1. (i) {2} (ii) {0, 1} (iii) {1, p} ⎧−11⎫2. (i) {0, –1, 1} (ii) ⎨⎬ (iii) {− 3, − 2, 2,3}⎩ 3 ⎭ 3. {1, 2, 22, 23, ...2P – 1,(2p –1} 4. (i) True (ii) False (iii) True (iv) True 7. (i) {2, 4, 6, 8, ... , 98} (ii) (1,4, 9, 16, 25, 36, 49, 64, 81,} ⎧13⎫8. (i) {4, 8, 12} (ii) {7, 8, 9} (iii) ⎨ ,1, ⎬ (iv) {0, 1, 2} ⎩ 22⎭ 9. (i) {4, 5, 6, ....10} (ii) {5} (iii) {1, 2, 3, 4, 5} 10. 11. 13. True 14. False 15. True 16. True 17. True 22. T = {10} 24. (i) 2 (ii) 3 (iii) 3 (iv) 9 25. 25 26. 20 27. (a) 3300 (b) 4000 28. (i) 6, (ii) 3, (iii) 9, (iv) 1, (v) 2, (vi) 6, (vii) 30, (viii) 20 29. C 30. B 31. B 32. D 33. C 34. D 35. B 36. B 37. C 38. C 39. C 40. A 41. B 42. B 43. C 44. [1,2] 45. 1 46. n (B) 47. A ∩ B′ 48. {φ, {1}, {2}, {1, 2} 49. {0, 1, 2, 3, 4, 5, 6, 8} 50.(i) {1,5, 9, 10 } (ii) { 1, 2,3, 5, 6, 7, 9, 10 } 51. A ∪ Β′ 52. (i) ↔(b) (ii) ↔(c) (iii) ↔ (a) (iv) ↔(f) (v) ↔(d) (vi) ↔ (e) 53. True 54. False 55. False 56. True 57. True 58. False 1. (i) {(– 1, 1), (–1, 3), (2, 1), (2, 3), (3, 1), (3, 3)} (ii) {(1, –1), (1, 2), (1, 3), (3, –1), (3, 2), (3, 3)} (iii) {(1, 1), (1, 3), (3, 1), (3, 3)} (iv) {(–1, –1), (–1, 2), (–1, 3), (2, –1), (2, 2), (2, 3), (3, –1), (3, 2), (3, 3)} 2. {(0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)} 3. (i) {(0, 3), (1, 3)} (ii) {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1,5)} 11 2 4. (i) a = and b = (ii) a = 0 and b = – 2335. (i) {(1, 4), (2, 3), (3, 2), (4, 1) } (ii) {(1, 1), (1, 2), (1,3),(2, 1), (2, 2), (3, 1)} (iii) { (4, 5), (5, 4), (5, 5)} 6. Domain of R = {0, 3, 4, 5} = Range of R 7. Domain of R1 = [–5, 5 ] and Range of R1 = [–3, 17 ] 8. R2 = {(0, 8), (8, 0) (0,– 8), (– 8, 0)} 9. Domain of R3= R and range of R3 = R+ ∪ {0} 10. (i) h is not a function (ii) f is a function (iii) g is a function (iv) s is a function(v) t is a constant function 11. (a) 6 (b) (c) 13 (d) t2 _ 4 (e) t + 513644 12. (a) x = 4 (b) x > 4 13. (i) (f + g) x = x2 + 2x + 2 (ii) (f – g) x = 2x – x2 ANSWERS 305 ⎛⎞f 2x +1 (iii) ( fg) x = 2x3 + x3 + 2x + 1 (iv) ⎜⎟x = 2gx +1⎝⎠14. (i) f = {(–1, 0), (0, 1), (3, 28), (7, 344), (9, 730)} 15. x = –1 , 16. Yes, α = 2 , β = −1 17. (i) R – {2nπ : n∈Z} (ii) R+ (iii) R (iv)R – {–1, 1} (v) R – {4} 18. (i)[ ,∞) (ii) (– ∞,1] (iii) [ 0 , ∞) (iv) [ – 2, 4]43 32 ⎧−2, −3≤xx <−2 ⎪19. () =⎨4, 2 x 2fx −≤< ⎪2,2 x 3⎩ ≤≤ 21. (i) (f + g) x = x + x (ii) (f – g ) x = x – x ⎛⎞3 f 1(iii) (fg) x = (iv) x = x2 ⎜⎟g⎝⎠ x 22. Domain of f = (5, ∞) and Range of f = R+ 24. D 25. D 26. B 27. C 28. B 29. B 30. A 31. C 32. C 33. A 34. B 35. A 36. {2, 3, 4, 5} 37. (a) ↔ (iii) (b) ↔ (iv)(c) ↔ (ii) (d) ↔(i) 38. False 39. False 40. True 41. False 42. True. 56 2cos x 4. 5.33 cos2x 1 8. 21+ 7π16. θ = 2nπ + 4 5ππ 18. θ = ,33 22. 1 326. 