Intermediate Algebra and Statistics UPCAT Review PDF
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Leopold Laset
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This document is a math review for the UPCAT exam, providing practice problems and explanations for intermediate algebra and statistics topics. It covers various topics, such as functions, polynomial functions and more.
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REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 39 | P a g e UPCAT Review – Volume 5 – Intermeiate Algebra Downloadable e-Book TM...
REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 39 | P a g e UPCAT Review – Volume 5 – Intermeiate Algebra Downloadable e-Book TM Copyright © 2015 Review Masters. REVIEW MASTERS, upcatreview.com, Online UPCAT Review and other related indicia are trademarks of KB Tutorial and Consulting Corp. All rights reserved worldwide This book cannot be redistributed without permission from KB Tutorial and Consulting Corp. More info at: https://www.upcatreview.com For INFO and UPDATES, check out upcatreview.com, the official website of Review Masters and the Online UPCAT Review® Keep updated about UPCAT materials, reviewers, tips and news TM Copyright © 2015 Review Masters - https://www.upcatreview.com REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com PREFACE Believe That You Can Pass the UPCAT! by Leopold Laset Do you sometimes find it hard to believe that your dream to pass the UPCAT can become a reality? If so, then there is something very important that you need to know. UPCAT is for dreamers like you. Every student who passed the UPCAT began thinking or dreaming of passing the UPCAT. Your near-perfect or perfect score in a quarterly test, your cellphone, PSP, or any gadget, your out- of-town (or out-of-country) vacation, your new pair of shoes, and any other stuff that you desired and now possess - are all the result of your ‘dream come true’. What this means is that throughout your lifetime, you have had an idea, you have desired for many things and worked hard for them, overcome problems and ultimately transformed your dream into reality. And if hundreds and thousands of students have been able to pass the UPCAT in the past, by starting with a dream, then it stands to reason, that you can do it too. Often we make the mistake of thinking that UPCAT is for a small number of bright students who have the brains and intelligence that we don’t possess. But this is simply not true. The fact that thousands of average students have brought their dreams of passing the UPCAT to fruition in the past demonstrates that the opportunity to qualify in the UPCAT is something that is available to each UPCAT aspirant – average or bright. Right now, hundreds of UPCAT dreamers are taking the steps necessary to achieve the goals of passing the UPCAT. Some are studying this early, some are joining community of fellow dreamers, and some are attending review classes. What is it that you need to do? In order to achieve your goal of passing the UPCAT, the only things you really need are: (1) A crystal clear picture that you already passed the UPCAT (2) An unshakeable determination to do whatever it takes to make your dream of passing the UPCAT a reality 1|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com As soon as you take these two steps, passing the UPCAT becomes achievable. If you need a help – you look for it. If you encounter a difficult concept – you find a way to understand it. If you can’t solve a math problem – you try and try and practice more. And gradually, step-by-step, you bring your UPCAT dream into reality to join the dreams of the thousands of UPCAT dreamers who have gone before you. So today I’d like to encourage you to believe in yourself and appreciate the fact that you live in a world where ‘dreams do come true’. Understand that thousands of students have made their UPCAT dream a reality in the past – Thousands more will make their UPCAT dream a reality in the near future and you CAN be one of them. 2|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com TABLE OF CONTENTS page REVIEW TEST CHAPTER 1. Functions --------------------- 5 CHAPTER 2. Polynomial Functions --------------------- 6 CHAPTER 3. Exponential Functions --------------------- 7 CHAPTER 4. Logarithmic Functions --------------------- 8 CHAPTER 5. Systems --------------------- 9 CHAPTER 6. Matrices and Determinants --------------------- 10 CHAPTER 7. Arithmetic Sequence and Series --------------------- 11 CHAPTER 8. Geometric Sequence and Series --------------------- 12 CHAPTER 9. Basic Counting Principle --------------------- 13 CHAPTER 10. Probability --------------------- 14 SOLUTIONS AND EXPLANATIONS CHAPTER 1. Functions --------------------- 16 CHAPTER 2. Polynomial Functions --------------------- 19 CHAPTER 3. Exponential Functions --------------------- 22 CHAPTER 4. Logarithmic Functions --------------------- 24 CHAPTER 5. Systems --------------------- 26 CHAPTER 6. Matrices and Determinants --------------------- 29 CHAPTER 7. Arithmetic Sequence and Series --------------------- 31 CHAPTER 8. Geometric Sequence and Series --------------------- 33 CHAPTER 9. Basic Counting Principle --------------------- 35 CHAPTER 10. Probability --------------------- 37 3|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com ANSWER SHEET – ALGEBRA 2 Correct Filling A B C D A B C D A B C D A B C D A B C D Incorrect Filling A B C D A B C D A B C D Please use No. 2 Pencil 4|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 1: Functions x 1 1. Find the domain of the following rational function: f(x) =. x2 A. {xx –2} B. {xx –1} C. {xx 1} D. {xx 2} x 1 2. Find the range of the following rational function: f(x) =. x2 A. {yy –2} B. {yy –1} C. {yy 1} D. {yy 2} When f(x) = -2x + 3, g(x) = x – 3, x R, find g(x) – f(x). 2 3. 2 2 2 2 A. x + 2x – 6 B. x – 2x C. x – 2x + 6 D. x + 2x -1 4. Find an expression for f (x) if f(x) = -2x + 3. 𝑥−3 −𝑥+3 𝑥 𝑥 A. B. C. − + 3 D. −3 2 2 2 2 When f(x) = 3x –2x +1, x R, find f(x+1). 2 5. 2 2 2 2 A. x +3 B. 3x – 2x + 6 C. 3x + 4x + 2 D. 3x –2x + 2 𝒙−𝟏 6. When f(x) = 2x + 5 and g(x) = , x R, find (g f)(x). 𝒙+𝟐 2𝑥+4 𝑥+4 𝑥+2 4 A. B. C. D. 𝑥 + 2𝑥+7 7 7 7 7. Let f be the function specified by following table. Find f(1) – f(-2) x -2 -1 0 1 2 f(x) 6 3 2 3 6 A. -3 B. 3 C. 9 D. 14 8. Which of the following is not a graph of a function? A. B. C. D. When f(x) = 2x + 5 and g(x) = x + 1, x R, find (f g)(0). 2 9. A. 1 B. 5 C. 6 D. 7 𝒙−𝟓 -1 10. If f(x) = , which of the following points is located on the graph of f (x)? 𝟑 A. (-4, 3) B. (-1, 2) C. (2, -1) D. (8, 1) 5|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 2: Polynomial Functions 3 2 11. The zeros of the polynomial function x –5x –x + 5 are: A. -5, -1 and 1 B. 5, -1, and -5 C. 5, 1, and -5 D. 5, 1 and -1 4 2 12. What is the remainder when f(x) = 4x – 5x +2x – 5 is divided by x – 1? A. -4 B. -8 C. 4 D. 8 3 2 13. If x -x – 9x +9 = 0, which of the following is not a root of the equation? A. -3 B. 1 C. 3 D. 9 14. Which of the following polynomial function having zeros 1 of multiplicity 2 and –2 and such that f(0) = 6? 2 A. f(x) = 2(x–1)(x+2) + 6 C. f(x) = 3(x–1) (x+2) B. f(x) = (x–1)((x–2) + 6 D. f(x) = 3(x+1)(x–2) 15. Which of the following polynomial function has a degree of 6? 3 2 2 2 6 A. f(x) = (x+2) (2x+1) (4x–1) C. f(x) = (x –5) (x –1)(5x+1) 6 5 2 3 2 6 B. f(x) = (x–3) (x+1) (x+6) D. f(x) = (x +4)(2x –1) (x ) 4 3 2 16. Which of the following is not a possible natural root of 2x –5x +21x +9x–3 = 0? A. ±3 B. ±1 C. ±1/2 D. ±2/3 3 2 17. What is the quotient when f(x) = x – x –10x - 8 is divided by x –1? 3 2 2 A. x -2x –12x – 6/(x–1) C. x –10x+8/(x–1) 2 2 B. x –10 – 18/(x–1) D. x –2x –12 – 10/(x-1) 18. The first polynomial is a factor of the second polynomial in all of the following except _____. 3 2 3 2 A. x–3; x +x –9x–9 C. x+1; x + x +x+1 3 3 2 B. x+4; x –12x +16 D. x–1; x +5x –6x –2 3 2 19. By the remainder theorem, if f(x) = x +3x –x–5, the following are true except _____. A. f(3) = 46 B. f(-2)= 1 C. f(-3) = 2 D. f(4) = 103 𝟐𝒙 20. Find the horizontal and vertical asymptotes of the equation of the equation y=. 𝟑𝒙−𝟏 A. x =1/3 and y=3/2 C. x =2/3 and y= -2 B. x =1/3 and y=2/3 D. x= 3/2 and y = 0 6|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 3: Exponential Functions 𝟑 21. √𝟑𝟐𝒚𝟐 in exponential form is _____. 2 2/3 2/3 2/3 A. 2y B. 2(2y) C. (2y) D. 32 y -3.5 22. Evaluate: 4 A. 14 B. 1/18 C. 17 D. 1/128 𝟑 𝟗 23. Simplify and express answer in exponential form: √𝒙𝟐 √𝒙𝟒 10/9 2/9 6/27 6/11 A. x B. x C. x D. x 3/5 24. Evaluate : (-32) A. -2 B. -4 C. -6 D. -8 𝟑 25. Evaluate: √√𝟐𝟓𝟔 3 3 3 6 A. 2 √2 B. √24 C. √16 D. √256 x 26. Solve for x: 32 = 4 A. 1/8 B. 2/5 C. 5/2 D. 8 2x-2 2x+5 27. Solve for x: 5 = 25 A. -7 B. -6 C. 0 D. 3 2x+1 x+2 28. Solve for x: 9 = 27 A. 1 B. 2 C. 3 D. 4 𝒙+𝟔 29. Solve for x: (𝟏𝟐) = 𝟏𝟔−𝒙 A. -8 B. -2 C. 2 D. 8 𝟐 +𝟒𝒙 30. Solve for x: 𝟑𝒙 𝟏 = 𝟐𝟕 A. 3, 1 B. 3, -1 C. -3, 1 D. -3, -1 7|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 4: Logarithmic Functions 31. The equation 𝒍𝒐𝒈𝒃 𝒎 = 𝒏 express in exponential form is _____. n b m n A. m = b B. m = n C. b = n D. b = m 32. The equation 𝟓𝟑/𝟐 = 𝟓√𝟓 express in logarithmic form is ____. 3 3 A. 𝑙𝑜𝑔5√5 5 = B. 𝑙𝑜𝑔5√5 (32) = 5 C. 𝑙𝑜𝑔3 5√5 = 5 D. 𝑙𝑜𝑔5 5√5 = 2 2 2 33. Evaluate: 𝒍𝒐𝒈𝟕 𝟕√𝟕 A. √7 B. 3/2 C. 2 D. 3 𝒍𝒏𝟓𝒆 34. Evaluate: 𝟒+𝟒𝒍𝒏𝟓 A. 0.25 B. 0.75 C. 1.25 D. 2.00 35. Solve for x: 𝒍𝒐𝒈𝟐 𝟗𝟒 = 𝒙 + 𝟑 𝟑 A. -1 B. -3 C. -5 D. -9 x x+1 36. Solve for x: 4 + 2 – 24 = 0 A. -6 B. -3 C. 2 D. 4 For numbers 37 and 38: Given: log5 2 = 0.4 log5 3 = 0.7 37. What is the value of log5 9/4 ? A. 0.3 B. 0.6 C. 0.9 D. 1.1 38. What is the value of log5 √𝟔 ? A. 0.30 B. 0.55 C. 0.90 D. 1.10 39. Which of the following is false? A. 4𝑙𝑜𝑔42 = 2 C. 