1.Algebra Booster

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Matrices and Determinants 7.81 62. Let Êa a a P a a a 11 12 13 = Á ˜ Á 21 22 23 ˜ Ëa31 a32 a33¯ ˆ , where |P| = 2 Ê 2 3 4 2 a11 2 a12 2 a ˆ 13 Á 3 4 5 ˜ Now, Q= Á2 a21 2 a22 2 a23˜ Á 4 5 6 ˜ Ë2 a 2 a 2 a ¯ fi 31 32 33 2 3 4 a11 a12 a13 3 4 5 21 22 23 4 5 6 a31 a32 a33 2 2 2 | Q| = 2 a 2 a 2 a 2 2 2 a a a 2 3 4 = 2 ◊2 ◊2 2a 2a 2a 2 2 2 2 3 4 2 = 2 ◊2 ◊2 ◊2◊2 = 2 12 a a a a a a a a a 11 12 13 11 12 13 21 22 23 2 2 2 a31 a32 a33 21 22 23 31 32 33 = 2 12 ◊ 2 = 2 13 63. (a) (N T MN) T = N T M T N a a a a a a a a a 11 12 13 21 22 23 31 32 33 T ÏÔ N MN : if M is symmetric = Ì T Ô Ó - N MN : if M is skew-symmetric T T T T T (b) ( MN - NM ) = N M - M N = NM - MN =-( MN - NM ) Thus, (MN – NM) is skew-symmetric. (c) ( MN) T = N T M T = NM π MN if M and N are symmetric. So, MN is not symmetric. (d) (adj M)(adj N) = adj(NM) π adj(MN) 64. We have Ê 2 3 4 n+ 2 w w w w ˆ Á 3 4 5 n+ 3 ˜ w w w w P = Á ˜ Á ˜ Á n+ 2 n+ 3 n+ 4 2n+ 4 Ëw w w w ˜ ¯ Now P 2 = P.P Ê 2 3 4 n+ 2 2 3 4 n 2 w w w … w ˆÊ + w w w … w ˆ Á 3 4 5 n+ 3˜Á 3 4 5 n+ 3˜ w w w … w w w w … w = Á ˜Á ˜ Á ˜Á ˜ ˜Á Á n+ 2 n+ 3 n+ 4 2n+ 4 n+ 2 n+ 3 n+ 4 2n+ 4 Ëw w w … w ˜Á ¯Ëw w w … w ˜ ¯ Ê 4 6 5 7 9 w + w + . w + w + w + ˆ Á 5 7 9 ˜ w + w + w + = Á ˜ Á ˜ Á n+ 4 n+ 6 2n+ 4 2n+ 6 Ëw + w + w + w ˜ ¯ Thus, P 2 is a null matrix if n is a multiple of 3. Êa cˆ 65. Let M = Á Ëc b ˜ where a, b, c, d Œ I. ¯ 2 Then | | a c M = = ab - c . c b If a = b = c, then |M| = 0 If c = 0, a, b π 0, then |M| π 0 If ab π square of integer, then |M| π 0 66. Given M 2 = N 4 fi M 2 – N 4 = O fi (M – N 2 )(M + N 2 ) = O As M, N commute. Also, M π N 2 fi det((M – N 2 )(M + N 2 )) = 0 As (M – N 2 ) is not null. fi Det(M 2 + N 2 ) = 0 Also, Det(M + N 2 ) = (DetM)(Det(M + N 2 )) = 0 There exists a non-null U such that (M 2 + MN 2 )U = O 67. (a) (Y 3 Z 4 – Z 4 Y 3 ) = (–Y 3 Z 4 – Z 4 Y 3 ) fi (Y 3 Z 4 – Z 4 Y 3 ) is skew-symmetric. (c) (X 4 Z 3 – Z 3 Z 4 ) T = ((X 4 Z 3 ) T – (Z 3 Z 4 ) T ) = (Z T ) 3 (X T ) 4 – (X T ) 4 (Z T ) 3 = (Z 3 X 4 – X 4 Z 3 ) = – (X 4 Z 3 – Z 3 X 4 ) (d) (X 23 + Y 23 ) T = – X 23 – Y 23 fi (X 23 + Y 23 ) is skew-symmetric 68. We have, 2 2 2 (1 + a) (1+ 2 a) (1+ 3 a) 2 2 2 (2 + a) (2+ 2 a) (2+ 3 a) =-648 a? 2 2 2 (3 + a) (3+ 2 a) (3+ 3 a) 2 2 2 (1 + a) (1+ 2 a) (1+ 3 a) 3+ 2a 3+ 4a 3+ 6a =-648a 5+ 2a 5+ 4a 5+ 6a (R 3 Æ R 3 – R 2 , R 2 Æ R 2 – R 1 ) 2 2 2 (1 + a) (1+ 2 a) (1+ 3 a) 3+ 2a 3+ 4a 3+ 6a =-648a 2 2 2 (R 3 Æ R 3 – R 2 )
7.82 Algebra <strong>Booster</strong> fi 2 2 (1 + a) a(2+ 3 a) a(2+ 5 a) 3 + 2a 2a 2a = -648a 2 0 0 2 (1 + a) (2+ 3 a) (2+ 5 a) a 3 + 2a 2 2 = -648a 2 0 0 fi 2a 2 (4 + 6a – 4 – 10a) = –64a fi a 2 ¥ –4a = –324a fi a 3 = 81a fi a(a 2 – 81) = 0 fi a = 0, (a 2 – 81) = 0 fi a = 0, ±9
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Algebra Booster with Problems & Sol
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Algebra Booster with Problems & Sol
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Dedicated to light of my life (Rush
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viii Preface I owe a special debt o
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x Contents Descartes Rule of Signs
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xii Contents Chapter 8 Probability
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1.2 Algebra Booster (ii) a 1 - k, a
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1.4 Algebra Booster fi a + (n + 1)d
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1.6 Algebra Booster (i) Maximum Val
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1.8 Algebra Booster 25. In an AP, (
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1.10 Algebra Booster b 86. If b = a
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1.12 Algebra Booster 145. Find the
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1.14 Algebra Booster and n= 0 2n 2n
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1.16 Algebra Booster 51. In a serie
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1.18 Algebra Booster 37. Observing
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1.20 Algebra Booster 7. If a, b and
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1.22 Algebra Booster (B) (C) (D) If
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1.24 Algebra Booster 22. No questio
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1.26 Algebra Booster 17 1 50. Ê ˆ
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1.28 Algebra Booster 8. 3 - 1 2 9.
