SAT 8

244

29. What is the radius of the circle x2 + y2 − 14x + 4y + 44 = 0?

(A) 2.2

(B) 3

(C) 6.6

(D) 9

(E) 44

30. Which of the following equations has the graph shown in Figure 3?

(A) y = cos 2x

(B) y = 2cos x

(C) y = 2cos 2x

(D)	y = 2 cos x
	2
(E)	y = cos x
	2

31. What is the range of the piecewise function

	x2			x	> 3
		2			?
f (x) =
	−		x + 11	x	≤ 3
		3
(A)	All real numbers
(B)	y ≥ 0
(C)	−3 ≤ y ≤ 3
(D)	y ≤ 3
(E)	y ≥ 9

32. If vector v = (−3, 7) and vector u = (2, −1), then vector u − v =

(A) (−5, 8)

(B) (−1, 6)

(C) (−8, 5)

(D) (−6, −7)

(E) (5, −8)

33. A cube has edges of length 3. If P and Q are points on its surface, what is the maximum straight-line distance from P to Q?

(A) 3 2

(B) 3 3

(C) 6

(D) 3 5

(E) 9

34. If a is any positive integer, then which of the following is not a true statement?

(A) 2a + 1 is always an odd integer.

(B) a is always a real number.

(C) −a is always an imaginary number.

(D) a3 is always an odd integer.

(E) The product of a and 1 always equals 1.

PART III / EIGHT PRACTICE TESTS

USE THIS SPACE AS SCRATCH PAPER

y

	4
	2
	π			x
–5		5	10	15 5(π)
	–2

–4

Figure 3

GO ON TO THE NEXT PAGE

a

PRACTICE TEST 2
35. If		n	=		1	+	1	, then n =
	x2	− 36			x − 6
							x + 6

(A)x

(B)2x

(C)2(x + 6)

(D)2(x − 6)

(E)2

36.If f (x) = x3 and h(x) is obtained by shifting f (x) down 4 units and right 2 units, then h(x) =

(A)(x − 2)3 + 4

(B)(x + 2)3 − 4

(C)(x − 2)3 − 4

(D)(x − 4)3 − 2

(E)4 − (x − 2)3

37.If the lines y1 = (n + 2)x + 10 and y2 = (n − 4)x + 2 are perpendicular, then n could equal which of the following?

(A)3.45

(B)3.83

(C)1

(D)0.5

(E)−6

38.The graph of y = −x4 + 12x − 18

(A)intersects the x-axis at exactly one point.

(B)intersects the x-axis at exactly two points.

(C)intersects the x-axis at exactly three points.

(D)intersects the x-axis at exactly four points.

(E)does not intersect the x-axis.

39.Which of the following is equivalent to the equation x2 + 5x + y2 − 1 = 0 in polar form?

(A)r2 + 5r cos θ − 1 = 0

(B)r = 5r cos θ + 1

(C)r2 = 5r cos θ

(D)r2 + 5r sin θ − 1 = 0

(E)r2 = −5r sin θ

40.If A is a 2 × 3 matrix, and B is a 3 × 4 matrix, then the product of A times 3B is a matrix of which of the following dimensions?

(A)6 × 4

(B)2 × 3

(C)2 × 4

(D)3 × 3

(E)2 × 12

41.(sec θ + tan θ)(sec θ − tan θ) =

(A)sec2 θ

(B)1

(C)tan2 θ − sec2 θ

(D)−1

(E)2sec2 θ − 1

245

USE THIS SPACE AS SCRATCH PAPER

GO ON TO THE NEXT PAGE

246

42.The linear regression model C = 13.2m + 20.5 relates the calories burned using a new exercise machine (C) to the number of minutes a person uses the machine (m). Which of the following statements about this model must be true?

I.When a person spends 22 minutes using the machine, the predicted number of calories burned is approximately 311.

II.There is relatively no correlation between C and m.

III.A person burns approximately 13.2 calories each minute he or she uses the machine.

(A)I only

(B)III only

(C)I and II only

(D)I and III only

(E)I, II, and III

43.If (x, y) is a point on the graph of f (x), then which of the following is a point on the graph of f −1(x)?

(A) (−x, −y)

(B) (−x, y)

(C) (y, x)

(D) (x, −y)

(E) (−y, −x)

44. The graph of y = −2x + 7 can be expressed as a set of parametric equations. If x = 1 − t, and y = f (t), then what does f (t) equal?

