Algebra 105

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Q1. Find k such that f(x) = x4 + kx3 + 2 has the factor x + 1.    a. -3    b. -2    c. 3    d. 2 Q2. Solve the inequality. (x - 5)(x2 + x + 1) > 0    a. (-∞, -1) or (1, ∞)    b. (-1, 1)    c. (-∞, 5)    d. (5, ∞) Q3. Use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval. f(x) = 8x3 - 10x2 + 3x + 5; [-1, 0]    a. f(-1) = -16 and f(0) = -5; no    b. f(-1) = -16 and f(0) = 5; yes    c. f(-1) = 16 and f(0) = -5; yes    d. f(-1) = 16 and f(0) = 5; no Q4. Solve the equation in the real number system. x4 - 3x3 + 5x2 - x - 10 = 0    a. {-1, -2}    b. {1, 2}    c. {-1, 2}    d. {-2, 1} Q5. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. f(x) = x2 - 2x - 5    a. maximum; 1    b. minimum; 1    c. maximum; - 6    d. minimum; - 6 Q6. Determine whether the rational function has symmetry with respect to the origin, symmetry with respect to the y-axis, or neither. f(x) =    a. symmetry with respect to the origin    b. symmetry with respect to the y-axis    c. neither Q7. Find the domain of the rational function. f(x) = .    a. {x|x ≠ -3, x ≠ 5}    b. {x|x ≠ 3, x ≠ -5}    c. all real numbers    d. {x|x ≠ 3, x ≠ -3, x ≠ -5} Q8. Find the domain of the rational function. g(x) =    a. all real numbers    b. {x|x ≠ -7, x ≠ 7, x ≠ -5}    c. {x|x ≠ -7, x ≠ 7}    d. {x|x ≠ 0, x ≠ -49} Q9. Find all zeros of the function and write the polynomial as a product of linear factors. f(x) = 3x4 + 4x3 + 13x2 + 16x + 4    a. f(x) = (3x - 1)(x - 1)(x + 2)(x - 2)    b. f(x) = (3x + 1)(x + 1)(x + 2i)(x - 2i)    c. f(x) = (3x - 1)(x - 1)(x + 2i)(x - 2i)    d. f(x) = (3x + 1)(x + 1)(x + 2)(x - 2) Q10. Use the graph to find the vertical asymptotes, if any, of the function.    a. y = 0    b. x = 0, y = 0    c. x = 0    d. none Q11. Find the power function that the graph of f resembles for large values of |x|. f(x) = (x + 5)2    a. y = x10    b. y = x25    c. y = x2    d. y = x5 Q12. Determine whether the rational function has symmetry with respect to the origin, symmetry with respect to the y-axis, or neither. f(x) =    a. symmetry with respect to the y-axis    b. symmetry with respect to the origin    c. neither Q13. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. f(x) = -x2 - 2x + 2    a. minimum; - 1    b. maximum; 3    c. minimum; 3    d. maximum; - 1 Q14. Find the power function that the graph of f resembles for large values of |x|. f(x) = -x2(x + 4)3(x2 - 1)    a. y = x7    b. y = -x7    c. y = x3    d. y = x2 Q15. A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 320 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?    a. 25,600 ft2    b. 19,200 ft2    c. 12,800 ft2    d. 6400 ft2 Q16. Find the indicated intercept(s) of the graph of the function. x-intercepts of f(x) =    a. (5, 0)    b.    c.    d. (-5, 0) Q17. State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. f(x) = 9x3 + 8x2 - 6    a. No; the last term has no variable    b. Yes; degree 5    c. Yes; degree 3    d. Yes; degree 6 Q18. Solve the equation in the real number system. x3 + 9x2 + 26x + 24 = 0    a. {-4, -2, -3}    b. {2, 4}    c. {3, 2, 4}    d. {-4, -2} Q19. Use the graph to find the vertical asymptotes, if any, of the function. <b </b    a. x = -3, x = 3, x = 0    b. x = -3, x = 3, y = 0    c. none    d. x = -3, x = 3 Q20. Use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval. f(x) = -2x4 + 2x2 + 4; [-2, -1]    a. f(-2) = 20 and f(-1) = 5; no    b. f(-2) = -20 and f(-1) = 4; yes    c. f(-2) = 20 and f(-1) = -4; yes    d. f(-2) = -20 and f(-1) = -4; no

