1. 4. 7. 10. 12. 14. 16. 18. 20. 22. EXERCISE 7.1 1 cos 2 2 x− 2. 1 sin 3 3 x 3. 31 ( ) 3 ax b a + 5. 31 4 cos 2 2 3 x x e− − 6. 3 C 3 x − x + 8. 3 2 C 3 2 ax bx cx + + + 9. 2 log 2 C 2 x x x+ − + 11. 7 3 2 2 2 2 8 7 x x+ + Cx + 13. 3 5 2 2 2 2 C 3 5 x x− + 15. 2 3sin + Cx x x e− + 17. tan x + sec x + C 19. 2 tan x – 3 sec x + C 21. A EXERCISE 7.2 12x e 2 43x e + x + C 3 23 x x + e + C 3 x 24 + 5x ++C 2 x 3 x + x + C 3 7 53 64 2 22x + x + 2x + C 75 23 10 3 x +3cos x + x2 + C 33 tan x – x + C C 1. log (1 + x2) + C 2. 31 (log| |) C 3 x + 3. log 1+log Cx + 4. cos (cos x) + C 5. 1 cos2( ) C 4 ax b a − + + 6. 3 2 2 ( ) C 3 ax b a + + 7. 5 2 2 ( 2) 5 x + 3 2 4 ( 2) C 3 x− + + ANSWERS 589 8. 3 2 2 1 (1 2 ) 6 x+ C+ 9. 3 2 2 4 ( 1) C 3 x + x + + 10. 2log 1 Cx − + 11. 2 3 8) Cx − + 12. 7 3 3 1 ( 1) 7 x − 4 3 3 1 ( 1) 4 x+ − C+ 13. 3 2 1 C 18(2 3 )x − + + 14. 1(log ) C 1 m x m − + − 15. 21 log | 9 4x | C 8 − − + 16. 2 31 C 2 x e + + 17. 2 1 2 x e − C+ 18. 1 tan Cx e − + 19. log( ) +C x x e e −+ 20. 2 21 log ( ) C 2 x x e e −+ + 21. 1 tan (2 3) C 2 x x− − + 22. 1 tan (7 4 ) C 4 x− − + 23. 1 21 (sin ) C 2 x − + 24. 1 log 2sin 3cos C 2 x x+ + 25. 1 C (1 tan x) + − 26. 2sin Cx + 27. 3 2 1 (sin 2 ) 3 x C+ 28. 2 1+sin Cx + 29. 21 (logsin ) C 2 x + 30. – log 1+cos x C+ 31. 1 C 1+cos x + x 1 x 1 32. − log cos x + sin x + C 33. − log cos x −sin x + C 22 22 13 13 34. 2 tan x + C 35. (1 + log x) + C 36. (x + log x) + C 33 1 −14 37. − cos(tan x ) + C 38. D 4 39. B EXERCISE 7.3 x 1 11 1. − sin (4 x +10) + C 2. − cos7 x + cos x + C 28 14 2 1 ⎡ 1 11 ⎤ 3. sin12 x + x + sin8 x + sin 4 x + C⎢ ⎥4 ⎣12 84 ⎦ 1 13 16144. − cos(2 x +1) + cos (2 x +1) + C 5. cos x − cos x + C 26 64 1 ⎡ 111 ⎤ 6. cos6 x − cos 4 x − cos2 x + C⎢ ⎥4 ⎣ 642 ⎦ 1 ⎡ 11 ⎤ x 7. sin 4 x − sin12 x + C 8. 2tan − x + C⎢ ⎥2 ⎣ 4 12 ⎦ 2 x 3x 11 9. x − tan + C 10. − sin 2 x + sin 4 x + C 2 84 32 3x 11 11. + sin 4 x + sin8 x + C 12. x – sin x + C 88 64 1 −+ C13. 2 (sinx + x cosα) + C 14. cos x +sin x 11 1 15. sec 3 2x − sec2 x + C 16. tan 3 x − tan x + x + C 62 3 17. sec x – cosec x + C 18. tan x + C 12 tan x + tan x + C 20. log 19. log cos x + sin x + C 2 21 cos( x − a)π xx log + C21. −+ C 22. sin ( a − b) cos( x −b)22 23. A 24. B EXERCISE 7.4 1 −13 log 2x + 1+ 4x 2 + C1. tan x +C 2. 2 1 15xlog + C –1 4. sin + C3. 