ANSWERS/HINTS EXERCISE 1.1 1. (i) 45 (ii) 196 (iii) 51 2. An integer can be of the form 6q, 6q + 1, 6q + 2, 6q + 3, 6q + 4 or 6q + 5. 3. 8 columns 4. An integer can be of the form 3q, 3q + 1 or 3q + 2. Square all of these integers. 5. An integer can be of the form 9q, 9q + 1, 9q + 2, 9q + 3, . . ., or 9q + 8. EXERCISE 1.2 1. (i) 22 × 5 × 7 (ii) 22 × 3 × 13 (iii) 32 × 52 × 17 (iv) 5 × 7 × 11 × 13 (v) 17 × 19 × 23 2. (i) LCM = 182; HCF = 13 (ii) LCM = 23460; HCF = 2 (iii) LCM = 3024; HCF = 6 3. (i) LCM = 420; HCF = 3 (ii) LCM = 11339; HCF = 1 (iii) LCM = 1800; HCF = 1 4. 22338 7. 36 minutes EXERCISE 1.4 1. (i) Terminating (ii) Terminating (iii) Non-terminating repeating (iv) Terminating (v) Non-terminating repeating (vi) Terminating (vii) Non-terminating repeating (viii) Terminating (ix) Terminating (x) Non-terminating repeating 2. (i) 0.00416 (ii) 2.125 (iv) 0.009375 (vi) 0.115 (viii) 0.4 (ix) 0.7 3. (i) Rational, prime factors of q will be either 2 or 5 or both only. (ii) Not rational (iii) Rational, prime factors ofq will also have a factor other than 2 or 5. EXERCISE 2.1 1. (i) No zeroes (ii) 1 (iii) 3 (iv) 2 (v) 4 (vi) 3 EXERCISE 2.2 1 1 1 31. (i) –2, 4 (ii) , (iii) − , 2 2 32 4 (iv) –2, 0 (v) − (vi) –1, 3 2. (i) 4x 2 – x – 4 (ii) 3x 2 − 32 x + 1 (iii) x 2 + (iv) x2 – x + 1 (v)4x2 + x + 1 (vi) x2 – 4x + 1 EXERCISE 2.3 1. (i) Quotient = x – 3 and remainder = 7x – 9 (ii) Quotient = x 2 + x – 3 and remainder = 8 (iii) Quotient = – x 2 – 2 and remainder = – 5x + 10 2. (i) Yes (ii) Yes (iii) No 3. –1, –1 4. g(x) = x 2 – x + 1 5. (i) p(x) = 2x2 – 2x + 14,g(x) = 2, q(x) = x2 – x + 7, r(x) = 0 (ii) p(x) =x3 + x2 + x + 1, g(x) = x2 – 1, q(x) = x + 1,r(x) = 2x + 2 (iii) p(x) =x3 + 2x2 –x + 2, g(x) =x2 – 1, q(x) = x + 2, r(x) = 4 There can be several examples in each of (i), (ii) and (iii). EXERCISE 2.4 (Optional)* 2. x3 – 2x2 – 7x + 14 3. a = 1, b = ± 2 4. – 5, 7 5. k = 5 and a = –5 EXERCISE 3.1 1. Algebraically the two situations can be represented as follows: x – 7y + 42 = 0; x – 3y – 6 = 0, where x and y are respectively the present ages of Aftab and his daughter. To represent the situations graphically, you can draw the graphs of these two linear equations. 2. Algebraically the two situations can be represented as follows: x + 2y = 1300; x + 3y = 1300, where x and y are respectively the costs (in `) of a bat and a ball. To represent the situations graphically, you can draw the graphs of these two linear equations. 