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1.30 Algebra Booster fi a(a 2 - d 2
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1.32 Algebra Booster On subtraction
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1.34 Algebra Booster 1 1 1 1 fi - =
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1.36 Algebra Booster 61. We have (a
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1.38 Algebra Booster fi 3 n > 14001
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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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6.38 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
Page 457 and 458: 7.30 Algebra Booster 43. If a 0 A
Page 459 and 460: 7.32 Algebra Booster 65. Let M be a
Page 461 and 462: 7.34 Algebra Booster 12. Then AB =
Page 463 and 464: 7.36 Algebra Booster Êa bˆÊ1 2ˆ
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Page 467 and 468: 7.40 Algebra Booster 2 2 2 1 0 = (1
Page 469 and 470: 7.42 Algebra Booster Thus, D1 ( d -
Page 471 and 472: 7.44 Algebra Booster 77. We have, 2
Page 473 and 474: 7.46 Algebra Booster 90. We know th
Page 475 and 476: 7.48 Algebra Booster 108. We have,
Page 477 and 478: 7.50 Algebra Booster 118. We have,
Page 479 and 480: 7.52 Algebra Booster 4. We have, 2
Page 481 and 482: 7.54 Algebra Booster = ([x] + [y] +
Page 483 and 484: 7.56 Algebra Booster It is given th
Page 485 and 486: 7.58 Algebra Booster 27. The given
Page 487 and 488: 7.60 Algebra Booster and 12 -3 5 D1
Page 489 and 490: 7.62 Algebra Booster fi Ê a ˆ Ê
Page 491 and 492: 7.64 Algebra Booster a b c b c a =
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Page 495 and 496: 7.68 Algebra Booster 1 1 1 1 = 2 n
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Page 499 and 500: 7.72 Algebra Booster fi (49 - 42)si
Page 501 and 502: 7.74 Algebra Booster fi p + q + r =
Page 503 and 504: 7.76 Algebra Booster 34. We have fi
Page 505 and 506: 7.78 Algebra Booster where a 2 + b
Page 507: 7.80 Algebra Booster 55. (iii) Let
Page 511 and 512: 8.2 Algebra Booster For example, th
Page 513 and 514: 8.4 Algebra Booster (ii) P(AB) £ P
Page 515 and 516: 8.6 Algebra Booster (ii) a black ca
Page 517 and 518: 8.8 Algebra Booster 64. If two dice
Page 519 and 520: 8.10 Algebra Booster ferred from th
Page 521 and 522: 8.12 Algebra Booster 168. The proba
Page 523 and 524: 8.14 Algebra Booster 28. A fair coi
Page 525 and 526: 8.16 Algebra Booster product is 0.7
Page 527 and 528: 8.18 Algebra Booster event that thr
Page 529 and 530: 8.20 Algebra Booster Questions aske
Page 531 and 532: 8.22 Algebra Booster The probabilit
Page 533 and 534: 8.24 Algebra Booster 72. A is targe
Page 535 and 536: 8.26 Algebra Booster 5, 6, 7. A car
Page 537 and 538: 8.28 Algebra Booster 137. Ê 9 m ˆ
Page 539 and 540: 8.30 Algebra Booster 24. 25. 3 10 1
Page 541 and 542: 8.32 Algebra Booster So, the probab
Page 543 and 544: 8.34 Algebra Booster (iv) Let D be
Page 545 and 546: 8.36 Algebra Booster Hence, the req
Page 547 and 548: 8.38 Algebra Booster 58. Here, S =
Page 549 and 550: 8.40 Algebra Booster 81. Here, S =
Page 551 and 552: 8.42 Algebra Booster The probabilit
Page 553 and 554: 8.44 Algebra Booster Thus, 1 36 =
Page 555 and 556: 8.46 Algebra Booster 128. Hence, th
Page 557 and 558: 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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