53 3 2 − x + x 2 − 4x + 5 1 1+ x 3 tan −12 x 2 + C 6. log + C5. 1− x 3 1 22 6 3 66 7. x 2 −1 − log x + x 2 −1 + C 8. log x + x + a 3 9. log tan x + tan 2 x +4 + C 10. log x +1+ x 2 + 2x 1 ⎛ 3x +1⎞⎛ x +3 ⎞−1 –1 tan + C sin + C11. ⎜⎟ 12. ⎜⎟⎝⎠62 ⎝ 2 ⎠ –1 ⎛ 2x −3⎞32 + Clog x – + x − 3x + 2 + C 14. sin ⎜13. ⎟⎝ 41 ⎠2 a +b log x – + (x − a)( x −b) + C15. 2 ANSWERS 591 + C + 2 + C 16. 2 17. x 2 −1 + 2log x + x 2 −1 + C22x + x −3 + C 5 11 −1 ⎛ 3x +1⎞ log 3x 2 + 2x +1 − tan ⎜ + C18. ⎟6 32 ⎝ 2 ⎠ 926 x 2 –9x + 20 + 34 log x −+ x − 9x + 20 + C19. 2 ⎛ x − 2⎞2 −1 –4x – x + 4sin + C20. ⎜⎟⎝ 2 ⎠ 21. x 2 + 2x +3 + log x +1+ x 2 + 2x +3 + C 1 2 x −1− 6 log x 2 − 2x −5 + log +C22. 2 6 x −1+ 6 23. 5 x 2 + 4x +10 − 7 log x + 2 + x 2 + 4x +10 + C 24. B 25. B EXERCISE 7.5 (x + 2) 2 1 x − 3 log + C log + C1. 2. x + 16 x + 3 3. log x −1 − 5log x − 2 + 4log x − 3 + C 1 3 4. log x −1 − 2log x − 2 + log x − 3 + C 2 2 x 3 x − log 1 −5. 4log x+2 − 2log x +1 + C 6. + log 2 4 1 112 −1 7. log x −1 − log ( x +1) + tan x + C 2 42 2 x −1 11 x + 1 4 log −+ C log −+8. 9. 9 x + 2 3( x −1) 2 x −1 x −1 5 1 12 x +1 − log x −1 − log 2x + 3 + C10. log 2 10 5 5 5 5 x +1 − log x + 2 + log x − 2 + C11. log 3 2 6 x 21 3 12. + log x +1 + log x −1 + C 22 2 1 13. – log x −1 + log (1 + x2) + tan–1x + C 2 x − 1 171 − tan 14. 3log x –2 ++ C 15. log x + 1 2x + 24 n1 x 2 –sin xlog + C +C16. 17. log n xn +1 1–sin x 22 −1 x −1 x 1 ⎛ x +1⎞ x + tan −3tan + C log + C18. 19. ⎜ 2 ⎟332 2 ⎝ x + 3⎠ 2x + C C −1 x + C x1 x 4 − 1 ⎛ e –1 ⎞ log + C log +C20. 21. ⎜ x ⎟44 x e⎝⎠ 22. B 23. A 1. 3. 5. 7. 9. 10. 11. 13. 15. 17. 19. 21. 23. EXERCISE 7.6 x 1 – x cos x + sin x + C 2. − cos3 x + sin3 x + C 39 22 xx xe (x2 – 2x + 2) + C 4. log x −+ C 24 22 33 xx xx log 2 x −+ C 6. log x −+ C 24 39 1 x 1− x 2 x 2 −1 x 1 −12 −1(2 x −1) sin x ++ C 8. tan x −+ tan x + C 4 4 222 –1 cos xx (2 x 2 −1) − 1− x 2 + C 44 –1 2 −1(sin x)2 x + 21 − x sin x − 2 x + C ⎡ 2 –1 ⎤– 1– x cos x + x + C 12. x tan x + log cos x + C ⎢ ⎥⎣ ⎦ 2 22 −12 21 x xx x tan x − log(1 + x ) + C 14. (log x) − log x ++ C 2 224 ⎛ x 3 ⎞ x 3 + x log x −− x + C x⎜⎟ 16. e sin x + C⎝ 3 ⎠ 9 x e x + C 18. ex tan + C 1+ x 2 xe x e + C+ C 20. 2(x −1) x 2x e (2sin x −cos x) + C 22. 2x tan–1x – log (1 + x2) + C5 A 24. B 1. 3. 4. 5. 6. 7. 8. 9. 