𝑙𝑜𝑔2 (𝑥 + 3)5 = 5[𝑙𝑜𝑔2 𝑥 + 𝑙𝑜𝑔2 3] 𝑥𝑦 2 B. 𝑙𝑜𝑔3 5 = 𝑙𝑜𝑔10 5 ÷ 𝑙𝑜𝑔10 3 D. 𝑙𝑜𝑔5 √ = 12[𝑙𝑜𝑔5 𝑥+2𝑙𝑜𝑔5 𝑦−𝑙𝑜𝑔5 𝑧] 𝑧 𝒍𝒐𝒈𝟑 (𝒙+𝟏) 40. Solve for x: =𝟐 𝒍𝒐𝒈𝟑 (𝒙−𝟏) A. 0 B. 1 C. 2 D. 3 8|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 5: Systems For numbers 41-43: Refer to the following graphs of systems of equation and the choices. I II III A. I only B. II only C. III only D. I and II 41. The system is consistent and the equation are independent. 42. The system is inconsistent. 43. The equations are dependent. x = 5 – 2y 44. Which of the following is the solution of the system ? 2x + 4y = 6 A. (1, 2) B. (11, -3) C. (10, 0) D. no solution 4 x 13 y 8 3 45. Which of the following is the solution of the system ? 1 2 x 34 y 52 A. (2, 3) B. (1, 20) C. (3, -28) D. (1, -4) x 13 y y 2 46. Which of the following is the solution of the system ? 1 4 x y x y A. (-1, 3/2) B. (0, 3) C. (1, 9/2) D. (3, -2) x + y = -1 47. Which of the following is the solution of the system y + z = 4 ? x+z=1 A. (1, -2, 6) B. (-1, 0, 4) C. (-2, 1, 3) D. (0, -1, 3) 48. Danica has twice as many P1 coin as P5 coin in her purse. If the total amount of her P1 and P5 coins is P21. How many of each type of coin does she have? A. Two P5 coins and eleven P1 coins C. Four P5 coins and two P1 coins B. Three P5 coins and six P1coins D. Five P5 coins and one P1 coin 3y = 2x 49. What value of m will make the system 6y - m – 4x = 0 , a dependent equation? A. 0 B. 2 C. 3 D. -4 𝐲 < 2𝑥 − 5 50. Which of the following is the solution of the system ? 𝐲 ≥ −𝐱 − 𝟐 A. B. C. D. 9|Page REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 6: Matrices and Determinants For numbers 51-53: 4 6 5 2 15 𝐴 = [3 4 7] 𝐵 = [2 −3 −7] 𝐶 = [−2 0 5] 𝐷 = [−4] 𝐸 = [ 10 ] 1 0 9 5 −5 51. Which of the following is the sum of the row matrices? A. [5 4 12] B. [1 4 2] C. [−5 4 2] D. [1 4 12] 52. Solve: 3D – E −9 21 17 −21 A. [−22] B. [−2] C. D. [−22] 20 10 8 10 53. Find the difference of matrix A and the additive inverse of Matrix C. A. [5 4 2] B. [5 4 12] C. [1 4 12] D. [1 4 2] 54. Find the value of x and y: x[3 5] + y[4 -2] = 2[-5 9] A. x= 3, y = 4 B. x = 5, y = -2 C. x = -5, y = 3 D. x = 2, y = -4 𝒙 −𝒙 −𝟏 −𝒚 𝒚 𝒙 55. Find the value of x and y: [−𝒚 −𝟏 𝒙 ] = [ 𝒙 −𝟏 −𝟏] 𝟏 −𝟏 𝒚 𝒚 𝒙 𝟏 A. x = -1, y = 1 B. x = -1, y = -1 C. x =1, y = -1 D. x = 1, y = 1 𝟓 −𝟒 56. Find the value of the determinant: | | 𝟑 𝟐 A. 120 B. 22 C. 6 D. -2 𝟒 𝟐 57. Find the value of x: | | = −𝟏𝟒 𝒙 −𝟔 A. 6 B. -5 C. -4 D. -14 𝟏 𝟐 𝟑 58. The product of AB if 𝑨 = [ ] and 𝑩 = [ ] is _____. 𝟑 −𝟏 −𝟒 2 3 −5 6 A. [ ] B. [ ] C. [ ] D. [ ] 1 10 13 −2 𝟏 −𝟏 59. The inverse of the matrix [ ] is _____? 𝟑 −𝟏 𝟏 𝟏 −2 3 −1 2 3 −2 − 𝟐 𝟐 A. [ ] B. [ ] C. [ ] D. [ ] 1 −1 1 −3 −1 1 − 𝟑 𝟐 𝟏 𝟐 𝟐 −𝟑 𝟏 60. Find the value of the determinant: |𝟏 −𝟏 𝟏| 𝟏 −𝟏 −𝟐 A. -3 B. -1 C. 6 D. 12 10 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 7: Arithmetic Sequence and Series 61. What is the next term in the sequence: 4, 7, 10, 13, ____? A. 15 B. 16 C. 17 D. 18 62. The first three terms of a sequence is -1, 4, 9… What is the 14th term of the sequence? A. 64 B. 69 C. 74 D. 79 63. The first three terms of a sequence are -3, 1, 5 … if the last term is 101. How many terms are there in the sequence? A. 24 B. 25 C. 26 D. 27 64. Which of the following sequence has a general term 4n – 3? A.. -3, 4, 11, 18 … B. -3, 1, 5, 9… C. 1, 5, 9, 13 … D. 5, 9 13, 18 … 65. The first and the third term of an arithmetic sequence is 10 and 4 respectively. What is the sum of the first ten terms, s10 of the sequence? A. -17 B. -18 C. -25 D. -35 66. What is the general term of an arithmetic sequence whose first three terms are 3, 9, 15 … A. 6n + 3 B. 6n – 3 C. 3(2n +3) D. 3(2n – 3) 67. The fourth and eight term of an arithmetic sequence is 27 and 51 respectively. What is the sixth term of the sequence? A. 39 B. 41 C. 43 D. 45 68. How many multiples of 4 is between 10 and 950? A. 225 B. 235 C. 250 D. 500 69. Find the sum of all odd integers between 1 and 99? A. 2200 B. 2300 C. 2400 D. 2500 70. Evaluate: ∑𝟕𝒏=𝟏(𝟑𝒏 + 𝟏) A. 91 B. 95 C. 99 D. 103 11 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 8: Geometric Sequence and Series 71. Find the 10th term of the sequence 32, -16, 8, -4, …. A. -1/2 B. -1/4 C. -1/8 D. -1/16 72. Find the 6th term of a geometric sequence if the first term, a1 is 27 and the common ratio, r is 1/3. A. 1/9 B. 1/3 C. 3 D. 9 73. Which of the following is the mean proportional (single geometric mean) between 8 and 32? A. 12 B. 16 C. 24 D. 28 74. What is the sum of the first six terms of the sequence 8, -4, 2, -1 ? A. 18 ¾ B. 18 ¼ C. 5 ¼ D. 5 ¾ 75. What is the general term of the sequence having 3, -6, 12, … as its first three terms? n 3 3 n-1 A. 3(-2) B. (2)𝑛 C. (−2)𝑛−1 D. 3(-2) 2 2 76. Find the first term of a geometric series if the sum up to the eight term, s 8 is 1020 and the common ratio, r is 2. A. 1 B. 2 C. 4 D. 8 77. Find the sum up to the seventh term, s 7 of the geometric series having 2 as its first term, a1 and 1458 as its last term with a common ratio, r, which is equal to 3. A. 2186 B. 2556 C. 2916 D. 3286 78. What is the sum of the infinite geometric series 1, -1/3, 1/9, …. A. 1/3 B. ½ C. 2/3 D. 3/4 𝟑𝒏 79. Evaluate: ∑∞ 𝒏=𝟏 𝟒𝒏−𝟏 A. 12 B. 4 C. 3 D. 3/4 80. A ball bounces from the ground 27 cm into the air. Each time it bounces, it rises 1/3 the height of the previous bounce. How far will the ball travel if it continues until it comes to rest. A. 81cm B. 67.5 cm C. 54 cm D. 40.5 cm 12 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 9: Basic Counting Principle 𝟔!