MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine algebraically whether the function is even, odd, or neither even nor odd. ) f(x) = -0.98x2 + x + 4 Neither Even Odd ) Find the asymptote(s) of the given function. 2) f(x) = 3x horizontal asymptotes(s) 3x2-2 2) None y = -2 y = 2 y = State how many complex and real zeros the function has. 3) f x = x4-8x3 + 82x2-8x +8 4 complex zeros; all 4 real 4 complex zeros; 2 real 4 complex zeros; real 4 complex zeros; none real 3) Perform the requested operation or operations. 4) f(x) = x - 7 ; g(x) = x Find f(g(x)). f(g(x)) = f(g(x)) = x - 7 x x - 7 f(g(x)) = (x - 7) x f(g(x)) = x - 7 4) Fill in the blanks to complete the statement. 5) The graph of y = -3x3 + 7 can be obtained from the graph of y = x3 by vertically stretching by a factor of? ; reflecting across the? -axis, and shifting vertically? units in the? direction. 5) -3; x; 7; downward 3; y; 7; upward 3; x; 7; upward 7; x; 3; upward Solve the problem. 6) The radius of a car wheel is 2 inches. How many revolutions per minute is the wheel making when the car is travelling at 40 mph. Round your answer to the nearest revolution. 6) 5529 rpm 560 rpm 26 rpm 9 rpm Find the zeros of the function. 7) f x = x3 + 2x2 + 47x , -4, and -5-3, -4, and -5 0, 3, 4, and 5 3, 4, and 5 7) A-

2 Match the equation with the appropriate graph. 6x 8) f(x) = x2-8) 0 y 0 y -0 0 x -0 0 x y 0 y -0 0 x -0 0 x -0-0 State the domain of the rational function. x - 9) f(x) = x2 + 2x 9) (-, ) (, ) (-, 2) (2, ) (-, -2) (-2, ) (-, -2) (-2, 0) (0, ) Evaluate the logarithm. 0) log ) Describe how to transform the graph of the basic function g(x) into the graph of the given function f(x). ) f(x) = - 9 ln (- x); g(x) = ln x Reflect across the x-axis and translate 9 units down. Reflect across the y-axis and vertically stretch by a factor of 9. Reflect across the x-axis and the y-axis and translate9 units down. Reflect across the x-axis and the y-axis and vertically stretch by a factor of 9. ) A-2

3 Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. 2) 2 log ax + 5 loga y - 3 loga x 2) loga x5 y5 loga y 5 x5/2 loga x y5 loga x3 y5 The graph is that of a function y = f(x) that can be obtained by transforming the graph of y = x. Write a formula for the function f. 3) 0 y 3) x -5-0 f(x) = -x + 8 f(x) = -x + 8 f(x) = - x + 8 f(x) = - x + 8 Solve the problem. 4) An open-top rectangular box has a square base and it will hold 0 cubic centimeters (cc). Each side has length x cm, and it has a height of y cm. Express the surface area as a function of the length x of a side of the base x2 404 x + x x + x 2 0 x + x 2 4) Rewrite the expression as a sum or difference or multiple of logarithms. 5) log5 4 3 s2r 5) 4 log log5 s - 2 log5 r 4 log log5 s - log5 r 4 log5 3-2 log5 s - log5 4 log5 3 - log5 s - log5 r Find all solutions to the equation. 6) 8 sin2x - 6 sin x + 4 = -4 π + 2nπ n = 0, ±, ±2,... nπ n = 0, ±, ±2,... 2 π 4 + 2nπ, 3π 4 + 2nπ n = 0, ±, ±2,... π + nπ n = 0, ±, ±2, ) A-3

4 Solve the problem. 7) The minute hand of a clock is 7 inches long. What distance does its tip move in 23 minutes? π in. π in. π in π in. 7) Find the exact value of the composition. 8) arccos[sin(π/6)] 2 π 3 π ) Evaluate without using a calculator by using ratios in a reference triangle. 9) tan (- 2π 3 ) 9) Find the value of the unique real number θ between 0 and 2π that satisfies the given conditions. 20) tan θ = - 7π and cos θ > 0 5π 3 π 6 5π 6 20) Simplify the expression. 2) sec x sin x - cos x sin x 2) tan x csc2 x cot x -tan x Write each expression in factored form as an algebraic expression of a single trigonometric function. 22) sec4x + sec2x tan2x - 2 tan4x sec2x + 2 tan2x - 4 sec2x 3 sec2x ) Find all solutions in the interval [0, 2π). 23) sec2x - 2 = tan2x 23) x = π 3 x = π 6 x = π 4 No solution 24) 7 tan3x - 2 tan x = 0 24) 0, π 3, 2π 3, π, 4π 3, 5π 3 0, π 5, π, 6π 5 π 3, 2π 3, 4π 3, 5π 3 0, π 3, π, 4π 3 A-4