10. EXERCISE 7.7 1 x 112 −1 −12 x 4 − x + 2sin + C 2. sin 2x + x 1− 4x + C 2 242 ( +2) 2x x + 4x + 6 + log x + 2+ x 2 + 4x + 6 + C 2 ( +2) x x 2 + 4x +1− 3 log x + 2+ x 2 + 4x +1 + C 2 2 5 ⎛ x + 2⎞ x + 2−12sin + 1− 4x − x + C⎜ ⎟2 ⎝ 5 ⎠ 2 ( +2) x x 2 + 4x −5 − 9 log x + 2+ x 2 + 4x − 5 + C 2 2 (2 x − 3) 13 ⎛ 2x −3⎞2 −11+ 3x − x + sin + C⎜ ⎟48 ⎝ 13 ⎠ 2x +3 3 x 2 + 3x − 9 log x ++ x 2 + 3x + C 4 8 2 x 23 x +9 + log x + x 2 + 9 + C 6 2 A 11. D EXERCISE 7.8 1 35 19 1. (b2 − a 2) 2. 3. 2 23 27 115 + e 8 4. 5. e − 6. 2 e 2 EXERCISE 7.9 3 64 1. 2 2. log 3. 23 1 4. 5. 0 6. e4 (e – 1)2 ANSWERS 595 7. 1 2 log 2 8. 2 1 log 2 3 ⎛ ⎞− ⎜ ⎟−⎝ ⎠ 9. π 2 10. π 4 11. 1 3 log 2 2 12. π 4 13. 1 2 log 2 14. 1 log6 5 + 13 tan 5 − 5 15. 1 2 (e – 1) 16. 5 5 3 5 – 9log log 2 4 2 ⎛ ⎞ −⎜ ⎟⎝ ⎠ 17. 4 2 1024 2 π π + + 18. 0 19. 3 3log 2 8 π + 20. 1 + 4 2 2 π π − 21. D 22. C EXERCISE 7.10 1 64 π 1. log 2 2. 3. – log 22 231 2 16 2 π 1 21 + 5 17 log 15 4 17 4 π e 2(e 2 − 2) 4. (2 +1) 5. 6. 7. 8. 9. D 84 10. B EXERCISE 7.11 πππ π 1.2.3. 4. 444 4 1 5. 29 6. 9 7. (n +1)( n + 2) π 16 2 π 1 π 8. log 2 9. 10. log 11. 8 15 222 12. π 13. 0 14. 0 15. a 16. – π log 2 17. 18. 5 20. 2 21. C MISCELLANEOUS EXERCISE ON CHAPTER 7 ⎡ 332 21 x + C ⎢(x + a)2 − (x + b)2log 1. 2. 3( a − b)2 1− x 2 ⎢⎣ 1 2(a − x) ⎛ 1 ⎞ 4 3. – + C 4. – 1+ + C⎜ ⎟⎝ 4a x ⎠x 11 1 36 65. 2 x − 3x + 6x − 6log(1 + x )+ C 1 13 x2 −1 6. − log x +1 + log ( x + 9) + tan + C 2 4 23 3 x 7. sin a log sin( x − a) + x cos a + C 8. + C 3 –1 ⎛ sin x⎞ 1 9. sin ⎜⎟ + C 10. − sin2 x + C⎝ 2 ⎠ 2 1 1cos( x + b) −14 + C 12. sin (x )+ C11. log 4sin ( a – b) cos( x + a) ⎛ 1+ ex ⎞ 11 x1 −1 13. log ⎜ x ⎟ + C 14. tan − x − tan +C⎝ 2+ e ⎠ 3 62 14 1 15. − cos x + C 16. log( x 4 +1) + C 44 [( ax +b)] n+1 –2 sin( x +αf )+ C + C17. 18. a ( +1) α sin xn sin 2(2 x −1) −12 x − x 2 19. sin x +− x + C π π 0 C ⎤ ⎥ + C ⎥⎦ 20. –2 1– x + cos −1 x + x − x 2 + C 21. ex tan x + C 22. 2log − 1 +1 3log 1 x x − + + 2 Cx + + 23. 11 cos 2 x x −⎡ −⎢⎣ 21 Cx ⎤− +⎥⎦ 24. 3 2 2 1 1 – 1 3 x ⎛ ⎞ +⎜ ⎟⎝ ⎠ 2 1 log 1 x ⎡ ⎛ ⎞ +⎜ ⎟⎢ ⎝ ⎠⎣ 2 C 3 ⎤ − +⎥⎦ 25. 2e π 26. 8 π 27. 6 π 28. 1 (2sin − 3 1) 2 − 29. 4 2 3 30. 1 log9 40 31. π 1 2 − 32. π (π 2) 2 − 33. 19 2 40. 21 1 3 e e ⎛ ⎞−⎜ ⎟⎝ ⎠ 41. A 42. B 43. D 44. B EXERCISE 8.1 1. 14 3 2. 16 4− 2 3. 32 8 3 − 2 4. 12π 5. 6π 6. π 3 7. 2 π 1 2 2 a ⎛ ⎞−⎜ ⎟⎝ ⎠ 8. 2 (4)3 9. 1 3 10. 9 8 11. 