Q1. Solve the equation 3^x = 81.

a. {27}

b. {5}

c. {4}

d. {3}

Q2. Write log ₂ (⁷m ³n)/k² as the sum and/or difference of logarithms. Express powers as factors.

a. (1/7) log ₂ m + (1/3) log ₂ n - 2log ₂ k

b. (1/7) log ₂ m (1/3) log ₂ n 2log ₂ k

c. 7log ₂ m + 3log ₂ n - 2log ₂ k

d. (7/2) log ₂ m + (3/2) log ₂ n - (2/2) log ₂ k

Q3. In 1990, the population of a country was estimated at 4 million. For any subsequent year the population, P(t) (in millions), can be modeled by the equation P(t) = 240/(5 + 54.99e^-0.0208t), where t is the number of years since 1990. Estimate the year when the population will be 21 million.

a. approximately the year 2093

b. approximately the year 2041

c. approximately the year 2088

d. approximately the year 2016

Q4. Find the domain of the composite function f g if f(x) = 28/x and g(x) = -8/(x - 7).

a. {x | x 7}

b. {x | x 0, x 7, x 4}

c. {x | x 7, x 0}

d. {x | x is any real number}

Q5. Find the inverse of the function and state its domain and range.

{(-3, 4), (-1, 5), (0, 2), (2, 6), (5, 7)}

a. {(3, 4), (1, 5), (0, 2), (-2, 6), (-5, 7)}; D = {3, 1, 0, -2, -5}; R = {2, 4, 5, 6, 7}

b. {(-3, -4), (-1, -5), (0, -2), (2, -6), (5, -7)}; D = {-3, -1, 0, 2, 5}; R = {-7, -6, -5, -4, -2}

c. {(4, -3), (5, -1), (2, 0), (6, 2), (7, 5)} D = {2, 4, 5, 6, 7}; R = {-3, -1, 0, 2, 5}

d. {(3, -4), (1, -5), (0, -2), (-2, -6), (-5, -7)}; D = {3, 1, 0, -2, -5}; R = {-7, -6, -5, -4, -2}

Q6. Find functions f and g so that f g = H(x) = (5 - 2x³)².

a. f(x) = 5 - 2x³ ; g(x) = x²

b. f(x) = (5 - 2x)³ ; g(x) = x²

c. f(x) = x² ; g(x) = 5 - 2x³

d. f(x) = x³ ; g(x) = (5 - 2x)²

Q7. Express 6log _b m - log _b n as a single logarithm.

a. log _b m⁶ log _b n

b. log _b (m⁶ - n)

c. log _b (m⁶/n)

d. log _b (6m/n)

Q8. Solve the equation 8^{x - 4} = 32^4x.

a. {-12/17}

b. {-4/3}

c. {-2}

d. {-4/17}

Q9. Write log ₇ (xy)/15 as the sum and/or difference of logarithms. Express powers as factors.

a. (1/2) log ₇ xy - (1/2) log ₇ 15

b. (1/2) log ₇ x (1/2) log ₇ y (1/2) log ₇ 15

c. (1/2) log ₇ x + (1/2) log ₇ y - (1/2) log ₇ 15

d. (1/2) log ₇ x + (1/2) log ₇ y - log ₇ 15

Q10. Change the exponential expression 5^-3 = 1/125 to an equivalent expression involving a logarithm.

a. log _-3 1/125 = 5

b. log ₅ 1/125 = -3

c. log ₅ -3 = 1/125

d. log _1/125 5 = -3

Q11. Find the amount that results from the investment of $12,000 invested at 5% compounded quarterly after a period of 8 years.

a. $5857.57

b. $17,729.47

c. $17,857.57

d. $17,637.10

Q12. Find the present value to get $6500 after 2 years at 5% compounded quarterly. Round to the nearest cent.

a. $5895.69

b. $614.91

c. $5885.09

d. $5958.65

Q13. Suppose that ln 2 = a and ln 5 = b. Use properties of logarithms to write ln 10 in terms of a and b.

a. ab

b. a - b

c. a + b

d. ln a + ln b

Q14. A size 6 dress in Country C is size 52 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 20). Find a formula for the inverse of the function described.

a. f ^-1(x) = (x - 20)/2

b. f ^-1(x) = x - 20

c. f ^-1(x) = (x/2) - 20

d. f ^-1(x) = (x/2) + 20

Q15. If f(x) = 4^x and g(x) = 12, find the point of intersection of the graphs of f and g by solving f(x) = g(x). Give an exact answer.

a. (log ₄ 12, 12)

b. (log ₄ 12, 4)

c. (log ₄ 12, 0)

d. (12, 12)

Q16. Solve the equation 3^{(1 + 2x)} = 243.

a. {81}

b. {2}

c. {-2}

d. {6}

Q17. Find the amount that results from the investment of $480 invested at 6% compounded quarterly after a period of 3 years.

a. $573.90

b. $93.9

c. $565.42

d. $571.69

Q18. Use the horizontal line test to determine whether the function is one-to-one.

a. Yes

b. No

Q19. Find the domain of the function f(x) = log ₅ (100 - x²).

a. (-10, 10)

b. (-, -10) (10, )

c. (-100, 100)

d. [-10, 10]

Q20. The half-life of silicon-32 is 710 years. If 100 grams is present now, how much will be present in 200 years? (Round your answer to three decimal places.)

a. 14.192

b. 82.263

c. 98.066

d. 0

Q21. Solve using elimination.