5 Find an exact value. 25) sin ) Write the expression as the sine, cosine, or tangent of an angle. 26) sin π 3 cos π 7 + cos π 3 sin π 7 26) sin -4π 2 cos 0π 2 cos -4π 2 sin 0π 2 Find the exact value by using a half-angle identity. 27) cos - π 8 27) Find all solutions in the interval [0, 2π). 28) tan x 2 = + cos x - cos x 28) 0 π 4, 5π 4 π 2 π 2, 3π 2 29) sin x 2 = 2 cos 2 x - 29) π 5, π, 9π 5 π 5, 9π 5 0, π π 3, π, 5π 3 State whether the given measurements determine zero, one, or two triangles. 30) A = 87, a = 26, b = 66 One Two Zero 30) 3) A = 63, a = 2, b = 23 Zero Two One 3) Find the component form and magnitude of the indicated vector. 32) Given that P = (-, 2) and Q = (-3, 3), find the component form and magnitude of the vector PQ. -4,, 7-2,, 5-2,, 5 2, -, 5 32) Find the unit vector in the direction of the given vector. Write your answer in the indicated form. 33) Let u = -2,-3. Find the unit vector in the direction of u, and write your answer as a linear combination of the standard unit vectors i and j i j 3 5 i j i + j -2 3 i j 33) A-5

6 Find a b. 34) a = 0i + 6j, b = -3i + 4j 34) 54-30, 24 7, 0-6 Find the angle between the given vectors to the nearest tenth of a degree. 35) u = 6 cos π 6 i + 6 sin π 6 j, v = cos 4π 3 i + sin 4π 3 j 35) Write the complex number in the form a + bi. 36) 6(cos 35 + i sin 35 ) i 6-6i i 3-3i 2 36) Find the product or quotient. Write the answer in standard form. 37) (8-7i)(6 + 3i) 37) 27-66i i -2i2-8i i Find the product or quotient, as indicated. Leave your answer in trigonometric form. 38) Find the product of z and z2. z = 7[cos (50 ) + i sin (50 )], z2 = 5(cos i sin -75 ) 38) (7 + 5) [cos (-25 ) + i sin (-25 )] 7 5 [cos (-25 ) + i sin (-25 )] 7 5 [cos (-8750 ) + i sin (-8750 )] 7 5 [cos (225 ) + i sin (225 )] Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. 39) (- 3 + i)6 39) i i 64i -64 Find the indicated roots. Write the answer in a + bi form. 40) Cube roots of -8i 40) -2i, 3 - i, 3 - i 2i, i, i 2i, i, 3 - i -2i, 3 + i, 3 + i A-6

7 Find the indicated roots. Write the answer in trigonometric form. 4) Fifth roots of 243(cos i sin 300 ) 4) 3(cos i sin 300 ), 3(cos i sin 372 ), 3(cos i sin 444 ), 3(cos 56 + i sin 56 ), 3(cos i sin 588 ) 3(cos 60 + i sin 60 ), 3(cos 50 + i sin 50 ), 3(cos i sin 240 ), 3(cos i sin 330 ), 3(cos i sin 375 ) 3(cos 60 + i sin 60 ), 3(cos 32 + i sin 32 ), 3(cos i sin 204 ), 3(cos i sin 276 ), 3(cos i sin 348 ) 3(cos i sin 300 ), 3(cos i sin 390 ), 3(cos i sin 480 ), 3(cos i sin 570 ), 3(cos 65 + i sin 65 ) Solve the system by substitution. 42) x + 3y = 27 2x + 3y = 36 42) (0, 5) (0, 9) No solution (9, 6) Solve the system by elimination. 43) -6x + 7y = -54-3x + 3y = -27 (9, ) (9, 0) No solution (8, ) 43) Find the matrix product, if possible. 44) ) Find the indicated matrix. 45) Let A = 3 and B = Find 2A + B. 45) Determine whether the matrices are inverses ) , ) No Yes A-7