(A) 2t − 7

(B) −2t + 5

(C) 2t + 9

(D) − 1 t − 7 2

(E) 2t + 5

45. If x0 = 5 and xn+1 = xn xn + 4 , then x4 =

(A)23.4

(B)65.4

(C)544.6

(D)12,709.8

(E)12,756.4

						1
	46. If f (x) = 6x, then f log7							=
						343
	(A)	−3
	(B)	−18
	(C)	1
		216
	(D)	1
		1,296
	(E)	216

PART III / EIGHT PRACTICE TESTS

USE THIS SPACE AS SCRATCH PAPER

GO ON TO THE NEXT PAGE

PRACTICE TEST 2

47.How long is the base of an isosceles triangle if the

other two sides measure 20 cm and each base angle measures 28°?

(A)12.7

(B)28.3

(C)30.6

(D)35.3

(E)88.6

48.A committee of 4 people is to be selected from a group of 8 women and 4 men. Assuming the selection is made randomly, what is the probability that the committee consists of 2 women and 2 men?

56

(A)

165

6

(B)

495

28

(C)

495

28

(D)

165

34

(E)

495

49.In terms of θ, what is the area of the triangle in Figure 4?

(A)2 sin2 θ

(B)4 sin θ cos θ

(C)1 tan θ

2

(D)2 sin θ cos θ

(E)1 θ

2

50.If f is a function whose graph is the line segment shown in Figure 5, then f −1(6.2) =

(A)0.44

(B)0.76

(C)0.89

(D)1.11

(E)1.77

247

USE THIS SPACE AS SCRATCH PAPER

a	2

θ

b

Figure 4

y

(2,14)

x

Figure 5

S T O P

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY.

DO NOT TURN TO ANY OTHER TEST IN THIS BOOK.

This page intentionally left blank

PRACTICE TEST 2

249

ANSWER KEY

1. B	11. C	21. B	31. E	41. B
2. C	12. E	22. A	32. E	42. D
3. A	13. D	23. D	33. B	43. C
4. D	14. D	24. C	34. D	44. E
5. D	15. E	25. B	35. B	45. E
6. E	16. C	26. C	36. C	46. C
7. D	17. A	27. D	37. B	47. D
8. C	18. D	28. B	38. E	48. A
9. A	19. C	29. B	39. A	49. D
10. B	20. E	30. D	40. C	50. C

ANSWERS AND SOLUTIONS

1. B Take the fifth root of both sides of the equation to solve for x:

x5			= 84.
1						1
(x5 )			= (84 )					.
	5						5
			4
x = 8						= 5.27803			≈ 5.28.
				5

2. C You will need to use the ex function on your calculator for this problem.

g(4) = 4−2 = 161 = 0.0625. f ( g(4)) = f (0.0625) = e2(0.0625).

=1.133148 ≈ 1.13.

3.A Transform the equation and isolate n :

m

m		=	1	.
			5
(3m − n)
3m − n = 5m.
−2m = n.
	−2 =		n		.
			m

4. D The LCD of the three fractions is x3.

1x + x22 + x33 =

x2 + 2x + 3 = x3 x3 x3

(x2 + 2x + 3) . x3

5. D The complex conjugate of 6 + i is 6 − i. To simplify the fraction, multiply the numerator and the denominator by the complex conjugate. Recall that i2 = −1.

1	=	1		×	6	− i
6 + i		6 + i			6	− i
	=		6 − i						=	6 − i
		(6 + i)(6 − i)								36 − i2
	=	6 − i				=	6 − i				.
		36 − (−1)
									37

250

6. E Factor by grouping terms:

x3 − 2x2 + 2x − 4 = (x3 − 2x2 ) + (2x − 4).

Now, distribute common factors:

(x3 − 2x2 ) + (2x − 4) = x2 (x − 2) + 2(x − 2)

=(x − 2)(x2 + 2).

7.D First, determine the product of the radicands. Then, take the cube root of each term.

3 20x2 y7 3 50x4 y2 =

3 [(20)(50)(x2 x4 )( y7 y2 )] =

3 1000x6 y9 =

1 6 9

10003 x3 y3 =

10x2 y3.

8.C This problem involves a rational expression. Find the critical points where the numerator and denominator equal zero. Then, determine which intervals satisfy the inequality.

x + 2 ≥ 0 x − 5

Critical points: x = −2 and x = 5.

Note that x ≠ 5, however, because the denominator cannot equal zero.

2

Test when x = 0. −5 ≥ 0 is not a true statement, so the

interval between −2 and 5 is not part of the solution. The solution is the interval x ≤ −2 or x > 5.

9. A “N varies jointly with the square of x and with y” is equivalent to “N is jointly proportional to the square of x and with y.” If N was inversely proportional to the square of x and with y, the x and y terms would be in the denominator.

The square of x translates to x2, not x , so N = kx2y is the correct answer.

PART III / EIGHT PRACTICE TESTS

10. B Complete the square to write the equation of the parabola in standard form.

f (x) = x2	− 6x + 5.
y = x2	− 6x + 5.
y − 5 = x2	− 6x.
y − 5 + 9 = x2	− 6x + 9.

y + 4 = (x − 3)2.