−𝟑! 81. What is the value of 𝟑! ? A. 119 B. 120 C. 719 D. 720 82. Evaluate: 5P2 A. 120 B. 60 C. 20 D. 10 83. Evaluate: 5C2 A. 120 B. 60 C. 20 D. 10 84. How many three digit numbers can be formed from the integers 4, 5, 6, 7, 8, and 9 if the number is even and repetition is not allowed? A. 120 B. 90 C. 75 D. 60 85. How many 4 digit zip code may be formed if the first digit can’t be a zero and the last digit is an odd number. A. 45102 B. 14102 C. 9103 D. 104 86. A company wants to hire five women and 3 men. In how many ways can the choice be made if seven women and three men are available? A. 10 B. 21 C. 35 D. 63 87. In how many ways can the letters in the word “REVIEW” be arranged? A. 15 B. 30 C. 120 D. 360 88. Find the number of ways in which three of 10 participating contestants be ranked first, second and third. A. 140 B. 240 C. 520 D. 720 89. In how many ways can seven students be seated at a round table if two of them must sit next to each other? A. 10 B. 20 C. 240 D. 1040 90. In how many ways can five different gifts be given among two girls and one boy if the two girls received 2 gifts and 1 gift for the boy? A. 10 B. 20 C. 30 D. 120 13 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 10: Probability 91. What is the probability that a card drawn from an ordinary deck of 52 cards is a face card? A. ¼ B. 1/13 C. 3/13 D. 4/13 92. A two digit number is chosen randomly. What is the probability that it is divisible by 9? A. 1/10 B. 1/9 C. 9/45 D. 4/50 93. A bag contains 3 red balls, 5 green balls and 4 blue balls. The probability of not getting a blue ball in the first draw is _____. A. 0 B. 1/3 C. 2/3 D. 1 94. Danny and Ricky each throw two dice. If Danny get a sum of 5, what is the probability that Ricky will get less? A. 1/6 B. 2/9 C. 2/3 D. 4/9 95. From a box containing 4 green marbles and 3 blue marbles, two marbles are drawn one at a time at random. The probability that both are blue is _____. A. 1/7 B. 3/14 C. 6/49 D. 9/49 96. In the problem above, find the probability that one is green and one is blue. A. 4/7 B. 2/7 C. 12/49 D. 24/49 97. In the problem in #95, find the probability that all are the same color. A. 2/7 B. 3/7 C. 2/49 D. 3/49 98. If five coins are tossed simultaneously, find the probability that they will have 3 heads A. 1/16 B. 3/32 C. 5/16 D. 5/32 99. The probability of getting a passing mark in an examination is 2/3. If 3 students are selected at random, what is the probability that at least one of them will get a passing mark? A. 1/27 B. 8/27 C. 19/27 D. 26/27 100. A card is chosen from a deck of playing cards. What is the probability that it is either black or a face card? A. 8/13 B. 10/13 C. 8/15 D. 19/26 14 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 15 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 1: Functions x 1 1. Find the domain of the following rational function: f(x) =. x2 A. {xx –2} B. {xx –1} C. {xx 1} D. {xx 2} The domain of a rational function is the set of all real numbers such that the denominator would not be equal to zero. Thus, for 𝒙+𝟏 𝒇(𝒙) = 𝒙−𝟐 x–20 or x2 The answer is D. x 1 2. Find the range of the following rational function: f(x) =. x2 A. {yy –2} B. {yy –1} C. {yy 1} D. {yy 2} First, let’s solve for x in terms of y: 𝒙+𝟏 𝒙+𝟏 𝒇(𝒙) = 𝒙−𝟐 𝒚= 𝒙−𝟐 𝒙𝒚 − 𝟐𝒚 = 𝒙 + 𝟏 𝒙𝒚 − 𝒙 = 𝟐𝒚 + 𝟏 𝒙(𝒚 − 𝟏) = 𝟐𝒚 + 𝟏 𝟐𝒚+𝟏 𝒙= 𝒚−𝟏 The denominator on the right side is y – 1. Since y – 1 0 y 1. The answer is C. When f(x) = -2x + 3, g(x) = x – 3, x R, find g(x) – f(x). 2 3. A. x2 + 2x – 6 B. x2 – 2x C. x2 – 2x + 6 D. x2 + 2x We simply subtract f(x) from g(x): 𝒈(𝒙) − 𝒇(𝒙) = 𝒙𝟐 − 𝟑 − (−𝟐𝒙 + 𝟑) = x2 – 3 + 2x – 6 or x2 + 2x – 6 The answer is A. -1 4. Find an expression for f (x) if f(x) = -2x + 3. 𝑥−3 −𝑥+3 𝑥 𝑥 A. 2 B. 2 C. − 2 + 3 D. 2 −3 We first solve x in terms of y: −𝒚+𝟑 𝒇(𝒙) = −𝟐𝒙 + 𝟑 𝒙= 𝟐 y = -2x + 3 2x = -y + 3 −𝒙+𝟑 −𝒙+𝟑 𝒚= 𝒇−𝟏 (𝒙) = 𝟐 𝟐 Replace x by y and y by x: or The answer is B. 16 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com When f(x) = 3x –2x +1, x R, find f(x+1). 2 5. 2 2 2 2 A. x +3 B. 3x – 2x + 6 C. 3x + 4x + 2 D. 3x –2x + 2 𝒇(𝒙 + 𝟏) = 𝟑(𝒙 + 𝟏)𝟐 − 𝟐(𝒙 + 𝟏) + 𝟏 Replace x in f(x) by x + 1: = 3(x2 + 2x + 1) – 2x – 2 + 1 = 3x2 + 6x + 3 – 2x – 2 + 1 or 3x2 + 4x + 2 The answer is C. 