SAT 8

zontal asymptote at y =

has vertical

274

22. A Use either synthetic or long division to divide x4 − 5x2 − 10x − 12 by x + 2. Remember to include a zero placeholder for the x3 term.

−2 1 0 −5 −10 −12

−2 4 2 16

1 −2 −1 −8 4

The remainder is 4.

23. D The maximum value of f (x) = 3 − (x + 1)2 is the y-coordinate of its vertex.

f (x) = 3 − (x + 1)2.

y − 3 = −(x + 1)2.

The vertex is (−1, 3), so the maximum value is 3.

24. A

sin θ = 12 cosθ.

tan θ = 12 .

tan−1 1 ≈ 26.57°

2

25. C The integers from 1 to 300 form an arithmetic sequence having 300 terms. n = 300, a1 = 1, and an = 300. Substitute these values into the formula for the sum of a finite arithmetic sequence to get:

Sn = 1 + 2 + 3 + 4 + 5 + . . . + 300.

Sn = 2n (a1 + an ).

Sn = 3002 (1 + 300).

Sn = 150(301) = 45,150.

26. A Complete the square to write the equation of the hyperbola in standard form.

9x2 − y2 − 36x − 6y + 18 = 0.

(9x2 − 36x) − (y2 + 6 y) = −18.

9(x2 − 4x + 4) − (y2 + 6y + 9) = −18 + 36 − 9.

9(x − 2)2 − (y + 3)2 = 9.

(x − 2)2 − (y + 3)2 = 1. 9

The center of the hyperbola is (2, −3).

PART III / EIGHT PRACTICE TESTS

27. D The function f (x)

= 3(x2 − 9)

x2 − 4 asymptotes at the zeroes of the denominator.

x2 − 4 = 0.

x = ±2.

Because the degree of the numerator equals the degree of the denominator, a horizontal asymptote

exists at y = 3 = 3. 1

Statements I and III are, therefore, true.

28. A The probability that all 5 protractors are in error of 1° or more is:

0.15(0.15)(0.15)(0.15)(0.15) ≈ 7.6 × 10−5.

29. B

(sin θ + sin(−θ))(cosθ + cos(−θ)) =

(sin θ + − sin θ)(cosθ + cosθ) =

(0)(2cosθ) = 0.

30. B

Factor the equation and solve for θ. 12 sin2 θ + 5 sin θ − 2 = 0.

(3 sin θ + 2)(4 sin θ − 1) = 0.

sin = − 23 or sin θ = 14 .

θ ≈ −41.8° or 14.5°

14.5° is the smallest positive value of θ.

31. D Graph the function to see that it has a hori- 52 .

Alternately, you can evaluate the function at a large value of x. For example, let x = 10,000:

f (10,000) =	5(10,000)	− 11	= 2.49807.
	2(10,000)	+ 11

The function approaches 5 as x gets infinitely large. 2

variance =

≈ 1.679.

PRACTICE TEST 3

32. D Take the logarithm of both sides of the equation to solve for ba.

7.8a = 3.4b.

a log 7.8 = b log 3.4. ba = loglog3.47.8 .

b

a

33. C Because the rectangular coordinates (x, y) are (6, 8):

tan θ = xy = 86 . tan−1 86 53.1°

34. D Translating the graph of f (x) = x3 6 units up and 2 units right results in:

f (x) = (x − 2)3 + 6.

Then, reflecting it over the x-axis results in: g(x) = −(x − 2)3 − 6.

Now, evaluate the function for x = −1. g(−1) = −(−1 − 2)3 − 6 = 27 − 6 = 21.

35.C

(6sin x)(3sin x) − (9cos x)(−2cos x) =

18sin2 x + 18cos2 x =

18(sin2 x + cos2 x) =

18(1) = 18.

