Sample Math 22 Exam Questions No Calculators Allowed

Transcription

1 No Calculators Allowed REVIEW 1. Write the expression as a power of x. (a) x x (b) x m+1 (x 2m+1 ) 2 (c) (x2 ) n x 5 x n 2. Simplify the expression. (a) (x1/2 y 3/2 ) 4 y 2 (b) (x 3 y) 2 y 4 (c) x 2 + x 3. Simplify. (a) 1 x x (x 2)(x + 1) ( 1 x 1 ) 2 (b) x 2 4. Perform the indicated operation. (a) (3x y) 2 (b) (ab + 2ab 2 ) 2 (c) (2x 3y) 3 5. Factor the expression. (a) 12x 2 y 4 3xy 5 + 9x 3 y 2 (b) ax 2 ay 2 (c) 25x 2 16y 6 (d) x 2 9x + 18 (e) 6x 2 + x 12 (f) 8x 3 y 3 (g) 3x 3 2x x 12 (h) ax 2 + bx 2 a b (i) 4x 2 12xy + 9y 2 p 2 1

2 6. Solve for x. (a) x 2 9x + 14 = 0 (b) 2x = 0 (c) 3x 2 + 5x 2 = 0 (d) x x = 2x 2x (e) x x 2 + x = 8 x 2 4 (f) x 1/2 2x 1/2 + x 3/2 = 0 7. Calculate and write the result in the form a + bi (a) (2 + 3i)(3 5i) (b) 3 2i 2 5i (c) 3i 3 + 5i 5 8. Solve the inequality. (a) 3x 3 < 8 (b) 3x (c) x2 6x 16 x 3 > 0 (d) 5 x 3 x 2 4x + 4 < 0 THE COORDINATE PLANE 9. Sketch the region given by the set. (a) {(x, y) : 2 < x 4, 3 y < 6} (b) {(x, y) : y 2 3} 10. Show that the triangle with vertices A(2, 0), B( 3, l) and C( 4, 3) is isosceles. 11. Find the point that is one-fourth of the distance from the point P ( 1, 3) to the point Q(7, 5) along the segment P Q. 12. Find an equation of the circle satisfying the given conditions. (a) Center (4, 5); tangent to the y-axis. (b) Center at the origin; passes through (3, 4) (c) End points of a diameter are (3, 7) and (5, 13) 13. Sketch the graph of the following circles. (a) x 2 2x + y 2 + 4y + 1 = 0 (b) 2x 2 4x + 2y 2 + 6y = 1 2

3 14. Determine whether the given points are on the graph of the equation (a) A(0, 2) (b) B(1, 2) (c) C(2, 2) x 2 + xy + y 2 = 5 SOLVING EQNS AND INEQUALITIES (GRAPHICALLY) 15. Use a graphical approach to estimate values of x that solve the following equation with an error less than x 3 + x 2 2x + 7 = Use a graphical approach to estimate values of x that solve the following inequality with an error less than x 2 x 10 < 0 LINES 17. Find an equation of the line that satisfies the given conditions. (a) Through the points (3, 6) and ( 2, 4) (b) Slope 4; y-intercept 3 (c) Through (2, 3); parallel to the line 2x + 3y = 9 (d) Through (4, 3); perpendicular to the line 3x = 2y The manager of a furniture factory finds that it cost $2,200 to produce 100 chairs and $4,800 to produce 300 chairs. (a) Find a linear equation relating the number of chairs x with their cost y (b) Sketch the graph. (c) Estimate the total cost of producing 180 chairs. (d) How many chairs can be produced with 6000 dollars? (e) What is the y-intercept and what does it represent? (f) What is the cost of producing the 250-th chair? FUNCTIONS 19. Complete the following table 3

4 Words Math. notat Substract three from x, then divide by eight. Square x, then add three. Square x, add two, then take square root. (x 5) 3 x + 7 (x 2 + 5) In the following diagram replace the question marks by numbers assuming that the function y = 3x x y???? 21. Find the linear function y = ax + b that corresponds to the mapping given in the following diagram. x y If f(x) = 3x 2 x + 5 then find: (a) f(3) 4

5 (b) f( 7) (c) f( 2) (d) f(b) (e) f(a) + f(b) (f) f(a + b) 23. Sketch the graph of the given function and then determine its range. (a) f(x) = 6 3x, 10 x 5 (b) f(x) = 2x 3 3x + 1, 5 x Determine the domain and range of the given function. (a) f(x) = 7 15x (b) f(x) = x + 3 x 2 4 (c) f(x) = x x 25. Sketch the graph of the function and then state interval on which the function is increasing and on which it is decreasing. f(x) = x 3 2x 2 x The doors of a building are parabolic arcs that can be modeled by the equation y = 0.12x x, 0 x 20, where the units of y are feet s. Can a box of dimensions 9ft 9ft 12ft pass through the doors? 27. For f(x) = 5x 2 + 2x and g(x) = 3 x+2, evaluate each of the following: (a) f(g(f(3))) (b) g(f(x)) (c) f g(x) 28. Sketch the graph of the function and then determine whether or not the function is one-to-one. (a) f(x) = x 2 2x + 5, 20 x 30 (b) f(x) = 1 + 3x 5 2x, x 5 2 (c) f(x) = x x, 0 < x < Find the inverse of the given function and sketch the graph of the function and its inverse. (a) f(x) = 3x (b) f(x) = 1 + 3x 5 2x 5

