Math 110 Final Exam All Sections including SL Center Instructors: Rosen, Paulsen, Robertson, Dolbin, Riddle, Lewis, Woller, Teuscher, Ross, Luckau

Transcription

1 Math 1 Final Exam All Sections including SL Center Instructors: Rosen, Paulsen, Robertson, Dolbin, Riddle, Lewis, Woller, Teuscher, Ross, Luckau Dec. 16-1, 017 Do not write on this exam. 1. Let b and c be real numbers. The polynomial x + bx + c has 1 3i as a zero. In which of the following intervals does b + c lie? (a) (, 7] (b) ( 7, ] (c) (, ] (d) (, 1] (e) ( 1, 0] (f) (0, 1] (g) (1, ] (h) (, ] (i) (, 7] (j) (7, ). Solve the inequality: x + x 3 x + x + 0 (a) [ 1, 1) [3, ) (b) (, ) [ 3, 1) [1, ) (c) (, 1) [1, 3) [, ) (d) ( 1, 1] (3, ] (e) (, 3] ( 1, 1] (f) (, ] [ 1, 1) (, ) (g) (, 3] (1, ) (3, ) (h) [, 3] ( 1, 1) (i) (, 1) ( 1, ) (j) None of the above 3. Bill is shopping, which he hates to do. He has been told to buy 3 shirts and pair of pants, all different from each other. The store offers 6 different kinds of shirts, and 7 different kinds of pants. To avoid boredom, Bill decides to compute the number of different ways in which he can purchase 3 different kinds of shirts, and different kinds of pants. What is the answer? (a) between 1 and 0 (b) between 1 and 30 (c) between 31 and 0 (d) between 1 and 80 (e) between 81 and (f) between 11 and 0 (g) between 1 and 360 (h) between 361 and 600 (i) between 601 and 1,000 (j) 1,001 or more. Which of the following are roots of x 3 + 3x 8x + 3? (a) 1, 1 and 3 (b) 3, and (c) 1, and 3 (d) 1, 1 and 3 (e) 1, 1 and 3 (f), 1 and 3 (g) 3, 1 and 3 (h), 1 and (i) 3, 1 and 1 (j) None of the above. Given x = 1 i is a solution to x 3x 3 + x + x = 0. Of the real solutions to this equation, the largest one is x = (a) (b) 3 (c) (d) 1 (e) 1 (f) (g) 3 (h) (i) (j) None of the above. 6. Find the domain of the function f(x) = x 1 x. (a) (, 1) (1, ) (b) [0, 1) (, ) (c) [0, ) (, ) (d) (, 1] (, ) (e) (0, ) (, ) (f) (, 1) (, ) (g) (, ) (, ) (h) [0, ) (, ) (i) (, 1] (, ) (j) None of the above

2 7. Which of the following is the graph of log (x + )? (a) (b) (c) (d) (e) (f) (g) (h) (i) Clearly none of the graphs above is correct.

3 8. Consider the function f(x) = x +. If g is the inverse function to f, then g(x) = x 3 (a) x 3 x (b) x x 3 (c) x+3 x (d) 3x+ x+1 (e) x+ 3x+1 (f) x+3 x 3 (g) x x+3 (h) 3x 1 x 9. Solve the equation 8 x = x for x. The correct value of x is in the interval: (a) (, ] (b) (, 3] (c) ( 3, ] (d) (, 1] (e) ( 1, 0] (f) (0, 1] (g) (1, ] (h) (, 3] (i) (3, ] (j) (, ). Use properties of logarithms to find the exact value of the expression log 16 log. (a) (b) (c) 8 (d) 0 (e) (f) 3 (g) 80 (h) 00 (i) None of the above. 11. I just deposited 00 into a savings account. The bank pays interest on my deposit using an annual interest rate of 3%, compounded quarterly. How much money will I have years from now? (a) 00 (1.03) (b) 00 (1.03) (c) 00 (1.03) 8 (d) 00 (1.007) 16 (e) 00 (1.8) (f) 00 (1.03) 16 (g) 00 (1.007) (h) 00 (1.1) (i) None of the above. 1. Which of these equations matches the figure below? (a) 9(x + 1) + (y + ) = 36 (b) (x + 1) + 6(y + ) = 1 (c) 9(x + 1) + (y + ) = 36 (d) (x + ) + 3(y + 1) = 1 (e) (x + 1) + (y + ) = 36 (f) (x 1) + 6(y ) = 1 (g) 3(x + 1) + (y + ) = 1 (h) 9(x 1) + (y ) = 1 (i) None of the above