2 nππ28. ± 28 30. C 34. B 38. A 42. A 46. C 50. C 54. A 58. B 62. tan β 22 ππ15. θ = nπ + (–1)n 4 − 4 π17. θ = 2nπ ± 3 πππ 19. x = ,,642 23 ⎛ 3 −11 ⎞ +25. ⎜ ⎟17 ⎝ 22 ⎠ π27. nπ ± 4 ππ 29. θ= π2n ±+ 412 31. D 32. D 33. C 35. C 36. B 37. C 39. B 40. D 41. D 43. D 44. C 45. B 47. C 48. C 49. B 51. B 52. C 53. C 55. B 56. A 57. B 159. D 60. 1 61. 8 163. [4 – 3(a2 – 1)2] , 2 −a2 4 64. x − x +1 65. 13 66. [–3, 3] 67. 2sin2A 68. True 69. False 70. False 71. True 72. False 73. True 74. True 75. True 76. (a) ↔ (iv) (b) ↔ (i) (c) ↔ (ii) (d) ↔ (iii) ANSWERS 307 (1) nn+ 1. P(n) : 2n< ∠n 2. P(n) : 1 + 2 + 3 + ... + n= 2 26. A 27. B 28. A 29. 4 30. False 21. 2n 2. –1 + i 3. (0, – 2) 4. 5 θ 31 5. (1, 0) 6. icot 11. –2i 12. –2i 22 2 ⎛10 ⎞ 2 13. 1:3 14. ⎜ ,0⎟ , 15. 1 18. 0⎝ 3 ⎠ 3 21. 2 ± i 2, − 2 ± i 2 22. –2 – i ⎛ 5π 5π⎞23. 2cos + isin ⎜⎟⎝⎠12 12 2 222 z + z25. (i) (a + b)( ) (ii) –15 1 2 1 i(iii) –2 (iv) 0 (v) − (vi) z (vii) 0122 (viii)6and 0 (ix) a circle (x) –2 3 + 2i 26. (i) F (ii) F (iii) T (iv) T (v) T (vi) T (vii) F (viii) F 27. (a) ↔ (v), (b) ↔ (iii), (c) ↔ (i), (d) ↔ (iv), (e) ↔ (ii), (f) ↔ (vi), (g) ↔ (viii) and (h) ↔ (vii) 28. 32. 35. 39. 43. 47. 1. 4. 7. 8. 9. 10. 11.12. 13. 14. 15. 17. 19. 23. 27. 31. EXEMPLAR PROBLEMS – MATHEMATICS −2 11−i 29. No25 25 2π1 33. 3 D 36. C B 40. A D 44. D C 48. C 1 ≤≤x1 2. [0,1] ∪[3,4]3 ⎡−34 22⎤[ 4,−−∪2] [2,6] 5. ⎢ , ⎥⎣33 ⎦ More than 2000. Between 7.77 and 8.77. (a2 +1)4 30. 24a +1 34. Real axis 37. B 41. A 45. B 49. C 31. – 2 3 + 2 i 38. A 42. B 46. B 50. A 3. ( −)∪(–3, 3)∪[5, ∞)−∞,56. No Solution More than 230 litres but less than 920 litres. Between 104 °F and 113 °F 41 cm. Between 8 km and 10 km No Solution xy 20,+≤ xy 8,+≤ No Solution. C D D (i) F (v) T 3 2 48, 0, 0x y x y+ ≤ ≥ ≥ 4, 5, 5, 0,x y x y x+ ≥ ≤ ≤ ≥ 0y ≥ 20. C 24. C 28. B (ii) F (vi) F 21. A 25. B 29. A (iii) T (vii) T 22. B 26. A 30. B (iv) F (viii) F 32. 1. 5. 9. 16. 21. 23. 25. 28. 32. 36. 40. 44. 48. 52. 56. 60. 61. 62. 63. 64. 1. (ix) T (x) F (xiii)F (xiv) T (i) ≤ (ii) ≥(v) > (vi) > 1440 2. 481 22 6. 3960 n−3C(r –2)!3! r −3 10. 14400 192 17. 190 18! 3 22. (a) 11C4(6!) (i) 14C9 (ii) 14C11 (i) 21, (ii) 441 (iii) 91 C B D B 1,51,200 35 False True (a) ↔ (ii) (a) ↔ (iii) (a) ↔ (iv) (a) ↔ (i) (a) ↔ (iii) ⎛⎞15 15 C10 ⎜⎟⎝⎠6 29. B 33. D 37. A 41. n = 7 45. 80 49. 7800 53. False 57. True (b) ↔ (iii) and (b) ↔ (i) (b) ↔ (iii) (b) ↔ (iii) (b) ↔ (i) ANSWERS 309(xi) T (xii) F (xv) T. (iii) > (iv) > (vii) < , > (viii) ≤. 3. 780 4. 144 7. 4,68000 8. 200 11. 112 15. r = 3 18. 8400 19. 3 20. 11 (b) 6C2 × 5C2 (c) 6C4 + 5C4 24. 2(20C5 × 20C6) 26. A 30. C 34. B 38. C 42. 0 46. 56 50. 64 54. True 58. False (c) ↔ (i) (c) ↔ (iv), (c) ↔ (ii), (c) ↔ (iv), (c) ↔ (ii) 8.3EXERCISE 2. k = ±3 3. –19 27. B 31. A 35. C 39. B 43. nr 47. 18 51. False 55. True 59. False (d) ↔ (ii) (d) ↔ (i) (d) ↔ (ii) 4. –3003 (310) (25) 189 17 5 5 x −21 195. (i) –252 (ii) ; 6. –252 7. –1365 8. 