8 12. A 13. B EXERCISE 8.2 29 −12 2 ⎛ 2π 3 ⎞ + sin −1. 2. ⎜⎟ 643 32⎝⎠ 21 3. 4. 4 5. 8 2 6. B 7. B Miscellaneous Exercise on Chapter 8 7 1. (i) (ii) 624.8 3 17 2. 3. 4. 9 5. 4 63 8 a 23 6. 3 7. 27 8. (π − 2) 3 m 2 ab 91 9. (π − 2) 10. 11. 2 12. 42 3 79π 9 −1 ⎛ 1 ⎞ 1 − sin +13. 7 14. 15. ⎜⎟ 2 84 ⎝ 3 ⎠ 32 16. D 17. C 18. C 19. B EXERCISE 9.1 1. Order 4; Degree not defined 2. Order 1; Degree 1 3. Order 2; Degree 1 4. Order 2; Degree not defined 5. Order 2; Degree 1 6. Order 3; Degree 2 7. Order 3; Degree 1 8. Order 1; Degree 1 9. Order 2; Degree 1 10. Order 2; Degree 1 11. D 12. A EXERCISE 9.2 11. D 12. D EXERCISE 9.3 1. y″ = 0 2. xy y″ + x (y′)² – y y′ = 0 3. y″ – y′– 6y = 0 4. y″ – 4y′ + 4y = 0 5. y″ – 2y′ + 2y = 0 6. 2xyy′ + x2 = y2 7. xy′ – 2y = 0 8. xyy″ + x(y′)² – yy′ = 0 9. xyy″ + x(y′)² – yy′ = 0 10. (x² – 9) (y′)² + x² = 0 11. B 12. C EXERCISE 9.4 1. 3. 5. 7. 9. 11. 12. 14. 16. 18. 20. 22. x y = 2 tan − x + C 2 y = 1 + Ae –x y = log (ex + e –x) + C y = ecx y = x sin–1x + 2+ C1– x 1 22 31 y = log ⎡(x +1) ( x +1) ⎤ −⎣⎦42 1 ⎛ x 2 −1⎞ 13 y = log ⎜⎟ − log 2 x 2 24⎝⎠ y = sec x y – x + 2 = log (x2 (y + 2)2) (x + 4)2 = y + 3 6.93% 2log 2 ⎛ 11 ⎞ log ⎜⎟⎝ 10 ⎠ 2. 4. 6. 8. 10. –1 tan x 13. 15. 17. 19. 21. 23. y = 2 sin (x + C) tan x tan y = C 3 x tan –1 y = x + + C 3 –4–4x + y = C tan y = C ( 1 – ex) +1 ⎛ y − 2 ⎞ cos = a⎜⎟⎝ x ⎠ 2y – 1 = ex ( sin x – cos x) y2 – x2 = 4 1 (63 t + 27) 3 Rs 1648 A EXERCISE 9.5 − y x x + Cx1. (x − y)2 =Cxe 2. y = x log 3. 5. 7. 9. 11. 12. 14. 16. MATHEMATICS –1 ⎛ y ⎞ 1 22 tan = log ( x + y ) +C 4. x2 + y2 = Cx⎜⎟⎝ x ⎠ 2 1 x + 2y +C 222log = log x 6. y + x + y =Cx22 x − 2y ⎡⎛ y ⎞⎤ ⎛ y ⎞y x 1 −cos =Csin xy cos = C 8. ⎢ ⎜⎟⎥ ⎜⎟⎣⎝ x ⎠⎦ ⎝ x ⎠xx cy = log y –1 10. yye C+ x = x y π log ( x2 + y2) + 2 tan–1 = + log 2 x2 ⎛ y ⎞ ex y + 2x = 3x2 y 13. cot ⎜⎟ =log ⎝ x ⎠ 2x⎛ y ⎞ 15. y = (x ≠ 0, x ≠ e)cos =log ex ⎜⎟ x 1−log x⎝⎠ C 17. D EXERCISE 9.6 1 1. y = (2sin x – cos x) + C e–2x 2. y = e–2x + Ce–3x 5 4 x 3. xy = C+ 4. y (sec x + tan x) = sec x + tan x – x + C 4 2 5. y = (tan x – 1) + C e–tanx 6. = x y (4log 21) C −− +x x 16 7. 2 log (1 log − = +y x ) C+x 8. 