𝒙−𝟏 6. When f(x) = 2x + 5 and g(x) = , x R, find (g f)(x). 𝒙+𝟐 2𝑥+4 𝑥+4 𝑥+2 4 A. B. C. D. 𝑥 + 2𝑥+7 7 7 7 𝟐𝒙+𝟓−𝟏 𝟐𝒙+𝟒 (𝒈o f)(𝒙) = 𝒈(𝒇(𝒙)) = 𝒈(𝟐𝒙 + 𝟓) = = 𝟐𝒙+𝟓+𝟐 𝟐𝒙+𝟕 The answer is A. 7. Let f be the function specified by following table. Find f(1) – f(-2) x -2 -1 0 1 2 f(x) 6 3 2 3 6 A. -3 B. 3 C. 9 D. 14 𝒇(𝟏) = 𝟑 𝒂𝒏𝒅 𝒇(−𝟐) = 𝟔 𝒇(𝟏) = − 𝒇(−𝟐) = 𝟑 − 𝟔 = -3 From the table, Thus, The answer is A. 8. Which of the following is not a graph of a function? A. B. C. D. Vertical Line Test A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point. Only letter B does NOT satisfy the Vertical Line Test. The answer is B. When f(x) = 2x + 5 and g(x) = x + 1, x R, find (f g)(0). 2 9. A. 1 B. 5 C. 6 D. 7 (𝒇o g)(𝟎) = 𝒇(𝒈(𝟎)) = 𝒇((𝟎)𝟐 + 𝟏) = 𝒇(𝟏) = 𝟐(𝟏) + 𝟓 = 𝟕 The answer is D. 17 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 𝒙−𝟓 -1 10. If f(x) = , which of the following points is located on the graph of f (x)? 𝟑 A. (-4, 3) B. (-1, 2) C. (2, -1) D. (8, 1) 𝒙−𝟓 𝒙−𝟓 𝒇(𝒙) = 𝒚= 𝟑𝒚 = 𝒙 − 𝟓 𝒙 = 𝟑𝒚 + 𝟓 𝟑 𝟑 We first solve x in terms of y: 𝒚 = 𝟑𝒙 + 𝟓 𝒇−𝟏 (𝒙) = 𝟑𝒙 + 𝟓 Replace x by y and y by x: or Let’s see if which among the choices satisfies the equation: A. (-4, 3) means x = -4 and y = 3: y = 3x + 5? 3 = 3(-4) + 5? No! B. (-1, 2) means x = -1 and y = 2 y = 3x + 5? 2 = 3(-1) + 5? Yes! Try also C and D and you’ll find that they do not satisfy the equation. The answer is B. ALGEBRA TIP: Studying math for 3 hours on one night is not as effective as studying for 1 hour on each of three nights. Either way the total is 3 hours of studying. Why is the second method better for your memory? The answer is time. Your brain needs time between learning sessions to continue to process the information subconsciously. This is why cramming the night before a math test doesn’t work. The best learning, and remembering, happens when you study every day, rather than once or twice a week. 18 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 2: Polynomial Functions 3 2 11. The zeros of the polynomial function x –5x –x + 5 are: A. -5, -1 and 1 B. 5, -1, and -5 C. 5, 1, and -5 D. 5, 1 and -1 x3 – 5x2 – x + 5 = (x3 – 5x2) – (x – 5) Regroup x2(x – 5) – (x – 5) Factor out x 2 (x – 5)(x2 – 1) Factor out x – 5 (x – 5)(x + 1)(x – 1) Factor x2 – 1 Zeros: x = 5, x = -1, x = 1 or 5, 1, and -1 The answer is D. 4 2 12. What is the remainder when f(x) = 4x – 5x +2x – 5 is divided by x – 1? A. -4 B. -8 C. 4 D. 8 According to the Remainder theorem: If a polynomial function f(x) is divided by x – c, the remainder is f(c). Thus if f(x) = 4x4 – 5x2 +2x – 5 is divided by x – 1, the remainder is: f(1) = 4(1)4 – 5(1)2 +2(1) – 5 = 4 – 5 + 2 – 5 = -4 The answer is A. 3 2 13. If x -x – 9x +9 = 0, which of the following is not a root of the equation? A. -3 B. 1 C. 3 D. 9 x3 -x2 – 9x +9 = 0 (x3 -x2) – (9x – 9) = 0 Regroup x2(x – 1) – 9(x – 1) = 0 Factor out x2 and 9 (x – 1)(x – 9) = 0 2 Factor out x – 1 (x – 1)(x + 3)(x – 3) Factor x2 – 9 Roots: x = 1, x = -3. x = 3 Therefore, the answer is D. 9. 14. Which of the following polynomial function having zeros 1 of multiplicity 2 and –2 and such that f(0) = 6? 2 A. f(x) = 2(x–1)(x+2) + 6 C. f(x) = 3(x–1) (x+2) B. f(x) = (x–1)((x–2) + 6 D. f(x) = 3(x+1)(x–2) Let’s examine each of the choices: C. Is f(1) = 0? f(1) = 3(1 – 1)2(1 + 2) = 0 Yes A. Is f(1) = 0? Is f(-2) = 0? f(1) = 2(1 – 1)(1 + 2) + 6 = 6 0 No f(-2) = 3(-2 – 1)2(-2 + 2) = 0 Yes Is f(0) = 6? B. Is f(1) = 0? f(0) = 3(0 – 1)2(0 + 2) = 6 Yes f(1) = (1 – 1)(1 – 2) + 6 = 6 0 No D. Is f(1) = 0? f(1) = 3(1+ 1)(1 – 2) = - 6 0 No The answer is C. Note: The term “multiplicity2” will give you a strong hint that C is the most probable answer without testing each of the choices because it is the only expression with a binomial squared (x -1)2. 19 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 15. Which of the following polynomial function has a degree of 6? 3 2 2 2 6 A. f(x) = (x+2) (2x+1) (4x–1) C. f(x) = (x –5) (x –1)(5x+1) 6 5 2 3 2 6 B. f(x) = (x–3) (x+1) (x+6) D. f(x) = (x +4)(2x –1) (x ) The degree of a polynomial function is the highest exponent of the function when written in expanded or general form. If the polynomial is written in factored form, it is the sum of all the exponents of the variable. Let’s add the exponents of each factored term or expression in each of the choices: A. 3 + 2 + 1 = 6 C. 2(2) + 6 + 1 = 11 B. 6 + 5 + 1 = 12 D. 2 + 3(2) + 6 = 14 The answer is A. 4 3 2 16. Which of the following is not a possible natural root of 2x –5x +21x +9x–3 = 0? A. ±3 B. ±1 C. ±1/2 D. ±2/3 According to the Rational Roots Theorem, the possible roots of a polynomial function are p/q where p are integral factors of the constant term and q are the integral factors of the leading coefficient. Thus, for the polynomial 2x4–5x3+21x2+9x–3 = 0, p = 3, 1 q = 2, 1 Therefore the possible roots are: 3/2, 1/2, 3, 1 The answer is D. 3 2 17. What is the quotient when f(x) = x – x –10x - 8 is divided by x –1? 