EXERCISE 1.1 1. (i), (iv), (v), (vi), (vii) and (viii) are sets. 2. (i) ∈ (ii) ∉ (iii) ∉ (vi) ∈ (v) ∈ (vi) ∉ 3. (i) A = {–3, –2, –1, 0, 1, 2, 3, 4, 5, 6 } (ii) B ={1, 2, 3, 4, 5} (iii) C = {17, 26, 35, 44, 53, 62, 71, 80} (iv) D = {2, 3, 5} (v) E = {T, R, I, G, O, N, M, E,Y} (vi) F={B, E, T, R} 4. (i) { x : x = 3n, n∈N and 1 ≤ n ≤ 4 } (ii) { x : x = 2n, n∈N and 1 ≤ n ≤ 5 } (iii) { x : x = 5n, n∈N and 1 ≤ n ≤ 4 } (iv) { x : x is an even natural number} (v) { x : x = n2, n∈N and 1 ≤ n ≤ 10 } 5. (i) A = {1, 3, 5, . . . } (ii) B ={0, 1, 2, 3, 4 } (iii) C = {–2, –1, 0, 1, 2 } (iv) D = { L, O,Y,A } (v) E = { February, April, June, September, November } (vi) F = {b, c, d, f, g, h, j } 6. (i) ↔ (c) (ii) ↔ (a) (iii) ↔ (d) (iv) ↔ (b) EXERCISE 1.2 1. (i), (iii), (iv) 2. (i) Finite (ii) Infinite (iii) Finite (iv) Infinite (v) Finite 3. (i) Infinite (ii) Finite (iii) Infinite (iv) Finite (v) Infinite 4. (i) Yes (ii) No (iii) Yes (iv) No 5. (i) No (ii) Yes 6. B= D, E = G EXERCISE 1.3 1. (i) ⊂ (ii) ⊄ (iii) ⊂ (iv) ⊄ (v) ⊄ (vi) ⊂ (vii) ⊂ 2. (i) False (ii) True (iii) False (iv) True (v) False (vi) True 3.(i), (v), (vii), (viii), (ix), (xi) 4. (i) φ, { a } (ii) φ, { a }, { b }, { a, b } (iii) φ, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } (iv) φ 5. 1 6. (i) (– 4, 6] (ii) (– 12, –10) (iii) [ 0, 7 ) (iv) [ 3, 4 ] 7. (i) { x : x ∈ R, – 3 < x < 0 } (ii) { x : x ∈ R, 6 ≤ x ≤ 12 } (iii) { x : x ∈ R, 6 < x ≤ 12 } (iv) { x R : – 23 ≤ x < 5 } 9. (iii) EXERCISE 1.4 1. (i) X ∪ Y = {1, 2, 3, 5 } (ii) A ∪ B = { a, b, c, e, i, o, u } (iii) A ∪ B = {x : x = 1, 2, 4, 5 or a multiple of 3 } (iv) A ∪ B = {x : 1 < x < 10, x ∈ N}(v) A ∪ B = {1, 2, 3 } 2. Yes, A ∪ B = { a, b, c } 3. B 4. (i) { 1, 2, 3, 4, 5, 6 } (ii) {1, 2, 3, 4, 5, 6, 7,8 } (iii) {3, 4, 5, 6, 7, 8 } (iv) {3, 4, 5, 6, 7, 8, 9, 10} (v) {1, 2, 3, 4, 5, 6, 7, 8 } (vi) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (vii) { 3, 4, 5, 6, 7, 8, 9, 10 } 5. (i) X ∩ Y = { 1, 3 } (ii) A ∩ B = { a } (iii) { 3 } (iv) φ (v) φ 6. (i) { 7, 9, 11 } (ii) { 11, 13 } (iii) φ (iv) { 11 } (v) φ (vi) { 7, 9, 11 } (vii) φ (viii) { 7, 9, 11 } (ix) {7, 9, 11} (x) { 7, 9, 11, 15 } 7. (i) B (ii) C (iii) D (iv) φ (v) { 2 } (vi) { x : x is an odd prime number } 8. (iii) 9. (i) {3, 6, 9, 15, 18, 21} (ii) {3, 9, 15, 18, 21 } (iii) {3, 6, 9, 12, 18, 21} (iv) {4, 8, 16, 20 ) (v) { 2, 4, 8, 10, 14, 16 } (vi) { 5, 10, 20 } (vii) {20} (viii) { 4, 8, 12, 16 } (ix) { 2, 6, 10, 14} (x) { 5, 10, 15 } (xi) {2, 4, 6, 8, 12, 14, 16} (xii) {5, 15, 20} 10. (i) { a, c } (ii) {f, g } (iii) { b , d } 11. Set of irrational numbers 12. (i) F (ii) F (iii) T (iv) T EXERCISE 1.5 1. (i) { 5, 6, 7, 8, 9} (ii) {1, 3, 5, 7, 9 } (iii) {7, 8, 9 } (iv) {5,7,9} (v) {1,2,3,4} (vi) { 1, 3, 4, 5, 6, 7, 9 } 2. (i) { d, e, f, g, h} (ii) { a, b, c, h } (iii) { b, d , f, h } (iv) { b, c, d, e ) 3. (i) { x : x is an odd natural number } (ii) { x : x is an even natural number } (iii) { x : x ∈ N and x is not a multiple of 3 } (iv) { x : x is a positive composite number and x = 1 ] (v) { x : x is a positive integer which is not divisible by 3 or not divisible by 5} (vi) { x : x ∈ N and x is not a perfect square } (vii) { x : x ∈ N and x is not a perfect cube } (viii) { x : x ∈ N and x ≠ 3 } (ix) { x : x ∈ N and x ≠ 2 } 9(x) { x : x ∈ N and x < 7 } (xi) { x : x ∈ N and x ≤ }2 6. A' is the set of all equilateral triangles. 7. (i) U (ii) A (iii) φ (iv) φ 7. False 12. We may take A = { 1, 2 }, B = { 1, 3 }, C = { 2 , 3 } 13. 325 14. 125 15. (i) 52, (ii) 30 16. 11 EXERCISE 2.1 1. x = 2 and y = 1 2. The number of elements in A × B is 9. 3. G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)} H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)} 4. (i) False P × Q = {(m, n), (m, m), (n, n), (n, m)} (ii) True (iii) True 5. A × A = {(– 1, – 1), (– 1, 1), (1, – 1), (1, 1)} A × A × A = {(–1, –1, –1), (–1, –1, 1), (–1, 1, –1), (–1, 1, 1), (1, –1, –1), (1, –1, 1), (1, 1, –1), (1, 1, 1)} 6. A = {a, b}, B = {x, y} 8. A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} A × B will have 24 = 16 subsets. 