23x 252 yx8 16 9. r = 6 11. 990 12. p =±2 14. n = 9 17 17. 18. (C) 19. (A) 20. (C)54 21. (D) 22. (B) 23. (B) 24. (C) (n +)n +2)1(30 1625. C15 26. 2 27. C8 28. n = 12 1120 a–6a4 a56 b1429. 30. 28 C31. 1 32. Third term27 14 33. 12 34. F 35. T 36. F 37. F 38. T 39. F 40. F 2. Rs 1400 3. Rs 8080 , Rs 83520 5. 12 days 156. 3420° 7. cm 8. 2480 m 9. Rs 7258 11. (i) 4n3 + 9n2 + 6n (ii) 4960 12. Tr= 6r – 1 17. D 18. C 19. A 20. B 21. C 22. B 23. B 24. A 25. D ab 26. A 27. or 28. First term + last termbc 29. 45 30. F 31. T 32. T 33. F 34. F 35. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (ii) 36. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (ii)(d) ↔ (iv) 1. x + y + 1 = 0 2. x – 4y + 3 = 0 3. 60° or 120° ANSWERS 311 xy4. x + y = 7 or +=1 5. (3, 1), (– 7, 11) 68 7. y –3 x – 2 + 3 = 0 8. 3x + 4y + 3 = 0 −89. a= ,b = 4 10. 8x – 5y + 60 = 0 3 11. 3x +y = 8 12. x – 7y – 12 = 0 2 13. 14. (1, 1)3 15. 15° or 75° 17. 9x – 20y + 96 = 0 18. 3x – 4y + 6 = 0 and 4x – 3y + 1 = 0 20. (0, 2 + 2 22. A 23. A 24. B 25. B 26. C 27. D 28. A 29. A 30. A 31. B 32. B 33. A 34. C 35. A 36. B 37. B 38. C 39. D 40. B 41. B 42. (1, – 2) 43. x + y + 1 = 0 44. 3x – y – 7 = 0, x + 3y – 9 = 0 45. opposite sides 46. 13 (x2 + y2) – 83 x + 64 y + 182 = 0 47. 4 x2 y2 = p2 (x2 + y2) 48. True 49. False 50. False 51. True 52. True 53. True 54. True 55. False 56. False 57. (a) ↔ (iii) (b) ↔ (i) and (c) ↔ (ii) 58. (a) ↔ (iv) (b) ↔ (iii) (c) ↔ (i), (d) ↔ (ii) 59. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (iv), (d) ↔ (ii) ⎛ab⎞ ,1. x2 + y2 – 2ax – 2ay + a2 = 0 3. ⎜⎟⎝⎠22 34. x2 + y2 – 2x – 4y + 1 = 0 5. 4 6. x2 + y2 + 4x + 4y + 4 = 0 7. (1, 2) 8. x2 + y2 – 2x + 4y – 20 = 0 9. k ± 8 10. x2 + y2 – 6x + 12y – 15 = 0 11. 2 4 3912. ecentricity = and foci (4, 0) and (–4, 0) 13.54 4x24 y2 14. += 1 15. 18 16. (2, 4) , (2, – 4)81 45 4cos θa 17. 2 18. x2 + 8y = 32 19. m = 1 20. x2 – y2 = 32sin θ 13 x2 y2421. 22. −= . 23. x2 + y2 – 2x + 2y = 47 2 459 24. x2 + y2 – 4x – 10y + 25 = 0 25. (x – 3)2 + (y + 1)2 = 38 26. x2 + y2 – 18x – 16y + 120 = 0 27. x2 + y2 – 8x – 6y + 16 = 0 28. (a) y2 = 12x – 36, (b) x2 = 32 – 8y, (c)4x2 + 4xy + y2 + 4x + 32y + 16 = 0 29. 3x2 + 4y2 – 36x = 0 30. 9x2 + 5y2 = 180 32. (a) 15x2 – y2 = 15 (b) 9x2 – 7y2 + 343 = 0, (c) y2 – x2 = 5 33. False 34. False 35. True 36. False 37. True 38. False 39. True 40. True ⎛ 45 ⎞ 2 41. (x – 3)2 + (y + 4)2 = ⎜⎟ 42. x2 + y2 – 46x + 22y = 0⎝⎠13 4x24 y2 43. 6 + 2 5, 2 5 44. += 1 15 y2 x2 45. 4x2 + 4xy + y2 + 4x + 32y + 16 = 0 46. – =1 and (0, ± 10).36 64 47. (C) 48. (C) 49. (C) 50. (C) ANSWERS 313 51. A 52. B 53. A 54. A 55. D 56. B 57. C 58. A 59. A 2. (i) 1st octant (ii) 4th octant (iii) viiith octant (iv) vth octant (v) 2nd octant (vi) 3rd octant (vii) viiith octant (viii) vith octant 3. (i) (3,0,0), (0,4,0), (0,0,2) (ii) (–5, 0, 0), (0,3,0), (0,0,7) (iii) (4,0,0), (0, –3, 0), (0,0,5) 4. (i) (3,4,0), (0,4,5), (3,0,5) (ii) (–5, 3, 0),(0,3,7), (–5, 0, 7) (iii) (4,–3, 0), (0,–3, –5), (4, 0, –5) 5. 