ANSWERS EXERCISE 1.1 1. (C) 2. (C) 3. (D) 4. (D) 5. (D) 6. (C) 7. (D) 8. (C) 9. (C) 10. (C) 11. (B) 12. (A) 13. (D) 14. (B) 15. (B) 16. (C) 17. (C) 18. (B) 19. (A) 20. (A) 21. (C) EXERCISE 1.2 1. Yes. Let x = 21, y = 2 be a rational number. Now x + y = 21 + 2 = 21 + 1.4142 ... = 22.4142 ... Which is non-terminating and non-recurring. Hence x + y is irrational. 2. No. 0 × 2 =0 which is not irrational . p23. (i) False. Although is of the form but here p, i.e., 2 is not an integer. 3q(ii) False. Between 2 and 3, there is no integer. (iii) False, because between any two rational numbers we can find infinitely many rational numbers. 2 p(iv) True. is of the form but p and q here are not integers.3 q(v) False, as (42 )2 =2 which is not a rational number. 12(vi) False, because = 42= which is a rational number. 3 15 5 = 5 =(vii) False, because which is p, i.e., 5 is not an integer.31 4. (i) Rational, as 196 =14 (ii) 318 = 9 2 , which is the product of a rational and an irrational number and so an irrational number. 9 1 =(iii) , which is the quotient of a rational and an irrational number and 27 3 so an irrational number. 28 2 =(iv) , which is a rational number. 343 7 (v) Irrational, − 0.4 =− 2 , which is the quotient of a rational and an irrational.10 12 2 (vi) = , which is a rational number. 75 7 (vii) Rational, as decimal expansion is terminating. (viii) (1+ 5)−(4 + 5)= –3 , which is a rational number. (ix) Rational, as decimal expansion is non-terminating recurring. (x) Irrational, as decimal expansion is non-terminating non-recurring. EXERCISE 1.3 1. Rational numbers: (ii), (iii) Irrational numbers: (i), (iv) 2. (i) –1.1, –1.2, –1.3 (ii) 0.101, 0.102, 0.103 515253 917 19 , , ,,(iii) (iv)7070 70 408080 3. (i) 2.1, 2.040040004 ... (ii) 0.03, 0.007000700007, ... 5 (iii) , 0.414114111 ... (iv) 0, 0.151151115 ... 12 (v) 0.151, 0.151551555 ... (vi) 1.5, 1.585585558 ... (vii) 3, 3.101101110 ... (viii) 0.00011, .0001131331333 ... (ix) 1, 1.909009000 ... (x) 6.3753, 6.375414114111 ... 1 8 47 1237. (i) (ii) (iii) (iv) (v)5 9 9 999 90 133 8 40 (vi) (vii) (viii)990 2475 99 76 8 3439. (i) 5 (ii) (iii) 168 2 (iv) (v)12 33 5(vi) 52 6 (vii) 0 (viii) 2 (ix) 3− 4 2 2 2 +10. (i) 3 (ii) 30 (iii) 232 (iv) 41 + 59 3 8 (v)74+ 3 (vi) 32 − 23 (vii)52+ 6 (viii) 92 15+ 946+(ix) 15 9 −5 11. (i) a = 11 (ii) a = (iii) b = (iv) a = 0, b = 111 6 12. 23 13. (i) 2.309 (ii) 2.449 (iii) 0.463 (iv) 0.414 (v) 0.318 14. (i) 6 (ii) (iii) 9 (iv) 5 2025 64 1 3– 3(v) (vi) –3 (vii) 16 EXERCISE 1.4 167 1. 2. 1 3. 2.063 4. 790 1 5. 98 6. 7. 