8 State how many complex and real zeros the function has. 47) f x = x4-63x complex zeros; none real 4 complex zeros; 2 real 4 complex zeros; all 4 real 4 complex zeros; real 47) Find the determinant of the given matrix. 48) ) Use Gaussian elimination to solve the system of equations. 49) x + y + z = 0 x - y + 4z = 3x + y + z = 6 49) (-, -2, 3) No solution (3, -2, -) (-, 3, -2) Identify which of the twelve basic functions listed below fit the description given. y = x, y = x2, y = x3, y = x, y = x, y = e x, y = x, y = ln x, y = sin x, y = cos x, y = int (x), y = + e-x 50) The three functions that are even y = x, y = x, y = x 3 y = x, y = x2, y = x3 50) y = cos x, y = sin x, y = x y = x2, y = cos x, y = x Find the partial fraction decomposition. 5) x + 2 x2 - = A x + + B x - 5) A = 2, B = 3 2 A = - 2, B = A = 2, B = A = - 2, B = ) 5 x(x2 + 3) = A x + Bx + C x ) A = 5 6, B = - 5 6, C = 0 A = - 5 6, B = 5 6, C = 0 A = 5 3, B = - 5 3, C = 0 A = - 5 3, B = 5 3, C = 0 A-8

9 Find the indicated matrix product or state that the product is undefined ) A = 0 0 0, B = ) AB Solve the problem. 54) The position of a weight attached to a spring is s(t) = -5 cos 3t. What are the frequency and period of the system? Frequency = 3 2π cycles per sec, period = 2π 3 sec 54) Frequency = 3 π cycles per sec, period = π 3 sec Frequency = 3 cycles per sec, period = 3 sec Frequency = π 3 cycles per sec, period = 3 π sec Graph the function on your calculator to determine the domain and range from the graph. 55) g(x) = + e-x ) Domain: (-, ); range: (-3, -2) Domain: (0, ); range: [-3, -2] Domain: (-, ); range: (-, -2] Domain: (-, ); range: (-, ) Using the given zero, find all other zeros of f(x). 56) 2 + 6i is a zero of f x = x4-6x3 + 24x2-624x i, 6 2-6i, 6, and i, i, 6, and -6 56) Find the domain of the given function. 57) f(x) = x4-8x2 (-, -9] [0] [9, ) (-, -9] [9, ) (-, -9) (9, ) (-, ) 57) A-9

10 Solve the problem. 58) A turntable rotates at 65 revolutions per minute. What is its angular speed in radians per second? 3 2 π radians/second 6 3 π radians/second 3 2 π radians/second 6 3 π radians/second 58) 59) The sprocket assembly on a bicycle consists of a chain and two sprockets, one on the pedal and the other on the rear wheel. If the sprocket on the pedal is 9 inches in diameter, the sprocket on the rear wheel is 2 inches in diameter, and the rear wheel is 26 inches in diameter, how fast is the bicycle traveling in miles per hour when the cyclist is pedaling at the rate of 0.9 revolutions per second? Round your answer to the nearest tenth mph. 59) 7.8 mph 20.8 mph 8.8 mph 22.8 mph Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 60) ) Convert the angle to degrees, minutes, and seconds. 6) ) Solve the problem. 62) The minimum length L of a highway sag curve can be computed by L = (θ 2 - θ )S 2 200(h + S tan α), where θ is the downhill grade in degrees (θ < 0 ), θ 2 is the uphill grade in degrees (θ 2 > 0 ), S is the safe stopping distance for a given speed limit, h is the height of the headlights, and α is the alignment of the headlights in degrees. Compute L for a 55-mph speed limit, where h =.9 ft, α =.0, θ = -5, θ 2 = 4, and S = 336 ft. Round your answer to the nearest foot. 62) 69 ft 638 ft 672 ft 654 ft 63) A ship travels 58 km on a bearing of 25, and then travels on a bearing of 5 for km. Find the distance from the starting point to the end of the trip, to the nearest kilometer. 63) 69 km 53 km 25 km 25 km 64) A fire is sighted due west of lookout A. The bearing of the fire from lookout B, 8.9 miles due south of A, is N W. How far is the fire from B (to the nearest tenth of a mile)? 64) 8. mi 7. mi 5. mi 6. mi A-0