9. A = {x, y, z} and B = {1,2} EXERCISE 1.6 1. 2 2. 5 3. 50 5. 30 6. 19 7. 25, 35 Miscellaneous Exercise on Chapter 1 1. A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C 2. (i) False (ii) False (iii) True (iv) False (vi) True 4. 42 8. 60 (v) False 10. A = {–1, 0, 1}, remaining elements of A × A are (–1, –1), (–1, 1), (0, –1), (0, 0), (1, –1), (1, 0), (1, 1) EXERCISE 2.2 1. R = {(1, 3), (2, 6), (3, 9), (4, 12)} Domain of R = {1, 2, 3, 4} Range of R = {3, 6, 9, 12} Co domain of R = {1, 2, ..., 14} 2. R = {(1, 6), (2, 7), (3, 8)} Domain of R = {1, 2, 3} Range of R = {6, 7, 8} 3. R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)} 4. (i) R = {(x, y) : y = x – 2 for x = 5, 6, 7} (ii) R = {(5,3), (6,4), (7,5)}. Domain of R = {5, 6, 7}, Range of R = {3, 4, 5} 5. (i) R = {(1, 1), (1,2), (1, 3), (1, 4), (1, 6), (2 4), (2, 6), (2, 2), (4, 4), (6, 6), (3, 3), (3, 6)} (ii) Domain of R = {1, 2, 3, 4, 6} (iii) Range of R = {1, 2, 3, 4, 6} 6. Domain of R = {0, 1, 2, 3, 4, 5,} 7. R = {(2, 8), (3, 27), (5, 125), (7, 343)} Range of R = {5, 6, 7, 8, 9, 10} 8. No. of relations from A into B = 26 9. Domain of R = ZRange of R = Z EXERCISE 2.3 1. (i) yes, Domain = {2, 5, 8, 11, 14, 17}, Range = {1} (ii) yes, Domain = (2, 4, 6, 8, 10, 12, 14}, Range = {1, 2, 3, 4, 5, 6, 7} (iii) No. 2. (i) Domain = R, Range = (– ∞, 0] (ii) Domain of function = {x : –3 ≤ x ≤ 3} Range of function = {x : 0 ≤ x ≤ 3} 3. (i) f (0) = –5 (ii) f (7) = 9 (iii) f (–3) = –11 412 4. (i) t (0) = 32 (ii) t (28) = (iii) t (–10) = 14 (iv) 1005 5. (i) Range = (– ∞, 2) (ii) Range = [2, ∞) (iii) Range = R Miscellaneous Exercise on Chapter 2 2. 2.1 3. Domain of function is set of real numbers except 6 and 2. 10. (i) Yes, (ii) No 11. No 12. Range of f = {3, 5, 11, 13 } EXERCISE 3.1 1. (i) 5π 36 (ii) 19π 72 – (iii) 4π 3 (iv) 26π 9 2. (i) 39° 22′ 30″ (ii) –229° 5′ 29″ (iii) 300° (iv) 210° 3. 12π 4. 12° 36′ 5. 20π 3 6. 5 : 4 7. (i) 2 15 (ii) 1 5 (iii) 7 25 EXERCISE 3.2 1. sin x =− 3 cosec 2 , x –= 2 sec 2 tan 3 , x , x=− = 3 cot , x = 1 3 2. 5 4 5 3 4 cosec cos sec tan cot 3 5 4 4 3 x , x – , x , x , x= = =− =− =− 3. 4 5 3 5 4sin cosec cos sec tan 5 4 5 3 3 x , x – , x , x , x=− = =− = − = 4. 12 13 5 12 5sin cosec cos tan cot 13 12 13 5 12 x , x – , x , x , x=− = = =− =− 4. Domain = [1, ∞), Range = [0, ∞) 5. Domain = R, Range = non-negative real numbers 6. Range = Any positive real number x such that 0 ≤ x < 1 7. (f + g) x = 3x – 2 8. a = 2, b = – 1 9. (i) No(f – g) x = – x + 4 ⎛⎞fx +13 x = , x ≠⎜⎟ g 2x − 32⎝⎠ (ii) No (iii) No 5 13 12 13 12 5. sin x = ,cosec x = ,cos x =− ,sec x =− , cot x =− 13 513 12 5 1 36. 7. 2 8. 3 9. 10. 12 2 EXERCISE 3.3 31+5. (i) (ii) 2 – 322 EXERCISE 3.4 π 4ππ π 5ππ1. , ,n π +, n ∈ Z 2. ,, 2nπ ± , n ∈ Z333 333 5π 11π 5π 7π 11π n 7π3. , ,n π + , n ∈ Z 4. , ,n π + (–1) , n ∈ Z666 666 nπ ππ5. x = or x = nπ, n ∈ Z 6. x = (2n +1) ,or 2 nπ ± , n ∈ Z3 43 n 7ππ7. xn= π +− or (2 (1) n +1) , n ∈ Z62 nπ nπ 3π nππ8. x = , or + , n ∈ Z 9. x = ,or n π± ,n ∈ Z228 33 Miscellaneous Exercise on Chapter 3 5258. , , 255 63 ,– ,– 29. 33 8 + 215 8 − 2 15 10. ,,4+ 15 44 1. 3 2. 5. 2 – 7 i 6. 2429. −− 26i 10.27 13. i 14. −2π 1. 2, 3 ⎛ 3π2cos +i sin4. ⎜⎝ 4 6. 3 (cos π + i sin π) 1. ± 3 i 2. −±3 11 i5. 6.2 1 221 i 9. 2 EXERCISE 5.1 0 3. i 4. 14 + 28 i 19 21i 17 5−− 7. + i 8. – 4510 33 −22107 43 53− i−i 11. + i 12.3 27 2525 1414 –72 i 2 EXERCISE 5.2 ⎛ –π –π⎞5π 2. 2, 3. 2cos + i sin⎜ ⎟6 ⎝ 44 ⎠ 3π⎞ ⎛−3π −3π⎞2cos +i sin⎟ 5. ⎜ ⎟4 ⎠⎝ 44 ⎠ ⎛ ππ⎞ ππ7. 2cos⎜+ i sin ⎟ 8. cos + i sin ⎝ 66 ⎠ 22 EXERCISE 5.3 −±1 7 i −±333 i–1± 7i3. 4.42 –2 1± 7 i −±17 i 2 ± 34 i 7. 8.2 22 −±1 7 i 10. 