Q1. Express y as a function of x. The constant C is a positive number. ln y = ln 4x + ln C    a. y = 4Cx    b. y = 4x + C    c. y = (4x)C    d. y = x + 4C Q2. The half-life of a radioactive element is 130 days, but your sample will not be useful to you after 80% of the radioactive nuclei originally present have disintegrated. About how many days can you use the sample?    a. 302    b. 287    c. 312    d. 297 Q3. The function f(x) = 300(0.5) x/60 models the amount in pounds of a particular radioactive material stored in a concrete vault, where x is the number of years since the material was put into the vault. Find the amount of radioactive material in the vault after 170 years. Round to the nearest whole number.    a. 425 pounds    b. 42 pounds    c. 235 pounds    d. 53 pounds Q4. The half-life of plutonium-234 is 9 hours. If 70 milligrams is present now, how much will be present in 6 days? (Round your answer to three decimal places.)    a. 0.689    b. 44.096    c. 0.001    d. 23.091 Q5. What annual rate of interest is required to triple an investment in 12 years?    a. 4.794%    b. 9.587%    c. 9.155%    d. 5.946% Q6. If 7-x = , what does 49x equal?    a. 4    b. 16    c. -16    d. -4 Q7. What principal invested at 8% compounded continuously for 4 years will yield $1190? Round the answer to two decimal places.    a. $864.12    b. $1188.62    c. $1638.78    d. $627.48 Q8. The function f(x) = 1 + 1.6 ln (x+1) models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where x is the number of consecutive days the basketball player has practiced for two hours. After how many days of practice can the basketball player make an average of 6 consecutive free throws?    a. 24 days    b. 80 days    c. 22 days    d. 78 days Q9. pH = -log10[H+] Find the [H+] if the pH = 8.4.    a. 3.98 x 10-8    b. 2.51 x 10-8    c. 3.98 x 10-9    d. 2.51 x 10-9 Q10. Express as a single logarithm. 3log6x + 5log6(x - 6)    a. log6x3(x - 6)5    b. log6x(x - 6)15    c. log6x(x - 6)    d. 15 log6x(x - 6) Q11. A local bank advertises that it pays interest on savings accounts at the rate of 3% compounded monthly. Find the effective rate. Round answer to two decimal places.    a. 3.44%    b. 3.40%    c. 36%    d. 3.04% Q12. Find the effective rate of interest. 50.11% compounded daily    a. 50.233%    b. 51.015%    c. 50.315%    d. 64.997% Q13. Solve the equation. log3x + log3(x - 24) = 4    a. {-3, 27}    b. {27}    c. No real solutions    d. {53} Q14. Express as a single logarithm. 40 log5 + log5(40x6) - log5 40    a. log5 x14/5    b. log5 x14    c. log5 x13/6    d. log5 x11/8 Q15. The half-life of silicon-32 is 710 years. If 100 grams is present now, how much will be present in 600 years? (Round your answer to three decimal places.)    a. 0    b. 0.286    c. 94.311    d. 55.668 Q16. The logistic growth function f(t) = describes the population of a species of butterflies tmonths after they are introduced to a non-threatening habitat. How many butterflies are expected in the habitat after 12 months?    a. 480 butterflies    b. 401 butterflies    c. 244 butterflies    d. 4800 butterflies Q17. Change the exponential expression to an equivalent expression involving a logarithm. 5x = 125    a. log125 x = 5    b. log5 125 = x    c. log125 5 = x    d. logx 125 = 5 Q18. A fossilized leaf contains 12% of its normal amount of carbon 14. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon 14.    a. 1031    b. 17,099    c. 20,040    d. 36,108 Q19. The function A = Aoe-0.0099x models the amount in pounds of a particular radioactive material stored in a concrete vault, where x is the number of years since the material was put into the vault. If 400 pounds of the material are initially put into the vault, how many pounds will be left after 40 years?    a. 350 pounds    b. 269 pounds    c. 114 pounds    d. 119 pounds Q20. The function f is one-to-one. Find its inverse. f(x) =    a. f -1(x) = x3 + 16    b. f -1(x) = x - 4    c. f -1(x) =    d. f -1(x) = x3 - 4

