SAT 8

316

28.The graph in Figure 3 represents a portion of the graph of which of the following functions?

(A)f (x) = 2 cos (x + π)

(B)f (x) = sin 2x

(C)f (x) = −2 sin x

(E) f (x) = 2 sin x

29.If x + 3 is a factor of x5 + 2x4 − kx3 + 3x2, then k =

(A)−3

(B)−2

(C)−1

(D)2

(E)3

30.By the Rational Root Test, how many possible rational roots does f (x) = 4x3 − x 2 + 10x − 6 have?

(A)3

(B)8

(C)14

(D)16

(E)24

31.What are the solutions to the system: xy + 6 = 0 and x − y = 5?

(A){(−3, 2), (−2, 3)}

(B){(−2, −3), (−3, −2)}

(C){(1, −6), (−6, 1)}

(D){(6, 1), (−1, −6)}

(E){(2, −3), (3, −2)}

32.$1,000 is invested at a rate of r % compounded quarterly. The value of the investment in t years can be

If the investment doubles in 12 years, what is the value of r?

(A)4.7

(B)5.8

(C)6.1

(D)10.2

(E)11.6

PART III / EIGHT PRACTICE TESTS

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y

2

x

5

–2

Figure 3

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(D)r r r

(E)v − w = u

34.The product of 212,121,212,121 and 33,333 contains how many digits?

(A)14

(B)15

(C)16

(D)17

(E)18v + w = u

35. What is the maximum value of the function f (x) = 1 x

2

(D)3

(E)infinity

36.In ABC in Figure 5, BC = 6.3 cm, x = 49°, and y = 22°. What is the length of side AB?

(A)2.4

(B)2.9

(C)3.1

(D)4.8

(E)12.7

37.The rectangle in Figure 6 is rotated about side WZ. What is the volume of the resulting solid?

(A)432

(B)108π

(C)432π

(D)72π

(E)330

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v

u

w

Figure 4

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ANSWERS AND SOLUTIONS

1. B First, cube both sides of the equation.

38x3 − 1 = 2.

8x3 − 1 = 23 = 8. 8x3 = 9.

x3 = 98 .

1

=9 3 ≈

x 1.04.

8

2.C Recall that factorial is represented by the “!” symbol.

12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

12! = 10 × 11 × 12 4!9! 1 × 2 × 3 × 4

=55.

3.A Parallel lines have the same slopes. The given line is written in slope-intercept form y = mx + b

3 . Answer A is the only

4

4. D

f (14, 3) = 12 (14) + 3 = 7 + 3 = 10.

Answer D is the only choice that also results in 10.

f (8, 6) = 12 (8) + 6 = 4 + 6 = 10

5. E

f (2) = 23 + 1 = 9.

f (f (2)) = f (9) = 93 + 1 = 730.

6. C Start by multiplying both sides by the LCD:

1 = h(x + 4) + k(x − 2).

Solving for k may not be immediately obvious. One way to solve for k is to substitute −4 for x, so the h term cancels out.

1 = h(−4 + 4) + k(−4 − 2).

1 = k(−6).

k = − 61 .

322

7. D Cosecant is the reciprocal function of the sine function. If csc θ = 3, then sin θ = 13 .

sin θ csc θ = 13 (3) = 1.

8. A

9x4 − 4 = 7.

9x4 = 11.

x4 = 119 .

1

x= ± 11 4 = ± 1.05.

9

9.B The boys’ team scored a total of 5(54) or 270 points in their first 5 games. The girls’ team scored a total of 6(59) or 354 points in their first 6 games. The average for all 11 games is:

10. D Because f (x) equals the square root of an expression, it must be a positive value and y ≥ 0 is part of the range. There is an upper limit on y, however. The maximum y-value occurs when x = 0.

f (0) = 16 − 02 = 4.

The range is between 0 and 4, inclusive.

11. A Because the two events are independent, the probability that Michelle does not hit the target and Mike does hit the target is the product of the two probabilities. The probability of Michelle missing the

12. B You know the length of the side adjacent to θ and the hypotenuse of the triangle, so use arccosine to solve for the angle measure.

PART III / EIGHT PRACTICE TESTS

13. A A logarithm is an exponent. Determine to

14. D Complex roots occur in conjugate pairs. If 5 + i is a root of the polynomial, then 5 − i is also a root. Use the three roots to determine the factors of the polynomial. Then multiply to determine the polynomial.

x[x − (5 + i)][x − (5 − i)] =

x[(x − 5) − i)][(5 − i) + i)] =

x[(x − 5)2 − i2 )] =

x(x2 − 10x + 25 + 1) =

x(x2 − 10x + 26) =

x3 − 10x2 + 26x.

15.D Recall that in the polar coordinate system the coordinates of a point are in the form (r, θ) where r is the directed distance from the pole and θ is the

directed angle. Because the point shown has polar coordinates (4, 40°), you don’t need to do any work to determine that r = 4.

16.C One way to solve this is to think of a right triangle. If cos θ = 23 , sin θ =

a2 + 22 = 32.

a2 = 9 − 4 = 5.

a = 5.

sin θ = 35 .

Recall that the double angle formula for sine is:

sin 2θ = 2 sin θ cosθ.

17. B The inverse of the logarithmic function is the exponential function. The inverse of f (x) = log8 x is, therefore, f −1(x) = 8x.

PRACTICE TEST 5

18. E The graph of y = tan x has asymptotes when cos x = 0. That means that asymptotes occur when

x equals all odd multiples of π . 2

The graph of f (x) = tan 2x is similar to the graph of

y = tan x with a period of π . Instead of an asymptote 2

π

22 , x = π4 .

(Graph the function on your calculator to confirm there is an asymptote at x = π4 . )

19. A Because 83 = 512, both sides of the equation can be written in base 8.

82− x = 512x+1.

82− x = 83( x+1).

Now, set the exponents equal to each other and solve for x.

2 − x = 3x + 3. −1 = 4x.

x= − 14 .

20.D Because the x2 and y2 terms have different signs, the graph is a hyperbola.

21.E f / g(−1) is the quotient of f (−1) and g(−1).

22. B Because there are 10 terms in the data set, the median is the average of the 5th and 6th terms. The median is, therefore, 3. The mode is 4, and the mean

is the sum of the data divided by 10 : 1030 = 3.

3 ≤ 3 ≤ 4.

mean ≤ median ≤ mode

23. A The cosine function is an even function, so it is symmetric with respect to the y-axis. To verify the symmetry, graph the equation using your graphing calculator.

323

24. C First, find the common difference between consecutive terms in the sequence. 232 − 7 = 92 .

The 15th term of the sequence is therefore:

an = a1 + (n − 1)d.

a15 = 7 + (15 − 1) 92 .

a15 = 7 + (14) 92 = 70.

Use the equation for the sum of a finite arithmetic sequence, Sn = n2 (a1 + an ), to determine the sum of the first 15 terms.

S15 = 152 (7 + 70).

S15 = 152 (77) = 577.5.

25.B

A = 2,800(1 − 0.22)2.5.

A = 2,800(0.78)2.5.

A = 1,505.

26.A

2x − 4 < 1.

−1 < 2x − 4 < 1.

3 < 2x < 5.

32 < x < 52 .

27. D The slope of the line with x-intercept (−3, 0) and y-intercept (0, −5) is:

The y-intercept is given as −5. In slope-intercept form, y = mx + b, the equation is therefore:

y = − 53 x − 5. 5x + 3y = −15.

28. C The graph shows the sine curve, y = sin x, reflected over the x-axis with an amplitude of 2. The function corresponding with the graph is, therefore, f (x) = −2 sin x.