1 = (1+ )y x − 2 1log sin C(1 )x x −+ + x 1 C 9. cot sin y x x x x = − + 10. (x + y + 1) = C ey 2 y C 11. =x + 12. x = 3y2 + Cy 3 y π 13. y = cos x – 2 cos2 x 14. y (1 + x2) = tan–1 x – 4 x15. y = 4 sin3 x – 2 sin2 x 16. x + y + 1 = e17. y = 4 – x – 2 ex 18. C 19. D Miscellaneous Exercise on Chapter 9 1. (i) Order 2; Degree 1 (ii) Order 1; Degree 3 (iii) Order 4; Degree not defined 2 y 2 − x 2 y′=3. 5. (x + yy′)² = (x – y)2 (1 + ( y′)2)4xy sec x 6. sin–1y + sin–1x = C 8. cos y = 2 π x 9. tan–1 y + tan–1(ex)= 10. ey = y +C 2 11. log x – y = x + y +1 12. ye 2 x = (2 x + C) 2 2x +1 , x ≠− 113. y sin x = 2x 2 − π (sin x ≠ 0) 14. y = log x +12 15. 31250 16. C 17. C 18. C EXERCISE 10.1 1. In the adjoining figure, the vector OP represents the required displacement. 2. (i) scalar (ii) vector (iii) scalar (iv) scalar (v) scalar (vi) vector 3. (i) scalar (ii) scalar (iii) vector (iv) vector (v) scalar 4. (i) Vectors a and b are coinitial (ii) Vectors b and d are equal (iii) Vectors a and c are collinear but not equal 5. (i) True (ii) False (iii) False (iv) False 1. 2. 3. 4. 6. 8. 10. 13. 16. EXERCISE 10.2 a = 3, b = 62, c =1 An infinite number of possible answers. An infinite number of possible answers. x = 2, y = 3 5. –7 and 6; ˆ ˆ–7i and 6 j ˆ4 ˆj k− − 7. 1 ˆ 6 i + 1 6 ˆj + 2 6 ˆk 1 ˆi + 1 ˆj + 1 ˆk 9. 1 ˆi + 1 ˆk 3 3 3 2 2 40 ˆi − 8 ˆj + 16 ˆk 12. 1 , 2 , 3 30 30 30 14 14 14 1 , 3 − − 2 2 , 3 3 15. (i) 1 ˆ 3 i− + 4 3 1 ˆˆ 3 j k+ (ii) ˆ3ˆ 3i k− + ˆ3 2i + ˆˆj k+ 18. (C) 19. (D) EXERCISE 10.3 π –1 ⎛ 5⎞ 1. 2. cos 3. 0⎜⎟4 ⎝ 7⎠ 16 2 22 2 ,4. 6. 7. 6 a +11 ab . –35 b 114 37 37 a =1, b =1 9.8. 10. 813 −3 12. Vector b can be any vector 13. 2 14. Take any two non-zero perpendicular vectors a and b –1 ⎛ 10 ⎞ cos 15. ⎜ ⎟ 18. (D)⎝ 102 ⎠ EXERCISE 10.4 2 2 1ˆ π 111ˆˆ ,2. ± i + j + k 3. ;,1. 19 2 3 3 3 3222 27 a = 0 or b = 05. 3, 6. Either 2 8. No; take any two nonzero collinear vectors 61 9. 10. 15 2 11. (B) 12. (C) 2 Miscellaneous Exercise on Chapter 10 31ˆ ˆ1. i + j 2 2 2 222. x – x , y – y , z − z ;(x − x ) + ( y − y ) + (z − z )21 2121 21 21 21 −5 33ˆ ˆ3. i + j 2 2 4. No; take a , b and c to represent the sides of a triangle. 1 3 3 23 10 ˆ ˆˆˆ ˆ± i − j + k6. 10 i + j 7.5. 3 2 2 22 22 22 1 ˆˆ ˆ8. 2 : 3 9. 3 a + 5 b 10. (3 i –6 j + 2k); 11 7 1 ˆˆ ˆ12. (160 i –5 j + 70 k) 13. λ = 1 16. (B)3 17. (D) 18. (C) 19. (B) EXERCISE 11.1 −11 111 − 96 − 2 3. ,,, , ± , ±1. 0, 2. ± 11 11 11 −2 −23 −2 −3 −245 −1 2 2 333 , , ; , , ; , ,5. 