3. Algebraically the two situations can be represented as follows: 2x + y = 160; 4x + 2 y = 300, where x and y are respectively the prices (in ` per kg) of apples and grapes. To represent the situations graphically, you can draw the graphs of these two linear equations. EXERCISE 3.2 1. (i) Required pair of linear equations is x + y = 10; x – y = 4, where x is the number of girls and y is the number of boys. To solve graphically draw the graphs of these equations on the same axes on graph paper. Girls = 7, Boys = 3. (ii) Required pair of linear equations is 5x + 7y = 50; 7x + 5y = 46, where x and y represent the cost (in `) of a pencil and of a pen respectively. To solve graphically, draw the graphs of these equations on the same axes on graph paper. Cost of one pencil = ` 3, Cost of one pen = ` 5 2. (i) Intersect at a point (ii) Coincident (iii) Parallel 3. (i) Consistent (ii) Inconsistent (iii) Consistent (iv) Consistent (v) Consistent 4. (i) Consistent (ii) Inconsistent (iii) Consistent (iv) Inconsistent The solution of (i) above, is given byy = 5 – x, where x can take any value, i.e., there are infinitely many solutions. The solution of (iii) above is x = 2, y = 2, i.e., unique solution. 5. Length = 20 m and breadth = 16 m. 6. One possible answer for the three parts: (i) 3x + 2y – 7 = 0 (ii) 2x + 3y – 12 = 0 (iii) 4x + 6y – 16 = 0 7. Vertices of the triangle are (–1, 0), (4, 0) and (2, 3). EXERCISE 3.3 1. (i) x = 9, y = 5 (ii) s = 9, t= 6 (iii) y = 3x – 3, where x can take any value, i.e., infinitely many solutions. (iv) x = 2, y = 3 (v) x = 0, y = 0 (vi) x = 2, y = 3 2. x = –2, y = 5; m = –1 3. (i) x – y = 26, x = 3y, where x and y are two numbers (x > y); x = 39, y = 13. (ii) x – y = 18, x + y = 180, where x and y are the measures of the two angles in degrees; x = 99, y = 81. (iii) 7x + 6y = 3800, 3x + 5y = 1750, where x and y are the costs (in `) of one bat and one ball respectively; x = 500, y = 50. (iv) x + 10y = 105, x + 15y = 155, wherex is the fixed charge (in `) and y is the charge (in ` per km); x =5, y = 10; ` 255. (v) 11x – 9y + 4 = 0, 6x – 5y + 3 = 0, where x and y are numerator and denominator of the 7 fraction; (x = 7, y = 9). 9 (vi) x – 3y – 10 = 0, x – 7y + 30 = 0, where x and y are the ages in years of Jacob and his son;x = 40, y = 10. 19 6 1. (i) x = , y = 55 (iv) x = 2, y = –3 2. (i) x – y + 2 = 0, 2x3 ⋅fraction; 5 (ii) x – 3y + 10 = 0, EXERCISE 3.4 95 (ii) x = 2, y = 1 (iii) x = , y = − 13 13 – y – 1 = 0, where x and y are the numerator and denominator of the x – 2y – 10 = 0, where x and y are the ages (in years) of Nuri and Sonu respectively. Age of Nuri (x) = 50, Age of Sonu (y) = 20. (iii) x + y = 9, 8x – y = 0, where x and y are respectively the tens and units digits of the number; 18. (iv) x + 2y = 40, x + y = 25, wherex and y are respectively the number of ` 50 and ` 100 notes; x = 10, y = 15. (v) x + 4y = 27, x + 2y = 21, where x is the fixed charge (in `) and y is the additional charge (in `) per day; x = 15, y = 3. EXERCISE 3.5 1. (i) No solution (ii) Unique solution; x = 2, y = 1 (iii) Infinitely many solutions (iv) Unique solution; x = 4, y = –1 2. (i) a = 5,b = 1 (ii) k = 2 3. x = –2, y = 5 4. (i) x + 20y = 1000, x + 26y = 1180, wherex is the fixed charges (in ` ) and y is the charges (in `) for food per day; x = 400, y = 30. (ii) 3x – y – 3 = 0, 4x – y – 8 = 0, where x and y are the numerator and denominator of the 5 fraction; ⋅ 12 (iii) 3x – y = 40, 2x – y = 25, where x and y are the number of right answers and wrong answers respectively; 20. (iv) u –v = 20, u + v = 100, where uand v are the speeds (in km/h) of the two cars;u = 60, v = 40. (v) 3x – 5y – 6 = 0, 2x + 3y – 61 = 0, wherex and y are respectively the length and breadth (in units) of the rectangle; length (x) = 17, breadth (y) = 9. EXERCISE 3.6 11 1 , 23 5 1. (i) x = , y = (ii) x = 4, y = 9 (iii) x = y = –2 (iv) x = 4, y = 5 (v) x = 1, y = 1 (vi) x = 1, y = 2 (vii) x = 3, y = 2 (viii) x = 1, y = 1 2. (i) u + v = 10, u – v = 2, where u and v are respectively speeds (in km/h) of rowing and current; u = 6, v = 4. 2 513 61 (ii) += , +=, where n and m are the number of days taken by 1 woman nm 4 nm 3 and 1 man to finish the embroidery work; n = 18, m = 36. 60 240 100 200 25 (iii) += 4, +=, where u and v are respectively the speeds uv uv 6 (in km/h) of the train and bus; u = 60, v = 80. EXERCISE 3.7 (Optional)* 1. Age of Ani is 19 years and age of Biju is 16 years or age of Ani 21 years and age of Biju 24 years. 2. ` 40, ` 170. Let the money with the first person (in `) be x and the money with the second person (in `) be y. x + 100 = 2(y – 100), y + 10 = 6 (x – 10) 3. 600 km 4. 36 5. ∠ A = 20°, ∠ B = 40°, ∠ C = 120° 6. Coordinates of the vertices of the triangle are (1, 0), (0, –3), (0, –5). c(a − b) − bc (a − b) + a 7. (i) x = 1, y = – 1 (ii) x = , y = 22 22 a − ba − b 2ab (iii) x = a, y = b (iv) x = a + b, y = − (v) x = 2, y = 1 a + b 8. ∠ A = 120°, ∠ B = 70°, ∠ C = 60°, ∠ D = 110° EXERCISE 4.1 1. (i) Yes (ii) Yes (iii) No (iv) Yes (v) Yes (vi) No (vii) No (viii) Yes 2. (i) 2x 2 + x – 528 = 0, where x is breadth (in metres) of the plot. (ii) x 2 + x – 306 = 0, where x is the smaller integer. (iii) x 2 + 32x – 273 = 0, where x (in years) is the present age of Rohan. (iv) u 2 – 8u – 1280 = 0, where u (in km/h) is the speed of the train. 1. (i) – 2, 5 11 (iv) , 44 2. (i) 9, 36 3. Numbers are 13 and 14. 5. 5 cm and 12 cm EXERCISE 4.2 3 5 ,(ii) – 2, (iii) − − 2 2 2 11 (v) , 10 10 (ii) 25, 30 4. Positive integers are 13 and 14. 6. Number of articles = 6, Cost of each article = Rs 15 EXERCISE 4.3 1 −1− 33 −1 + 33 331. (i) , 3 (ii) , (iii) − , − 2 44 22 (iv) Do not exist 3 − 13 3 + 13 2. Same as 1 3. (i) , (ii) 1, 2 4. 7 years 22 5. Marks in mathematics = 12, marks in English = 18; or, Marks in mathematics = 13, marks in English = 17 6. 