1.Algebra Booster

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Preface ALGEBRA BOOSTER with Problems & Solutions for JEE Main and Advanced is meant for aspirants preparing for the entrance examinations of different technical institutions, especially NIT/IIT/BITSAT/IISc. In writing this book, I have drawn heavily from my long experience of teaching at National Level Institutes. After many years of teaching, I have realised the need of designing a book that will help the readers to build their base, improve their level of mathematical concepts and enjoy the subject. This book is designed keeping in view the new pattern of questions asked in JEE Main and Advanced Exams. It has eight chapters. Each chapter has the concept booster followed by a large number of exercises with the exact solutions to the problems as given below: Level - I : Questions based on Fundamentals Level - II : Mixed Problems (Objective Type Questions) Level - III : Problems for JEE Advanced Exam Level - IV : Tougher problems for JEE Advanced Exams (0.......9) : Integer type Questions Passages : Comprehensive link passages Matching : Matrix Match Reasoning : Assertion and Reasoning Previous years’ papers : Questions asked in past IIT-JEE Exams Remember friends, no problem in mathematics is difficult. Once you understand the concept, they will become easy. So please don’t jump to exercise problems before you go through the Concept <strong>Booster</strong> and the Objectives. Once you are confident in the theory part, attempt the exercises. The exercise problems are arranged in a manner that they gradually require advanced thinking. I hope this book will help you to build your base, enjoy the subject and improve your confidence to tackle any type of problem easily and skilfully. My special thanks goes to Mr. M.P. Singh (IISc Bangalore), Mr. Manoj Kumar (IIT, Delhi), Mr. Nazre Hussain (B. Tech.), Dr. Syed Kashan Ali (MBBS) and Mr. Shahid Iqbal, who have helped, inspired and motivated me to accomplish this task. As a matter of fact, teaching being the best learning process, I must thank all my students who inspired me most for writing this book. I would like to convey my affectionate thanks to my wife, who helped me immensely and my children who bore with patience my neglect during the period I remained devoted to this book. I also convey my sincere thanks to Mr. Biswajit Das of McGraw Hill Education for publishing this book in such a beautiful format.
viii Preface I owe a special debt of gratitude to my father and elder brother, who taught me the first lesson of Mathematics and to all my learned teachers—Mr. Swapan Halder, Mr. Jadunandan Mishra, Mr. Mahadev Roy and Mr. Dilip Bhattacharya, who instilled the value of quality teaching in me. I have tried my best to keep this book error-free. I shall be grateful to the readers for their constructive suggestions toward the improvement of the book. Rejaul Makshud M. Sc. (Calcutta University, Kolkata)
Page 2 and 3: Algebra Booster with Problems & Sol
Page 4 and 5: Algebra Booster with Problems & Sol
Page 6: Dedicated to light of my life (Rush
Page 11 and 12: x Contents Descartes Rule of Signs
Page 13 and 14: xii Contents Chapter 8 Probability
Page 15 and 16: 1.2 Algebra Booster (ii) a 1 - k, a
Page 17 and 18: 1.4 Algebra Booster fi a + (n + 1)d
Page 19 and 20: 1.6 Algebra Booster (i) Maximum Val
Page 21 and 22: 1.8 Algebra Booster 25. In an AP, (
Page 23 and 24: 1.10 Algebra Booster b 86. If b = a
Page 25 and 26: 1.12 Algebra Booster 145. Find the
Page 27 and 28: 1.14 Algebra Booster and n= 0 2n 2n
Page 29 and 30: 1.16 Algebra Booster 51. In a serie
Page 31 and 32: 1.18 Algebra Booster 37. Observing
Page 33 and 34: 1.20 Algebra Booster 7. If a, b and
Page 35 and 36: 1.22 Algebra Booster (B) (C) (D) If
Page 37 and 38: 1.24 Algebra Booster 22. No questio
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Page 41 and 42: 1.28 Algebra Booster 8. 3 - 1 2 9.
