Math Test-3 SCORE 95 Percent

1.      The graph of the exponential function f with base b approaches, but does not touch, the __________-axis. This axis, whose equation is __________, is a __________ asymptote.

A. x; y = 0; horizontal

B. x; y = 1; vertical

C. -x; y = 0; horizontal

D. x; y = -1; vertical

 

 

2.      Evaluate the following expression without using a calculator.

8log8 19

A. 17

B. 38

C. 24

D. 19

 

3.      Write the following equation in its equivalent exponential form.
 
log6 216 = y

A. 6y = 216

B. 6x = 216

C. 6logy = 224

D. 6xy = 232

 

 

4.      Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.

A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t

B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t

C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t

D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln

 

 

5.      The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds?

A. 10 grams after 10 seconds; 6 grams after 20 seconds

B. 12 grams after 10 seconds; 7 grams after 20 seconds

C. 4 grams after 10 seconds; 1 gram after 20 seconds

D. 8 grams after 10 seconds; 4 grams after 20 seconds

 

 

6.      Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.

3 ln x – 1/3 ln y

A. ln (x / y1/2)

B. lnx (x6 / y1/3)

C. ln (x3 / y1/3)

D. ln (x-3 / y1/4)

 

 

7.      Write the following equation in its equivalent exponential form.
 
5 = logb 32

A. b5 = 32

B. y5 = 32

C. Blog5 = 32

D. Logb = 32

 

 

8.      Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

ex+1 = 1/e

A. {-3}

B. {-2}

C. {4}

D. {12}

 

 

9.      Use properties of logarithms to expand the following logarithmic expression as much as possible.
 
Logb (√xy3 / z3)

A. 1/2 logb x - 6 logb y + 3 logb z

B. 1/2 logb x - 9 logb y - 3 logb z

C. 1/2 logb x + 3 logb y + 6 logb z

D. 1/2 logb x + 3 logb y - 3 logb z

 

 

 

10.  Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.

log2 96 – log2 3

A. 5

B. 7

C. 12

D. 4

 

 

11.  Consider the model for exponential growth or decay given by A = A0ekt. If k __________, the function models the amount, or size, of a growing entity. If k __________, the function models the amount, or size, of a decaying entity.

A. > 0; < 0

B. = 0; ≠ 0

C. ≥ 0; < 0

D. < 0; ≤ 0

 

 

12.  Approximate the following using a calculator; round your answer to three decimal places.

3√5

A. .765

B. 14297

C. 11.494

D. 11.665

 

 

13.  Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

31-x = 1/27

A. {2}

B. {-7}

C. {4}

D. {3}

 

14.  Find the domain of following logarithmic function.

f(x) = ln (x - 2)2

A. (∞, 2) (-2, -∞)

B. (-∞, 2) (2, ∞)

C. (-∞, 1) (3, ∞)

D. (2, -∞) (2, ∞)

 

15.  Write the following equation in its equivalent logarithmic form.

2-4 = 1/16

A. Log4 1/16 = 64

B. Log2 1/24 = -4

C. Log2 1/16 = -4

D. Log4 1/16 = 54

 

 

16.  Find the domain of following logarithmic function.

f(x) = log5 (x + 4)

A. (-4, ∞)

B. (-5, -∞)

C. (7, -∞)

D. (-9, ∞)

 

 

17.  Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.

ex = 5.7

A. {ln 5.7}; ≈1.74

B. {ln 8.7}; ≈3.74

C. {ln 6.9}; ≈2.49

D. {ln 8.9}; ≈3.97

 

 

18.  Approximate the following using a calculator; round your answer to three decimal places.

e-0.95

A. .483

B. 1.287

C. .597

D. .387

 

 

19.  You have $10,000 to invest. One bank pays 5% interest compounded quarterly and a second bank pays 4.5% interest compounded monthly. Use the formula for compound interest to write a function for the balance in each bank at any time t.

A. A = 20,000(1 + (0.06/4))4t; A = 10,000(1 + (0.044/14))12t

B. A = 15,000(1 + (0.07/4))4t; A = 10,000(1 + (0.025/12))12t

C. A = 10,000(1 + (0.05/4))4t; A = 10,000(1 + (0.045/12))12t

D. A = 25,000(1 + (0.05/4))4t; A = 10,000(1 + (0.032/14))12t

 

 

20.  Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.

32x + 3x - 2 = 0

A. {1}

B. {-2}

C. {5}

D. {0}

 

21.  Solve each equation by the substitution method.

x + y = 1
x2 + xy – y2 = -5

 

A. {(4, -3), (-1, 2)}

B. {(2, -3), (-1, 6)}

C. {(-4, -3), (-1, 3)}

D. {(2, -3), (-1, -2)}

 

 

22.  Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, 6), (1, 4), (2, 9)

A. y = 2x2 - x + 3

B. y = 2x2 + x2 + 9

C. y = 3x2 - x - 4

D. y = 2x2 + 2x + 4

 

23.  Solve the following system.

x + y + z = 6
3x + 4y - 7z = 1
2x - y + 3z = 5

 

A. {(1, 3, 2)}

B. {(1, 4, 5)}

C. {(1, 2, 1)}

D. {(1, 5, 7)}

 

24.  On your next vacation, you will divide lodging between large resorts and small inns. Let x represent the number of nights spent in large resorts. Let y represent the number of nights spent in small inns.

Write a system of inequalities that models the following conditions:

You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average $200 per night and small inns average $100 per night. Your budget permits no more than $700 for lodging.