3 2 2 A. x -2x –12x – 6/(x–1) C. x –10x+8/(x–1) 2 2 B. x –10 – 18/(x–1) D. x –2x –12 – 10/(x-1) By Synthetic Division, -1 | 1 -1 -10 -8 -1 0 10 1 0 -10 -18 Thus, the quotient is x2 – 10 with a remainder of -18 or x2 – 10 – 18/(x – 1) The answer is B. 18. The first polynomial is a factor of the second polynomial in all of the following except _____. 3 2 3 2 A. x–3; x +x –9x–9 C. x+1; x + x +x+1 3 3 2 B. x+4; x –12x +16 D. x–1; x +5x –6x –2 According to the Factor Theorem: x – c is a factor of the polynomial function f(x) if f(c) = 0. A. f(3) = (3)3+(3)2–9(3)–9 = 27+9-27-9 = 0 Yes B. f(-4) = (-4)3–12(-4)+16 = -64+48+16 = 0 Yes C. f(-1) = (-1)3+(-1)2+(-1)+1 = -1+1-1+1 = 0 Yes D. f(1) = (1)3+5(1)2–6(1)–2 = 1+5-6-2 = -2 No The answer is D. 20 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 3 2 19. By the remainder theorem, if f(x) = x +3x –x–5, the following are true except _____. A. f(3) = 46 B. f(-2)= 1 C. f(-3) = 2 D. f(4) = 103 A. f(3) = 33+3(3)2 –3–5 = 27+27-3-5 = 46 True B. f(-2) = (-2)3+3(-2)2 –(-2)–5 = -8+12+2-5 = 1 True C. f(-3) = (-3)3+3(-3)2 –(-3)–5 = -27+27+3-5 = -2 False D. f(4) = 43+3(4)2 –4–5 = 64+48-4-5 = 103 True The answer is C. 𝟐𝒙 20. Find the horizontal and vertical asymptotes of the equation of the equation y=. 𝟑𝒙−𝟏 A. x =1/3 and y=3/2 C. x =2/3 and y= -2 B. x =1/3 and y=2/3 D. x= 3/2 and y = 0 To find the vertical asymptote, equate the denominator to 0, then solve for x: 3x – 1 = 0 3x = 1 x = 1/3 Thus, the vertical asymptote is x = 1/3. To find the horizontal asymptote, solve first for x in terms of y, then equate the denominator of the resulting rational expression to 0: y = 2x/(3x – 1) y(3x – 1) = 2x Multiply by 3x - 1 3xy – y = 2x Distribute y 3xy – 2x = y Transpose x(3y – 2) = y Factor out x x = y/(3y – 2) Divide by 3y – 2 3y – 2 = 0 Equate 3y – 2 to 0 3y = 2 Transpose y = 2/3 Divide by 3 Thus, the horizontal asymptote is y = 2/3. The answer is B. ALGEBRA TIP: Students who do not review After 20 minutes, they remember only 60% of what they learned. After 1 day, they remember only 30%. Students who review immediately and review again later They remember 80% of what they learned for long periods of time. 21 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 3: Exponential Functions 𝟑 21. √𝟑𝟐𝒚𝟐 in exponential form is _____. 2 2/3 2/3 2/3 A. 2y B. 2(2y) C. (2y) D. 32 y 𝟏⁄ 𝟏⁄ 𝟑 𝟑 𝟑 𝟑 √𝟑𝟐𝒚𝟐 = √𝟐𝟓 𝒚𝟐 (𝟑𝟐 = 𝟐𝟓 ) √𝟐𝟓 𝒚𝟐 = (𝟐𝟓 𝒚𝟐 ) 𝟑 ( √𝒂 = 𝒂 𝟑) 𝟏⁄ 𝟏⁄ 𝟏⁄ 𝟏⁄ 𝟏⁄ (𝟐𝟓 𝒚𝟐 ) 𝟑 = [(𝟐𝟑 )(𝟐𝟐 𝒚𝟐 )] 𝟑 (𝟐𝟓 = 𝟐𝟑 𝟐𝟐 ) [(𝟐𝟑 )(𝟐𝟐 𝒚𝟐 )] 𝟑 = (𝟐𝟑 ) 𝟑 (𝟐𝟐 𝒚𝟐 ) 𝟑 〈(𝒂𝒃)𝒏 = 𝒂𝒏 𝒃𝒏 〉 𝟏⁄ 𝟏 𝟏⁄ 𝟏⁄ 𝟐⁄ 𝟏⁄ 𝒏⁄ (𝟐𝟑 ) 𝟑 (𝟐𝟐 𝒚𝟐 ) ⁄𝟑 = 𝟐[(𝟐𝒚)𝟐 ] 𝟑 〈𝒂𝒏 𝒃𝒏 = (𝒂𝒃)𝒏 〉 𝟐[(𝟐𝒚)𝟐 ] 𝟑 = 𝟐(𝟐𝒚) 𝟑 〈(𝒂𝒏 ) 𝒎 = 𝒂 𝒎〉 The answer is B. -3.5 22. Evaluate: 4 A. 14 B. 1/18 C. 17 D. 1/128 𝟕⁄ 𝟏 𝟑.𝟓 𝟏 𝒏 𝟏 𝟑.𝟓 𝟏 𝟐 𝟒−𝟑.𝟓 = ( ) 𝒂−𝒏 = ( ) ( ) = ( ) 𝟑. 𝟓 = 𝟕/𝟐 𝟒 𝒂 𝟒 𝟒 𝟕⁄ 𝟏⁄ 𝟏⁄ 𝟕 𝟏 𝟐 𝟏 𝟐 𝟕 𝒏⁄ 𝟏⁄ 𝒏 𝟏 𝟐 𝟕 𝟏 𝟏⁄ ( ) = [( ) ] 𝒂 𝒎 = (𝒂 𝒎) [( ) ] = (√ ) 𝒂 𝟐 = √𝒂 𝟒 𝟒 𝟒 𝟒 𝟕 𝟏 𝟏 𝟕 𝟏 𝟏 𝟏 𝟕 𝟏 (√ ) = ( ) √ = ( ) = 𝟐𝟕 = 128 𝟒 𝟐 𝟒 𝟐 𝟐 𝟏𝟐𝟖 The answer is D. 𝟑 𝟗 23. Simplify and express answer in exponential form: √𝒙𝟐 √𝒙𝟒 10/9 2/9 6/27 6/11 A. x B. x C. x D. x 𝟑 𝟗 𝟐⁄ 𝟒⁄ 𝒒 𝒑 𝟐⁄ 𝟒⁄ 𝟐⁄ +𝟒⁄ ) ⁄𝒒 √𝒙𝟐 √𝒙𝟒 = 𝒙 𝟑 ∙𝒙 𝟗 √𝒂𝒑 = 𝒂 𝒙 𝟑 ∙𝒙 𝟗 = 𝒙( 𝟑 𝟗 𝒂𝒎 𝒂𝒏 = 𝒂𝒎+𝒏 𝟐⁄ +𝟒⁄ ) 𝟔+𝟒 𝟔+𝟒 𝟏𝟎⁄ 𝒙( 𝟑 𝟗 = 𝒙 𝟗 𝑳𝑪𝑫 = 𝟗 𝒙 𝟗 = 𝒙 𝟗 The answer is A. 3/5 24. Evaluate : (-32) A. -2 B. -4 C. -6 D. -8 𝟑⁄ 𝟏⁄ 𝟑 𝒏⁄ 𝟏⁄ 𝒏 𝟏⁄ 𝟑 𝟓 𝟑 𝟏⁄ 𝒒 (−𝟑𝟐) 𝟓 = [(−𝟑𝟐) 𝟓] 𝒂 𝒎 = (𝒂 𝒎) [(−𝟑𝟐) 𝟓] = ( √−𝟑𝟐) 𝒂 𝒒 = √𝒂 𝟓 𝟑 𝟓 ( √−𝟑𝟐) = (−𝟐)𝟑 √−𝟑𝟐 = −𝟐 (−𝟐)𝟑 = −𝟖 The answer is D. 22 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 𝟑 25. Evaluate: √√𝟐𝟓𝟔 3 3 3 6 A. 2 √2 B. √24 C. √16 D. √256 𝟑 √√𝟐𝟓𝟔 = 𝟑√𝟏𝟔 √𝟐𝟓𝟔 = 𝟏𝟔 𝟑 𝟑 √𝟏𝟔 = √𝟖 ∙ 𝟐 = 𝟐 √𝟐 𝟑 𝟑 √𝟖 = 𝟐 The answer is A. x 26. Solve for x: 32 = 4 A. 1/8 B. 2/5 C. 5/2 D. 8 𝟑𝟐 = 𝟒 𝒙 𝟐𝟓 = (𝟐𝟐 )𝒙 𝒐𝒓 𝟐𝟐𝒙 𝟑𝟐 = 𝟐𝟓 𝒂𝒏𝒅 𝟒 = 𝟐𝟐 2x = 5 Equate the exponents x = 5/2 Divide by 2 The answer is C. 2x-2 2x+5 27. Solve for x: 5 = 25 A. -7 B. -6 C. 0 D. 3 𝟓𝟐𝒙−𝟐 = 𝟐𝟓 𝟐𝒙+𝟓 𝟓𝟐𝒙−𝟐 = (𝟓𝟐 )𝟐𝒙+𝟓 𝒐𝒓 𝟓𝟒𝒙+𝟏𝟎 𝟐𝟓 = 𝟓𝟐 2x – 2 = 4x + 10 Equate the exponents 2x – 4x = 10 + 2 Transpose -2x = 12 Simplify x = -6 Divide by -2 The answer is B. 2x+1 x+2 28. Solve for x: 9 = 27 A. 1 B. 2 C. 3 D. 4 𝟗𝟐𝒙+𝟏 = 𝟐𝟕 𝒙+𝟐 (𝟑𝟐 )𝟐𝒙+𝟏 = (𝟑𝟑 )𝒙+𝟐 𝟗 = 𝟑𝟐 𝒂𝒏𝒅 𝟐𝟕 = 𝟑𝟑 𝟑𝟒𝒙+𝟐 = 𝟑𝟑𝒙+𝟔 𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒆 4x + 2 = 3x + 6 (Equate the exponents) 4x – 3x = 6 – 2 (Transpose) x = 4 (Simplify) The answer is D. 𝒙+𝟔 29. Solve for x: (𝟏𝟐) = 𝟏𝟔−𝒙 A. -8 B. -2 C. 2 D. 8 𝟏 𝒙+𝟔 𝟏 (𝟐) = 𝟏𝟔−𝒙 (𝟐−𝟏 )𝒙+𝟔 = (𝟐𝟒 )−𝒙 = 𝟐−𝟏 & 16 = 𝟐𝟒 𝟐−𝒙−𝟔 = 𝟐−𝟒𝒙 (𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒆) 𝟐 -x - 6 = -4x (Equate the exponents) -x + 4x = 6 (Transpose) 3x = 6 (Simplify) x=2 (Divide by 3) The answer is C. 