4 13. Which of the following conics is represented by the equation 3x + y + 6x + y = (a) Circle (b) Ellipse (c) Parabola (d) Hyperbola (e) None of these 1. Let H(x) = x + x 6 x. What is the domain of H? + x + (a) (, ) ( 3, 1) (, ) (b) (, 6) ( 6, 1) (1, ) (, ) (, ) (c) (, 3) ( 3, ) (, ) (d) (, ) (, 3) ( 3, 1) ( 1, ) (, ) (e) (, ) (, 3) (3, ) (f) (, ) (, 1) (1, ) (, 6) (6, ) (g) (, 1) (1, ) (, ) (h) (, ) (, 1) (1, 3) (3, ) (, ) (i) (, ) (, 1) ( 1, ) (j) None of the above. 1. There were animals in Susan s barn - 3 roosters, pigs and rabbits. Jim chose one of the animals at random. Then, Amy chose one of the remaining animals, at random. After that Ben chose one of the remaining animals, at random. The probability that Jim chose a rooster and Amy chose a pig and Ben chose a rabbit, is closest to: (a) 0.0 (b) 0.11 (c) 0.17 (d) 0. (e) 0.33 (f) 0.1 (g) 0.0 (h) 0.6 (i) 0.7 (j) Solve the system of equations below. Which of the following numbers is the value of x closest to? y + z = 3 x y z = 1 x y + z = If (a) (b) 3 (c) (d) 1 (e) 0 (f) 1 (g) (h) 3 (i) 3x x x 3 = A x 3 + B x + 1 then B is in the interval: (a) (, 6] (b) ( 6, ] (c) (, ] (d) (, 1] (e) ( 1, 0] (f) (0, 1] (g) (1, ] (h) (, ] (i) (, 6] (j) (6, ) 18. Find the sum of the infinite geometric series The sum is in the interval: (a) (, ] (b) (, 30] (c) (30, 3] (d) (3, 0] (e) (0, ] (f) (, 0] (g) (0, ] (h) (, 60] (i) (60, 6] (j) (6, ) 19. In the expansion of (x 3) there is a term of the form ax 3 where a is a real number. To what interval does a belong? (a) (, 70] (b) ( 70, 300] (c) ( 300, ] (d) (, 0] (e) ( 0, 0] (f) (0, 0] (g) (0, ] (h) (, 300] (i) (300, 70] (j) (70, )

5 0. A math department has 3 different sections of College Algebra to teach, numbered 1,,3,. They have 7 professors who can teach the subject. We have computed the number of different ways in which they can they assign 1 of the professors to teach section 1, another professor to section, another professor to section 3, and another professor to section. In which of the intervals listed below does the correct answer lie? (a) (, 0] (b) (0, 0] (c) (0, 70] (d) (70, ] (e) (, 0] (f) (0, 00] (g) (00, 800] (h) (800, 1300] (i) (1300, 000] (j) (000, ) 1. Every day my business produces a profit of $1 with probability 3, and produces a loss of $1 with probability 1 3. Based on that information, I just computed the probability that in of the last 6 days my business produced a profit of $1, and on 1 of the last 6 days it produced a loss of $1. In which of the following intervals does this probability lie? (a) [0, 0.0] (b) (0.0, 0.1] (c) (0.1, 0.1] (d) (0.1, 0.] (e) (0., 0.3] (f) (0.3, 0.] (g) (0., 0.] (h) (0., 0.6] (i) (0.6, 0.8] (j) (0.8, 1]. Select the function that best describes the given graph. (a) f(x) = + ( x) (b) f(x) = + (1 x) (c) f(x) = 1 + (1 x) (d) f(x) = 1 + (1 x) (e) f(x) = + (1 x) (f) f(x) = + 1 (1 x) (g) f(x) = (1 x) (h) f(x) = 1 + (1 x) (i) f(x) = + (1 x) (j) Clearly, none of the above is accurate Solve the following equation algebraically for x: log 3 ( x ) = log 3 + log 3 (3 + x). The correct value of x is in the interval: (a) (, ] (b) (, 3] (c) ( 3, ] (d) (, 1] (e) ( 1, 0] (f) (0, 1] (g) (1, ] (h) (, 3] (i) (3, ] (j) (, ). If (3 x ) x+ = 1 7 then there are possible values of x. The sum of the values is in the interval: (a) (, 6] (b) ( 6, ] (c) (, ] (d) (, 1] (e) ( 1, 0] (f) (0, 1] (g) (1, ] (h) (, ] (i) (, 6] (j) (6, )

6 . Consider the hyperbola 1 16 (x + ) 1 9 (y 1) = 1. What are the foci? (a) (, 6) and (, ) (b) (, 1 7) and (, 1 + 7) (c) ( 7, 1) and (3, 1) (d) ( 1, 1) and ( + 1, 1) (e) (, 6) and (, ) (f) (, 6) and (, ) (g) ( 7, 1) and ( + 7, 1) (h) (, 1 1 ) and (, ) (i) ( 7, 1) and ( + 7, 1) (j) None of the above. 6. If x + x 3 + = 0, then to which of the following intervals does x belong? Hint: Use the Intermediate Value Theorem. (a) (, ] (b) (, 3] (c) ( 3, ] (d) (, 1] (e) ( 1, 0] (f) (0, 1] (g) (1, ] (h) (, 3] (i) (3, ] (j) (, ) 7. Which of the following is the graph of f(x) = x +x x +x+3? (a) (b) (c) (d) (e) (f) (Continued on the next page)

7 (g) (h) (i) 8 6 (j) 8. Consider the system of equations x x + y =, x y =. How many distinct ordered pairs (x, y) are there, that solve this system of equations? (a) 0 (b) 1 (c) (d) 3 (e) (f) (g) 6 (h) 7 (i) More than Consider the binomial expansion of To which of the following intervals does the sum belong? ( x x) 6. Find the sum of the first three coefficients. (a) (, 600) (b) [ 600, 00) (c) [ 00, 60) (d) [ 60, 0) (e) [ 0, 0) (f) [0, 0) (g) [0, 60) (h) [60, 00) (i) [00, 600) (j) [600, ) 30. Assume that a = log 3 and b = log 3. Express the following as a function of a and b: log 3 (0 ): (a) a + 1 b (b) a + 3 b (c) a + 1 b (d) a + b (e) a + b (f) a 1 b (g) a + b (h) a + 3 b (i) a 3 b (j) None of the above.

8 Answers 1. j. e 3. h. i. f 6. c 7. c 8. d 9. h. b 11. d 1. c 13. d 1. i 1. a 16. g 17. f 18. g 19. i 0. e,h 1. e. e 3. c. b. c 6. d 7. a 8. e 9. g 30. h