1.3 EXERCISE 1. (b,b), (c,c), (a,c) 2. [-5,5] 3. 4x24x –1 1 x 34. fx 2 5. f −1 {( ,,,db )( , cd ba)( , ac ),(, )} 43 26. ffx x –6x 10 x –3 x 7. 2, –1 8. (i) represents function which is surjective but not injective (ii) does not represent function.9. fog2,5 , 5,2 ,1,5 12. (i) f is not function (ii) g is function (iii) h is function (iv) k is not function1 14. ,1317. Domain of R = {1,2,3,4, ..... 20} and Range of R = {1,3,5,7,9, ..... 39}. R is neither reflective, nor symmetric and nor transitive.21. (i) f is one-one but not onto , (ii) g is neither one-one nor onto (iii) h is bijective, (iv) k is neither one-one nor onto.22. (i) transitive (ii) symmetric (iii) reflexive, symmetric and transitive (iv) transitive.23. ⎡⎣(2,5 )⎤⎦ ={(1,4 , 2,5 , 3,6 , 4,7 (5,8),(6,9) )( )( )( ) } 25. 26. 27. 31. 35. 39. 43. 47. 49. 50. 51. 53. 57. 61. 1. 7. (i) fog x 4x2 –6 x 1 (ii) gof x 2x26x –1 43 2(iii) fof x x 6x 14 x 15 x 5 (iv) gog x 4–9 (ii)& (iv) (i) B C B C x B 28. C 29. B 30. D 32. B 33. A 34. C 36. B 37. D 38. A 40. B 41. A 42. A 44. B 45. D 46. A 48. R = 3,8 , 6,6 , (9,4), (12,2) = 1,1 , 1,2 , 2,1 ,(2,2),(2,3), (3,2), (3,3), (3,4), (4,3), (4,4), (5,5) } {()()( )fog =( )( )() 2,5 , 5,2 ,1,5 } R {()( )( ) gof = 1,3,3,1, 4,3 }and{ x 13fofof x 2 52. f –1 ()x =+4– x)7 (3x 1 False 54. False 55. False 56. False True 58. False 59. False 60. True False 62. False 2.3 EXERCISE –π π0 2. – 1 4. 5. – 12 3 14 –33 0, –1 8. 11. ,15 44 an – a–14 113. tan – x 17. 19. 1+aa34 1 n 20. C 21. D 22. B 23. D 24. A 25. A 26. B 27. C 28. A 29. B 30. A 31. D 32. D 33. B 34. A 35. C 36. A 37. A 2π 2π38. 39. 40. 3 41. φ3 5 π 2π42. 43. 44. 0 45. 13 3 46. –2π,2π 47. xy > – 1 48. π–cot –1 x 49. False 50. False 51. True 52. True 53. True 54. False 55. True 3.3 EXERCISE 1. 28 × 1, 1 × 28, 4 × 7, 7 × 4, 14 × 2, 2 × 14. If matrix has 13 elements then its order will be either 13 × 1 or 1 × 13. 2. (i) 3×3, (ii) 9, (iii) x2–, a 1a ya 0, 23 31 12 19 1422 3. (i) (ii) –1 202 ex sin x ex sin 2 x 2 x 2 xe sin xe sin 2 x4. 5. a = 2, b = 2 6. Not possible3x 3xe sin xe sin 2 x⎡52 –2 ⎤ ⎡0–1 1⎤ += 3Y =7. (i) XY ⎢⎥ (ii) 2X −⎢ ⎥12 01 –11 –10 –18 ⎣⎦ ⎣⎦ ⎡−5 –2 2⎤ (iii) Z=⎢ ⎥–12 0–1⎣⎦ 8. x = 4 –2 –3 –1 –1A11. 