36. A There are 8 possible outcomes when a coin is tossed three times:

TTT TTH

HHH HHT

THH HTT

THT HTH

Three out of the 8 have exactly two heads, so the

probability is 3. 8

275

37. B Exchange the x and y values and solve for the inverse of f.

f (x) = 3 2x3 − 5.

	y		=	3	2x3 − 5.
	x		=	3	2y3 − 5.
	x3 + 5		= y3 .
2
	y =			3	x3 + 5				= f −1(x).
						2
f −1(2.5)			=	3	2.53		+ 5			≈ 2.18
						2

38. C Recall that a function is even if it is symmetric with respect to the y-axis and f(−x) = f(x).

Graph the functions on your calculator to determine which one has y-axis symmetry. f(x) = −x2 + 1 is the correct answer choice.

2π

39. D All of the functions have a period of π or 2

with the exception of y = 1 cos 2πx + 1. It has a period 2

of 2π = 1. Answer D is the correct answer choice. 2π

40. B First, determine the mean of the 9 temperatures.

Mean

= sum

9

= 16 + 19 + 21 + 22 + 28 + 28 + 30 + 34 + 36 9

= 2349 = 26.

Now use the deviation of each temperature from the mean to determine the variance.

−102 + − 72 + −52 + − 42

+ 22 + 22 + 42 + 82 + 102

9

=378 = 42

9

The standard deviation is the square root of the variance.

σ = 42 ≈ 6.48

276

41. D The graph has 4 zeroes, so it could be the graph of either the 4th degree polynomial in statement I or the 6th degree polynomial in statement III. Because the leading coefficient in statement II is negative, the graph of F(x) would fall to the left and right.

x

42.E Since f(x) = ln e 2 must be greater than 100: 2x > 100.

x > 200.

The smallest possible integer greater than 200 is 201.

43. D Use the change of base formula to rewrite the right side of the equation.

log3 (x − 9) = log9 (x − 3).

log3 (x − 9) =

log3 (x − 9) =

Now, solve for x.

log3 (x − 3) . log3 9

log3 (x − 3) .

2

2log3 (x − 9) = log9 (x − 3).

(x − 9)2 = x − 3.

x2 − 18x + 81 = x − 3.

x2 − 19x + 84 = 0.

(x − 7)(x − 12) = 0.

x = 7 or 12.

7 is an extraneous solution, however, because the log of a negative number is undefined. x = 12 is the only solution.

44. D

−11 + 4[n] = 5.

4[n] = 16.

[n] = 4.

4 is the greatest integer less than or equal to n, so n must be on the interval 4 ≤ n < 5.

PART III / EIGHT PRACTICE TESTS

45. B Let w = the width, l = the length, and h = the height of the prism. Using the three given areas:

wh = 8.

wl = 10.

lh = 20.

Use substitution to solve for the variables. Because wh = 8, w = 8h.

wl = 8h l = 10.

l = 10h = 5h . 8 4

Substitute this value of l into the third equation to get:

lh = 54h (h) = 20. 5h2 = 80.

h2 = 16.

h = 4.

If h = 4, then w(4) = 8, and w = 2. 2l = 10, so l = 5.

The volume is, therefore, 5 × 2 × 4 = 40 units3.

46. E

tan 4x = 2 cot 4x.

	tan 4x =	2		.
		tan 4x
	(tan 4x)2 = 2.
	tan 4x =	2.
	tan−1 2 ≈ 54.7° .
	x =	54.7	≈ 13.7° .
		4

On the interval 0 ≤ x ≤ 180, x = 13.7°, 76.3°, 103.7°, and 166.3°.

PRACTICE TEST 3

47. C Complete the square to write the equation of the ellipse in standard form.

x2 + 4y2 − 2x + 16 y + 13 = 0.

(x3 − 2x) + 4( y2 + 4 y) = −13

(x2 − 2x + 1) + 4(y2 + 4y + 4) = −13 + 1 + 16.

(x − 1)2 + 4(y + 2)2 = 4.

(x − 1)2 + (y + 2)2 = 1. 4

a = 4 = 2. The length of the major axis is 2a, or 4 units.

48. D The number of possible committees is given

by 15C3 × 28C3.

12!3!15! × 25!3!28! = 455 × 3,276 = 1,490,580.