5 6. 11 9. (2,–4, 16) 11. (–2, –2, –1) 12. (1, 1, –2) 13. (–3, 4, –7), (7, 2, 5) and (–3, 12, 17) 14. (4, 7, 6) 15. (4, –5, 1), (3, –2, –1) 16. a = –2, b = –8, c = 2 ⎛ 713 ⎞ 17. ⎜ , ,9 ⎟ 18. 2:1 externally⎝ 22 ⎠ 19. vertices are (3,4,5), (–1,6,–7), (1,2,3) and centroid is (1,4, 20. 1:3 externally 21. (2,0,0), (2,2,0), (0,2,0), (0,2,2) (0,0,2) (2,0,2), (0,0,0), (2,2,2) 22. A 23. B 24. A 26. A 27. B 28. B 30. A 31. B 32. A 34. A 35. Three cordinates planes 37. given point 38. Eight 39. (0, y, z) 41. (0, 0, z) 42. x = 0 , y = 0 43. z- cordinates 44. (y, z cordinates) 45. yz-plane 46. x-axis 48. a = 5 or –3 49. (1, 1, –2) 1 )325. B 29. A 33. D 36. Three pairs 40. x = 0 47. 333 50. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (ii) (d) ↔ (vi) (e) ↔ (iv) (f) ↔ (v) (g) ↔ (viii) (h) ↔ (vii) (i) ↔ (x) (j) ↔ (ix) 1. 5. 9. 13. 17. 21. 25. 29. 31. 33. 35. 36. 37. EXEMPLAR PROBLEMS – MATHEMATICS 1 –216 2. 2 3. 2 x 4. 32 3 3 6. 5 (a + 2) 3 7. 7 8. 822 81 10. 1 11. 0 12.5 15 7 31 14. n = 5 15. 16.2 74 2m2 18. 1 19. 2 20. 3 n 2 22. 2 23. 1 24. 2 a cos a 13 4 26. 27. 0 28. k = 42 8 1 3332 + 2x+ 1– x2x 3– – + 32 30. 24x xx 22 32. 23 sec x + 5sec x+ 3tan x+ 3x 2tan xsec x 55 – 40x –15x2 54– x cos x+ 5 sec sin x+1 x22 34.5–7 x + 9 2() sin x x cosec x(2– xcot x)2 2 2(ax + cot x)( –qsin x)+( p + qcos x)( 2ax – cos ec x ) )bccos x + adsin x + db 2 2cos2 x(cd cos 38.+ x 39. 2x –730–433 )2( )( x )( x + 5 3241. sin 2xcos 2 x4 243. –2sin ( x +1)x 21 45. x –3 3 47. sec ( tan x +1xx ) 1 49. –4 50. 2 54. C 55. A 58. A 59. C 62. B 63. D 66. B 67. B 70. A 71. A 74. C 75. A 2378. m = 79. y3 ANSWERS 315 40. x2 cos x + 2xsin x – 2sin2 x –2ax + b() 42. 22(ax + bx + c)ad – bc 44. )2(cx + d 46. cos x – xsin x 2α 48. 22α – β 52. k = 6 53. c = 1 56. A 57. B 60. C 61. D 64. C 65. D 68. D 69. A 72. A 73. B 76. D 77. 1 80. 1 1. (i) to (v) and (viii) to (x) are statements. 2. (i) p : Number 7 is prime (ii) p : Chennai is in India q : Number 7 is odd q : Chennai is capital of Tamil Nadu (iii)p : 100 is divisble by 3 (iv) p : Chandigarh is capital of Haryana q : 100 is divisible by 11 q : Chandigarh is the capital of U.P r : 100 is divisible by 5 (v) p : 7 is a rational number (vi) p : 0 is less than every positive integer q : 7 is an irrational number q : 0 is less than every negative integer (vii) p : plants use sunlight for photosynthesis q : plants use water for photosynthesis r : plants use carbondioxide for photosynthesis (viii) p : two lines in a plane intersect at one point q : two lines in a plane are parallel (ix) p : a rectangle is a quadrilateral q : a rectangle is a 5- sided polygons. 