22 Miscellaneous Exercise on Chapter 5 307 +599i1. 2 – 2 i 3. 442 ⎛ 3π 3π⎞ ⎛ 3π 3π⎞5. (i) 2cos +i sin⎜⎟ , (ii) 2cos +i sin⎜⎟⎝ 44 ⎠⎝ 44 ⎠24 2 52214i 8. ± i 9. ± i6. ± i 7. 1 ±33 2 2727 321 −2 13π 12. (i) , (ii) 0 13.10. , 14. x = 3, y = – 32 5 24 15. 2 17. 1 18. 0 20. 4 EXERCISE 6.11. (i) {1, 2, 3, 4} (ii) {... – 3, – 2, –1, 0, 1,2,3,4,}2. (i) No Solution (ii) {... – 4, – 3}3. (i) {... – 2, – 1, 0, 1} (ii) (–∞, 2)4. (i) {–1, 0, 1, 2, 3, ...} (ii) (–2, ∞)5. (–4, ∞) 6. (– ∞, –3) 7. (–∞, –3] 8. (–∞, 4] 9. (– ∞, 6) 10. (–∞, –6) 11. (–∞, 2] 12. (– ∞, 120] 13. (4, ∞) 14. (–∞, 2] 15. (4, ∞) 16. (–∞, 2] 17. x < 3, 18. x ≥ –1, 219. x > – 1, 20. x ≥ – ,7 21. Greater than or equal to 35 22. Greater than or equal to 82 23. (5,7), (7,9) 24. (6,8), (8,10), (10,12) 25. 9 cm 26. Greater than or equal to 8 but less than or equal to 22 EXERCISE 6.2 1. 2. 3. 4. 5. 6. EXERCISE 6.3 1. 2. 5. 6. 11. 12. 13. 14. 15. 1. [2, 3] 4. (– 23, 2] 7. 8. 9. 10. [– 7, 11] Miscellaneous Exercise on Chapter 6 2. (0, 1] 3. [– 4,2] ⎛ – 80 – 10 ⎤⎡ 11⎤5. ⎜ , ⎥ 6. ⎢1, ⎥⎝ 33 ⎦⎣ 3 ⎦ 11. Between 20°C and 25°C 12. More than 320 litres but less than 1280 litres. 13. More than 562.5 litres but less than 900 litres. 14. 9.6 ≤ MA ≤ 16.8 EXERCISE 7.1 1. (i) 125, (ii) 60. 2. 108 3. 5040 4. 336 5. 8 6. 20 1. 5. 1. 5. 9. 11. 1. 5. 9. 1. 4. 8. 1. 2. 3. 4. 5. 6. 9. 12. EXERCISE 7.2 (i)40320, (ii) 18 2. 30, No 3. 28 4. 64 (i)30, (ii) 15120 EXERCISE 7.3 504 2. 4536 3. 60 4. 120, 48 56 6. 9 7. (i) 3, (ii) 4 8. 40320 (i)360, (ii) 720, (iii) 240 10. 33810 (i)1814400, (ii) 2419200, (iii) 25401600 EXERCISE 7.4 45 2. (i) 5, (ii) 6 3. 210 4. 40 2000 6. 778320 7. 3960 8. 200 35 Miscellaneous Exercise on Chapter 7 3600 2. 1440 3. (i) 504, (ii) 588, (iii) 1632 907200 5. 120 6. 50400 7. 420 ×48C4 22C74C19. 2880 10. +22C10 11. 151200 EXERCISE 8.1 1–10x + 40x2 – 80x3 + 80x4 – 32x5 324020 5 x5 5xx3 x5 x3 x 8 32 64 x6 –576 x5 + 2160 x4 – 4320 x3 + 4860 x2 – 2916 x + 729 5 243 x + 35 81 x + 10 27 x + 10 9x + 3 5 3x + 5 1 x 6x 46x 215 x 20 2 15 x 4 6 x 6 1 x 884736 7. 11040808032 8. 104060401 9509900499 10. (1.1)10000 > 1000 11. 8(a3b + ab3); 40 6 2(x6 + 15x4 + 15x2 + 1), 198 EXERCISE 8.2 6 122 r1. 1512 2. – 101376 3. ()r Cr .y−1 .x − r 12 24−rr4. −1 r C .x .y 5. – 1760 x9y3 6. 18564 −105 35 () r 7. 8 x9 ; 48 x12 8. 61236 x5y5 10. n = 7; r = 3 12. m = 4 Miscellaneous Exercise on Chapter 8 91. a = 3; b = 5; n = 6 2. a = 3. 1717 5. 396 6 6. 2a8 + 12a6 – 10a4 – 4a2 + 2 7. 0.9510 8. n = 10 2 3416 8 3216 xxx9. +−+− 4x +++− 52 34x x x x 2 216 10. 27x6 – 54ax5 + 117a2x4 – 116a3x3 + 117a4x2 – 54a5x + 27a6 EXERCISE 9.1 12345 1. 3, 8, 15, 24, 35 2. ,,,, 3. 2, 4, 8, 16 and 322 3456 1115 74. − ,,, and 5. 25, –125, 625, –3125, 156256626 6 3921 75 496. ,, , 21and 7. 65, 93 8.222 2 128 3609. 729 10. 23 11. 3, 11, 35, 107, 323; 3+11+35+107 + 323 + ... −−11 −1 −1 ⎛−1⎞ ⎛−1⎞ ⎛−1 ⎞ ⎛−1 ⎞−1,,,, ;– 1++++ + ... 12. ⎜⎟⎜⎟⎜⎟⎜ ⎟2 6 24 120 ⎝ 2 ⎠⎝ 6 ⎠⎝ 24 ⎠⎝120 ⎠ 448 MATHEMATICS 35 81213. 2, 2, 1, 0, –1; 2 + 2 + 1 + 0 + (–1) + ... 14. ,, , and 23 5 EXERCISE 9.2 1. 1002001 2. 98450 4. 5 or 20 6. 4 n 179 7. (5n + 7) 8. 2q 9. 10. 02 321 13. 27 14. 11, 14, 17, 20 and 23 15. 1 16. 14 17. Rs 245 18. 9 EXERCISE 9.3 55 1. 20 , 2n 2. 3072 4. – 21872 1 )20101 5. (a) 13th , (b) 12th, (c) 9th 6. ± 1 7. ⎡⎣ −( . ⎤ ⎦6 ⎛ n ⎞ n 32n7 ⎣1−− 1 x( 31+ )⎜32 −1⎟ ⎡( a) ⎦⎤ x (− )8. 9. 10.⎜⎟2 2⎝⎠ 1+ a 1− x 3 11 52 255211. 22 +(3 −1) 12. r = or ; Terms are ,, 1 or ,, 1 2 25 5225 16 16 n −13. 4 14. ;2; (21) 15. 205977 4816 80 n 8−−−16. , 32 64 18. ( 1), , ,... or 4 ,− 8 16 , − , ,.. 10 −− n 333 819 19. 496 20. rR 21. 3, –6, 12, –24 26. 9 and 27 –127. n = 30. 120, 480, 30 (2n) 31. Rs 500 (1.1)10 2 32. x2 –16x + 25 = 0 EXERCISE 9.4 n (+1)(n + 2)(n + 3)nn1. (n +1)( n + 2) 2.3 4 3. ( ) ( )213 5 1 6 n n n n+ + + 4. 1 n n + 5. 2840 6. 