersection of a right cylinder and a plane EXCEPT:
(A) A line
(B) A circle
(C) An ellipse
(D) A parabola
(E) A triangle

8. If f(x) =
(A) 0.1
(B) 1.2
(C) 1.3
(D) 1.8
(E) 2.3

x and g(x) =

1
x + 1, then f(g(2.3)) =
2

9. If x mod y is the remainder when x is divided by

y, then (61 mod 7) (5 mod 5) =
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

4 0 4 | Cracking the SAT Math 2 Subject Test

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MATHEMATICS LEVEL 2Continued

10. Which of the following must be true?
I. sin () = sin
II. cos () = cos
III. tan () = tan , where tan is defined

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(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

11. If for all real numbers x, a function f(x) is defined

2, x 13
by f (x) =
, then f(15) f(14) =
4, x = 13
(A) 2 (B) 0 (C) 1 (D) 2 (E) 4

x5
= 25, then x =
25
(A) 1.00
(B) 1.90
(C) 2.19
(D) 3.62
(E) 5.00

12. If

13. If the ratio of sec x to csc x is 1:4, then the ratio of

tan x to cot x is
(A) 1:16
(B) 1:4
(C) 1:1
(D) 4:1
(E) 16:1

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Practice Test 1 | 4 0 5

MATHEMATICS LEVEL 2Continued

USE THIS SPACE FOR SCRATCHWORK.
y
(6, 3)

(0, 3)
J

(0, 0)

(6, 0)

Figure 2

14. In Figure 2, rectangle J contains all points (x, y).

What is the area of a rectangle that contains all
points (2x, y 1) ?
(A) 12
(B) 18
(C) 24
(D) 36
(E) 48

15. In right triangle ABC, B measures 90, C measures 27, and AB = 9. What is the length of the
hypotenuse of ABC ?
(A) 4.1
(B) 10.1
(C) 17.7
(D) 19.8
(E) 21.2

16. Which of the following is a zero of

f(x) = x2 + 6x 12 ?
(A) 15.16
(B) 7.58
(C) 0.67
(D) 3.16
(E) 7.58

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4 0 6 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

17. If sin x = m and 0 < x < 90, then tan x =
(A)
(B)
(C)

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1
m2
m
1 _ m2

1 _ m2
m

m
(D) _ 2
1 m
m2
(E)
1 _ m2

18. If logy 2 = 8, then y =

(A) 0.25
(B) 1.04
(C) 1.09
(D) 2.83
(E) 3.00

19. If sin =

7
(A) _
9
2
(B) _
3
2
(C)
3
7
(D)
9

and , then cos (2) =

3
4
4

(E) 1

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Practice Test 1 | 4 0 7

MATHEMATICS LEVEL 2Continued

20. If f(x) = x _ 1 , for all x > 0, then f 1 ( x ) =
_

USE THIS SPACE FOR SCRATCHWORK.

(A) (x + 1)2
(B) x2 + 2
(C) x2 + 1
(D) (x 1)2
(E) (x + 2)2

21. When 4x2 + 6x + L is divided by x + 1, the

remainder is 2. Which of the following is the
value of L ?
(A) 4
(B) 6
(C) 10
(D) 12
(E) 15

22. What is the length of the major axis of the ellipse

given by the equation
(A) 3.2
(B) 4.5
(C) 8.9
(D) 10.0
(E) 20.0

x 2 y2
+ =1 ?
10 20

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4 0 8 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

23. If f(x) = [x], where [x] is the greatest integer less

USE THIS SPACE FOR SCRATCHWORK.

than or equal to x, which of the following is a

x
graph of f _ 1 ?
2

(A)

(B)

(C)

(D)

2
2

(E)

y
6
4
2
6

2
4
6

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Practice Test 1 | 4 0 9

MATHEMATICS LEVEL 2Continued

24. Which of the following is equal to the positive
value of sec (cos1 (0.3527)) ?
(A) 0.01
(B) 0.94
(C) 1.69
(D) 2.84
(E) 69.35

USE THIS SPACE FOR SCRATCHWORK.

25. If f(x) = x2 + 5x + 6, for what value of x will f(x)

have its minimum value?
(A) 3
5
(B) _
2
(C) 2
(D) 0
5
(E)
2

26. If the 20th term of an arithmetic sequence is 20

and the 50th term is 100, what is the first term of
the sequence?
(A) 33.33
(B) 30.67
(C) 1.00
(D) 2.00
(E) 2.67

27. The polar equation r sin = 1 defines the graph

of
(A) a line
(B) a circle
(C) an ellipse
(D) a parabola
(E) a hyperbola

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4 1 0 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

28. For which of the following functions f is f 1 a
function?

USE THIS SPACE FOR SCRATCHWORK.

I. f(x) = x2
II. f(x) = x3
III. f(x) = x
(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II, and III

29. What is lim

x 1

x3 _ x
?
x +1

(A) 2
(B) 1
(C) 1
(D) 2
(E) The limit does not exist.

30. If f(x) =
(A)

e7 x + 3
, and g(f(x)) = x, then g(x) =
2

ln 2 x 3
7

(B)

2x 3
e7

(C)

2x 3
7

(D) 7 ln (2x
(E)

3)

(2 x 3 ) ln e
7

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Practice Test 1 | 4 11

MATHEMATICS LEVEL 2Continued

31. A cube is inscribed in a sphere of radius 6. What
is the volume of the cube?

USE THIS SPACE FOR SCRATCHWORK.