3. (i) Compound statement is true and its component statements are : p : 57 is divisible by 2 and q : 57 is divisble by 3 (ii) component statement is true and its component statements are : p : 24 is multiple of 4 and q : 24 is multiple of 6 (iii) component statement is false and is component statements are p :All living things have two eyes q :All living things have two legs (iv) component statement is true and its component statements are : p : 2 is an number ; q : 2 is a prime number 4. (i) The number 17 is not prime (ii) 2 + 7 ≠ 6 (iii) Violet are not blue (iv) 5 is not a rational number (v) 2 is a prime number (vi) There exists a real number which is not an irrational number (vii) Cow has not four legs (viii) A leap year has not 366 days (ix) There exist similar triangles which are not congruent (x) Area of a circle is not same as the perimeter of the circle 5. (i) p ∧ q where p : Rahul passed in Hndi; q : Rahul passed in English (ii) p ∧ q where p : x is even integer ; q : y is even integer (iii) p ∧ q ∧ r where p : 2 is factor of 12; q : 3 is factor of 12; r : 6 is factor of 12 (iv) p ∨ q where p : x is an odd integer ; q : x +1 is an odd integer (v) p ∨ q where p : a number is divisible by 2, q : it is divisibe by 3 (vi) p ∨ q where p : x = 2 is a root of 3x2 – x –10 = 0, q : x = 3 is a root of 3x2 –x –10 = 0 ANSWERS 317 (vii) p ∨ q where p : student can take Hindi as an optional paper and q : student can take English as an optional paper. 6. (i) It is false that all rational numbers are real and complex (ii) It is false that all real numbers are rational or irrational (iii) x = 2 is not a root of the quadratic equation x2– 5x + 6 = 0 or x = 3 is not a root of the quadratic equation x2–5x + 6 = 0 (iv) A triangle has neither 3-sides nor 4-sides (v) 35 is not a prime number and it is not a complex number (vi) It is false that all prime integers are either even or odd (vii) x is not equal to x and it not eqaul to –x (viii) 6 is not divisible by 2 or it is not divisible by 3. 7. (i) If the number is odd number then its square is odd number (ii) If you take the dinner then you will get sweet dish (iii) If you will not study then you will fail (iv) If an integer is divisible by 5 then its unit digits are 0 or 5 (v) If the number is prime then its square is not prime (vi) If a,b and c are in A.P then 2b = a + c. 8. (i) The unit digit of an integer is zero if and only if it is divisible by 5. (ii) A natural number n is odd if and only if it is not divisible by 2. (iii) A triangle is an equilateral triangle if and only if all three sides of triangle are equal. 