The vertex is (3, −4). You can also verify this answer by graphing the function on your graphing calculator.

11. C Because cot θ = 1.33, tan θ =			1	= 0.75.
		1.33
tan−1(0.75)	≈ 36.87°.
sin 36.87°	≈ 0.6.

Alternatively, recognize that for cot θ to equal 1.33, the triangle has sides measuring 6–8–10. cot θ = 86 = 1.33. sin θ = 106 = 0.6.

12. E Set the polynomial equal to zero and factor. 8x3 − 2x2 − 3x = 0.

x(8x2 − 2x − 3) = 0. x(2x + 1)(4x − 3) = 0. x = 0 or − 12 or x = 34 .

13. D Because the events are independent, find the product of their probabilities.

The probability that it will NOT snow the day after tomorrow is:

1 − 15 = 45 .

The probability that it will snow tomorrow but not the day after tomorrow is:

23 × 45 = 158 .

PRACTICE TEST 2

14. D

a b c = a(c − b).

2 1 x = 2(x − 1) = 10. x − 1 = 5.

x = 6.

3

15. E Raise each side of the equation to the 2 power to solve for x:

3

(x − 1)2 = 25.

		3
	2		2				3
					= 252.
(x − 1)	3
		3			3
x − 1 =	25					= (52 )				= 53.
				2					2

x= 125 + 1 = 126.

16.C The double angle formula for cosine is: cos2θ = cos2 θ − sin2 θ.

Because cos 2θ = 34 , cos2 θ − sin2 θ also equals 34

1	=	1	= 4 .
cos2 θ − sin2 θ		3	3
		4

17. A Recall that π radians = 180°. To convert the

given angle to degrees, multiply it by								180	.
								π
	5π	×	180		= −	5(180)	= −75°
	12			π		12

18. D Factor the numerator and denominator. Then, simplify the expression and evaluate it when x = − 32 .

f (x) =

4x2 − 9

=

(2x − 3)(2x + 3)

= 2x − 3.

2x + 3

2x + 3

When x = −

3

−

3

− 3 = −3

− 3

= −6.

,

2

2

2

19. C Recall that complex zeroes occur in conjugate pairs. If i is a zero of the polynomial p(x), then −i is also a zero. One factor of the polynomial is (x − i) (x + i).

(x − i)(x + i) = x2 + ix − ix − i2

=x2 − i2 = x2 − (−1) = x2 + 1.

251

20. E Recognize that 2 − x = −(x − 2).

f (x) = (x − 2)(2 − x)−1 = − x − 2 − 1. x − 2

The function f (x) = (x − 2)(2 − x)−1 equals −1 for all values of x except x = 2.

f (1) = f (−1) =	f (4) = f (0) =		1
		f
			2

= f (−2) = −1. Statements I, II, and III are true. 21. B

f (−3) = 5(−3) − 1 = −16.

For g(f(−3)) to equal 18, g(−16) must equal 18. There are many possible functions that would have an output of 18 when −16 is the input. Of the possible answers, Answer B is the only one that works.

g(−16) = −(−16) + 2 = 16 + 2 = 18.

22. A One way to determine the y-intercept is to set x equal to zero and solve for y.

	y2		−		x2	= 1.
	9			16
	y2		−		0	= 1.
	9			16
					y2	= 1.
				9
					y2	= 9.

y = ±3.

The point (0, −3) is one possible answer.

23.D

Distance = (x2 − x1)2 + ( y2 − y1)2 + (z2 − z1)2

=(3 − 0)2 + (0 − 2)2 + (0 − 5)2

=32 + (−2)2 + (−5)2

=9 + 4 + 25

=38 ≈ 6.16.

252

24. C Note that the car is decreasing in value. The value of the car can be calculated using the equation:

A= 24,500(1 − 0.07)5.

A= 24,500(0.93)5.

A≈ $17,044.

25.B The inverse of the exponential function f (x) = 4x is the logarithmic function f −1(x) = log4 x.

f−1( p)f −1(q) = log4 p log4 q.

26.C If logb 6 = 36, then b36 = 6. Take the 36th root of each side of the equation.

1		1
(b36 )		= 6		.
	36		36
		1
	b = 6				= 1.05.
			36

27.D First, find the angle whose cosine is 0.75. cos−1(0.75) = 41.4096°.

Then, determine the tangent of the angle. tan 41.4096° ≈ 0.88.

28. B Find the difference between the two fractions.

5	−	3	=	10 −	9	=	1 .
6		4		12			12

121 of the class is passing with a grade of C− or better but not a grade of B− or better.

29. B Complete the square to get the standard form of the equation of the circle.

x2 + y2 − 14x + 14 y + 44 = 0.