17 17 17 17 17 17 42 42 42 EXERCISE 11.2 4. r = iˆ + 2ˆj +3 kˆ +λ (3 iˆ + 2ˆj − 2kˆ ) , where λ is a real number 5. r = 2 iˆ − ˆj + 4 kˆ +λ ( iˆ + 2ˆj − kˆ) and cartesian form is x − 2 y + 1 z − 4 == 12 −1 x + 2 y − 4 z + 5 6. == 3 56 ˆˆˆ ˆˆˆ7. r = (5i − 4 j + 6 k) +λ (3 i + 7 j + 2k) 8. Vector equation of the line: r =λ (5 iˆ − 2ˆj + 3kˆ ); x yz Cartesian equation of the line: == 5 − 23 9. Vector equation of the line: r = 3iˆ − 2ˆj − 5kˆ +λ (11 )ˆk x − 3 y + 2 z + 5 Cartesian equation of the line: == 0 011 −1 ⎛ 19 ⎞ −1 ⎛ 8 ⎞ 10. (i) θ = cos ⎜⎟ (ii) θ = cos ⎜ ⎟ ⎝ 21⎠ 53⎝ ⎠ −1 ⎛ 26 ⎞ −1 ⎛ 2 ⎞ 11. (i) θ = cos (ii) θ = cos⎟ ⎜⎟⎜ 9 38 ⎠ ⎝ 3 ⎠⎝ 70 32 12. p = 14. 15. 2 29 11 2 38 16. 17. 19 29 EXERCISE 11.3 1 11 1 , , ; 333 3 2 3 −15 8 1. (a) 0, 0, 1; 2 (b) , , ;(c) (d) 0, 1, 0; 14 14 14 14 5 ⎛ 3 iˆ +5ˆj − 6 kˆ ⎞ 2. r ⋅ ⎜⎟ = 7⎜ ⎟70 ⎝ ⎠ 3. (a) x + y – z = 2 (b)2x + 3y – 4 z = 1 (c) (s – 2t) x + (3 – t) y + (2s + t) z = 15 ⎛ 24 36 48 ⎞⎛ 18 24 ⎞ 4. (a) ⎜ ,, ⎟ (b) ⎜ 0, , ⎟⎝ 29 29 29 ⎠ ⎝ 25 25 ⎠ ⎛ 1 11 ⎞⎛ − 8 ⎞ (c) ⎜ ,, ⎟ (d) ⎜ 0, ,0⎟ ⎝ 3 33 ⎠⎝ 5 ⎠ 5. (a) [ r − (iˆ − 2 kˆ)] ( ⋅ iˆ + ˆj − kˆ) = 0; x + y – z = 3 (b) [ r − (iˆ + 4ˆj + 6 kˆ)] ( ⋅ iˆ − 2ˆj + kˆ) = 0; x – 2y + z + 1 = 0 6. (a) The points are collinear. There will be infinite number of planes passing through the given points. (b) 2x + 3y – 3z = 5 5 7. , 5, –5 8. y = 3 9. 7x – 5y + 4z – 8 = 0 2 ˆˆ10. r ⋅(38 i + 68 j + 3kˆ)= 153 11. x – z + 2 = 0 −1 ⎛ 15 ⎞ 12. cos ⎜⎝ −1 ⎛ 2 ⎞ 13. (a) cos (b) The planes are perpendicular⎜⎟⎝ 5 ⎠ (c) The planes are parallel (d) The planes are parallel (e) 45o 3 13 14. (a) (b)13 3 (c) 3 (d)2 Miscellaneous Exercise on Chapter 11 x yz o3. 90° 4. == 5. 0100 −10 6. k = 7. ˆˆˆ ˆˆˆ7 r = i + 2 j + 3 k +λ ( i + 2 j − 5 k ) 8. x + y + z = a + b + c 9. 9 ⎛ 17 −13 ⎞⎛ 17 23 ⎞ 10. ⎜ 0, , ⎟ 11. ⎜ , 0, ⎟ 12. (1, – 2, 7)⎝ 22 ⎠⎝ 33 ⎠ 3 11 7 13. 7x – 8y + 3z + 25 = 0 14. p = or or 263 15. y – 3z + 6 = 0 16. x + 2y – 3z – 14 = 0 17. 33 x + 45y + 50 z – 41 = 0 18. 13 ˆˆˆ ˆˆˆ19. r = i + 2 j + 3k +λ (− 3i + 5 j + 4k) ˆˆˆ ˆˆˆ20. r = i + 2 j − 4k +λ (2 i + 3 j + 6k) 22. D 23. B EXERCISE 12.1 1. Maximum Z = 16 at (0, 4) 2. Minimum Z = – 12 at (4, 0) 235 ⎛ 20 45 ⎞ 3. Maximum Z = at ⎜ , ⎟ 19⎝ 19 19 ⎠ ⎛ 3 1⎞ ,4. Minimum Z = 7 at ⎜⎟⎝ 2 2⎠ 5. Maximum Z = 18 at (4, 3) 6. Minimum Z = 6 at all the points on the line segment joining the points (6, 0) and (0, 3). 