120 m, 90 m 7. 18, 12 or 18, –12 8. 40 km/h 9. 15 hours, 25 hours 10. Speed of the passenger train = 33 km/h, speed of express train = 44 km/h 11. 18 m, 12 m EXERCISE 4.4 22 3 ± 3 ,1. (i) Real roots do not exist (ii) Equal roots; (iii) Distinct roots; 33 2 62. (i) k = ± 2 (ii) k = 6 3. Yes. 40 m, 20 m 4. No 5. Yes. 20 m, 20 m EXERCISE 5.1 1. (i) Yes. 15, 23, 31, . . . forms an AP as each succeeding term is obtained by adding 8 in its preceding term. 3V ⎛ 3 ⎞2 (ii) No. Volumes are V, V, L (iii) Yes. 150, 200, 250, . . . form an AP. , ⎜ ⎟ 4 ⎝ 4 ⎠ ⎛ 8 ⎞⎛ 8 ⎞2 ⎛ 8 ⎞3 , ,,1 + 10000 1 + 10000 1 + L(iv) No. Amounts are 10000 ⎜⎟⎜⎟ ⎜⎟⎝ 100⎠⎝ 100 ⎠⎝ 100 ⎠ 2. (i) 10, 20, 30, 40 (ii) – 2, – 2, – 2, – 2 (iii) 4, 1, – 2, – 5 11(iv) –1, − , 0, (v) – 1.25, – 1. 50, – 1.75, – 2.0 22 3. (i) a = 3, d = – 2 (ii) a = – 5, d = 4 14 ,(iii) a = d = (iv) a = 0.6, d = 1.1 33 19 ,4. (i) No (ii) Yes. d = ;4, 5 22 (iii) Yes. d = – 2; – 9.2, –11.2, – 13.2 (iv) Yes. d = 4; 6, 10, 14 (v) Yes. d = 2; 3 + 4 2,3 + 5 2,3 + 6 2 (vi) No 111 (vii) Yes. d = – 4; – 16, – 20, – 24 (viii) Yes. d = 0; − , − , − 222 (ix) No (x) Yes. d = a; 5a, 6a, 7a (xi) No (xii) Yes. d = (xiii) No (xiv) No (xv) Yes. d = 24; 97, 121, 145 EXERCISE 5.2 1. (i) a = 28 (ii) d = 2 (iii) a = 46 (iv) n = 10 (v) a = 3.5 nn2. (i)C (ii)B 3. (i) 14 (ii) 18 , 8 (iii) 61, 8(iv) –2 ,0,2,4 (v) 53 ,23 ,8,–7 4. 16th term 5. (i) 34 (ii) 27 2 6. No 7. 178 8. 64 9. 5th term 10. 1 11. 65th term 12. 100 13. 128 14. 60 15. 13 16. 4, 10, 16, 22, . . . 17. 20th term from the last term is 158. 18. –13, –8, –3 19. 11th year 20. 10 EXERCISE 5.3 33 1. (i) 245 (ii) –180 (iii) 5505 (iv) 20 1 2. (i) 1046 (ii) 286 (iii) – 8930 2 7 ,3. (i) n = 16, S = 440 (ii) d = S = 273 (iii) a = 4, S = 246 n1312335 85 ,(iv) d = –1, a = 8 (v) a = − a = (vi) n = 5, a = 34109 n33 54 (vii) n = 6, d = (viii) n = 7, a = – 8 (ix) d = 6 5 (x) a = 4 n 4. 12. By puttinga = 9, d = 8, S = 636 in the formula S = [2 a + (n − 1) d], we get a quadratic2 53 equation 4n 2 + 5n – 636 = 0. On solving, we getn = − , 12 . Out of these two roots only4 one root 12 is admissible. 8 5. n = 16, d = 6. n= 38, S = 6973 7. Sum = 1661 3 8. S51 = 5610 9. n2 10. (i) S15 = 525 (ii) S15 = – 465 11. S1 = 3, S2 = 4; a2 = S2 – S1 = 1; S3 = 3, a3 = S3 – S2 = –1, a = S – S = – 15; a = S – S = 5 – 2n.10 109nnn – 112. 4920 13. 960 14. 625 15. ` 27750 16. Values of the prizes (in ` ) are 160, 140, 120, 100, 80, 60, 40. 17. 234 18. 143 cm 19. 16 rows, 5 logs are placed in the top row. By putting S = 200, a = 20,d = –1 in the formula n S = [2 a + (n − 1) d], we get, 41n – n 2 = 400. On solving, n = 16, 25. Therefore, the 2 number of rows is either 16 or 25. a = a + 24d = – 4 25i.e., number of logs in 25th row is – 4 which is not possible. Therefore n = 25 is not possible. For n = 16, a = 5. Therefore, there are 16 rows and 5 logs placed in the top16row. 20. 