Page 43 and 44: 1.30 Algebra Booster fi a(a 2 - d 2
Page 45 and 46: 1.32 Algebra Booster On subtraction
Page 47 and 48: 1.34 Algebra Booster 1 1 1 1 fi - =
Page 49 and 50: 1.36 Algebra Booster 61. We have (a
Page 51 and 52: 1.38 Algebra Booster fi 3 n > 14001
Page 53 and 54: 1.40 Algebra Booster and 2 sin n n=
Page 55 and 56: 1.42 Algebra Booster 102. We have,
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1.44 Algebra Booster 115. It is giv
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1.46 Algebra Booster Now, Ê a + b
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1.48 Algebra Booster 136. (i) We ha
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1.50 Algebra Booster 152. Let t n 3
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1.52 Algebra Booster fi 161. As we
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1.54 Algebra Booster 177. We have (
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1.56 Algebra Booster 193. We know t
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1.58 Algebra Booster Thus, (1 + x)(
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1.60 Algebra Booster Alternate meth
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1.62 Algebra Booster fi 1007d = a -
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1.64 Algebra Booster 2 2 fi Ê 1 1
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1.66 Algebra Booster and b + br = 9
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1.68 Algebra Booster = 1 ◊ cos(1
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1.70 Algebra Booster 6. We have 1 1
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1.72 Algebra Booster fi fi Dividing
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1.74 Algebra Booster fi 7 4 4 4 (1
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1.76 Algebra Booster n 35. We have
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1.78 Algebra Booster 5. We know tha
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1.80 Algebra Booster Hence, the val
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1.82 Algebra Booster Since the equa
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1.84 Algebra Booster 2 S2 = = 3 1 1
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1.86 Algebra Booster 37. Let a and
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1.88 Algebra Booster fi fi Ê n Ê
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CHAPTER 2 Quadratic Equations and E
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Quadratic Equations and Expressions
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CHAPTER 3 Logarithm 1. INTRODUCTION
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Logarithm 3.3 = log 2 (4) = log 2 (
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Logarithm 3.5 x fi Ê1ˆ Á = 3 Ë
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Logarithm 3.7 fi fi 10 log10x log10
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Logarithm 3.9 16. If a 2 + b 2 = 7a
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Logarithm 3.11 2 log 2 + log 3 y =
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Logarithm 3.13 (Q) (R) (S) The valu
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Logarithm 3.15 1 1 13. { , 2 4} 14.
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Logarithm 3.17 Also, (a - b) = log
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Logarithm 3.19 Again, log a b = 2 f
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Logarithm 3.21 Now, log( a + c) + l
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Logarithm 3.23 fi Ê1 2 ˆ log 3/4
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Logarithm 3.25 fi 2x 2 = 12 fi x 2
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Logarithm 3.27 fi fi 1 2 x 2 = 3 3,
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Logarithm 3.29 = log 10 (64 ¥ 31)
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Logarithm 3.31 fi 2x = 8, 4 = 2 3 ,
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4.2 Algebra Booster EXAMPLE 2: The
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4.4 Algebra Booster (i) If z lies i
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4.6 Algebra Booster (iii) w 3 = 1 (
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4.8 Algebra Booster Note The sum of
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4.10 Algebra Booster In an equilate
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4.12 Algebra Booster 3. Straight Li
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4.14 Algebra Booster (vii) We consi
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4.16 Algebra Booster 55. If |z 1 +
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4.18 Algebra Booster 8. The area of
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4.20 Algebra Booster 53. The number
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4.22 Algebra Booster 49. Find the r
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4.24 Algebra Booster 30. Resolve z
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4.26 Algebra Booster 1. The value o
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4.28 Algebra Booster 12. Show that
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4.30 Algebra Booster X¢ P(-1, 0) Y
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4.32 Algebra Booster 41. (d) 42. (c
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4.34 Algebra Booster HINTS AND SOLU
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4.36 Algebra Booster Hence, the val
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4.38 Algebra Booster p q fi = = l(s
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4.40 Algebra Booster p fi < 2q< p 2
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4.42 Algebra Booster 2 2Êq1- q2ˆ
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4.44 Algebra Booster x◊ 3 p + y
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4.46 Algebra Booster 92. We have x