7. Minimum Z = 300 at (60, 0); Maximum Z = 600 at all the points on the line segment joining the points (120, 0) and (60, 30). 8. Minimum Z = 100 at all the points on the line segment joining the points (0, 50) ANSWERS 607 9. 10. and (20, 40); Maximum Z = 400 at (0, 200) Z has no maximum value No feasible region, hence no maximum value of Z. EXERCISE 12.2 ⎛ 1⎞⎛ 8 ⎞ 1. Minimum cost = Rs 160 at all points lying on segment joining ⎜ ,0 ⎟ and ⎜ 2, ⎠⎟ .⎝ 3 ⎠⎝ 2 2. Maximum number of cakes = 30 of kind one and 10 cakes of another kind. 3. (i) 4 tennis rackets and 12 cricket bats (ii) Maximum profit = Rs 200 4. 3 packages of nuts and 3 packages of bolts; Maximum profit = Rs 73.50. 5. 30 packages of screws A and 20 packages of screws B; Maximum profit = Rs 410 6. 4 Pedestal lamps and 4 wooden shades; Maximum profit = Rs 32 7. 8 Souvenir of types A and 20 of Souvenir of type B; Maximum profit = Rs 160. 8. 200 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150000. 9. Minimise Z = 4x + 6y subject to 3x + 6y ≥ 80, 4x + 3y ≥ 100, x ≥ 0 and y ≥ 0, where x and y denote the number of units of food F1 and food F2 respectively; Minimum cost = Rs 104 10. 100 kg of fertiliser F1 and 80 kg of fertiliser F2; Minimum cost = Rs 1000 11. (D) Miscellaneous Exercise on Chapter 12 1. 40 packets of food P and 15 packets of food Q; Maximum amount of vitamin A = 285 units. 2. 3 bags of brand P and 6 bags of brand Q; Minimum cost of the mixture = Rs 1950 3. Least cost of the mixture is Rs 112 (2 kg of Food X and 4 kg of food Y). 5. 40 tickets of executive class and 160 tickets of economy class; Maximum profit = Rs 136000. 6. From A : 10,50, 40 units; From B: 50,0,0 units to D, E and F respectively and minimum cost = Rs 510 7. From A: 500, 3000 and 3500 litres; From B: 4000, 0, 0 litres to D, E and F respectively; Minimum cost = Rs 4400 8. 40 bags of brand P and 100 bags of brand Q; Minimum amount of nitrogen = 470 kg. 9. 140 bags of brand P and 50 bags of brand Q; Maximum amount of nitrogen = 595 kg. 10. 800 dolls of type A and 400 dolls of type B; Maximum profit = Rs 16000 EXERCISE 13.1 1. ( ) ( )2 1 P E|F , P F|E 3 3 = = 2. ( ) 16 P A|B 25 = 3. (i) 0.32 (ii) 0.64 (iii) 0.98 4. 11 26 5. (i) 4 11 (ii) 4 5 (iii) 2 3 6. (i) 1 2 (ii) 3 7 (iii) 6 7 7. (i) 1 (ii) 0 8. 