(A) 36 3 (B) 36 (C) 216 (D) 192 3 (E) 216 3

32. A right circular cone has height h and radius r.

If the cone is cut into two pieces by a plane that
passes through the midpoint of the height and is
parallel to the base, then the volume of the larger
of the two resulting solids is
2
(A) r h
6
2
(B) r h
3
2
(C) r h
2
2
2
r
h
(D)
3
2
7
r
h
(E)
24

2
1
33. If ex 1 and e x = x , then x =
3
(A) 1.73
(B) 0.55
(C) 1.00
(D) 1.10
(E) 1.73

34. If the graph of the equation y = 2x2 6x + c is

tangent to the x-axis, then the value of c is
(A)3 (B)3.5 (C)4 (D)4.5 (E)5

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4 1 2 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

35. If x = i 1, then x2 + 2x + 2 =

USE THIS SPACE FOR SCRATCHWORK.

(A) 2i + 4
(B) 4 + 2i
(C) 0
(D) i
(E) 2
y

O
Figure 3

36. The curve shown in Figure 3 could represent a

portion of the graph of which of the following
functions?
(A) y = ex
(B) y = ex
(C) y = 100 x
(D) y = x2 3x + 2
(E) xy = 3

37. If two coins are removed at random from a purse

containing three nickels and eight dimes, what is
the probability that both coins will be dimes?
(A)

32
64
14
49
(B)
(C) 28(D)
(E)
55
121
55
110
55

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Practice Test 1 | 4 1 3

MATHEMATICS LEVEL 2Continued

USE THIS SPACE FOR SCRATCHWORK.

38. A function g(x) is odd if g(x) = g(x) for all x

and even if g(x) = g(x) for all x. Which of the
following is the graph of a function that is both
odd and even?
(A)

(B)

(C)

(D)

(E)

39. Points A and B lie on the edge of a circle with

center O. If the circle has a radius of 5, and if the
measure of AOB is 70, what is the length of
chord AB ?
(A) 2.9
(B) 4.7
(C) 5.0
(D) 5.7
(E) 9.4

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4 1 4 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

USE THIS SPACE FOR SCRATCHWORK.

3
4

Figure 4

40. If the graph of y = f(x) is shown in Figure 4, then

which of the following could be true?

4

(B) f(x) = cot x
4

(C) f(x) = tan x +

(D) f(x) = cot x +

(E) f(x) = tan x +

4
(A) f(x) = tan x

41. Vectors v and w have components (3, 4) and

(12, 5), respectively. If z = (v + w), then z has
components
(A) (9, 9)
(B) (5, 13)
(C) (5, 13)
(D) (9, 9)
9 9
(E) ,
2 2

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Practice Test 1 | 4 1 5

MATHEMATICS LEVEL 2Continued

42. If f(x) =

1
2

x2
2

, then for which of the follow-

USE THIS SPACE FOR SCRATCHWORK.

ing values of x does f(x) = 0.33 ?

(A) 0.62
(B) 0.71
(C) 1.36
(D) 3.93
(E) 4.95

43. The system of equations given by

2x + 3y = 7

10x + cy = 3
has solutions for all values of c EXCEPT
(A) 15
(B) 3
(C) 3
(D) 10
(E) 15

xy
for all x, y, f(a, b) = 15, f(b, c) =
3
20, and f(a, c) = 10, which of the following could

44. If f(x, y) =

be the product of a, b, and c ?

(A) 18.26
(B) 54.77
(C) 284.60
(D) 1,800.00
(E) 3,000.00

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4 1 6 | Cracking the SAT Math 2 Subject Test

MATHEMATICS LEVEL 2Continued

45. If x > 0 and y > 1, then logx y =
I. logx y2
2

USE THIS SPACE FOR SCRATCHWORK.

II. logx y

y
III. logx
2

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only

46. Carlos is filling a spherical balloon with water.

If he increases the volume of the balloon from
4,188.79 cubic centimeters to 14,137.167 cubic
centimeters in 12 seconds, then what is the average rate at which he has increased the balloons
surface area?
(A) 130.9 square centimeters per second
(B) 314.159 square centimeters per second
(C) 829.031 square centimeters per second
(D) 1,570.796 square centimeters per second
(E) 9,948.377 square centimeters per second

47. What is the value of 6 _ 3i ?

(A) 3
(B) 3 2
(C) 3 5
(D) 9
(E) 15

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Practice Test 1 | 4 1 7

MATHEMATICS LEVEL 2Continued

48. The menu of a certain restaurant lists 10 items
in column A and 20 items in column B. A family
plans to share 5 items from column A and 5 items
from column B. If none of the items are found in
both columns, then how many different combinations of items could the family choose?

USE THIS SPACE FOR SCRATCHWORK.

(A)
25
(B)
200
(C) 3,425
(D) 3,907,008
(E) 5.63 1010

49. y varies directly as the square of x. When y = 2.5,

x = 0.5. If y = 80, then x could equal
(A)
(B)
(C)
(D)
(E)

2 2
8
10
16
64

50. Seven blue marbles and six red marbles are held in
a single container. Marbles are randomly selected
one at a time and not returned to the container. If
the first two marbles selected are blue, what is the
probability that at least two red marbles will be
chosen in the next three selections?
(A)

19
5
5
6
2
(B) (C) (D) (E)
33
33
11
11
3

STOP
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY.
DO NOT WORK ON ANY OTHER TEST IN THIS BOOK.
4 1 8 | Cracking the SAT Math 2 Subject Test