7 15 13. A = [– 1 2 1] ⎡9612 ⎤⎡12 9 ⎤⎢ ⎥15. AB= BA =7 816 ⎢⎥⎢ ⎥12 15 ⎣⎦ ⎢4510 ⎥⎣⎦ ⎡–2 0 ⎤⎡21⎤X = ,Y =19. ⎢ ⎥⎢⎥–1–3 22⎣ ⎦⎣⎦ 24. A = [– 4] 1 ⎡7-3 ⎤ 37. (i) ⎢⎥(ii) not possible2251⎣⎦38. x = 2, y = 4 or x = 4, y = 2, z = – 6, w = 4 –24 –10 39. –28 –38 41. a = 2, b = 4, c = 1, d = 3 ⎡18 8 ⎤ 43. ⎢⎥16 18 ⎣⎦ 45. a = – 2, b = 0, c = – 3 10. – 2, – 14 11A=12. 10 18. x = 1, y = 2 ⎡⎤k ⎡kk ⎤ ,20. ⎢⎥⎢ ⎥etc.2k 2k 2k⎣⎦⎣ ⎦where k is a real number 30. True when AB = BA 187 –195 A340. –156 148 1–2 –5 42. 34 0 44.True for all real values of α 1 11 x =± , y =± , z =±50. 2 63 ⎡−7 −9 10 ⎤⎡ 3 −11 ⎤ ⎢⎥ ⎢⎥−12 −15 17 −15 6 −551. (i) ⎢⎥ (ii) inverse does not exist (iii) ⎢⎥ ⎢ 1 1–1 ⎥⎢ 5 −22 ⎥⎣⎦ ⎣⎦ ⎡ 5 ⎤⎡ −3⎤22 01⎢⎥⎢ ⎥22⎢⎥⎢ ⎥31⎢⎥⎢ ⎥2 −1 +−10⎢⎥⎢ ⎥2252. ⎢⎥⎢ ⎥53 3 −1⎢⎥⎢ ⎥20⎢22 ⎥⎢ 22 ⎥⎣⎦⎣ ⎦ 53. A 54. D 55. B 56. D 57. D 58. D 59. A 60. B 61. C 62. D 63. A 64. A 65. D 66. D 67. A 68. Null matrix 69. Skew symmetric matrix 70. – 1 71. 0 72. Rectangular matrix 73. Distributive 74. Symmetrix matrix 75. Symmetrix matrix 76. (i)B A (ii)kA (iii) k A –B 77. Skew Symmetric matrix 78. (i) Skew symmetric matrix (ii) neither symmetric nor skew symmetric matrix 79. Symmetric matrix 80. AB = BA 81. does not exist 82. False 83. False 84. False 85. True 86. True 87. False 88. False 89. True 90. False 91. False 92. False 93. False 94. True 95. False 96. True 97. False 98. True 99. False 100. True 101. True 4.3 EXERCISE 1. x3 – x2 + 2 2. a2 (a + x + y + z) 3. 2x3y3z3 4. 3 (x + y + z) (xy + yz + zx) 5. 16 (3x + 4) 6. (a + b + c)3 n π⎛⎞θ or nπ+–1 12. =nπ ()⎜⎟13. x = 0, – 12 18. x = 0, y = – 5, z = – 36⎝⎠ 19. x = 1, y = 1, z = 1 20. x = 2, y = – 1, z = 4 24. C 25. C 26. B 27. D 28. C 29. A 30. A 31. A 32. C 33. D 34. D 35. D 1 36. B 37. C 38. 27 A 39. A 140. Zero 41. 42. (A–1)2 43. 92 44. Value of the determinant 45. x = 2 y = 7 46. (y – z) (z – x) (y – x + xyz) 47. Zero 48. True 49. False 50. False 51. True 52. True 53. True 54. False 55. True 56. True 57. True 58. True 5.3 EXERCISE 1. Continuous at x =1 2. Discontinuous 3. Discontinuous 4. Continuous 5. Discontinuous 6. Continuous 7. Continuous 8. Discontinuous 719. Continuous 10. Continuous 11. k = 12. k = 22 13. k = –1 14. k =±1 16. a = 1, b = –1 –5 117. Discontinuous at x = – 2 and x 18. Discontinuous at x = 1, and 22220. Not differentiable at x = 2 21. Differentiable at x = 0 cos x22. Not differentiable at x = 2 25. –(log2) sin 2 x⋅22⋅ 158x ⎡ 8 ⎤26. 