277

	49. B
		π				π
	f				= cos arctan			≈ 0.9308.
			8			8

g(0.9308) = tan(arccos 0.9308) ≈ 0.393.

50. A Don’t waste time trying to determine what function fits the graph. Simply use the graph to determine the y value when x = 5.

f (5) = 2.

f (f (5)) = f (2) = 0.

This page intentionally left blank

PRACTICE TEST 3

279

DIAGNOSE YOUR STRENGTHS AND WEAKNESSES

Check the number of each question answered correctly and “X” the number of each question answered incorrectly.

Algebra	1	2	3	7	11	14	32	43	44		Total Number Correct
9 questions
Solid Geometry	4	20	45	Total Number Correct
3 questions
Coordinate Geometry		12	17	26	33	34	47	Total Number Correct
6 questions
Trigonometry	5	16	24	29	30	35	46	Total Number Correct
7 questions
Functions	6	8	15	18	22	23	27	37	38	39		41		42	49	50		Total Number Correct
14 questions
Data Analysis,
Statistics, and
Probability	21	28	36	40	Total Number Correct
4 questions
Numbers and
Operations	9	10	13	19	25	31	48	Total Number Correct
7 questions

Number of correct answers − 14 (Number of incorrect answers) = Your raw score

___________________________ − 14 (_____________________________) = ________________

280	PART III / EIGHT PRACTICE TESTS

Compare your raw score with the approximate SAT Subject Test score below:

		SAT Subject Test
	Raw Score	Approximate Score
Excellent	43–50	770–800
Very Good	33–43	670–770
Good	27–33	620–670
Above Average	21–27	570–620
Average	11–21	500–570
Below Average	< 11	< 500

PRACTICE TEST 4

281

PRACTICE TEST 4

Treat this practice test as the actual test and complete it in one 60-minute sitting. Use the following answer sheet to fill in your multiple-choice answers. Once you have completed the practice test:

1.Check your answers using the Answer Key.

2.Review the Answers and Solutions.

3.Fill in the “Diagnose Your Strengths and Weaknesses” sheet and determine areas that require further preparation.

This page intentionally left blank

PRACTICE TEST 4

283

PRACTICE TEST 4

MATH LEVEL 2

ANSWER SHEET

Tear out this answer sheet and use it to complete the practice test. Determine the BEST answer for each question. Then, fill in the appropriate oval using a No. 2 pencil.

1.

A

B

C

D

E

21.

A

B

C

D

E

41.

A

B

C

D

E

2.

A

B

C

D

E

22.

A

B

C

D

E

42.

A

B

C

D

E

3.

A

B

C

D

E

23.

A

B

C

D

E

43.

A

B

C

D

E

4.

A

B

C

D

E

24.

A

B

C

D

E

44.

A

B

C

D

E

5.

A

B

C

D

E

25.

A

B

C

D

E

45.

A

B

C

D

E

6.

A

B

C

D

E

26.

A

B

C

D

E

46.

A

B

C

D

E

7.

A

B

C

D

E

27.

A

B

C

D

E

47.

A

B

C

D

E

8.

A

B

C

D

E

28.

A

B

C

D

E

48.

A

B

C

D

E

9.

A

B

C

D

E

29.

A

B

C

D

E

49.

A

B

C

D

E

10.

A

B

C

D

E

30.

A

B

C

D

E

50.

A

B

C

D

E

11.

A

B

C

D

E

31.

A

B

C

D

E

12.

A

B

C

D

E

32.

A

B

C

D

E

13.

A

B

C

D

E

33.

A

B

C

D

E

14.

A

B

C

D

E

34.

A

B

C

D

E

15.

A

B

C

D

E

35.

A

B

C

D

E

16.

A

B

C

D

E

36.

A

B

C

D

E

17.

A

B

C

D

E

37.

A

B

C

D

E

18.

A

B

C

D

E

38.

A

B

C

D

E

19.

A

B

C

D

E

39.

A

B

C

D

E

20.

A

B

C

D

E

40.

A

B

C

D

E