3n (n + 1) (n + 3) 7. ( ) ( )1 2 2 12 nn n+ + 8. ( ) ( )21 3 23 34 12 nn n n + + + 9. ( ) ( ) ( )12 1 2 2 1 6 nn n n+ + + − 10. ( ) ( )2 1 2 1 3 n n n+ − Miscellaneous Exercise on Chapter 9 2. 5, 8, 11 4. 8729 5. 3050 6. 1210 7. 4 8. 160; 6 9. ± 3 10. 8, 16, 32 11. 4 12. 11 21. (i) ( )50 510 1 81 9 n n−− , (ii) ( )2 2 110 3 27 nn −− − 22. 1680 23. ( )2 3 5 3 n n n+ + 25. ( )22 9 13 24 n n n+ + 27. Rs 16680 28. Rs 39100 29. Rs 43690 30. Rs 17000; 20,000 31. Rs 5120 32. 25 days EXERCISE 10.1 1. 121square unit. 2 2. (0, a), (0, – a) and (− )30a, or (0, a), (0, – a), and ( )30a, 3. (i) 2 1y y ,− (ii) 2 1x − x 4. 15 0 2 , ⎛ ⎞ ⎜ ⎟⎝ ⎠ 5. 1 2 − 7. – 3 8. x = 1 10. 135° 11. 1 and 2, or 1 2 and 1, or – 1 and –2, or 1 2 − and – 1 14. 1 2 , 104.5 Crores EXERCISE 10.2 1. y = 0 and x = 0 2. x – 2y + 10 = 0 3. y = mx 4. ( ) (31 x+ − ) (3 1 4y− = )3 1– 5. 2x + y + 6 = 0 6. x − 3 2y + 3 0= 7. 5x + 3y + 2 = 0 8. 3 10 xy+= 9. 3x – 4y + 8 = 0 10. 5x – y + 20 = 0 11. (1 + n)x + 3(1 + n)y = n +11 12. x + y = 5 13. x + 2y – 6 = 0, 2x + y – 6 = 0 14. 3 2 0 and xy+− = 3 2 0xy+ + = 15. 2x – 9y + 85 = 0 16. ( )192 L C 20 124 942 90 . .= − + 17. 1340 litres. 19. 2kx + hy = 3kh. EXERCISE 10.3 1. (i) 1 10 0; 7 7y x , ,=− + − (ii) 5 52 2 ;3 3y x , ,=− + − (iii) y = 0x + 0, 0, 0 2. (i) 14 6 46 x y+= ,, ; (ii) 31 2; 3 2 2 x y ,,+ = − − 2 (iii) 2 3y ,=− intercept with y-axis = 2 3 − and no intercept with x-axis. 3. (i) x cos 120° + y sin 120° = 4, 4, 120° (ii) x cos 90° + y sin 90° = 2, 2, 90°; (iii) x cos 315° + y sin 315° = 22 , 22 , 315° 4. 5 units 5. (– 2, 0) and (8, 0) 6. (i) 65 units, (ii) 17 1 2 pr l + units. 7. 3x – 4y + 18 = 0 8. y + 7x = 21 9. 30° and 150° 10. 22 9 12. ( ) (32 2x+ + )3 1 8 3 1– y = + (or ) (3 2 1 2x− + + )3 1 8 3 y –= + ⎛ 68 49 ⎞ 1513. 2x + y = 5 14. ⎜ , −⎟ 15. m = ,c = ⎝ 25 25 ⎠ 22 17. y – x = 1, 2 Miscellaneous Exercise on Chapter 10 1. (a) 3, (b) ± 2, (c) 6 or 1 2. 7π ,1 6 ⎛ 8 ⎞⎛ 32 ⎞3. 2x − 3y = 6, − 3x + 2 y = 6 4. ⎜0,− ⎟⎜ , 0, ⎟⎝ 3 ⎠⎝ 3 ⎠ sin ( – θ) 5. 6. x =− 5 7. 2x – 3y + 18 = 0– θ 22 22sin 8. k2 square units 9. 5 11. 3x – y = 7, x + 3y = 9 23 5 units12. 13x + 13y = 6 14. 1 : 2 15. 18 16. The line is parallel to x - axis or parallel to y-axis 152±17. x = 1, y = 1. 18. (–1, – 4). 19. 7 ⎛13 ⎞21. 18x + 12y + 11 = 0 22. ⎜ ,0⎟ 24. 119x + 102y = 125⎝ 5 ⎠ EXERCISE 11.1 1. x2 + y2 – 4y = 0 2. x2 + y2 + 4x – 6y –3 = 0 3. 36x2 + 36y2 – 36x – 18y + 11 = 0 4. x2 + y2 – 2x – 2y = 0 5. x2 + y2 + 2ax + 2by + 2b2 = 0 6. c(–5, 3), r = 6 117. c(2, 4), r = 65 8. c(4, – 5), r = 53 9. c ( , 0) ; r = 44 10. x2 + y2 – 6x – 8y + 15 = 0 11. x2 + y2 – 7x + 5y – 14 = 0 12. x2 + y2 + 4x – 21 = 0 & x2 + y2 – 12x + 11 = 0 13. x2 + y2 – ax – by = 0 14. x2 + y2 – 4x – 4y = 5 15. Inside the circle; since the distance of the point to the centre of the circle is less than the radius of the circle. EXERCISE 11.2 1. F (3, 0), axis -x - axis, directrix x = – 3, length of the Latus rectum = 12 2. F (0, 3 ), axis -y - axis, directrix y = – 23 , length of the Latus rectum = 63. F (–2, 0), axis -x - axis, directrix x = 2, length of the Latus rectum = 8 4. F (0, –4), axis -y - axis, directrix y = 4, length of the Latus rectum = 16 2 5 55. F ( , 0) axis -x - axis, directrix x = – , length of the Latus rectum = 102 2 –9 96. F (0, ) , axis -y - axis, directrix y = , length of the Latus rectum = 94 4 7. y2 = 24x 8. x2 = – 12y 9. y2 = 12x 10. y2 = –8x 11. 2y2 = 9x 12. 