A.

y ≥ 1
x + y ≥ 5
x ≥ 1
300x + 200y ≤ 700

B.

y ≥ 0
x + y ≥ 3
x ≥ 0
200x + 200y ≤ 700

C.

y ≥ 1
x + y ≥ 4
x ≥ 2
500x + 100y ≤ 700

D.

y ≥ 0
x + y ≥ 5
x ≥ 1
200x + 100y ≤ 700

 

25.  Solve each equation by either substitution or addition method.

x2 + 4y2 = 20
x + 2y = 6

 

A. {(5, 2), (-4, 1)}

B. {(4, 2), (3, 1)}

C. {(2, 2), (4, 1)}

D. {(6, 2), (7, 1)}

 

 

26.  Solve the following system by the addition method.

{4x + 3y = 15
{2x – 5y = 1

A. {(4, 0)}

B. {(2, 1)}

C. {(6, 1)}

D. {(3, 1)}

 

 

27.  Let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers.

The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.

A. x + y = 7; x - y = -1; 3 and 4

B. x + y = 7; x - y = -1; 5 and 6

C. x + y = 7; x - y = -1; 3 and 6

D. x + y = 7; x - y = -1; 2 and 3

 

28.  Solve the following system.

x = y + 4
3x + 7y = -18

 

A. {(2, -1)}

B. {(1, 4)}

C. {(2, -5)}

D. {(1, -3)}

 

 

29.  Write the partial fraction decomposition for the following rational expression.

6x - 11/(x - 1)2

A. 6/x - 1 - 5/(x - 1)2

B. 5/x - 1 - 4/(x - 1)2

C. 2/x - 1 - 7/(x - 1)

D. 4/x - 1 - 3/(x - 1)

 

 

 

30.  Perform the long division and write the partial fraction decomposition of the remainder term.

x4 – x2 + 2/x3 - x2

A. x + 3 - 2/x - 1/x2 + 4x - 1

B. 2x + 1 - 2/x - 2/x + 2/x + 1

C. 2x + 1 - 2/x2 - 2/x + 5/x - 1

D. x + 1 - 2/x - 2/x2 + 2/x – 1

 

 

31.  Write the partial fraction decomposition for the following rational expression.

4/2x2 - 5x – 3

A. 4/6(x - 2) - 8/7(4x + 1)

B. 4/7(x - 3) - 8/7(2x + 1)

C. 4/7(x - 2) - 8/7(3x + 1)

D. 4/6(x - 2) - 8/7(3x + 1)

 

 

32.  Write the form of the partial fraction decomposition of the rational expression.
7x - 4/x2 - x - 12

A. 24/7(x - 2) + 26/7(x + 5)

B. 14/7(x - 3) + 20/7(x2 + 3)

C. 24/7(x - 4) + 25/7(x + 3)

D. 22/8(x - 2) + 25/6(x + 4)

 

 

33.  Write the partial fraction decomposition for the following rational expression.
 
x + 4/x2(x + 4)

A. 1/3x + 1/x2 - x + 5/4(x2 + 4)

B. 1/5x + 1/x2 - x + 4/4(x2 + 6)

C. 1/4x + 1/x2 - x + 4/4(x2 + 4)

D. 1/3x + 1/x2 - x + 3/4(x2 + 5)

 

 

 

34.  Perform the long division and write the partial fraction decomposition of the remainder term.

x5 + 2/x2 - 1

A. x2 + x - 1/2(x + 1) + 4/2(x - 1)

B. x3 + x - 1/2(x + 1) + 3/2(x - 1)

C. x3 + x - 1/6(x - 2) + 3/2(x + 1)

D. x2 + x - 1/2(x + 1) + 4/2(x - 1)

 

35.  Solve the following system.

2x + 4y + 3z = 2
x + 2y - z = 0
4x + y - z = 6

 

A. {(-3, 2, 6)}

B. {(4, 8, -3)}

C. {(3, 1, 5)}

D. {(1, 4, -1)}

 

 

36.  Write the form of the partial fraction decomposition of the rational expression.
5x2 - 6x + 7/(x - 1)(x2 + 1)

A. A/x - 2 + Bx2 + C/x2 + 3

B. A/x - 4 + Bx + C/x2 + 1

C. A/x - 3 + Bx + C/x2 + 1

D. A/x - 1 + Bx + C/x2 + 1

 

37.  Solve each equation by the substitution method.

x2 - 4y2 = -7
3x2 + y2 = 31

 

A. {(2, 2), (3, -2), (-1, 2), (-4, -2)}

B. {(7, 2), (3, -2), (-4, 2), (-3, -1)}

C. {(4, 2), (3, -2), (-5, 2), (-2, -2)}

D. {(3, 2), (3, -2), (-3, 2), (-3, -2)}

 

 

 

38.  Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, -4), (1, -2), (2, 5)

A. y = 2x2 + x - 6

B. y = 2x2 + 2x - 4

C. y = 2x2 + 2x + 3

D. y = 2x2 + x – 5

 

 

 

39.  Write the partial fraction decomposition for the following rational expression.

x2 – 6x + 3/(x – 2)3

A. 1/x – 4 – 2/(x – 2)2 – 6/(x – 2)

B. 1/x – 2 – 4/(x – 2)2 – 5/(x – 1)3

C. 1/x – 3 – 2/(x – 3)2 – 5/(x – 2)

D. 1/x – 2 – 2/(x – 2)2 – 5/(x – 2)3

 

40.  Solve the following system.

3(2x+y) + 5z = -1
2(x - 3y + 4z) = -9
4(1 + x) = -3(z - 3y)

 

A. {(1, 1/3, 0)}

B. {(1/4, 1/3, -2)}

C. {(1/3, 1/5, -1)}

D. {(1/2, 1/3, -1)}