1.Algebra Booster

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Algebra <strong>Booster</strong> with Problems & Solutions for JEE Main and Advanced Rejaul Makshud M. Sc. (Calcutta University, Kolkata) McGraw Hill Education (India) Private Limited CHENNAI McGraw Hill Education Offices Chennai New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto
McGraw Hill Education (India) Private Limited Published by McGraw Hill Education (India) Private Limited 444/1, Sri Ekambara Naicker Industrial Estate, Alapakkam, Porur, Chennai - 600 116 Algebra <strong>Booster</strong> Copyright © 2017, McGraw Hill Education (India) Private Limited. No part of this publication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers. The program listings (if any) may be entered, stored and executed in a computer system, but they may not be reproduced for publication. This edition can be exported from India only by the publishers, McGraw Hill Education (India) Private Limited ISBN (13): 978-93-5260-250-6 ISBN (10): 93-5260-250-1 Information contained in this work has been obtained by McGraw Hill Education (India), from sources believed to be reliable. However, neither McGraw Hill Education (India) nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw Hill Education (India) nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw Hill Education (India) and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. Typeset at The Composers, 260, C.A. Apt., Paschim Vihar, New Delhi 110 063 and text and cover printed at Cover Designer: Creative Designer visit us at: www.mheducation.co.in
Page 2 and 3: Algebra Booster with Problems & Sol
Page 6: Dedicated to light of my life (Rush
Page 9 and 10: viii Preface I owe a special debt o
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
Page 39 and 40: 1.26 Algebra Booster 17 1 50. Ê ˆ
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=
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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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6.40 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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