6 30. Sketch the graph of the given function. Include x- and y-intercepts (if any) and identify asymptote (if any). (a) f(x) = 3x4 + 4x + 1 x 2 4 (b) f(x) = 4x2 + 2x 1 x (c) f(x) = 2x2 + 3x + 2 x + 5 (d) f(x) = x2 9 x 2 5x + 3 APPLIED FUNCTIONS 31. You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Sketch a rough graph of the temperature of the pie as a function of time. 32. A rectangle has a perimeter of 40 ft. Express the area A of the rectangle as a function of the length x of one of its sides. 33. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window. 34. The resistance R of a wire varies directly as its length L and inversely as the square of its diameter d. (a) A wire 1.2 m long and m in diameter has a resistance of 140 ohms. Write an equation for this variation and find the constant of proportionality. (b) Find the resistance of a wire made of the same material that is 3 m long and has a diameter of m. 35. A farmer has 650 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? 36. A triathion champion is on an island 6 miles off the coast of California. She wants to get her bike on the shore, 8 miles from the point on the shore nearest the island. She can swim at 5 mph and run at 13 mph. Toward what point on the shore should she swim to minimize her total time? POLYNOMIALS 37. Use long division to find the quotient and remainder. x 4 + 3x 3 2x + 3 x 2 2x Use synthetic division to find the quotient and remainder. 5x 3 3x + 3 x + 2 6

7 39. Find a polynomial with integer coefficients that has degree 3 and zeros 2, -3 and 2/ Show that the polynomial does not have any rational roots. x 3 x For what values of k does the quation have at least one rational root? x 3 + kx + 2 = Show that the following polynomial has exactly one positive root and no more than 3 negative roots. 2x 8 + k 2 x 6 + 3x 5 + 3x 2 10x Show that 2i is a solution of the given equation and then find all its solutions. 44. Find all solutions. Give exact values. 2x 4 + 9x = 0 (x 2 4x) 2 3(x 2 4x) + 2 = Find all rational roots and then find the irrational roots, if any. Give exact values. x 3 x 2 5x Find all roots. Identify and approximate irrational roots with an error less that Find the factorization of the polynomial. x 4 + 8x + 16 = 2x 3 + 8x 2 x 5 + 3x 3 + 2x EXPONENTIAL AND LOGARITHMIC FUNCTIONS 48. Solve for x. Give exact values. (a) log 2 16 = x (b) log x 81 = 3 (c) log x = 2 log x 49. Use a calculator to evaluate log Use the Laws of Logarithms to rewrite the follwong expression so that it involves no products, quotients or powers. log a3 b c 2 7

8 51. Rewrite the expression as a single logarithm. 3 log x 2 log y + 5 log z 52. Sketch the graph of the given function. Identify domain, range and assymptotes. (a) f(x) = log 3 x 4 (b) f(x) = 3 + e x Solve for x. Give exact values. (a) 3 x = 7 (b) 2 3x+1 = 3 x 2 (c) x 2 2 x 2 x = 0 (d) e 2x + 4e x 5 = 0 (e) log 3 + log x = log 5 + log(x 2) (f) log 5 x + log 5 (x + 1) = log Solve for x. Give your answers with an error less that (a) x 3 = log(x + 1) (b) 4 x = x (c) log(x 2) + log(9 x) < 2 (d) 2 < 10 x < The population of a certain city is 800,000 in 1997 and is growing at the relative growth rate of 13% per year. (a) Estimate the population in the year (b) When will the city reach a population of 2,000,000? 56. Seven years ago an account was open with $2,000. Thepresent value of this account is $4,000. Determine theinterest rate on the investment assuming that it wascompounded continuously. LINEAR SYSTEMS 57. Solve the linear system. Use the indicated method, if any. (a) (b) (c) 3x + 5y = 17 7x + 9y = 29 x + y + z = 2 2x 2y + 2z = 4 4x + y 3z = 1 3x + 5y + 2z = 22 x + 2y z = 9 2x z = 2 (using Gaussian elimination) 8

9 58. A chemist has two large containers of sulfuric acid solution, with different concentration of acid in each container. Blending 300 ml, of the first solution and 600 ml of the second gives a mixture that is 15% acid, whereas 100 ml, of the first mixed with 500 ml, of the second gives a 12.5% acid mixture. What is the concentration of sulfuric acid in each ot the original containers? 59. The drawer of a cash register contains 30 coins; pennies, nickels, dimes, and quarters. The total value of the coins is $3.31. The total number of pennies and nickels combined is the same as the total number of dimes and quarters combined. The total value of the quarters is five times the total value of the dimes. How many coins of each type are in the drawer? CONIC SECTIONS 60. Determine the type of curve represented by the equation. Find the foci and vertices (if any), and sketch the graph. (a) x 2 + 6x = 9y 2 (b) x y 9 = 1 (c) x y2 9 = 1 (d) 36x 2 4y 2 36x 8y = Find an equation for the conic section with the given properties. (a) The parabola with focus F (0, 1) and directrix y = 1 (b) The ellipse with center C(0, 4), foci F 1 (0, 0) and F 2 (0, 8), and major axis of length 10. (c) The hyperbola with vertices V (0, ±2) and asymptotes 2y = ±x. 9