1 6 9. 1 10. (a) 1 3 , (b) 1 9 11. (i) 1 2 , 1 3 (ii) 1 2 , 2 3 (iii) 3 4 , 1 4 12. (i) 1 2 (ii) 1 3 13. 5 9 14. 1 15 15. 0 16. C 17. D EXERCISE 13.2 3 25 44 1. 2. 3. 25 102 91 4. A and B are independent 5. A and B are not independent 6. E and F are not independent 11 7. (i) p = (ii) p = 10 5 8. (i) 0.12 (ii) 0.58 (iii) 0.3 (iv) 0.4 3 9. 10. A and B are not independent8 11. (i) 0.18 (ii) 0.12 (iii) 0.72 (iv) 0.28 7 16 20 40 12. 13. (i) , (ii) , (iii)8 81 81 81 21 111 14. (i) , (ii) 15. (i) , (ii) 16. (a) , (b) , (c)32 532 17. D 18. B EXERCISE 13.3 1. 1 2 2. 2 3 3. 9 13 4. 12 13 5. 22 133 6. 4 9 7. 1 52 8. 1 4 9. 13. 2 9 A 10. 14. 8 11 C 11. 5 34 12. 11 50 EXERCISE 13.4 1. (ii), (iii) and (iv) 2. X = 0, 1, 2; yes 3. X = 6, 4, 2, 0 4. (i) X 0 1 2 1 1 1 P(X) 4 2 4 (ii) X 0 1 2 3 P(X) 1 8 3 8 3 8 1 8 (iii) X 0 1 2 3 4 P(X) 1 16 1 4 3 8 1 4 1 16 5. (i) (ii) X 0 1 2 4 4 1 P(X) 9 9 9 X 0 1 25 11 P(X) 36 36 6. 7. X 0 1 2 3 4 P(X) 256 625 256 625 96 625 16 625 1 625 X 0 1 2 9 6 1 P(X) 16 16 16 1 317 8. (i) k = (ii) P(X < 3) = (iii) P(X > 6) = 10 10 100 3 (iv) P(0 < X < 3) = 10 11 1 9. (a) k = (b) P(X < 2) = , P(X ≤ 2) =1, P(X ≥ 2) = 62 2 1 14 10. 1.5 11. 12. 33 13. Var(X) = 5.833, S.D = 2.415 14. X 14 15 16 17 18 19 20 21 P(X) 2 15 1 15 2 15 3 15 1 15 2 15 3 15 1 15 Mean = 17.53, Var(X) = 4.78 and S.D(X) = 2.19 15. E(X) = 0.7 and Var (X) = 0.21 16. B 17. D EXERCISE 13.5 3 763 1. (i) (ii) (iii)32 64 64 25 ⎛ 29 ⎞⎛ 19 ⎞9 2. 3. ⎜ ⎟⎜ ⎟216 ⎝ 20 ⎠⎝ 20 ⎠ 1 45 243 4. (i) (ii) (iii)1024 512 1024 5. (i) (0.95)5 (ii) (0.95)4 × 1.2 (iii) 1 – (0.95)4 × 1.2 (iv) 1 – (0.95)5 4 20 ⎛ 9 ⎞ ⎛ 1 ⎞ 20 20 ⎡20C + C + ... + C ⎤6. ⎜ ⎟ 7. ⎜⎟⎣ 12 13 20 ⎦⎝ 10 ⎠⎝ 2 ⎠ 11 9. 243 50 49 49 ⎛ 99 ⎞ 1 ⎛ 99 ⎞ 149 ⎛ 99 ⎞ 10. (a) 1−⎜⎟ (b) ⎜⎟ (c) 1− ⎜⎟⎝ 100 ⎠ 2 ⎝ 100 ⎠ 100 ⎝ 100 ⎠ 7 ⎛ 5⎞ 5 35 ⎛ 5⎞ 4 22 × 93 11. ⎜⎟ 12. ⎜⎟ 13. 11 12 ⎝ 6⎠ 18 ⎝ 6⎠ 10 14. C 15. A Miscellaneous Exercise on Chapter 13 1. (i)1 (ii)0 11 2. (i) (ii)32 20 3. 21 10 10 r 10 −r4. 1− ∑ C (0.9) (0.1)r r = 7 646⎛ 2 ⎞⎛ 2 ⎞⎛ 2 ⎞ 864 5. (i) ⎜⎟ (ii) 7⎜⎟ (iii) 1− ⎜⎟ (iv)⎝ 5 ⎠⎝ 5 ⎠⎝ 5 ⎠ 3125 612 MATHEMATICS 6. 10 9 5 2 × 6 7. 625 23328 8. 2 7 9. 4 31 2 9 3 ⎛ ⎞ ⎜ ⎟⎝ ⎠ 10. n ≥ 4 11. 11 216 12. 1 2 8 , , 15 5 15 13. 14 29 14. 3 16 15. 17. (i) A 0.5 (ii) 0.05 18. C 16. 19. 16 31 B —•—

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