370 m EXERCISE 5.4 (Optional)* 1. 32nd term 2. S = 20, 76 3. 385 cm 164. 35 5. 750 m3 EXERCISE 6.1 1. (i) Similar (ii) Similar (iii) Equilateral (iv) Equal, Proportional 3. No EXERCISE 6.2 1. (i) 2 cm (ii) 2.4 cm 2. (i) No (ii) Yes (iiii) Yes 9. Through O, draw a line parallel to DC, intersecting AD and BC at E and F respectively. EXERCISE 6.3 1. (i) Yes. AAA, Δ ABC ~ Δ PQR (ii) Yes. SSS, Δ ABC ~ Δ QRP (iii) No (iv) Yes. SAS, Δ MNL ~ Δ QPR (v) No (vi) Yes. AA, Δ DEF ~ Δ PQR 2. 55°, 55°, 55° 14. Produce AD to a point E such that AD = DE and produce PM to a point N such that PM = MN. Join EC and NR. 15. 42 m EXERCISE 6.4 1. 11.2 cm 2. 4 : 1 5. 1 : 4 8. C 9. D EXERCISE 6.5 1. (i) Yes, 25 cm (ii) No (iii) No (iv) Yes, 13 cm 6. a 9. 6 m 10. 6 7m 11. 300 61km 12. 13 m 17. C EXERCISE 6.6 (Optional)* 1. Through R, draw a line parallel to SP to intersect QP produced at T. Show PT = PR. 6. Use result (iii) of Q.5 of this Exercise. 7. 3 m, 2.79 m EXERCISE 7.1 1. (i)22 (ii)42 (iii)2 a 2+ b2 2. 39; 39 km 3. No 4. Yes 5. Champa is correct. 6. (i) Square (ii) No quadrilateral (iii) Parallelogram 7. (– 7, 0) 8. – 9, 3 9. ± 4,QR = 41,PR = 82,9 2 10. 3x + y – 5 = 0 EXERCISE 7.2 ⎛ 5⎞⎛ 7 ⎞ 1. (1, 3) 2. 2, − ; 0, −⎜⎟⎜ ⎟⎝ 3⎠⎝ 3 ⎠ 3. 61m; 5th line at a distance of 22.5 m 4. 2 : 7 ⎛ 3 ⎞5. 1:1 ; ⎜ − ,0⎟ 6. x = 6, y = 3 7. (3, – 10) ⎝ 2 ⎠ ⎛ 2 20⎞⎛ 7 ⎞⎛ 13⎞8. − , − 9. 1, , (0,5), 1, 10. 24 sq. units⎜⎟ ⎜ − ⎟ ⎜⎟⎝ 77 ⎠⎝ 2 ⎠⎝ 2 ⎠ EXERCISE 7.3 21 1. (i) sq. units (ii) 32 sq. units 2. (i) k = 4 (ii) k = 3 23. 1 sq. unit; 1 : 4 4. 28 sq. units EXERCISE 7.4 (Optional)* 1. 2 : 9 2. x + 3 y – 7 = 0 3. (3, – 2) 4. (1, 0), (1, 4 ) 5. (i) (4, 6), (3, 2), (6, 5); taking AD and AB as coordinate axes 9 (ii) (12, 2), (13, 6), (10, 3); taking CB and CD as coordinate axes. sq. units, 29 sq. units; areas are the same in both the cases. 2156. sq. units; 1 : 16 32⎛ 7 9 ⎞⎛ 11 11⎞ 7. (i) D , (ii)P ,⎜⎟ ⎜⎟⎝ 22 ⎠⎝ 33 ⎠ ⎛ 11 11⎞⎛ 11 11⎞(iii) Q , , R , (iv) P, Q, R are the same point. ⎜ ⎟⎜ ⎟⎝ 33 ⎠⎝ 33 ⎠ ⎛ x + x + xy + y + y ⎞12 312 3(v) 8. Rhombus⎜ , ⎟⎝ 33 ⎠ EXERCISE 8.1 7 24 247 ,, 25 25 2525 7 3 1517 1. (i) sinA = cosA = (ii) sinC = cosC = , 4 7 178 2. 0 3. cosA = ,tanA= 4. sinA = secA= 512 512 13 5. sin θ= , cos θ = , tan θ= , cot θ= , cosec θ = 13 13 12 5 5 49 49 7. (i) (ii) 8. Yes 64 64 12 512 ,, 13 13 5 9. (i) 1 (ii)0 10. sinP = cosP= tanP = 11. (i) False (ii) True (iii) False (iv) False (v) False EXERCISE 8.2 32 − 6 43 − 243 67 1. (i) 1 (ii) 2 (iii) (iv) (v)8 11 12 2. (i) A (ii) D (iii) A (iv) C 3. ∠ A = 45°, ∠ B = 15° 4. (i) False (ii) True (iii) False (iv) False (v) True EXERCISE 8.3 1. (i) 1 (ii) 1 (iii) 0 (iv) 0 3. ∠ A = 36° 5. ∠ A = 22° 7. cos 23° + sin 15° EXERCISE 8.4 2 1 1 1+cotA ,,1. sinA = tanA = secA = 2 cot A cotA 1+cot A ,1, 22. cosA = tanA = sec A–1 sec A sec A , cosecA = sec 2A –1 3. (i) 1 (ii)1 4. (i) B (ii) C (iii) D (iv) D EXERCISE 9.1 1. 