2142 EXERCISE 2.1 1. (C) 2. (B) 3. (A) 4. (D) 5. (B) 6. (A) 7. (D) 8. (C) 9. (B) 10. (B) 11. (D) 12. (C) 13. (B) 14. (D) 15. (D) 16. (B) 17. (D) 18. (D) 19. (C) 20. (C) 21. (C) EXERCISE 2.2 1. Polynomials: (i), (ii), (iv), (vii) because the exponent of the variable after simplification in each of these is a whole number. 2. (i) False, because a binomial has exactly two terms. (ii) False, x3 + x + 1 is a polynomial but not a binomial. (iii) True, because a binomial is a polynomial whose degree is a whole number ≥ 1, so, degree can be 5 also. (iv) False, because zero of a polynomial can be any real number. (v) False, a polynomial can have any number of zeroes. It depends upon the degree of the polynomial. (vi) False, x5 + 1 and – x5 + 2x + 3 are two polynomials of degree 5 but the degree of the sum of the two polynomials is 1. EXERCISE 2.3 1. (i) One variable (ii) One variable (iii) Three variable (iv) Two variables 2. (i) 1 (ii) 0 (iii) 5 (iv) 7 1 13. (i) 6 (ii) (iii) –1 (iv)5 5 4. (i) 1 (ii) 0 (iii) 3 (iv) –16 5. Constant Polynomial : (v) Linear Polynomials : (iii), (vi), (x) Quadratic Polynomials : (iv), (viii), (ix) Cubic Polynomials : (i), (ii), (vii) 28. (i) 16a2 + b2 + 4c2 – 8ab – 4bc + 16ac 6. (i) 10x (ii) x20 + 1 (iii) 2x2 – x – 1 7. 61, –143 8. 31 4 − 9. (i) –3, 3, – 39 (ii) – 4, –3, 0 10. (i) False (ii) True (iii) False (iv) True (v) True 11. (i) 4 (ii) 1 2 (iii) 7 2 (iv) 0 12. 0 13. x3 + x2 + x + 1, 2 14. (i) 0 (ii) 62 (iii) 3 2 (iv) 136 27 − 15. (i) No (ii) No 17. (i) 19. 1 20. 3 2 21. –2 22. 2 23. (i) (x + 6) (x + 3) (ii) (3x –1) (2x + 3) (iii) (x – 5) (2x + 3) (iv) 2(7 + r) (6 – r) 24. (i) (x – 2) (x + 3) (2x – 5) (ii) (x – 1) (x – 2) (x – 3) (iii) (x + 1) (x – 2) (x + 2) (iv) (x – 1) (x + 1) (3x – 1) 25. (i) 1092727 (ii) 10302 (iii) 998001 26. (i) (2x + 5)2 (ii) (3y – 11z)2 (iii) 13– 6x⎛ ⎞⎜ ⎟⎝ ⎠ 5 6 x ⎛ ⎞+⎜ ⎟⎝ ⎠ 27. (i) 3(x – 1) (3x – 1) (ii) (3x – 2) (3x – 2) (ii) 9 a2 + 25b2 +c2 – 30ab +10bc – 6ac (iii) x2 + 4y2 + 9z2 – 4xy – 12yz + 6xz 29. (i) (3x + 2y – 4z) (3x + 2y – 4z) (ii) (–5x + 4y + 2z) (–5x + 4y + 2z) (iii) (4x – 2y + 3z) (4x – 2y + 3z) 30. 29 1 yy2 y3 31. (i) 27a3 – 54a2b + 36ab2 – 8b3 (ii) 3 + 2 ++ xx 3x 27 164 1 (iii) 64 −+ 2– 3x 3x 27 x ⎛ 1 ⎞⎛ 1 ⎞⎛ 1 ⎞32. (i) (1– 4a) (1 – 4a) (1 – 4a) (ii) ⎜2 p + ⎟⎜⎝ 2 p + ⎟⎜ 2 p +⎟⎝ 5 ⎠ 5 ⎠⎝ 5 ⎠ 3x33. (i) +8y3 (ii) x6 – 1 8 2 2a34. (i) (1+ 4x) (1– 4x + 16x2) (ii) ( –2 b)(a + 2 ab + 2b ) 35. 8x3– y3 + 27z3 + 18xyz 36. (i) (a –2b – 4c) (a2+4b2+16c2 + 2ab – 8bc + 4ac) 222(ii) ( 2a + 2b –3c)(2a + 4b + 9c –2 2ab + 6bc + 32ac ) 5 37. (i) − 12 (ii) – 0. 018 38. 