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1 Final Exam Math Precalculus NAME DATE ************************************************************************************* MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 1) f(x) = -x2 + 2x - 8 A) maximum; 1, - 7 B) minimum; 1, - 7 C) maximum; - 7, 1 D) minimum; - 7, 1 1) 2) f(x) = 3x2-3x 2) A) maximum; 1 2, B) maximum; - 1 2, C) minimum; 1 2, D) minimum; - 1 2, Use synthetic division and the Remainder Theorem to find the indicated function value. 3) f(x) = 6x4 + 3x3 + 2x2-7x + 20; f(3) A) 584 B) 1646 C) 780 D) 388 3) A-1

2 Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function. 4) f(x) = 5x3-3x2 + x A) 2 or 0 positive zeros, 1 negative zero B) 3 or 1 positive zeros, 2 or 0 negative zeros C) 2 or 0 positive zeros, no negative zeros D) 3 or 1 positive zeros, 1 negative zero 4) Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. x2 5) log 2 5) y6 A) 1 3 log 2 ( x y ) B) 2 log 2 x - 6 log 2 y C) 2 log 2 x + 6 log 2 y D) 6 log 2 y - 2 log 2 x Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. 6) 8 ln x ln y 6) A) ln x 8 y4 B) ln x 8 4 y C) ln x8 4 y D) ln x8y4 Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places 7) log A) B) C) D) ) A-2

3 Solve the exponential equation. Express the solution set in terms of natural logarithms. 8) 4x + 4 = 52x + 5 A) {ln 5 - ln 4} B) ln 5 5 C) 5 ln 5-4 ln 4 ln 4-2 ln D) {7 ln 5-5 ln 4} 8) Solve the problem. 9) The ph of a solution ranges from 0 to 14. An acid has a ph less than 7. Pure water is neutral and has a ph of 7. The ph of a solution is given by ph = - log x where x represents the concentration of the hydrogen ions in the solution in moles per liter. Find the hydrogen ion concentration if the ph = 2.4. A) 2.51 x 10-2 B) 3.98 x 10-2 C) 2.51 x 10-3 D) 3.98 x ) Use Newton's Law of Cooling, T = C + (T0 - C)ekt, to solve the problem 10) A casserole baked at 375 F is taken out of the oven into a kitchen that is 74 F. After 6 minutes, the temperature of the casserole is F. After how many minutes will its temperature be 240 F? Round your answer to the nearest minute. A) 21 min B) 18 min C) 12 min D) 24 min 10) Solve the problem. 11) What is the domain of the cosine function? A) all real numbers, except integral multiples of π (180 ) B) all real numbers from -1 to 1, inclusive C) all real numbers, except odd multiples of π 2 (90 ) 11) D) all real numbers A-3

4 12) What is the range of the sine function? A) all real numbers from -1 to 1, inclusive B) all real numbers greater than or equal to 0 C) all real numbers greater than or equal to 1 or less than or equal to -1 D) all real numbers 12) 13) The total sales in dollars of some small businesses fluctuates according to the equation S = A + B sin π x, where x is the time in months, with x = 1 corresponding to January, A = 8500, 6 13) and B = Determine the month with the greatest total sales and give the sales in that month. A) December; $11,400 B) September; $5600 C) June; $8500 D) March; $11,400 Use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. 14) tan θ = A) radians B) radians C) radians D) radians 14) Solve the problem. 15) The current I, in amperes, flowing through a particular ac (alternating current) circuit at time t seconds is 15) I = 220 sin 55πt - π 4 What is the period of the current? A) second B) 2 π second C) 55π seconds D) second A-4

5 Find the exact value of the expression. 16) tan-1 (- 3) 16) A) π 6 B) - π 6 C) - π 3 D) π 3 Find the exact value by using a difference identity. 17) tan 105 A) -2-3 B) C) D) ) Use trigonometric identities to find the exact value. tan 5 + tan 25 18) 1 - tan 5 tan 25 18) A) 2 B) 1 2 C) 3 D) ) tan tan (-20 ) 1 + tan 100 tan (-20 ) 19) A) B) C) - 3 D) -2 A-5