𝒙𝟐 +𝟒𝒙 𝟏 30. Solve for x: 𝟑 = 𝟐𝟕 A. 3, 1 B. 3, -1 C. -3, 1 D. -3, -1 𝟐 +𝟒𝒙 𝟏 𝟐 +𝟒𝒙 𝟏 𝟑 𝟏 𝟏 𝟑 𝟐 +𝟒𝒙 𝟏 𝟑 𝟑𝒙 = 𝟐𝟕 𝟑𝒙 = ( ) 𝟑 〈 𝟐𝟕 = ( ) 〉 𝟑𝒙 𝟑 = 𝟑−𝟑 〈( ) = 𝟑−𝟑 〉 𝟑 x2 + 4x = -3 (Equate the exponents) x2 + 4x + 3 = 0 (Transpose) (x + 3)(x + 1) = 0 (Factor) x = -3, x = -1 (Equate each factor to 0) The answer is D. 23 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 4: Logarithmic Functions 31. The equation 𝒍𝒐𝒈𝒃 𝒎 = 𝒏 express in exponential form is _____. n b m n A. m = b B. m = n C. b = n D. b = m The logarithmic equation n = logbm in exponential form is m = bn. The answer is A. 32. The equation 𝟓𝟑/𝟐 = 𝟓√𝟓 express in logarithmic form is ____. 3 3 A. 𝑙𝑜𝑔5√5 5 = B. 𝑙𝑜𝑔5√5 (32) = 5 C. 𝑙𝑜𝑔3 5√5 = 5 D. 𝑙𝑜𝑔5 5√5 = 2 2 2 𝟓 𝟑/𝟐 = 𝟓√𝟓 ↔ 𝟑⁄𝟐 = 𝐥𝐨𝐠 𝟓 𝟓√𝟓 Since the exponential form m = bn in logarithmic equation is n = logbm , The answer is D. 33. Evaluate: 𝒍𝒐𝒈𝟕 𝟕√𝟕 A. √7 B. 3/2 C. 2 D. 3 Since the logarithmic equation n = logbm in exponential form is m = bn. 𝟏⁄ 𝟑⁄ 𝒏 = 𝒍𝒐𝒈𝟕 𝟕√𝟕 ↔ 𝟕√𝟕 = 𝟕𝒏 ↔ 𝟕∙𝟕 𝟐 = 𝟕𝒏 ↔ 𝟕 𝟐 = 𝟕𝒏 Thus, n = 3/2. The answer is B. 𝒍𝒏𝟓𝒆 34. Evaluate: 𝟒+𝟒𝒍𝒏𝟓 A. 0.25 B. 0.75 C. 1.25 D. 2.00 𝒍𝒏𝟓𝒆 𝒍𝒏𝟓+𝒍𝒏𝒆 𝒍𝒏𝟓+𝒍𝒏𝒆 𝒍𝒏𝟓+𝟏 𝟏 𝟒+𝟒𝒍𝒏𝟓 = 𝟒(𝟏+𝒍𝒏 𝟓) 𝟒(𝟏+𝒍𝒏 𝟓) = 𝟒(𝟏+𝒍𝒏 𝟓) = 𝟒 = 𝟎. 𝟐𝟓 The answer is A. 35. Solve for x: 𝒍𝒐𝒈𝟐 𝟗𝟒 = 𝒙 + 𝟑 𝟑 A. -1 B. -3 C. -5 D. -9 𝟐 𝒙+𝟑 𝟗 𝟐 𝒙+𝟑 𝟑 𝟐 𝟐 𝒙+𝟑 𝟐 −𝟐 𝒍𝒐𝒈𝟐 𝟗𝟒 = 𝒙 + 𝟑 ↔ ( ) = ↔ ( ) = ( ) ↔ ( ) = ( ) 𝟑 𝟑 𝟒 𝟑 𝟐 𝟑 𝟑 Thus, x + 3 = -2 x = -2 – 3 = -5 The answer is C. x x+1 36. Solve for x: 4 + 2 – 24 = 0 A. -6 B. -3 C. 2 D. 4 4x + 2x+1 – 24 = 0 22x + 212x – 24 = 0 x Let y = 2 y2 + 2y – 24 = 0 (y + 6)(y – 4) = 0 y = -6 or y = 4 2x = -6 or 2x = 4 2x = 4 or 2x = 22 Thus, x = 2. The answer is C. 24 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com For numbers 37 and 38: Given: log5 2 = 0.4 log5 3 = 0.7 37. What is the value of log5 9/4 ? A. 0.3 B. 0.6 C. 0.9 D. 1.1 log5 9/4 = log5 9 – log5 4 log5 32 – log5 22 2log5 3 – 2log5 2 2(0.7) – 2(0.4) 1.4 – 0.8 = 0.6 The answer is B. 38. What is the value of log5 √𝟔 ? A. 0.30 B. 0.55 C. 0.90 D. 1.10 𝟏⁄ log5 √𝟔 = log5 𝟔 𝟐 ½log5 6 ½log5 (3*2) ½(log5 3 + log5 2) ½(0.7 + 0.4) ½(1.1) = 0.55 The answer is B. 39. Which of the following is false? A. 4𝑙𝑜𝑔42 = 2 C. 𝑙𝑜𝑔2 (𝑥 + 3)5 = 5[𝑙𝑜𝑔2 𝑥 + 𝑙𝑜𝑔2 3] 𝑥𝑦 2 B. 𝑙𝑜𝑔3 5 = 𝑙𝑜𝑔10 5 ÷ 𝑙𝑜𝑔10 3 D. 𝑙𝑜𝑔5 √ = 12[𝑙𝑜𝑔5 𝑥+2𝑙𝑜𝑔5 𝑦−𝑙𝑜𝑔5 𝑧] 𝑧 C. 𝑙𝑜𝑔2 (𝑥 + 3)5 = 5[𝑙𝑜𝑔2 𝑥 + 𝑙𝑜𝑔2 3] A. 4𝑙𝑜𝑔42 = 2 𝑭𝒂𝒍𝒔𝒆 𝑠𝑖𝑛𝑐𝑒 𝑙𝑜𝑔𝑏 𝑥 𝑛 = 𝑛 log 𝑏 𝑥 , 𝑏𝑢𝑡 D. 𝐥𝐨𝐠 𝒃 (𝒙 + 𝒚) ≠ 𝐥𝐨𝐠 𝒃 𝒙 + 𝐥𝐨𝐠𝒃 𝒚 𝑻𝒓𝒖𝒆 𝑠𝑖𝑛𝑐𝑒 𝑏 𝑙𝑜𝑔𝑏 𝑥 = 𝑥 B. B. 𝑙𝑜𝑔3 5 = 𝑙𝑜𝑔10 5 ÷ 𝑙𝑜𝑔10 3 𝑥𝑦 2 𝑙𝑜𝑔 𝑥 D. 𝑙𝑜𝑔5 √ = 12[𝑙𝑜𝑔5 𝑥+2𝑙𝑜𝑔5𝑦−𝑙𝑜𝑔5 𝑧] 𝑧 𝑻𝒓𝒖𝒆 𝑠𝑖𝑛𝑐𝑒 𝑙𝑜𝑔𝑦 𝑥 = 10 1 𝑙𝑜𝑔10 𝑦 𝑻𝒓𝒖𝒆 𝑠𝑖𝑛𝑐𝑒 𝑙𝑜𝑔𝑏 √𝑥 = 𝑙𝑜𝑔𝑏 𝑥 ⁄2 = 1⁄2 ∙ 𝑙𝑜𝑔𝑏 𝑥 , 𝑙𝑜𝑔𝑏 𝑥𝑦 = log 𝑏 𝑥 + log 𝑏 𝑦 , and The answer is C. 𝑚 𝑙𝑜𝑔𝑏 = log 𝑏 𝑚 − log 𝑏 𝑛 𝑛 𝒍𝒐𝒈𝟑 (𝒙+𝟏) 40. Solve for x: =𝟐 𝒍𝒐𝒈𝟑 (𝒙−𝟏) A. 0 B. 1 C. 2 D. 3 𝒍𝒐𝒈𝟑 (𝒙+𝟏) = 𝟐 𝐥𝐨𝐠 𝟑 (𝒙 + 𝟏) = 𝟐 𝐥𝐨𝐠 𝟑 (𝒙 − 𝟏) 𝐥𝐨𝐠 𝟑 (𝒙 + 𝟏) = 𝐥𝐨𝐠 𝟑 (𝒙 − 𝟏)𝟐 x + 1 = (x – 1)2 𝒍𝒐𝒈𝟑 (𝒙−𝟏) x + 1 = x2 – 2x + 1 0 = x2 – 2x – x + 1 – 1 x2 – 3x = 0 x(x – 3) = 0 x = 0 or x = 3 x cannot be 0 because x – 1 becomes -1. The answer is D. 25 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 5: Systems For numbers 41-43: Refer to the following graphs of systems of equation and the choices. I II III A. I only B. II only C. III only D. I and II 41. The system is consistent and the equation are independent. There are three possibilities for the manner in which the graphs of two linear equations could meet--the lines could intersect once, not intersect at all (be parallel), or intersect an infinite number of times (in which case the two lines are actually the same). If the two equations describe lines that intersect once (I), the system is independent and consistent. If the two equations describe parallel lines (II), and thus lines that do not intersect, the system is independent and inconsistent. If the two equations describe the same line (III), and thus lines that intersect an infinite number of times, the system is dependent and consistent. The answer is A. 42. The system is inconsistent. See explanation in the previous item. The answer is B. 43. The equations are dependent. See explanation in the first item. The answer is C. 44. Which of the following is the solution of the system x = 5 – 2y ? 2x + 4y = 6 A. (1, 2) B. (11, -3) C. (10, 0) D. no solution 𝒙 = 𝟓 − 𝟐𝒚 𝟐𝒙 + 𝟒𝒚 = 𝟔 −𝟐𝒙 = −𝟐(𝟓 − 𝟐𝒚) Multiply the first equation by -2 𝟐𝒙 + 𝟒𝒚 = 𝟔 −𝟐𝒙 = −𝟏𝟎 + 𝟒𝒚 𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒆 − 𝟐 𝒊𝒏 𝒕𝒉𝒆 𝟏𝒔𝒕 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒂𝒏𝒅 𝒕𝒓𝒂𝒏𝒔𝒑𝒐𝒔𝒆 𝟒𝒚 𝒊𝒏 𝒕𝒉𝒆 𝟐𝒏𝒅, 𝒕𝒉𝒆𝒏 𝒂𝒅𝒅 𝒕𝒉𝒆 𝒕𝒘𝒐 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 𝟐𝒙 = 𝟔 − 𝟒𝒚 0 = -4 The resulting equation is FALSE; thus, there is no solution. 4 x 13 y 8 3 The answer is D. 45. Which of the following is the solution of the system ? 1 2 x 34 y 52 26 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com A. (2, 3) B. (1, 20) C. (3, -28) D. (1, -4) 𝟏 𝟖 𝟏 𝟖 𝟒𝒙 + 𝒚 = 𝟒𝒙 + 𝒚 = 𝟑 𝟑 𝟑 𝟑 𝟏 𝟑 𝟓 𝟏 𝟑 𝟓 𝒙+ 𝒚=− −𝟖 ( 𝒙 + 𝒚) = −𝟖 (− ) 𝟐 𝟒 𝟐 𝟐 𝟒 𝟐 Multiply the second equation by -8 𝟏 𝟖 𝟒𝒙 + 𝒚 = 𝟑 𝟑 Distribute -8 in the 1st equation, then add the two equations −𝟒𝒙 − 𝟔𝒚 = 𝟐𝟎 𝟑 𝟏𝟕 𝟑 𝟔𝟖 𝟑 − (− 𝒚) = − ( ) Multiply both sides by − 𝟏𝟕 𝟑 𝟏𝟕 𝟑 𝟏𝟕 y = -4 Only D satisfies the resulting value for y. You may verify by solving for the corresponding value of x. It should be x = 1. The answer is D. x 13 y y 2 46. Which of the following is the solution of the system ? 