3(x – 2y) (2y –3z) (3z – x) 39. (i) 0 (ii) 0 40. One possible answer is: Length = 2a – 1, Breadth = 2a + 3 EXERCISE 2.4 1. –1 2. a = 5; 62 5. –120x2 y – 250y3 6. x3– 8y3– z3– 6xyz EXERCISE 3.1 1. (B) 2. (C) 3. (C) 4. (A) 5. (D) 6. (A) 7. (C) 8. (C) 9. (D) 10. (C) 11. (C) 12. (D) 13. (B) 14. (B) 15. (B) 16. (D) 17. (B) 18. (D) 19. (B) 20. (C) 21. (B) 22. (C) 23. (C) 24. (A) EXERCISE 3.2 1. (i) False, because if ordinate of a point is zero, the point lies on the x-axis. (ii) False (1, –1 ), lies in IV quadrant and (–1, 1) lies in II quadrant. (iii) False, because in the coordinates of a point abscissa comes first and then the ordinate . (iv) False, because a point on the y-axis is of the form (0, y). (v) True, because in the II quadrant, signs of abscissa and ordinate are –, +, respectively. EXERCISE 3.3 1. P(1, 1), Q(–3, 0), R(–3, –2), S(2,1), T(4, –2), O(0,0) 2. Trapezium 4. (i) Collinear (ii) Not collinear (iii) Collinear 5. (i) II (ii) III (iii) II (iv) I 6. (i) P(3, 2), R(3, 0), Q(3, –1) (ii) 0 7. II, IV, x-axis, I, III 8. C, D, E, G 10. (7, 0), (0, –7) 11. (i) (0, 0) (ii) (0,– 4) (iii) (5, 0) EXERCISE 3.4 1. C(–2, – 4) 2. (0, 0), (–5, 0), (0, –3) 3. (4, 3) 4. (i) A, L and O (ii) G, I and O (iii) D and H 5. (i) (2, 1), (ii) (5, 7) EXERCISE 4.1 1. (C) 2. (A) 3. (A) 4. (A) 5. (D) 6. (B) 7. (C) 8. (A) 9. (B) 10. (A) 11. (C) 12. (B) 13. (A) 14. (C) 15. (C) 16. (B) 17. (C) 18. (C) 19. (D) EXERCISE 4.2 1. True, since (0, 3) satisfies the equation 3x + 4y = 12. 2. False, since (0, 7) does not satisfy the equation. 3. True, since (–1, 1) and (–3, 3) satisfy the given equation and two points determine a unique line. 4. True, since this graph is a line parallel to y-axis at a distance 3 units (to the right) from it. 5. False, since the point (3, – 5) does not satisfy the given equation. 6. False, since every point on the graph of the equation represents a solution. 7. False, since the graph of a linear equation in two variables is always a line. EXERCISE 4.3 1. Graph of each equation is a line passing through (0, 0). 2. (2, 3) 3. Any line parallel to x-axis and at a distance of 3 units below it is given by y = –3 5 4. x + y = 10 5. y = 3x 6. 3 7. (i) one (ii) Infinitely many solutions 8. (i) (4, 0) (ii) (0, 2) 9. c =− , x ≠ 0 10. y = 3x; y = 15. 82xx EXERCISE 4.4 2. The graph cuts the x-axis at (3, 0) and the y-axis at (0, 2). 3. The graph cuts the x-axis at (2, 0) and the y-axis ⎜0, ⎟ .⎛ 3 ⎞⎝ 2 ⎠ ⎛−160 ⎞ ° 4. (i) 30°C (ii) 95°F (iii) 32°F, ⎜⎟ C⎝ 9 ⎠ (iv) – 40 5. (i) 104°F (ii) 343°K 6. y = mx, where y denotes the force, x denotes the acceleration and m denotes the constant mass. (i) 30 Newton (ii) 36 Newton EXERCISE 5.1 1. (A) 2. (C) 3. (B) 4. (A) 5. (A) 6. (A) 7. (A) 8. (B) 9. (B) 10. (D) 11. (A) 12. (B) 13. (A) 14. (C) 15. (B) 16. (A) 17. (C) 18. (C) 19. (A) 20. (A) 21. (C) 22. (B) EXERCISE 5.2 1. False, it is valid only for the figures in the plane. 