2x2 = 25y EXERCISE 11.3 201. F (± 20,0); V (± 6, 0); Major axis = 12; Minor axis = 8 , e = ,6 16Latus rectum = 3 212. F (0, ± 21); V (0, ± 5); Major axis = 10; Minor axis = 4 , e = ;5 8Latus rectum = 5 73. F(± 7, 0); V (± 4, 0); Major axis = 8; Minor axis = 6 , e = ;49Latus rectum = 2 34. F (0, ± 75); V (0,± 10); Major axis = 20; Minor axis = 10 , e = ;2Latus rectum = 5 135. F(± 13,0); V (± 7, 0); Major axis =14 ; Minor axis = 12 , e = ;7 72Latus rectum = 7 36. F (0, ±10 3); V (0,± 20); Major axis =40 ; Minor axis = 20 , e = ;2Latus rectum = 10 227. F (0, ± 4 2 ); V (0,± 6); Major axis =12 ; Minor axis = 4 , e = ;3 4Latus rectum = 3 15F0,±8. ( 15 ) ; V (0,± 4); Major axis = 8 ; Minor axis = 2 , e = ;4 1Latus rectum = 2 59. F(± 5 ,0); V (± 3, 0); Major axis = 6 ; Minor axis = 4 , e = ;3 8Latus rectum = 3 22 2222xy xyxy10. += 1 11. +=1 12. +=1 25 9 144169 36 20 22 2222xy xyxy13. +=1 14. +=1 15. +=1 9 4 1 5 169144 22 2222xy xyxy16. +=1 17. +=1 18. += 1 64100 167 259 22 22xy 22xy19. +=1 20. x + 4y = 52 or +=1 1040 5213 EXERCISE 11.4 5 9 1. Foci (± 5, 0), Vertices (± 4, 0); e = ; Latus rectum = 42 2. Foci (0 ± 6), Vertices (0, ± 3); e = 2; Latus rectum = 18 3. Foci (0, ± 13 ), Vertices (0, ± 2); e = 13; Latus rectum = 92 5 644. Foci (± 10, 0), Vertices (± 6, 0); e = ; Latus rectum = 33 214 614 455. Foci (0,± ), Vertices (0,± ); e = ; Latus rectum = 553 3 65 496. Foci (0, ± 65 ), Vertices (0, ± 4); e = ; Latus rectum = 42 22 2222xy yxyx7. −=1 8. −=1 9. −=1 4 5 2539 916 22 2222xy yxxy10. −=1 11. −=1 12. −=1 16 9 25144 2520 22 2222xy x 9y yx13. −=1 14. −=1 15. −=1 412 49343 5 5 Miscellaneous Exercise on Chapter 11 1. Focus is at the mid-point of the given diameter. 2. 2.23 m (approx.) 3. 9.11 m (approx.) 4. 1.56m (approx.) 22 22xy xy5. +=1 6. 18 sq units 7. +=1 819 259 8. 83a EXERCISE 12.1 1. y and z - coordinates are zero 2. y - coordinate is zero 3. I, IV, VIII, V, VI, II, III, VII 4. (i) XY - plane (ii) (x, y, 0) (iii) Eight EXERCISE 12.2 1. (i)2 5 (ii) 43 (iii) 2 26 (iv)2 5 4. x – 2z = 0 5. 9x2 + 25y2 + 25z2 – 225 = 0 EXERCISE 12.3 ⎛−41 27 1. (i) ⎜ ,, ⎟⎞ (ii) (− 817 3 , ) 2. 1 : 2⎝ 555 ⎠ , 3. 2 : 3 5. (6, – 4, – 2), (8, – 10, 2) Miscellaneous Exercise on Chapter 12 1. (1, – 2, 8) 2. ,, 3. a = – 2, b = −16 , c = 23 4. (0, 2, 0) and (0, – 6, 0) 2 2 22 k– 109 5. (4, – 2, 6) 6. x + y + z − 2x − 7 y + 2z = 2 EXERCISE 13.1 1. 6 2. π⎛ ⎜⎝ − 22 7 ⎞ ⎟⎠ 3. π 4. 19 2 5. 1 2 − 6. 5 7. 11 4 8. 108 7 9. b 10. 2 11. 1 12. 1 4 − 13. a b 14. a b 15. 1 π 16. 1 π a +117. 4 18. 19. 0 20. 1b 21. 0 22. 2 23. 3, 6 24. Limit does not exist at x = 1 25. Limit does not exist at x = 0 26. Limit does not exist at x = 0 27. 0 28. a = 0, b = 4 29. lim f (x) = 0 and lim f (x) =(a – a1) (a – a2) ... (a – a )x→a1 x→axlim30. x→af (x) exists for all a ≠ 0. 31. 2 lim lim32. For f (x) to exists, we need m = n; f (x) exists for any integral valuex→0 x→1 of m and n. EXERCISE 13.2 1. 20 2. 99 3. 1 −2−2 4. (i) 3x2 (ii) 2x – 3 (iii) 3 (iv)x (x −1)2 n−1 n−22 n−3 n−16. nx + ( − x + a ( − 2) + ...an 1) nx + a ab− 7. (i) 2x −− (ii) 4ax ax(+ b) (iii)ab2 ( x − b)2 nn−1 nnnx − anx − x + a xa8. (−)2 9. (i) 2 (ii) 20x3 – 15x2 + 6x – 4 (iii) 4 (52x) (iv) 15x4 + 5−3 24+xx –2 x(3x– 2)–12 36 – (v) 5 +10 (vi) ()2 (3x–1)2 10. – sin x xx x +1 11. (i) cos 2x (ii) sec x tan x (iii) 5sec x tan x – 4sin x (iv) – cosec x cot x (v) – 3cosec2 x – 5 cosec x cot x (vi) 5cos x+ 6sin x (vii) 2sec2 x – 7sec x tan x Miscellaneous Exercise on Chapter 13 1. (i) – 1 (ii) 2 1 x (iii) cos (x + 1) (iv) πsin 8 x ⎛ ⎞− −⎜ ⎟⎝ ⎠ 2. 1 3. 2 qr x ps 4. 2c (ax+b) (cx + d) + a (cx + d)2 5. ad cx 2 bc d 6. ( )2 2 01 1 ,x , x– − ≠ 7. 2 2ax ax bx b c 2 8. ( ) 2 22 2apx bpx ar bq px qx r − − + − + + 9. 2apx 2bpx ax b 2 bq ar 10. 5 4a x 3 2b x sin x 11. 2 x 12. ( ) 1n na ax b −+ 13. ( ) ( ) ( ) ( )1 1n m ax b cx d mc ax b na cx d− −+ + ⎡ + + + ⎤⎣ ⎦ 14. cos (x+a) 1 15. – cosec3 x – cosec x cot2 x 16. 1sin x 17. sin x 2 cos x 2 18. 2 2sec tan sec 1 x x x 19. n sinn–1 x cos x 20. 2 cos sin cos bc x ad x cd x bd 21. 2 cos cos a x 22. 3x 5 cos x x 3 sin x x 20 sin x 12cos x 23. 2 sinx x sin x 2 cos x x 24. qsin x 2ax sin x p cos q x 2a x cos x 25. tan2 x x cos x x tan x 1 sin x 26. 35 15 cosx x 2 28 cos 28 sin 3 7 cos x x x x x 15sin x 27. 29. 30. 1. 2. π 2x cos 2 sin xx cos x 1tan xx sec x 4 28. 