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4.48 Algebra Booster fi fi fi fi Ê
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4.50 Algebra Booster Putting z 3 =
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4.52 Algebra Booster i e p 122. Let
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4.54 Algebra Booster fi fi 3(2 + co
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4.56 Algebra Booster 19. Let z = r(
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4.58 Algebra Booster 2 1 w w = + +
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4.60 Algebra Booster fi Ê 2p 4p 6p
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4.62 Algebra Booster A B 50. Given
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4.64 Algebra Booster fi 2 2 (3x + y
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4.66 Algebra Booster fi fi |(x - 4)
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4.68 Algebra Booster Comparing the
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4.70 Algebra Booster = |(cos (3q) -
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4.72 Algebra Booster Thus, 18. Let
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4.74 Algebra Booster 2 Ê a ˆ = 2
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4.76 Algebra Booster 1È Ê3pˆ Ê5
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4.78 Algebra Booster 12 Ê2k + 1ˆ
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4.80 Algebra Booster 16. We have, s
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4.82 Algebra Booster 3 fi x =± 2 T
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4.84 Algebra Booster fi fi 4 Ê3z -
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4.86 Algebra Booster 53. fi 2 2 2 (
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4.88 Algebra Booster Now, 1 iq -iq
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CHAPTER 5 Permutations and Combinat
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Permutations and Combinations 5.3 (
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Permutations and Combinations 5.5 =
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Permutations and Combinations 5.7 5
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Permutations and Combinations 5.9 1
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Permutations and Combinations 5.11
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6.2 Algebra Booster 5. Subtracting
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6.4 Algebra Booster Proof 1. ( a +
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6.6 Algebra Booster EXERCISES LEVEL
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6.8 Algebra Booster 80. Find the su
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6.10 Algebra Booster 10. In the exp
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6.12 Algebra Booster 11. Find the c
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6.14 Algebra Booster 64. Find the c
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6.16 Algebra Booster 43. Find the s
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6.18 Algebra Booster 3. Match the f
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6.20 Algebra Booster (a) 0 (c) (-1)
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6.22 Algebra Booster LEVEL IV COMPR
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6.24 Algebra Booster 30 2 30 2 2 30
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6.26 Algebra Booster Now, (33 43 -
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6.28 Algebra Booster 57. We have, 1
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6.34 Algebra Booster Comparing the
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6.36 Algebra Booster 1 1 1 1 + + +
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6.38 Algebra Booster 127. We have,
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6.40 Algebra Booster n n 2 n 3 7. W
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6.42 Algebra Booster n r n r n r n
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6.44 Algebra Booster n n Â Ck k =
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6.46 Algebra Booster A1 Ê 1 1 1 1
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6.48 Algebra Booster Multiplying Eq
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6.50 Algebra Booster Ê12 ˆ Middle
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6.52 Algebra Booster 72. Let t n 1
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6.54 Algebra Booster Putting x = 1,
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6.56 Algebra Booster Â Â Thus, 2I
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6.58 Algebra Booster fi fi 2n 2n Ê
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6.60 Algebra Booster Cn ( ,4) 36. L
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6.62 Algebra Booster Thus, S = t 1
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6.64 Algebra Booster 7. We have 10
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6.66 Algebra Booster = ( 49 C 4 + 4
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6.68 Algebra Booster Differentiatin
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6.72 Algebra Booster 51. Let the th
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7.2 Algebra Booster (vii) Identity
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7.4 Algebra Booster (xii) If A is s
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7.6 Algebra Booster where Proof: fi
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7.8 Algebra Booster Replace A by ad
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7.10 Algebra Booster 10. Unitary ma
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7.12 Algebra Booster 46. Expand the
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7.14 Algebra Booster 88. If A be a
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7.16 Algebra Booster 7. For non-zer