2. False, boundaries of the solids are surfaces. 3. False, the edges of surfaces are line. 4. True, one of the Euclid’s axioms. 5. True, because of one of Euclid’s axioms. 6. False, statements that are proved are theorms. 7. True, it is an equivalent version of Euclid’s fifth postulate. 8. True, it is an equivalent version of Euclid’s fifth postulate. 9. True, these geometries are different from Euclidean geometry. EXERCISE 5.4 1. Answer this question on the same manner as given in the solution of Sample Question 1 in (E). 3. No 4. No 5. Consistent EXERCISE 6.1 1. (C) 2. (D) 3. (A) 4. (A) 5. (D) 6. (A) 7. (C) 8. (B) EXERCISE 6.2 1. x + y must be equal to 180°. For ABC to be a line, the sum of the two adjacent angles must be 180°. 2. No, angle sum will be less than 180°. 3. No, angle sum cannot be more than 180°. 4. None, angle sum cannot be 181°. 5. Infinitely many triangles. sum of the angles of every triangle is 180°. 6. 136°. 7. No, each of these will be a right angle only when they form a linear pair. 8. Each will be a right angle. Linear pair axiom . 9. l || m because 132° + 48° = 180°, p is not parallel to q, because 73° + 106° ≠ 180°. 10. No, they are parallel EXERCISE 6.3 7. 90° 8. 40°, 60, 80° EXERCISE 7.1 1. (C) 2. (B) 3. (B) 4. (C) 5. (A) 6. (B) 7. (B) 8. (D) 9. (B) 10. (A) 11. (B) EXERCISE 7.2 1. QR; They will be congruent by ASA. 2. RP; They will be congruent by AAS. 3. No; Angles must be included angles. 4. No; Sides must be corresponding sides. 5. No; Sum of the two sides = the third side. 6. No; BC = PQ. 7. Yes; They are corresponding sides. 8. PR; Side opposite the greater angle is longer. 9. Yes; AB + BD >ADand AC + CD > AD. 10. Yes; AB + BM >AM and AC + CM > AM. 11. No; Sum of two sides is less than the third side. 12. Yes, because in each case the sum of two sides is greater than the third side. EXERCISE 7.4 1. 60°, 60°, 60° 3. It is defective to use ∠ABD = ∠ACD for proving this result. 19. ∠B will be greater. EXERCISE 8.1 1. (D) 2. (B) 3. (C) 4. (C) 5. (D) 6. (C) 7. (D) 8. (C) 9. (B) 10. (D) 11. (C) 12. (C) 13. (C) 14. (C) EXERCISE 8.2 1. 6 cm, 4 cm; Diagonals of a parallelogram bisect each other. 2. No; Diagonals of a parallelogram bisect each other. 3. No; Angle sum must be 360°. 4. Trapezium. 5. Rectangle. 6. No; Diagonals of a rectangle need not be perpendicular. 7. No; sum of the angles of a quadrilateral is 360°. 8. 3.5 cm, as DE = AC.1 2 9. Yes; because BD = EF and CD = EF. 10. 55°, ∠F = ∠A and ∠A = ∠C. 11. No; Angle sum of a quadrilateral is 360°. 