8 ⎢log8 −⎥ 27. 2 28. 55xlog x log log xx ⎣ x ⎦ xa cos x sin 2 x – n–12 230. n 2ax b sin ax bxc cos ax bx c29. 2 x 2 x –1 sin tan x 1 sec 2 x 131. 2 x 1 –12232. 2xcos x 2xsin 2 x sin 2 x 33. 2 xx 1 2 cos x cos x 34. sin x –sin x.logsin x 35. sinmx xcos nx (–n tan x + mcotsin x 36. x 1 x 22 x 33 9x2 34 x 29 1137. – 1 38. 39. 40. – 122 –3 – x 23at 141. 2 42. 22 43. 4244.1– x ax 1– xt –1 −2θ ⎛ -θ3+θ2+θ+1⎞ 45. e⎜⎟ 46. cot θ 47. 1θ3+θ2+θ–1⎝⎠ 1 tan x – x 1−48. t 51. 52. 2 53.3 sin x 2 2xy2– y3 cos xy – yy sec xy tan xy54. 22 55. xy cos xy – xy sec xy tan xy – x –xy –4x3–4 xy 2 56. 57. 23 64. –2sin ycos 3 yy 4yx + 4 y – x 70. Not applicable since f is not differentiable at x = 1 x) 71. ,–2 79. p 3, q 5 84. C 88. A 92. A 96. B ⎛ 31+⎞ 100. ⎜⎟ 2⎝⎠ 104. True 3. 8 m/s 7. 0.018πcm3 71 3 72. (2, –4) 77. ,78. ,024 2 ⎛ 2 tan x ⎞ x sec xlog x ++82. xtanx ⎜⎟ 83. D⎝ x ⎠ 2 x2 +1 85. B 86. A 87. A 89. C 90. B 91. B 93. A 94. B 95. A 2 –1 97. x x –1 98. 99.3x 2 101. – 1 102. False 103. True 105. True 106. False 6.3 EXERCISE π4. ( 2– 2 )v unit/sec. 5. θ= 6. 31.923 28. 2 m/s towards light, –1 m/s3 9. 2000 litres/s, 3000 litre/s 11. 2x3 – 3x + 1 −1 ⎛ 12. k2 = 8 14. (4, 4) 15. tan ⎜⎟ 17. x + 3y =± 87⎝⎠ 18. (3, 2), (–1, 2) 23. (1, – 16), max. slope = 12 26. x = 1 is the point of local maxima; local maximum = 0 x = 3 is the point of local minima; local minimum = – 28 x = 0 is the point of inflection. 27. Rs 100 30. 6cm, 12 cm, 864 cm3 2 ⎛ 2π⎞31. 1:1 33. Rs 1920 34. x3 ⎜1+⎟3 ⎝ 27 ⎠ 35. C 36. B 37. A 38. C 39. D 40. A 41. A 42. D 43. B 44. B 45. C 46. B 47. D 48. A 49. B 50. C 51. A 52. C 53. B 54. C 55. B 56. A 57. B 58. B 59. C 60. (3, 34) 61. x+ y= 0 62. –,–1 63. (1, ) 64. 2 ab 7.3 EXERCISE 23xx3. – x+3log x+1 +c 4. c5.log x sin xc32 53xtan x tan x6. tan +C 7. c8. x+ c2 53 ⎤xx ⎡xx x 9. –2cos 2sin c 10. 2 ⎢ – +x– log x+1 ⎥+c 22 ⎣32 ⎡ 2 ⎤⎡ 3xx 4 a –1 ⎛⎞−⎢cos +1− ⎥+c ⎢x3/4 –log1 +x411. ⎜⎟2 12.aa⎥ 3 ⎢⎢ ⎝⎠⎣ ⎦⎣ 3 –1⎛ 1 ⎞2 1 –13x13. 1++c 14. sin c⎜ 2 ⎟ 343 ⎝ x⎠ 1 14t–3sin c15. 23 16. 3 x2 9–log xx29 c ⎦ ⎤ ⎥+c ⎥⎦ x –1 2 x –1+ 5–2x + x2 + c17. 