(x2 − 14x) + ( y2 + 4 y)	= −44.
(x2 − 14x + 49) + ( y2 + 4 y + 4)	= −44 + 49 + 4.
(x2 − 7)2 + ( y + 2)2	= 9.
Its radius is: 9 = 3 units.

30. D The figure is the graph of the cosine function with amplitude 2 and period 4π. Therefore, the equation is,

y = 2 cos 2π x = 2 cos	x .
4π	2

PART III / EIGHT PRACTICE TESTS

31.E The graph is linear when x ≤ 3. Because the function only looks at the ray formed when x ≤ 3, the ray starts at (3, 9) and extends upward in the negative direction of x. When x > 3, the graph is a parabola whose lowest point is (3, 9). All y values greater than or equal to 9 satisfy the range.

32.E Vector v = (−3, 7) and vector u = (2, −1).

u − v = (2 − (−3),−1 − 7).

u − v = (5,−8).

33. B The maximum straight line distance is the distance between opposite vertices of the cube. Recall that distance between opposite vertices of any rectangular prism is:

Distance = l2 + w2 + h2

=32 + 32 + 32

=27 = 3 3.

34.D The statement “a3 is always an odd integer” is not always true. One counterexample is 23 = 8.

35.B Multiply both sides of the equation by the LCD, (x + 6)(x − 6), to get:

n= (x + 6) + (x − 6).

n= 2x.

36.C Shifting f (x) down 4 units results in: h(x) = f (x) − 4.

h(x) = x3 − 4.

Then, shifting the function right 2 units results in:

h(x) = (x − 2)3 − 4.

37. B The slopes of perpendicular lines are opposite reciprocals. If y1 = (n + 2)x + 10 and y2 = (n − 4)x + 2 are perpendicular, then:

(n + 2)(n − 4) = −1.

n2 − 2n − 8 = −1.

n2 − 2n − 7 = 0.

Use the Quadratic Formula to determine that n = 3.828 and n = −1.828 are solutions for n. 3.8 is the correct answer choice.

PRACTICE TEST 2

38.E Graph the equation y = −x4 + 12x − 18 on your calculator to observe that it never intersects the x-axis. Answer E is the best answer choice.

39.A In the polar coordinate system, x2 + y2 = r2 and x = r cos θ. Substituting these values into the given equation results in:

x2 + 5x + y2 − 1 = 0.

r2 − 5r cosθ − 1 = 0.

40.C The product of a 2 × 3 matrix and a 3 × 4 matrix is a 2 × 4 matrix. Recall that the number of columns in the first matrix must be equal to the number of rows in the second matrix for matrix multiplication to be defined.

41.B

(secθ + tan θ)(secθ − tan θ) = sec2 θ − tan2 θ.

Recall that one of the Pythagorean Identities is 1 + tan2 θ = sec2 θ. Substituting 1 + tan2 θ for sec2 θ results in:

1 + tan2 θ − tan2 θ = 1.

42.D Based on the model C = 13.2m + 20.5, there is a positive correlation between the calories burned and the number of minutes a person uses a machine. (If datapoints were graphed, they would cluster about a line with a slope of approximately 13.2.) Statements I and III are true based on the linear model, but II is not true. Answer D is the correct answer choice.

43.C The inverse function f−1(x) is the reflection of f (x) over the line y = x. If (x, y) is a point on the graph of f (x), then (y, x) is point on the graph of the inverse.

44.E Substitute x = 1 − t into the equation for y to solve for y in terms of t.

y = f (t) = −2(1 − t) + 7.

y= −2 + 2t + 7.

y= 2t + 5.

253

45. E The function xn+1 = xn (xn + 4) is recursive. Because you are given the first term of the sequence, you can define the other terms using it.

x0 = 5.

x1 = 5 9 = 15.

x2 = 15 19 = 65.38.

x3 = 65.83 69.38 = 544.62.

x4 = 544.62 548.62 ≈ 12,756.4.

46. C First, evaluate log7 3431 :

	log7	1	= −3 since 7−3 =	1	.
		343		343
	If f (x) = 6x , then f (−3) = 6−3 =					1	.
						216

47. D If each base angle measures 28°, the vertex angle of the isosceles triangle measures 180 − 2(28) = 124°. Let x = the length of the base. By the Law of Sines:

sin 28° = sin124° . 20 x

xsin 28° = 20sin 124°.

x = 35.3 cm.

48. A First, determine the number of possible ways to choose 4 people from a group of 12.

12 C4 = 4!8!12! = 495

Then, determine how many possible ways to choose 2 women from 8 and 2 men from 4.

8 C2 = 2!6!8! = 28.

4 C2 = 2!2!4! = 6.

The probability that the committee consists of 2 women and 2 men is, therefore,

28(6) = 168 = 56 . 495 495 165