1 4 x y x y A. (-1, 3/2) B. (0, 3) C. (1, 9/2) D. (3, -2) 𝟏 𝟏 𝒙+ 𝒚=𝒚−𝟐 𝒙 + 𝒚 − 𝒚 = −𝟐 𝟑 𝟑 𝟏 𝟏 Transpose variable terms. 𝒙+𝒚=𝒙+𝒚 𝒙−𝒙+𝒚−𝒚=𝟎 𝟒 𝟒 𝟐 𝒙− 𝒚 = −𝟐 𝟑 𝟑 Simplify. It appears that from the 2nd equation, x = 0. Among the choices, only B − 𝒙 =𝟎 𝟒 satisfies the resulting value of x. You may still verify if y = 3. The answer is B. x + y = -1 47. Which of the following is the solution of the system y + z = 4 ? x+z=1 A. (1, -2, 6) B. (-1, 0, 4) C. (-2, 1, 3) D. (0, -1, 3) In this case, it is easier if we work backwards: A. (1, -2, 6): Is x + y = 1 + (-2) = -1? Yes! C. (-2, 1, 3): Is x + y = -2 + 1 = -1? Yes! Is y + z = -2 + 6 = 4? Yes! Is y + z = 1 + 3 = 4? Yes! Is x + z = 1 + 6 = 1? No! Is x + z = -2 + 3 = 1? Yes! B. (-1, 0, 4): Is x + y = -1 + 0 = -1? Yes! D. (0, -1, 3): Is x + y = 0 + (-1) = -1? Yes! Is y + z = 0 + 4 = 4? Yes! Is y + z = -1 + 3 = 4? No! Is x + z = -1 + 4 = 1? No! Is x + z = 0 + 3 = 1? No! The answer is C. 48. Danica has twice as many P1 coin as P5 coin in her purse. If the total amount of her P1 and P5 coins is P21. How many of each type of coin does she have? A. Two P5 coins and eleven P1 coins C. Four P5 coins and two P1 coins 27 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com C. Three P5 coins and six P1coins D. Five P5 coins and one P1 coin “Danica has twice as many P1 coin as P5 coin in her purse.” This first statement eliminates choices A, C and D. We are left with choice B. Does it satisfy the second statement “If the total amount of her P1 and P5 coins is P21.”? B: 3*P5 + 6*P1 = P15 + P6 = P21 Yes! The answer is B. 3y = 2x 49. What value of m will make the system 6y - m – 4x = 0 , a dependent equation? A. 0 B. 2 C. 3 D. -4 As explained in the first item of this chapter, if two equations describe the same line, then they are dependent. If we multiply the first equation by 2, the result will be: 6y = 4x If we transpose 4x, we have: 6y – 4x = 0 There is no constant term, thus m must be equal to 0. The answer is A. 𝐲 < 2𝑥 − 5 50. Which of the following is the solution of the system ? 𝐲 ≥ −𝐱 − 𝟐 A. B. C. D. The line sloping upward to the right is y = 2x – 5. Since the first inequality is y < 2x – 5, its graph includes all the points below but excluding the line y = 2x – 5. Since choices C and D indicate that the line y = 2x – 5 is included in the graph, they should be eliminated. We are left with choices A and B. The line sloping downward to the right is y = -x – 2. Since the second inequality is y -x – 2, its graph includes all the points above and including the line y = -x – 2. Between choices A and B, A satisfies the given condition. The answer is A. ALGEBRA TIP: Remember this general rule of thumb: you must have the SAME NUMBER of equations as unknowns. If there are three variables and only two equations provided, chances are you will not be able to solve the problem. If there are two variables AND two equations given, chances are you will be able to solve it. 28 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com CHAPTER 6: Matrices and Determinants For numbers 51-53: 4 6 5 2 15 𝐴 = [3 4 7] 𝐵 = [2 −3 −7] 𝐶 = [−2 0 5] 𝐷 = [−4] 𝐸 = [ 10 ] 1 0 9 5 −5 51. Which of the following is the sum of the row matrices? A. [5 4 12] B. [1 4 2] C. [−5 4 2] D. [1 4 12] The row matrices are A and C. Thus, 𝑨 + 𝑪 = [𝟑 𝟒 𝟕] + [−𝟐 𝟎 𝟓] = [(𝟑 + −𝟐) (𝟒 + 𝟎) (𝟕 + 𝟓)] = [𝟏 𝟒 𝟏𝟐] The answer is D. 52. Solve: 3D – E −9 21 17 −21 A. [−22] B. [−2] C. D. [−22] 20 10 8 10 𝟐 𝟏𝟓 𝟔 𝟏𝟓 𝟔 − 𝟏𝟓 −𝟗 𝟑𝑫 − 𝑬 = 𝟑 [−𝟒] − [ 𝟏𝟎 ] = [−𝟏𝟐] − [ 𝟏𝟎 ] = [−𝟏𝟐 − 𝟏𝟎] = [−𝟐𝟐] 𝟓 −𝟓 𝟏𝟓 −𝟓 𝟏𝟓 − −𝟓 𝟐𝟎 The answer is A. 53. Find the difference of matrix A and the additive inverse of Matrix C. A. [5 4 2] B. [5 4 12] C. [1 4 12] D. [1 4 2] 𝑨 − 𝑪′ =? Solve first for C’: 𝑪 = [−𝟐 𝟎 𝟓], thus 𝑪′ = [𝟐 𝟎 −𝟓] 𝑨 − 𝑪′ = [𝟑 𝟒 𝟕] − [ 𝟐 𝟎 −𝟓] = [𝟑 − 𝟐 𝟒−𝟎 𝟕 − −𝟓] = [𝟏 𝟒 𝟏𝟐] The answer is C. 54. Find the value of x and y: x[3 5] + y[4 -2] = 2[-5 9] A. x= 3, y = 4 B. x = 5, y = -2 C. x = -5, y = 3 D. x = 2, y = -4 𝒙 [𝟑 𝟓] + 𝒚[𝟒 −𝟐] = 𝟐[−𝟓 𝟗] [𝟑𝒙 𝟓𝒙] + [𝟒𝒚 −𝟐𝒚] = [−𝟏𝟎 𝟏𝟖] [𝟑𝒙 + 𝟒𝒚 𝟓𝒙 − 𝟐𝒚] = [−𝟏𝟎 𝟏𝟖] Thus: 3x + 4y = -10 5x – 2y = 18 3x + 4y = -10 Multiply the second equation by 2 then add the two equations: 10x – 4y = 36 13x = 26 Dividing both sides by 13, we get x=2 The only choice with x = 2 is D. Verify by solving for y using the first equation: y = -4 The answer is D. 29 | P a g e REVIEW MASTERS LEAGUE, INC. MATH REVIEW ©2015 www.upcatreview.com 𝒙 −𝒙 −𝟏 −𝒚 𝒚 𝒙 55. Find the value of x and y: [−𝒚 −𝟏 𝒙 ] = [ 𝒙 −𝟏 −𝟏] 𝟏 −𝟏 𝒚 𝒚 𝒙 𝟏 A. x = -1, y = 1 B. x = -1, y = -1 C. x =1, y = -1 D. x = 1, y = 1 Equating the 3rd column, 1st row, we get x = -1. Equating the 1st column, 3rd row, we get y = 1. The answer is A. 𝟓 −𝟒 56. Find the value of the determinant: | |