10 m 2. 8 3m 3. 3m,2 3m 4. 10 3m + 1m 5. 40 3m 6. 19 3m 7. 20( 3 − 1) m 8. 0.8 ( ) 29. 16 m 10. 20 3m, 20m, 60m 11. 10 3m,10m 12. 7( 3 + 1m )3 13. 75( 3 −1)m 14. 58 3m 15. 3 seconds 1. Infinitely many 2. (i) One (ii) 1. A 7. 8 cm 1. 28 cm EXERCISE 10.1 Secant 2. B (iii) Two (iv) Point of contact EXERCISE 10.2 3. A 12. AB = 15 cm, AC = 13 cm EXERCISE 12.1 2. 10 cm 3. D 6. 3 cm 3. Gold : 346.5 cm2; Red : 1039.5 cm2; Blue : 1732.5 cm2; Black : 2425.5 cm2; White : 3118.5 cm2. 4. 4375 5. A EXERCISE 12.2 154132 77 2 2 1. cm 2 2. cm 3. cm 7 4. (i) 28.5 cm2 5. (i) 22 cm 83 (ii) 235.5 cm2 ⎛ ⎞4413 2(ii) 231 cm2 (iii) ⎜ 231 − ⎟ cm ⎜⎟4⎝⎠ 6. 20.4375 cm2 ; 686.0625 cm2 7. 88.44 cm2 385 28. (i) 19.625 m2 (ii) 58.875 cm2 9. (i) 285 mm (ii) mm 4 22275 2 158125 10. cm 11. cm 2 12. 189.97 km2 28 126 13. ` 162.68 14. D EXERCISE 12.3 4523 2 154 1. cm 2. cm 2 3. 42 cm2 28 3 ⎛ 22528 ⎞ 2⎛ 660 ⎞ 6822 − 7683 cm 4. + 363 cm 5. cm 6. ⎜ ⎟⎜ ⎟ ⎝ 7 ⎠⎝ 7 ⎠ 7 7. 42 cm2 8. (i) 2804 m (ii) 4320 m2 7 9. 66.5 cm2 10. 1620.5 cm2 11. 378 cm2 77 2 49 212. (i) cm (ii) cm 13. 228 cm2 88 2563082 214. cm 15. 98 cm2 16. cm 3 7 EXERCISE 13.1 1. 160 cm2 2. 572 cm2 3. 214.5 cm2 4. Greatest diameter = 7 cm,surface area = 332.5 cm2 5. 1 l 2 (π + 24) 6. 220 m2 7. 44 m2, Rs 22000 4 8. 18 cm2 9. 374 cm2 EXERCISE 13.2 1. π cm3 2. 66 cm3. Volume of the air inside the model = Volume of air inside (cone + cylinder + cone) ⎛ 11 ⎞22 2 = π rh 1 +π rh +π rh1 ⎟ , where r is the radius of the cone and the cylinder, h is⎜ 2 1⎝ 33 ⎠the height (length) of the cone and h2 is the height (length) of the cylinder. 1 2Required Volume = π r (h1 + 3h2 + h1 ). 3 3. 338 cm3 4. 523.53 cm3 5. 100 6. 892.26 kg 7. 1.131 m3 (approx.) 8. Not correct. Correct answer is 346.51 cm3. EXERCISE 13.3 1. 2.74 cm 2. 12 cm 3. 2.5 m 4. 1.125 m 5. 10 6. 400 7. 36cm;12 13 cm 8. 562500 m2 or 56.25 hectares. 9. 100 minutes EXERCISE 13.4 23 221. 102 cm 2. 48 cm2 3. 710 cm 3 7 4. Cost of milk is` 209 and cost of metal sheet is` 156.75. 5. 7964.4 m EXERCISE 13.5 (Optional)* 1. 1256 cm; 788g (approx) 2. 30.14 cm3; 52.75 cm2 423. 1792 5. 782 cm 7 EXERCISE 14.1 1. 8.1 plants. We have used direct method because numerical values of xi and fi are small. 2. ` 145.20 3. f = 20 4. 75.9 5. 57.19 6. ` 211 7. 0.099 ppm 8. 12.38 days 9. 69.43 % EXERCISE 14.2 1. Mode = 36.8 years, Mean = 35.37 years. Maximum number of patients admitted in the hospital are of the age 36.8 years (approx.), while on an average the age of a patient admitted to the hospital is 35.37 years. 