6 Use the graph to complete the identity. 20) cos 2x cos 5x + sin 2x sin 5x =? y 20) 3 x A) y = sin 3x B) y = sin 7x C) y = cos 7x D) y = cos 3x 21) sin 3x cos 3 2 x - cos 3x sin 3 2 x =? 21) y 3 x A) y = cos 9 2 x B) y = sin 3 2 x C) y = sin 9 2 x D) y = cos 3 2 x Use a graph in a [-2π, 2π, π ] by [-3, 3, 1] viewing rectangle to complete the identity. 2 22) 1-2 cos 2x 2 sin x - 1 =? A) 2 sin x + 1 B) cos x - 1 C) sin x + 1 D) 2 cos x ) A-6

7 Fill in the blank using the word product, sum, quotient, or difference. 23) The formula sin α - sin β = 2 sin α - β cos α + β can be used to change a 2 2 into the of a sine and a cosine expression. A) difference, quotient B) sum, product C) sum, quotient D) difference, product of two sines 23) Express the sum or difference as a product. 24) cos 9x - cos 3x A) 2 cos 3x B) 2 cos 6x cos 3x C) -2 sin 6x sin 3x D) -2 cos 6x sin 3x 24) Solve the equation on the interval [0, 2 π). 25) cos x + 2 cos x sin x = 0 A) 7π 6, 11π 6 C) π 2, 7π 6, 3π 2, 11π 6 B) 0, π 2, 7π 6, 3π 2 D) 7π 6, 11π 6, 2π 25) Find a. If necessary, round your answer to the nearest hundredth. 26) 26) A) B) 6.82 C) 9.19 D) 6.03 A-7

8 Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. 27) 27) A) A = 109, B = 32, C = 39 B) A = 39, B = 32, C = 109 C) A = 39, B = 109, C = 32 D) A = 109, B = 39, C = 32 Solve the problem. 28) A plane flying a straight course observes a mountain at a bearing of 31.3 to the right of its course. At that time the plane is 8 kilometers from the mountain. A short time later, the bearing to the mountain becomes How far is the plane from the mountain when the second bearing is taken (to the nearest tenth of a km)? A) 11.4 kilometers B) 10.2 kilometers C) 6.3 kilometers D) 4.2 kilometers 28) 29) Two sailboats leave a harbor in the Bahamas at the same time. The first sails at 23 mph in a direction 330. The second sails at 30 mph in a direction 210. Assuming that both boats maintain speed and heading, after 2 hours, how far apart are the boats? A) 88.8 miles B) 75 miles C) 92.1 miles D) 64.7 miles 29) Polar coordinates of a point are given. Find the rectangular coordinates of the point. 30) (-1, 360 ) A) (-1, 0) B) (0, 1) C) (1, 0) D) (0, -1) 30) A-8

9 The rectangular coordinates of a point are given. Find polar coordinates of the point. Express θ in radians. 31) (6, -6 3) A) 6, 5π 3 B) 12, 5π 3 C) 12, 11π 6 D) 6, 11π 6 31) Find the absolute value of the complex number. 32) z = -7 A) 14 B) 0 C) -7 D) 7 32) Write the complex number in rectangular form. 33) -5(cos i sin 120 ) A) i B) i C) i D) i 33) Find the product of the complex numbers. Leave answer in polar form. 34) z1 = 5(cos 20 + i sin 20 ) z2 = 4(cos 10 + i sin 10 ) A) 9(-cos i sin 200 ) B) 20(cos i sin 200 ) C) 20(cos 30 + i sin 30 ) D) 9(cos 30 + i sin 30 ) 34) A-9

10 Find the quotient z 1 of the complex numbers. Leave answer in polar form. z2 35) z1 = 30(cos 40 + i sin 40 ) z2 = 5(cos 7 + i sin 7 ) A) 25 cos i sin B) 6(cos 47 + i sin 47 ) 7 35) C) 25(cos 33 - i sin 33 ) D) 6(cos 33 + i sin 33 ) Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form. 36) 2(cos 15 + i sin 15 ) 4 A) i B) 16i C) i D) 8 + 8i 36) Use the dot product to determine whether the vectors are parallel, orthogonal, or neither. 37) v = i + 2j, w = i - 3j A) parallel B) orthogonal C) neither 37) Perform the matrix row operation (or operations) and write the new matrix. 38) R1 + R A) C) B) D) ) A-10