5–2 x + x + 2log 2 1 ⎧⎪1 x2 –1 – log x21 c 19. ⎨log 18. log 4 ⎪4 ⎩ xa 2 a2 −1 ⎛ x – a ⎞ xsin−1 x– 20. 2– x sin +ax + c⎜⎟ 21. 1+ x 1− x ⎫⎪ 1 −1⎬ – tan x +c ⎪ 2⎭ 22 ⎝ a ⎠ 1– x2 1 sin 2 x sin xc22. – 23. tan x – cot x – 3x + c2 21 x3 24. sin c 25. 2 sin x + x + c3 a3 1 −12 2626. sec (x ) +c 27.23 log m 28. e2 –1 29. tan 1 e – 30. 2 31. π4 m –1 2 −12tan 32. 2–1 33. 34.3 23 1 x –2 3 –1 x+ tan + c35. log 7 x + 2 73 1 xx 36. 22 a tan 1 b tan 1 c 37. π a – ba b x –3log c tan–111 39. xe x + c x –1 6 x 23 38. +log 1− x2 ⎡ x –1 xx –1 x ⎤ 3a tan − + tan + c40. ⎢ ⎥ 41.aaa a⎣ ⎦ 2 −3x −3xe 3e42. [sin 3 x−cos3 x]+ [sin x−3cos x]+ c 24 40 ⎛⎞1 tan x–1 1 tan x– tan −1 ⎜ ⎟+ log 43. + c⎜ ⎟2 2 tan x 22 tan x+ 2 tan x+1⎝ ⎠ π ⎛ a2 +b2 ⎞ 3 π21 π 144. ⎜ 33 ⎟ 45. log3 46. log 47. log 4 ab⎝⎠ 8 22 42 48. A 49. C 50. A 51. C 52. D 53. C 54. D 55. D 56. D 57. A 58. D 59. e–1 ex 1 –1 –1 2cos xtan 60. c 61. 62. c 63. 0 x 4 2 233 8.3 EXERCISE 1 42 161. sq.units 2. p sq. units 3. 10 sq.units 4. sq.units 23 3 279 325. sq.units 6. sq. units 7. sq. units 8. 2π sq.units22 34 π 216 asq.units 9. 10. 96 sq.units 11. sq.units 12. sq. units3 34 19 ⎡ 8⎤13. sq. units 14. sq. units 15. 9 sq.units 16. 2 ⎢π−⎥sq.units 62 ⎣ 3⎦ 17. 4 sq.units 18. 15 sq. units 19. 4 32+π)a2 sq. units23 ( 1520. 6 sq.units 21. sq. units 22. 8 sq.units 23. 15 sq.units2 24. C 25. D 26. A 27. B 28. A 29. A 30. D 31. A 32. B 33. A 34. C 9.3 EXERCISE 26 xydy e 91. 2–2– k 2. 20 3.dx 2 21 ⎛x–1 ⎞ 2+kx– xyx–1 =log 4. () ⎜ ⎟ 5. yc=.e2 ⎝x+1 ⎠ mx −ax x6. (amye =+ce) e+y+ 1 = 0+) 7. (x– c22 – xx 18. = e2 9. y tan x 10. x yy2 c 11. ykx2 3 2 2 dy dy 22 dy13. (1 – x)2 −x–2 =0 14. x– y –2xy 0 dxdx dx 4x3 y−⎛⎞15. y 2 16. tan 1 ⎜⎟=log x+c31 xx⎝⎠ x−1 −1tan y 2 tan y tan–1 ⎛⎞+log yc=17. 2xe =e +c 18. ⎜⎟y⎝⎠ –cos2 x 3 x– y 23 y219. x yke 20. xy 3 e 21. ysin x= 2 + 2 2 1–1 2 xyy +xy ′ – y tan x)+log 1 +y)=c22. ′′ ()y′=0 23. ( 2 (2 dy 2sin x 2cos xx log xx 24. x–1 y–2 0 25. y –cos x 2– dx xx 39 –2cx – y xc26. x sin y cos y sin y ce 27. log 1 tan xy 2 3sin 2 x 2cos 2 x 2228. y – ce 3x 29. 