2. 65.625 hours 3. Modal monthly expenditure = ` 1847.83, Mean monthly expenditure = ` 2662.5. 4. Mode : 30.6, Mean = 29.2. Most states/U.T. have a student teacher ratio of 30.6 and on an average, this ratio is 29.2. 5. Mode = 4608.7 runs 6. Mode = 44.7 cars EXERCISE 14.3 1. Median = 137 units, Mean = 137.05 units, Mode = 135.76 units. The three measures are approximately the same in this case. 2. x = 8, y = 7 3. Median age = 35.76 years 4. Median length = 146.75 mm 5. Median life = 3406.98 hours 6. Median = 8.05, Mean = 8.32, Modal size = 7.88 7. Median weight = 56.67 kg 1. Daily income (in `) Less than 120 Less than 140 Less than 160 Less than 180 Less than 200 EXERCISE 14.4 Cumulative frequency 12 26 34 40 50 Draw ogive by plotting the points : (120, 12), (140, 26), (160, 34), (180, 40) and (200, 50) 2. Draw the ogive by plotting the points : (38, 0), (40, 3), (42, 5), (44, 9), (46, 14), (48, 28), n(50, 32) and (52, 35). Here =17.5. Locate the point on the ogive whose ordinate is 17.5. 2 The x-coordinate of this point will be the median. 3. Production yield Cumulative (kg/ha) frequency More than or equal to 50 100 More than or equal to 55 98 More than or equal to 60 90 More than or equal to 65 78 More than or equal to 70 54 More than or equal to 75 16 Now, draw the ogive by plotting the points : (50, 100), (55, 98), (60, 90), (65, 78), (70, 54) and (75, 16). 2015-16 EXERCISE 15.1 1. (i) 1 (ii) 0, impossible event (iii) 1, sure or certain event (iv) 1 (v) 0, 1 2. The experiments (iii) and (iv) have equally likely outcomes. 3. When we toss a coin, the outcomes head and tail are equally likely. So, the result of an individual coin toss is completely unpredictable. 4. B 5. 0.95 6. (i)0 (ii)1 35 7. 0.008 8. (i) (ii)88 5 8 13 517 9. (i) (ii) (iii) 10. (i) (ii)1717 17 918 5 113 11. 12. (i) (ii) (iii) (iv) 1 13 824 11 1 13. (i) (ii) (iii)22 2 13 3 111 14. (i) (ii) (iii) (iv) (v) (vi)26 13 2652452 11 11 15. (i) (ii) (a) (b) 0 16. 54 12 115 911 17. (i) (ii) 18. (i) (ii) (iii)5 19 10 10 5 1 π 31 51 19. (i) (ii) 20. 21. (i) (ii) 3 624 36 36 22. (i) Sum on 2 dice 2 3 4 5 6 7 8 9 10 11 12 Probability 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 (ii) No. The eleven sums are not equally likely. 3 23. ; Possible outcomes are : HHH, TTT, HHT, HTH, HTT, THH, THT, TTH. Here, THH 4means tail in the first toss, head on the second toss and head on the third toss and so on. 1125 24. (i) (ii) 3636 25. (i) Incorrect. We can classify the outcomes like this but they are not then ‘equally likely’. Reason is that ‘one of each’ can result in two ways — from a head on first coin and tail on the second coin or from a tail on the first coin and head on the second coin. This makes it twicely as likely as two heads (or two tails). (ii) Correct. The two outcomes considered in the question are equally likely. EXERCISE 15.2 (Optional)* 18 41. (i) (ii) (iii) 525 5 2. 