2 2 1tan xsin x x sec x 1 sec 2 xx tan x. 1 sec x tan x sin x nx cos x sinn 1x EXERCISE 14.1 (i) This sentence is always false because the maximum number of days in a month is 31. Therefore, it is a statement. (ii) This is not a statement because for some people mathematics can be easy and for some others it can be difficult. (iii) This sentence is always true because the sum is 12 and it is greater than 10. Therefore, it is a statement. (iv) This sentence is sometimes true and sometimes not true. For example the square of 2 is even number and the square of 3 is an odd number. Therefore, it is not a statement. (v) This sentence is sometimes true and sometimes false. For example, squares and rhombus have equal length whereas rectangles and trapezium have unequal length. Therefore, it is not a statement. (vi) It is an order and therefore, is not a statement. (vii) This sentence is false as the product is (–8). Therefore, it is a statement. (viii) This sentence is always true and therefore, it is a statement. (ix) It is not clear from the context which day is referred and therefore, it is not a statement. (x) This is a true statement because all real numbers can be written in the form a + i × 0. The three examples can be: (i) Everyone in this room is bold. This is not a statement because from the context it is not clear which room is referred here and the term bold is not precisely defined. (ii) She is an engineering student. This is also not a statement because who ‘she’ is. (iii) “cos2θ is always greater than 1/2”. Unless, we know what θ is, we cannot say whether the sentence is true or not. EXERCISES 14.2 1. (i) Chennai is not the capital of Tamil Nadu. (ii) 2 is a complex number. (iii) All triangles are equilateral triangles. (iv) The number 2 is not greater than 7. (v) Every natural number is not an integer. 2. (i) The negation of the first statement is “the number x is a rational number.” which is the same as the second statement” This is because when a number is not irrational, it is a rational. Therefore, the given pairs are negations of each other. (ii) The negation of the first statement is “x is an irrational number” which is the same as the second statement. Therefore, the pairs are negations of each other. 3. (i) Number 3 is prime; number 3 is odd (True). (ii) All integers are positive; all integers are negative (False). (iii) 100 is divisible by 3,100 is divisible by 11 and 100 is divisible by 5 (False). EXERCISE 14.3 1. (i) “And”. The component statements are: All rational numbers are real. All real numbers are not complex. (ii) “Or”. The component statements are: Square of an integer is positive. Square of an integer is negative. (iii) “And”. the component statements are: The sand heats up quickily in the sun. The sand does not cool down fast at night. (iv) “And”. The component statements are:2x = 2 is a root of the equation 3x – x – 10 = 0 2x = 3 is a root of the equation 3x – x – 10 = 0 2. (i) “There exists”. The negation is There does not exist a number which is equal to its square. (ii) “For every”. The negation is There exists a real number x such that x is not less than x + 1. (iii) “There exists”. The negation is There exists a state in India which does not have a capital. 3. No. The negation of the statement in (i) is “There exists real number x and y for which x + y ≠ y + x”, instead of the statement given in (ii). 