a. x = 5, y = 9; x = -5, y = 9; x = 5, y = -9; x = -5, y = -9 or (5, 9), (-5, 9), (5, -9), (-5, -9)

b. x = 5, y = 9; x = 9, y = 5; x = -5, y = -9; x = -9, y = -5 or (5, 9), (9, -5), (-5, -9), (-9, -5)

c. x = -5, y = -9; x = -9, y = -5 or (-5, -9), (-9, -5)

d. x = 5, y = -9; x = 5, y = 9 or (5, -9), (5, 9)

Q22. Solve the system of equations. If the system has no solution, say that it is inconsistent.

a. y = -2x + 7, where x is any real number or {(x, y) | y = -2x + 7, where x is any real number}

b. y = 2x + 7, where x is any real number or {(x, y) | y = 2x + 7, where x is any real number}

c. x = -2y + 7, where y is any real number or {(x, y) |x = -2y + 7, where y is any real number}

d. inconsistent

Q23. Write the partial fraction decomposition of the rational expression (4x³ + 4x²)/(x² + 5)².

a. (4x - 4)/(x² + 5) + (-20x + 20)/(x² + 5)²

b. (4x + 4)/(x² + 5) + (-20x - 20)/(x² + 5)²

c. (4x + 4)/(x² + 5) + (20x - 20)/(x² + 5)²

d. (4x + 4)/(x² + 5) + (20x + 20)/(x² + 5)²

Q24. Solve the system of equations by elimination.

a. x = 10, y = 0; (10, 0)

b. x = 10, y = 10; (10, 10)

c. x = 0, y = 10; (0, 10)

d. x = 0, y = 0; (0, 0)

Q25. Graph the inequality x + y < -5.

a.

b.

c.

d.

Q26. Solve the system of equations by substitution.

a. x = 2, y = 9; (2, 9)

b. x = 3, y = 9; (3, 9)

c. x = 3, y = 8; (3, 8)

d. x = 2, y = 8; (2, 8)

Q27. Solve the linear programming problem. Minimize z = 11x + 6y + 7 subject to: x = 0, y = 0, x + y = 1.

a. minimum: 7

b. minimum: 18

c. minimum: 24

d. minimum: 13

Q28. A dietitian needs to purchase food for patients. She can purchase an ounce of chicken for $0.25 and an ounce of potatoes for $0.03. Let x = the number of ounces of chicken and y = the number of ounces of potatoes purchased per patient. Write the objective function that describes the total cost per patient per meal.

a. z = 3y + 25y

b. z = 0.25x + 0.03y

c. z = 25x + 3y

d. z = 0.03x + 0.25y

Q29. Solve the system of equations using substitution.

a. x = -6, y = 7; x = -7, y = 6 or (-6, 7), (-7, 6)

b. x = -6, y = -7; x = -7, y = -6 or (-6, -7), (-7, -6)

c. x = 6, y = -7; x = 7, y = -6 or (6, -7), (7, -6)

d. x = 6, y = 7; x = 7, y = 6 or (6, 7), (7, 6)

Q30. Write the partial fraction decomposition of the rational expression (125 - 19x)/(x³ - 10x² + 25x).

a. 5/x + -5/(x - 5) + 12/(x - 5)²

b. -5/x + 5/(x - 5) + 6/(x - 5)²

c. 5/x + 6/(x - 5) + -5/(x - 5)²

d. 5/x + -5/(x - 5) + 6/(x - 5)²

Q31. Write a system of linear inequalities that has the given graph.

a.

b.

c.

d.

Q32. Find the value of the determinant.

a. 45

b. 27

c. 15

d. -27

Q33. Find the value of the determinant.

a. 224

b. 24

c. -72

d. 72

Q34. Solve the system of equations using Cramer's Rule if it is applicable.

a. x = -4, y = 3; (-4, 3)

b. x = -3, y = -4; (-3, -4)

c. x = 4, y = 3; (4, 3)

d. x = 3, y = 4; (3, 4)

Q35. Find the inverse of the matrix. Be sure to check your answer.

a.

b.

c.

d.

Q36. Use properties of determinants to find the value of the second determinant, if the value of the first is known.

a. 9

b. -18

c. 18

d. -9

Q37. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

a. x = -2, y = -1, z = 0; (-2, -1, 0)

b. x = -2, y = 0, z = -1; (-2, 0, -1)

c. x = 0, y = -2, z = -1; (0, -2, -1)

d. x = 0, y = -1, z = -2; (0, -1, -2)

Q38. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

a. x = 2, y = 3, z = -7; (2, 3, -7)

b. x = 1, y = 3, z = -4; (1, 3, -4)

c. x = 5, y = 13/2, z = -3; (5, 13/2, -3)

d. x = 1, y = -3, z = -16; (1, -3, -16)

Q39. Solve the system of equations.

a. x = -4, y = 4, z = -3; (-4, 4, -3)

b. x = -3, y = 4, z = -4; (-3, 4, -4)

c. x = -4, y = -3, z = 4; (-4, -3, 4)

d. inconsistent

Q40. Write the partial fraction decomposition of the rational expression (14x + 1)/[(x - 1)(x² + x + 1)].

a. 5/(x - 1) + (4x - 5)/(x² + x + 1)

b. 5/(x - 1) + (-5x + 4)/(x² + x + 1)

c. 5/(x - 1) + -5/(x + 1) + 4/(x - 1)

d. -5/(x - 1) + (5x + 4)/(x² + x + 1)