11 65) The maximum monthly average temperature in Smithville is 87 in July, and the minimum is 37 in January. Determine f x = a cos b x - c + d so that f x models the monthly average temperature in Smithville, where x is the month and x = corresponds to January. 65) f x = 25 cos π 6 x f x = 25 cos π 6 x f x = 25 cos π 2 x f x = 62 cos π 6 x Find all solutions to the equation. 66) tan x = 3.7 (Use a calculator. Express your answer in radians, as a decimal rounded to the nearest thousandth.) 66) nπ n = 0, ±, ±2, nπ, nπ n = 0, ±, ±2, nπ n = 0, ±, ±2, nπ, nπ n = 0, ±, ±2,... Solve. 67) Points A and B are on opposite sides of a lake. A point C is 00.7 meters from A. The measure of angle BAC is 74 50', and the measure of angle ACB is determined to be 30 20'. Find the distance between points A and B (to the nearest meter). 67) 25 m 5 m 53 m 27 m Solve the problem. 68) A tower is supported by a guy wire 505 ft long. If the wire makes an angle of 39 with respect to the ground and the distance from the point where the wire is attached to the ground and the tower is 229 ft, how tall is the tower? Round your answer to the nearest tenth. 68) ft ft ft ft Provide an appropriate response. 69) Under which of the following conditions do we know that two triangles are congruent? (More than one may apply.) 69) (i) Three sides of one triangle are equal to the corresponding sides of the second triangle. (ii) Three angles of one triangle are equal to the corresponding angles of the second triangle. (iii) Two angles and the included side of one triangle are equal to the corresponding parts of the second triangle. (i), (iii) (i), (ii), (iii) (i) only (iii) only A-

12 Solve the problem. 70) Determine the resultant effect of three people pulling on a car as shown in the drawing. 70) a = 4.0 lb, b = 28.0 lb, c = 20.0 lb, d = 22, e = 6 Round results to an appropriate number of significant digits. 282 lb lb 0 70 lb lb -9 7) Find the work done by a force F of 35 pounds acting in the direction 2, 3 in moving an object 8 feet from (0, 0) to (8, 0). 7) foot-pounds foot-pounds 560 foot-pounds foot-pounds Solve the system graphically. 72) y = e2x+3 2x + y = 3 (0.75, -4.5) (-0.75, 4.5) (-0.75, -4.5) (0.75, 4.5) 72) Write the augmented matrix for the system. 73) -2x + 8y + 8z = 36 4x - 2y + 4z = -0-2x + 2y + 7z = 7 73) A-2

13 Find a matrix A and a column matrix B that describe the following tables involving credits and tuition costs. Find the matrix product AB, and interpret the significance of the entries of this product. 74) Credits College A College B Student 4 Student Cost Tuition College A $86 College B $65 AB = Tuition for Student 2 is $059 and tuition for Student is $906. AB = Tuition for Student is $053 and tuition for Student 2 is $893. AB = Tuition for Student 2 is $053 and tuition for Student is $893. AB = Tuition for Student is $059 and tuition for Student 2 is $ ) Write the system of equations associated with the augmented matrix. Do not solve ) x + 6y + 2z = -2 8x + 3z = 4 8x + 9z = 2 6x + 6y + 2z = -2 8x + 3z = 4 8x + 9y = 2 6x - 6y + 2z = -2 8x + 3z = -4 8x + 9y = -2 75) Solve the system of equations. 76) x +.5y = x + z = 8.5 y -2.75z = 4 76) No solution (5, 3, -4) (6, 3, -8.5) (5, 3, -3) Find the partial fraction decomposition. 2 77) x2(x + 3) = A x + B x2 + C x ) A = - 2 9, B = 2 3, C = 2 9 A = - 2 9, B = - 2 3, C = 2 9 A = 2 9, B = 2 3, C = A = 2 9, B = - 2 3, C = A-3

14 Find the indicated matrix product or state that the product is undefined ) A = 0 0 0, B = ) AB Find the inverse of A if it has one, or state that the inverse does not exist ) A = ) Inverse does not exist A-4

15 Answer Key Testname: PRACTICESEMPROB ) B 2) D 3) B 4) A 5) C 6) B 7) B 8) B 9) D 0) B ) D 2) B 3) A 4) B 5) B 6) A 7) A 8) B 9) D 20) C 2) A 22) D 23) D 24) A 25) D 26) D 27) B 28) C 29) A 30) C 3) B 32) B 33) D 34) D 35) A 36) D 37) D 38) B 39) D 40) C 4) C 42) D 43) B 44) A 45) B 46) A 47) B 48) C 49) C 50) D A-5

16 Answer Key Testname: PRACTICESEMPROB 5) D 52) C 53) D 54) A 55) A 56) A 57) A 58) A 59) C 60) D 6) C 62) D 63) D 64) C 65) A 66) A 67) C 68) B 69) A 70) A 7) A 72) B 73) A 74) D 75) B 76) B 77) A 78) A 79) C A-6