4. (i) Exclusive (ii) Inclusive (iii) Exclusive EXERCISE 14.4 1. (i) A natural number is odd implies that its square is odd. (ii) A natural number is odd only if its square is odd. (iii) For a natural number to be odd it is necessary that its square is odd. (iv) For the square of a natural number to be odd, it is sufficient that the number is odd (v) If the square of a natural number is not odd, then the natural number is not odd. 2. (i) The contrapositive is If a number x is not odd, then x is not a prime number. The converse is If a number x in odd, then it is a prime number. (ii) The contrapositive is If two lines intersect in the same plane, then they are not parallel The converse is If two lines do not interesect in the same plane, then they are parallel (iii) The contrapositive is If something is not at low temperature, then it is not cold The converse is If something is at low temperature, then it is cold (iv) The contrapositive is If you know how to reason deductively, then you can comprehend geometry. The converse is If you do not know how to reason deductively, then you can not comprehend geometry. (v) This statement can be written as “If x is an even number, then x is divisible by 4”. The contrapositive is, If x is not divisible by 4, then x is not an even number. The converse is, If x is divisible by 4, then x is an even number. 3. (i) If you get a job, then your credentials are good. (ii) If the banana tree stays warm for a month, then it will bloom. (iii) If diagonals of a quadrilateral bisect each other, then it is a parallelogram. (iv) If you get A+ in the class, then you do all the exercises in the book. 4. a (i) Contrapositive (ii) Converse b (i) Contrapositive (ii) Converse EXERCISE 14.5 5. (i) False. By definition of the chord, it should intersect the circle in two points. (ii) False. This can be shown by giving a counter example. A chord which is not a dimaeter gives the counter example. (iii) True. In the equation of an ellipse if we put a = b, then it is a circle (Direct Method) (iv) True, by the rule of inequality (v) False. Since 11 is a prime number, therefore 11 is irrational. Miscellaneous Exercise on Chapter 14 1. (i) There exists a positive real number x such that x–1 is not positive. (ii) There exists a cat which does not scratch. (iii) There exists a real number x such that neither x > 1 nor x < 1. (iv) There does not exist a number x such that 0 < x < 1. 2. (i) The statement can be written as “If a positive integer is prime, then it has no divisors other than 1 and itself. The converse of the statement is If a positive integer has no divisors other than 1 and itself, then it is a prime. The contrapositive of the statement is If positive integer has divisors other than 1 and itself then it is not prime. (ii) The given statement can be written as “If it is a sunny day, then I go to a beach. The converse of the statement is If I go to beach, then it is a sunny day. The contrapositive is If I do not go to a beach, then it is not a sunny day. (iii) The converse is If you feel thirsty, then it is hot outside. The contrapositive is If you do not feel thirsty, then it is not hot outside. 3. (i) If you log on to the server, then you have a password. (ii) If it rains, then there is traffic jam. (iii) If you can access the website, then you pay a subscription fee. 4. (i) You watch television if and only if your mind in free. (ii) You get an A grade if and only if you do all the homework regularly. (iii) A quadrilateral is equiangular if and only if it is a rectangle. 5. The compound statement with “And” is 25 is a multiple of 5 and 8 This is a false statement. The compound statement with “Or” is 25 is a multiple of 5 or 8 This is true statement. 7. Same as Q1 in Exercise 14.4 EXERCISE 15.1 1. 3 2. 8.4 3. 2.33 4. 7 5. 6.32 6. 16 7. 3.23 8. 5.1 9. 157.92 10. 11.28 11. 10.34 12. 7.35 EXERCISE 15.2 n +1 n2 −11. 9, 9.25 2. , 3. 16.5, 74.25 4. 19, 43.42 12 5. 100, 29.09 6. 64, 1.69 7. 107, 2276 8. 27, 132 9. 93, 105.52, 10.27 10. 5.55, 43.5 EXERCISE 15.3 1. B 2. Y 3. (i) B, (ii) B 4. A 5. Weight Miscellaneous Exercise on Chapter 15 1. 4, 8 2. 6, 8 3. 24, 12 5. (i) 10.1, 1.99 (ii) 10.2, 1.98 6. Highest Chemistry and lowest Mathematics 7. 