2 x – y 3x13 30. y –1 x 12x 0 31. ke2x 1– xy 1 x – y ⎛⎞x 32. xy 1 33. log ⎜⎟=cx 34. D 35. C y⎝⎠ 36. A 37. C 38. B 39. C 40. C 41. D 42. A 43. C 44. D 45. B 46. B 47. C 48. C 49. D 50. A 51. A 52. B 53. B 54. B 55. B 56. C 57. B 58. A 59. A 60. C 61. C 62. D 63. C 64. C 65. A 66. D 67. D 68. C 69. C 70. A 71. A 72. A 73. C 74. B 75. A 76. (i) not defined (ii) not defined (iii) 3 dy pdy pdy ∫1 ⎛∫1 ⎞(iv) += Q xe =⎜Q1 ×e ⎟dy +cpy (v) ∫dx ⎝⎠ 2x 2 23(vi) y cx (vii) 31yx 4xc4 – x sin x cos x(viii) xy = Ae –y (ix) yce – 22 ex (x) x = c sec y (xi) x 77. (i) True (ii) True (iii) True (iv) True (v)False (vi) False (vii) True (viii) True (ix)True (x) True (xi) True 10.3EXERCISE 11 1   j6k1. 2ij k2ij k32 2. (i) 3 –2 (ii) 37 ba 1 jk G = 3– 3. –2i3 –6 4. c5. k= –2 6. 2 ijk72 23 –6 –1 1 7. ,, ;4,6 , ij–12 k8. 2i 4j 4k 9. cos 777 156 10. Area of the parallelograms formed by taking any two sides represented by ,andabcas adjacent are equal 2 274 11. 12. 13.217 2 GGGGGG  abbcca n GGGGGG 16. 17.abbcca 2 GGG 118. 5i2 jk 23 19. C 20. D 21. C 22. B 23. D 24. A 25. D 26. D 27. D 28. A 29. C 30. A 31. C 32. C 33. B 34. If aand bare equal vectors GG2 235. 0 36. 37. k∈]–1,1 [k≠ –1 38. ab42 G39. 3 40. a 41. True 42. True 43. True 44. False 45. False 11.3 EXERCISE jk ˆ 2. iy ˆ (3 – 2 + 6) 1. 5+5 iˆ 2ˆ+5 (x–1) + ( ˆ +2) j+ (z – 3)kˆ = λ ˆjj ˆ kˆ 3. (–1, – 1, – 1) 19–1 ⎛⎞ cos 4. ⎜⎟7. x + y + 2z = 19 8. x + y + z = 921⎝⎠ 9. 3x – 2y + 6z – 27 = 0 xyz xyz11. == and == 12–1 –1 1–2 14. ax + by + cz = a2 + b2 + c2 15. 15° or 75° 17. 7 18. 6 19. ˆˆj ˆ ˆˆ kˆ(x – 3)j + y + (z–1)k = λ(–2i + j +3 ) 20. 18x + 17y + 4z = 49 21. 14 22. 51x + 15y – 50z + 173 = 0 24. 4x +2y – 4z – 6 = 0 and –2x + 4y + 4z – 6 = 0 26. iˆˆj3+8 +3 31. A 35. D 22–1 38. ,,333 40. (x – 3) iˆ + 42. True 46. True 1. 42 5. 196 10. 21x + 9y – 3z – 51 = 0 12. 60° 14. (1, 1) 16. (2, 6, –2) 35 kˆ,– 3 iˆ – 7 ˆj +6kˆ 29. D 30. D 32. D 33. D 34. A xyz36. C 37. ++=1 234 39. (x – 5) iˆ( y 4) ˆj (z –6) ˆ++ + k ( y – 4) ˆj (z 7) kˆ =λ iˆˆjkˆ++ (–2 – 5 + 13) 43. True 47. True 2. 4 6. 43 9. Minimum value = 3 44. False 48. False 12.3EXERCISE 3. 47 7. 21 10. Maximum = 9, minimum = 3 ˆˆj ˆ=λ(3i + 7 + 2) k 41. x + y – z = 2 45. False 49. True 4. – 30 8. 