9. (i) If x ≠3 then x ≠ y or y ≠ 3 (ii) If n is not an integer then it is not a natural number. (iii) If the triangle is not equilateral then all three sides of the triangle are not equal (iv) If xy is not positive integer then either x or y is not negative integer. (v) If natural number n is not divisible by 2 and 3 then n is not divisible by 6. (vi) The weather will not be cold if it does not snow. 10. (i) If the rectangle R is rhombus then it is square. (ii) If tomorrow is Tuesday then today is Monday. (iii) If you must visit Taj Mahal you go to Agra. (iv) If the triangle is right angle then sum of squares of two sides of a triangle is equal to the square of third side. (v) If the triangle is equilateral then all three anlges of triangle are equal.(vi) If 2x= 3ythen x:y =3:2 (vii) If the opposite angles of a quadrilaterals are supplementary then S is cyclic. (viii) If xis neither positive nor negative than xis 0. (ix) If the ratio of corresponding sides of two triangles are equal then trianges are similar. 11. (i) There exists (ii) For all (iii) There exists (iv) For every (v) For all (vi) There exists (vii) For all (viii)There exists (ix) There exists (x) There exists 17..C 18. D 19. B 20. D 21. C 22. B 23. A 24. B 25. C 26. A 27. C 28. B 29. A 30. C 31. B 32. A 33. C 34. A 35. C 36. D 37. (i), (ii) and (iv) are statement; (iii) and (v) are not statements. 1. 0.32 2. 1.25 3. 2 –1 4 n n 4. 4 n 5. 2 –1 12 n 6. 3.87 7. 2 2 2 1 1 2 2 1 2 1 2 2 1 2 1 2 () ( ) ( ) ( ) ns ns nnx x nn nn + −+ + + 8. 5.59 9. 7 10. 1.38 11. Mean = 2.8, SD = 1.12 12. 8.9 13. 5000, 251600 14. Mean = 5.17, SD = 1.53 15. Mean = 5.5, Var. = 4.26 16. 0.99 17. 7.08 18. Mean = 239 40 , SD = 2.85 ANSWERS 319 ( −1) dn19. Var. = 1.16gm, S.D = 1.08 gm 20. Mean = a+ ,2 n2 −1S.D = d 12 21. Hashina is more intelligent and consistent 22. 10.24 23. Mean = 42.3, Var. 43.81 24. B 25. B 26. B 27. C 28. A 29. C 30. C 31. A 32. C 33. A 34. D 35. D 36. A 37. D 38. A 39. A 40. SD 41. 0, less 42. 11 43. Independent 44. Minimum 45. Least 46. greater than or equal 12 1. 2.3. 0.55672 3 5k−144. (a) 5k–1 elements (b) 5. 6. 0.9349 7. (a) 0.65 (b) 0.55 (c) 0.8 (d) 0 (e) 0.35 (f) 0.2 258. (a) 0.35 (b) 0.77 (c) 0.51 (d) 0.57 9. (a) (b)99 11 110. (a)p(John promoted) = , p(Rita promoted) = , p(Aslam promoted) = ,84 2 11 p(Gurpreet promoted) = (b) P(A) =84 11. (a) 0.20 (b) 0.17 (c) 0.45 (d) 0.13 (e) 0.15 (f) 0.51 12. (a) S= B,, , WW, BB, WW, B,W,BBWBBB, BWWBWWB,WW{ 121 212 121 21122 21} 12 (b) (c)63 5 2840 2 2 251513. (a) (b) (c) 14. (a) (b) (c) (d)143143143 1431432626 7 15. 16. (a) p(A) = .25, p(B) = .32 , p(A∩Β) = .17 (b) p(A∪B) = .40 (c) .40 (d) .68 133 517. (a) (b) (c) (d) 18. A 19. B13 242636 20. C 21. C 22. D 23. A 24. A 25. C 26. B 27. C 28. C 29. B 30. False 31. False 32. False 33. True 34. True 35. False 36. True 37. 0.15 38. 0.3 39. E ={ 2, 4, 6} 40. 0.2 41. 0.2 42. (a) ↔ (iv) (b) ↔ (v) (c) ↔ (i) (d) ↔ (iii) (e) ↔ (ii) 43. (a) ↔ (iv) (b) ↔ (iii) (c) ↔ (ii) (d) ↔ (i)

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