20, 3.036 EXERCISE 16.1 1. {HHH, HHT, HTH, THH, TTH, HTT, THT, TTT} 2. {(x, y) : x, y = 1,2,3,4,5,6} or {(1,1), (1,2), (1,3), ..., (1,6), (2,1), (2,2), ..., (2,6), ..., (6, 1), (6, 2), ..., (6,6)} 3. {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT} 4. {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} 5. {H1, H2, H3, H4, H5, H6, T} 6. {XB, XB, XG, XG, YB, YG, YG, YG}7. {R1, R2, R3, R4, R5, R6, W1, W2, W3, W4, W5, W6, B1, B2, B3, B4, B5, B6} 8. (i) {BB, BG, GB, GG} (ii) {0, 1, 2} 9. {RW, WR, WW} 10. [HH, HT, T1, T2, T3, T4, T5, T6} 11. {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN} 12. {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66} 13. {(1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,3)} 14. {1HH, 1HT, 1TH, 1TT, 2H, 2T, 3HH, 3HT, 3TH, 3TT, 4H, 4T, 5HH, 5HT, 5TH, 5TT, 6H, 6T} 15. {TR, TR, TB, TB, TB, H1, H2, H3, H4, H5, H6}1 21233451 212316. {6, (1,6), (2,6), (3,6), (4,6), (5,6), (1,1,6), (1,2,6), ..., (1,5,6), (2,1,6). (2,2,6), ..., (2,5,6), ..., (5,1,6), (5,2,6), ... } EXERCISE 16.2 1. No. 2. (i) {1, 2, 3, 4, 5, 6} (ii) φ (iii) {3, 6} (iv) {1, 2, 3} (v) {6} (vi) {3, 4, 5, 6}, A∪B = {1, 2, 3, 4, 5, 6}, A∩B = φ, B∪C = {3, 6}, E∩F = {6}, D∩E = φ, A – C = {1, 2,4,5}, D – E = {1,2,3}, E∩ F′ = φ, F′ = {1, 2} 3. A = {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)} B = {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)} C ={(3,6), (6,3), (5, 4), (4,5), (6,6)} A and B, B and C are mutually exclusive. 4. (i) A and B; A and C; B and C; C and D (ii) A and C (iii) B and D 5. (i) “Getting at least two heads”, and “getting at least two tails” (ii) “Getting no heads”, “getting exactly one head” and “getting at least two heads” (iii) “Getting at most two tails”, and “getting exactly two tails” (iv) “Getting exactly one head” and “getting exactly two heads” (v) “Getting exactly one tail”, “getting exactly two tails”, and getting exactly three tails” There may be other events also as answer to the above question. 6. A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} B = {(1, 1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)} C = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)} (i) A′ = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)} = B (ii) B′ = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} = A (iii) A∪B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (2,1), (2,2), (2,3), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} = S (iv) A ∩ B = φ (v) A – C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} (vi) B ∪ C = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)} (vii) B ∩ C = {(1,1), (1,2), (1,3), (1,4), (3,1), (3,2)} (viii) A ∩ B′ ∩ C′ = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} 7. (i) True (ii) True (iii) True (iv) False (v) False (vi) False EXERCISE 16.3 31. (a) Yes (b) Yes (c) No (d) No (e) No 2. 4 121 5 1113. (i) (ii) (iii) (iv) 0 (v) 4. (a) 52 (b) (c) (i) (ii) 23 66 52132 11 35. (i) (ii) 6.1212 5 7. Rs 4.00 gain, Rs 1.50 gain, Re 1.00 loss, Rs 3.50 loss, Rs 6.00 loss. 1 13P ( Winning Rs 4.00) = , P(Winning Rs 1.50) = , P (Losing Re. 1.00) = 16 48 11P (Losing Rs 3.50) = , P (Losing Rs 6.00) = .4 16 131711 3 178. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)882888888 9 6719. 10. (i) (ii) 11. 11 13 13 38760 12. (i) No, because P(A∩B) must be less than or equal to P(A) and P(B), (ii) Yes 7413. (i) (ii) 0.5 (iii) 0.15 14.15 5 15. (i) 5 (ii) 3 16. No 17. (i) 0.58 (ii) 0.52 (iii) 0.7488 18. 0.6 19. 0.55 20. 0.65 1911 221. (i) (ii) (iii) 30 30 15 Miscellaneous Exercise on Chapter 16 20 30 1313C C C.C5 5311. (i) 60C (ii) 1− 60C 2. 52C5 54 9990 9990 115 999 C2 C103. (i) (ii) (iii) 4. (a) (b) 10000 (c) 10000 2 2 6 1000 C2 C10 1716 25. (a) (b) 6.3333 3 47. (i) 0.88 (ii) 0.12 (iii) 0.19 (iv) 0.34 8. 5 333 19. (i) (ii) 10.83 8 5040

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