122 336 1233 447 2344 558 2344 558 3455 669 3455 669 6788 9912 11 5 (i) (ii) (iii)29 12 x 3. 10 4. , x = 3 5. 8 12 EXERCISE A1.1 1. (i) Ambiguous (ii) True (iii) True (iv) Ambiguous (v) Ambiguous 2. (i) True (ii) True (iii) False (iv) True (v) True 3. Only (ii) is true. 4. (i) If a > 0 and a2 > b2, then a > b. (ii) If xy > 0 and x2 = y2, then x = y. (iii) If (x + y)2 = x2 + y2 and y ≠ 0, then x = 0. (iv) The diagonals of a parallelogram bisect each other. EXERCISE A1.2 1. A is mortal 2. ab is rational 3. Decimal expansion of 17 is non-terminating non-recurring. 4. y = 7 5. ∠ A = 100°, ∠ C = 100°,∠ D = 180° 6. PQRS is a rectangle. 7. Yes, because of the premise. No, because 3721 =61 which is not irrational. Since the premise was wrong, the conclusion is false. EXERCISE A1.3 1. Take two consecutive odd numbers as 2n + 1 and 2n + 3 for some integer n. EXERCISE A1.4 1. (i) Man is not mortal. (ii) Line l is not parallel to line m. (iii) The chapter does not have many exercises. (iv) Not all integers are rational numbers. (v) All prime numbers are not odd. (vi) Some students are lazy. (vii) All cats are black. (viii) There is at least one real number x, such that = – 1.x(ix) 2 does not divide the positive integer a. (x) Integers a and b are not coprime. 2. (i) Yes (ii) No (iii) No (iv) No (v) Yes EXERCISE A1.5 1. (i) If Sharan sweats a lot, then it is hot in Tokyo. (ii) If Shalini’s stomach grumbles, then she is hungry. (iii) If Jaswant can get a degree, then she has a scholarship. (iv) If a plant is alive, then it has flowers. (v) If an animal has a tail, then it is a cat. 2. (i) If the base angles of triangle ABC are equal, then it is isosceles. True. (ii) If the square of an integer is odd, then the integer is odd. True. (iii) If x = 1, then x2 = 1. True. (iv) If AC and BD bisect each other, then ABCD is a parallelogram. True. (v) If a + (b + c) = (a + b) + c, then a, b and c are whole numbers. False. (vi) If x + y is an even number, then x and y are odd. False. (vii) If a parallelogram is a rectangle, its vertices lie on a circle. True. EXERCISE A1.6 1. Suppose to the contrary b ≤ d. 3. See Example 10 of Chapter 1. 6. See Theorem 5.1 of Class IX Mathematics Textbook. EXERCISE A2.2 1 1. (i) (ii) 160 5 2. Take 1 cm2 area and count the number of dots in it. Total number of trees will be the product of this number and the area (in cm2). 3. Rate of interest in instalment scheme is 17.74 %, which is less than 18 %. EXERCISE A2.3 1. Students find their own answers.

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