11 Solve the system of equations using matrices. Use Gaussian elimination with back -substitution. 39) x + y + z - w = 6 2x - y + 3z + 4w = -4 4x + 2y - z - w =-13 -x - 2y + 4z + 3w = 12 39) A) {( 1 4, - 1 3, - 1 5, 1 )} B) {(-4, 3, 5, -2)} 2 C) {(- 1 4, 1 3, 1 5, - 1 )} D) {(4, -3, -5, 2)} 2 Find the standard form of the equation of the ellipse and give the location of its foci. 40) 10 y 40) x A) x y 2 36 = 1 foci at (- 13, 0) and ( 13, 0) C) x y 2 36 = 1 foci at (-7, 0) and (7, 0) B) x y 2 49 = 1 foci at (- 13, 0) and ( 13, 0) D) x y 2 36 = 1 foci at (- 13, 0) and ( 13, 0) A-11

12 Find the vertices and locate the foci for the hyperbola whose equation is given. 41) x y = 1 A) vertices: (0, -11), (0, 11) foci: (- 265, 0), ( 265, 0) C) vertices: (-12, 0), (12, 0) foci: (- 265, 0), ( 265, 0) B) vertices: (-11, 0), (11, 0) foci: (- 265, 0), ( 265, 0) D) vertices: (-11, 0), (11, 0) foci: (-12, 0), (12, 0) 41) 42) x y = 1 A) vertices: (-11, 0), (11, 0) foci: (- 221, 0), ( 221, 0) C) vertices: (-10, 0), (10, 0) foci: (- 221, 0), ( 221, 0) B) vertices: (-11, 0), (11, 0) foci: (-10, 0), (10, 0) D) vertices: (0, -11), (0, 11) foci: (- 221, 0), ( 221, 0) 42) Find the standard form of the equation of the parabola using the information given. 43) Focus: (14, 0); Directrix: x = -14 A) y2 = -56x B) y2 = 14x C) x2 = 56y D) y2 = 56x 43) Use the Binomial Theorem to expand the binomial and express the result in simplified form. 44) (x2 + 4y)4 A) x8 + 16x6y + 96x4y x2y y4 B) x4 + 16x3y + 96x2y xy y4 C) x8 + 4x6y + 96x4y x2y y4 D) x8 + 16x6y + 96x4y2 + 16x2y y4 44) A-12

13 Solve the problem. 45) A restaurant offers a choice of 4 salads, 7 main courses, and 4 desserts. How many possible 3-course meals are there? A) 28 possible meals B) 112 possible meals C) 224 possible meals D) 15 possible meals 45) 46) How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? No digit can be used more than once. A) 151,200 B) 302,400 C) 210 D) ) 47) From 10 names on a ballot, a committee of 3 will be elected to attend a political national convention. How many different committees are possible? A) 360 B) 604,800 C) 120 D) ) Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. 48) lim x 0 x3-6x + 8 x - 2 A) does not exist B) 0 C) 4 D) -4 48) 49) lim x 7 x - 7 x ) A) does not exist B) C) D) 0 A-13

14 50) lim x 1 x3 + 5x2 + 3x - 9 x - 1 A) 16 B) -16 C) does not exist D) 0 50) A-14

15 Answer Key Testname: UNTITLED1 1) A 2) C 3) A 4) A 5) B 6) B 7) D 8) C 9) D 10) B 11) D 12) A 13) D 14) B 15) B 16) C 17) A 18) D 19) C 20) D 21) B 22) A 23) D 24) C 25) C 26) A 27) D 28) C 29) C 30) A 31) B 32) D 33) B 34) C 35) D 36) A 37) C 38) A 39) B 40) A 41) B 42) A 43) D 44) A 45) B 46) D 47) c 48) D 49) 50) B A A-15