47 11. Maximise Z = 50x 60y, subject to: 2x+ y≤ 20, x+ 2y≤ 12, x+ 3y≤ 15, x≥ 0, y≥ 0 12. Minimise Z 400x 200y, subject to: 5x 2 y 30 2xy 15 , 0, yx yx 0 13. Maximise Z = 100x 170ysubject to : 3x 2y 3600, x 4y1800, x 0, y 0 14. Maximise Z = 200x120ysubject to : xy 300, 3 xy 600, yx 100, x 0, y 0 15. Maximise Z = xy, subject to 2x+ 3y≤ 120, 8x+ 5y≤ 400, x≥ 0, y≥ 0 16. Type A : 6, Type B : 3; Maximum profit = Rs. 480 17. 2571.43 18. 138600 19. 150 sweaters of each type and maximum profit = Rs 48,000 2 1020. 54 km. 21. 37 11 22. Model X : 25, Model Y : 30 and maximum profit = Rs 40,000 23. Tablet X : 1, Tablet Y : 6 24.Factory I : 80 days, Factory II : 60 days 25. Maximum : 12, Minimum does not exist 26. B 27. B 28. A 29. D 30. C 31. D 32. D 33. A 34. B 35. Linear constraints36. Linear 37. Unbounded 38. Maximum 39. Bounded 40. Intersection 41. Convex 42. True 43. False 44. False 45. True 13.3EXERCISE 251. Independent 2. not independent 3. 1.1 4. 56 15 75. P(E) = , P(F) : ,P(G)= ,no pair is independent12 18 36 3115 337. (i) , (ii) 2, (iii) 4, (iv) 8. ,4 8 410 9. (i) E1 and E2 occur (ii) E1 does not occur, but E2 occurs (iii) Either E1 or E2, or both E1 and E2 occurs (iv) Either E1 or E2 occurs, but not both 1233 110. (i) , (ii) 12. 13. Rs 0.50 14.3182 10 85 715. Expectation = Rs 0.65 16. 17.153 15 5 15718. 19. 20. 21.9 270725 16 128 4547 98 22. 23. 1– 24. (i) .1118 (ii) .44758192 10 8 14125. (i) 15, (ii) ,, (iii) 1 26. 0.7 (approx.) 27. 0.1815 15 128. 29. 0 1 2 P (X) .54 .42 .04 X2 108 9⎛ 49 ⎞ 45(49) 59(49) 31. (i) ⎜⎟ (ii) 10 (iii) 10⎝ 50 ⎠ (50) (50) 19 p –132. 33. 34.3 44 n –1 35. X 1 2 3 4 5 6 P(X) 36 36 36 36 36 36 1 665 775 36. p = 37. 38.2 324 7776 7 11 2939. not independent 41. (i) , (ii) 42. (i) , (ii)18 18 1111 7 1143. (i) 0.49, (ii) 0.65, (iii) .314 44. 45.11 21 1 110 546. 47. 48.3 221 11 149. (i) , (ii) 5.2, (iii) 1.7 (approx.) 50. (i) 3, (ii) 19.0550 1551. (i) 4.32, (ii) 61.9, (iii) 52. 1022 2153. Mean 13 , S.D. = 0.377 54. 2 55.Mean = 6, Variance = 3 56. C 57. A 58. D 59. C 60. C 61. D 62. B 63. D 64. C 65. D 66. D 67. D 68. C 69. D 70. D 71. D 72. C 73. C 74. C 75. B 76. B 77. D 78. C 79. A 80. D 81. B 82. C 83. C 84. A 85. B 86. A 87. C 88. D 89. D 90. A 91. B 92. D 93. D 94. False 95. True 96. False 97. False 98. True 99. True 100. True 101. True 102. False 103. True 110 1104. 105. 106.3 9 10 2107. Σ piix– (Σpii x)2 108. independent