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7.18 Algebra Booster 2x + 4p p + 6a
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7.20 Algebra Booster 4. Prove that
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7.22 Algebra Booster 4. If S r = a
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7.24 Algebra Booster 2 2 2 2 2 2 a
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7.26 Algebra Booster The value of (
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7.28 Algebra Booster 17. The determ
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7.30 Algebra Booster 43. If a 0 A
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7.32 Algebra Booster 65. Let M be a
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7.34 Algebra Booster 12. Then AB =
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7.36 Algebra Booster Êa bˆÊ1 2ˆ
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7.38 Algebra Booster 1 a a 47 The g
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7.40 Algebra Booster 2 2 2 1 0 = (1
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7.42 Algebra Booster Thus, D1 ( d -
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7.46 Algebra Booster 90. We know th
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7.54 Algebra Booster = ([x] + [y] +
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7.56 Algebra Booster It is given th
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7.58 Algebra Booster 27. The given
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7.60 Algebra Booster and 12 -3 5 D1
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7.62 Algebra Booster fi Ê a ˆ Ê
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7.64 Algebra Booster a b c b c a =
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7.66 Algebra Booster 2bc -2c -2b 2
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7.68 Algebra Booster 1 1 1 1 = 2 n
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7.70 Algebra Booster 3 m m m 3 m =
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7.72 Algebra Booster fi (49 - 42)si
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7.74 Algebra Booster fi p + q + r =
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7.76 Algebra Booster 34. We have fi
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7.78 Algebra Booster where a 2 + b
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7.80 Algebra Booster 55. (iii) Let
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7.82 Algebra Booster fi 2 2 (1 + a)
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8.2 Algebra Booster For example, th
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8.4 Algebra Booster (ii) P(AB) £ P
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8.6 Algebra Booster (ii) a black ca
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8.8 Algebra Booster 64. If two dice
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8.10 Algebra Booster ferred from th
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8.12 Algebra Booster 168. The proba
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8.14 Algebra Booster 28. A fair coi
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8.16 Algebra Booster product is 0.7
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8.18 Algebra Booster event that thr
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8.20 Algebra Booster Questions aske
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8.22 Algebra Booster The probabilit
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8.24 Algebra Booster 72. A is targe
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8.26 Algebra Booster 5, 6, 7. A car
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8.28 Algebra Booster 137. Ê 9 m ˆ
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8.30 Algebra Booster 24. 25. 3 10 1
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8.32 Algebra Booster So, the probab
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8.34 Algebra Booster (iv) Let D be
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8.36 Algebra Booster Hence, the req
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8.38 Algebra Booster 58. Here, S =
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8.40 Algebra Booster 81. Here, S =
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8.42 Algebra Booster The probabilit
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8.44 Algebra Booster Thus, 1 36 =
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8.46 Algebra Booster 128. Hence, th
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8.48 Algebra Booster tively and let
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8.50 Algebra Booster Hence, the pro
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8.52 Algebra Booster Ê 5 1 ˆ = 1-
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8.54 Algebra Booster We are interes
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8.56 Algebra Booster 193. Clearly,
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8.58 Algebra Booster 15. Let E be t
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8.60 Algebra Booster = P( A or CA o
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8.62 Algebra Booster 20. Out of 2 c
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8.64 Algebra Booster Clearly, a = 5
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8.66 Algebra Booster = (0.17) ¥ (0
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8.68 Algebra Booster 1 Ê13ˆÊ1ˆ
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8.70 Algebra Booster 46. We have, P
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8.72 Algebra Booster 3 2 3 3 2 1 1
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8.74 Algebra Booster and the number
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8.76 Algebra Booster 88. Let G = Or
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1.Algebra Booster

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