Q1. The Family Fine Arts Center charges $21 per adult and $12 per senior citizen for its performances. On a recent weekend evening when 486 people paid admission, the total receipts were $6894. How many who paid were senior citizens?    a. 208 senior citizens    b. 368 senior citizens    c. 118 senior citizens    d. 278 senior citizens Q2. Use the elimination method to solve the system.    a. x = 11, y = -11    b. x = -2, y = 9    c. x = 9, y = -9    d. x = -9, y = 9 Q3. Verify that the values of the variables listed are solutions of the system of equations. x = 0, y = -5, z =-4    a. solution    b. not a solution Q4. Find the inverse of the matrix.    a.    b.    c.    d. Q5. Find the inverse of the matrix. A =    a.    b.    c.    d. Q6. Solve the system.    a. inconsistent (no solution)    b. (-9, 0)    c. (0, 0)    d. y = - - 9, where x is any real number Q7. Perform the indicated operations and simplify. Let A = , B = , and C = . Find AB + BC.    a.    b.    c.    d. Q8. Use the properties of determinants to find the value of the second determinant, given the value of the first. Given > = 3, find the value of >.    a. -24    b. -3    c. 24    d. 3 Q9. Use Cramer's rule to solve the linear system.    a. x = 2, y = 4    b. x = 4, y = 2    c. x = -2, y = 4    d. x = -4, y = -2 Q10. An 8-cylinder Crown Victoria gives 18 miles per gallon in city driving and 21 miles per gallon in highway driving. A 300-mile trip required 15.5 gallons of gasoline. How many whole miles were driven in the city?    a. 153 miles    b. 168 miles    c. 147 miles    d. 132 miles Q11. A retired couple has $190,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 7%. The rest they want to invest in AA bonds yielding 10% per year. How much should they invest in each to realize exactly $17,200 per year?    a. $120,000 at 7% and $70,000 at 10%    b. $130,000 at 7% and $60,000 at 10%    c. $130,000 at 10% and $60,000 at 7%    d. $140,000 at 10% and $50,000 at 7% Q12. Solve the system of equations.    a. x = 0, y = 1, z = 0    b. x = 1, y =1, z = 1    c. x = 0, y = 1, z = -1    d. x = 0, y = 0, z = 1 Q13. Use the properties of determinants to find the value of the second determinant, given the value of the first. = -44 = ?    a. -44    b. 44    c. 88    d. -88 Q14. Find the inverse of the matrix.    a. No inverse    b.    c.    d. Q15. Solve the system using the inverse method.    a. x = -36, y = -38, z = -14    b. x = -14, y = 88, z = -32    c. x = -60, y = 82, z = -34    d. x = 8, y = 24, z = 10 Q16. Solve the system using the inverse method.    a. x = 1, y = -2    b. x = -2, y = 1    c. x = -1, y = 2    d. x = 2, y = -1 Q17. Write the partial fraction decomposition of the rational expression.    a. +    b. +    c. +    d. + Q18. Perform the indicated operation, whenever possible. +    a.    b.    c.    d. Q19. A flat rectangular piece of aluminum has a perimeter of 62 inches. The length is 15 inches longer than the width. Find the width.    a. 31 inches    b. 38 inches    c. 8 inches    d. 23 inches Q20. Verify that the values of the variables listed are solutions of the system of equations. x = 5, y = 5    a. not a solution    b. solution