12. Yes, Angle sum of a quadrilateral is 360°. 13. 145° 14. 4 cm EXERCISE 8.3 1. 84° 2. 135° each 3. 120°, 60°, 120°, 60° 4.120°, 60°, 120°, 60° EXERCISE 8.4 2. 4 cm. EXERCISE 9.1 1. (A) 2. (D) 3. (D) 4. (C) 5. (C) 6. (A) 7. (B) 8. (D) 9. (B) 10. (B) EXERCISE 9.2 1. False, since ar (AXCD) = ar (ABCD) – ar (BCX) = 48 – 12 = 36 cm2 2. True, SR = (13) 2 − (5) 2 = 12, ar (PAS) = 1 ar (PQRS) = 30 cm 3. False, because area of Δ QSR = 90 cm2 and area of Δ ASR < area of Δ QRS. 2 ar BDE 3 (BD) 2 (BC) 2 1 = ==4. True, ar ABC 3(BC) 2 (BC) 2 4 4 1 5. False, because ar (DPC) = ar (ABCD) = ar (EFGD)2EXERCISE 9.3 3. (i) 90 cm2 (ii) 45 cm2 (iii) 45 cm2 7. 12 cm2 EXERCISE 10.1 1. (D) 2. (A) 3. (C) 4. (B) 5. (D) 6. (A) 7. (C) 8. (B) 9. (C) 10. (D) EXERCISE 10.2 1. True. Because the distances from the centre of two chords are equal. 2. False. The angles will be equal only if AB = AC. 3. True. Because equal chords of congruent circles subtend equal angles at the respective centres. 4. False. Because a circle through two points cannot pass through a point which is collinear to these two points. 5. True. Because AB will be the diameter. 6. True. As ∠C is right angle, AC2 + BC2 = AB2. 7. False, as ∠A + ∠C = 90°+ 95° = 185° ≠ 180°. 8. False, because there can be many points D such that ∠BDC = 60° and each such point cannot be the centre of the circle through A,B,C. 9. True. Angles in the same segment. 10. True. ∠B = 180° –120° = 60° , ∠CAB = 90° – 60° = 30°. 1. 1:1 9. 17. 40° 19. 13. x = 30°, y = 15° 1. (B) 2. EXERCISE 10.3 60° 14. 30° 15. 100° 16. 50° 278 20. ∠BOC = 66°, ∠AOC = 54° EXERCISE 10.4 14. 30° EXERCISE 11.1 (A) 3. (D) EXERCISE 11.2 210°1. True. As 52.5° = and 210° =180° + 30° which can be constructed.4 12. False. As 42.5° = ×°85 and 85° cannot be constructed. 2 3. False. As BC + AC must be greater than AB which is not so. 4. True. AsAC – AB < BC, i.e., AC < AB + BC. 5. False. As ∠B + ∠C = 105° + 90° = 195° >180°. 6. True. As ∠B + ∠C = 60° + 45° = 105° < 180°. 2. Yes. 1. (A) 2. (D) 6. (B) 7. (C) EXERCISE 11.3 EXERCISE 12.1 3. (C) 8. (A) EXERCISE 12.2 4. (A) 5. (D) 9. (B) 1. False, area of the triangle is 12 cm2. ××2. True, area of the triangle = 1 44 =8 cm 2 3. True, Each of equal side = 3 cm. 4. False, area of the triangle 16 3 cm 2 . 5. True, the other diagonal will be 12 cm. 6. False, the area of the parallelogram is 35 cm2. 7. False, area is the sum of all the six equilateral triangles. 8. True, area = 306 m2. 9. True, area of the triangle = 12 105 cm2 . 2 EXERCISE 12.3 1. Rs 10500 2. Rs 84, 000 3. 300 3cm 4. 32 2cm2 ) 25. 180 cm2 6. 600 15 m2 7. 2100 15 m2 8. 24 (6 +1 cm 9. Rs 960 10. 114 m2 EXERCISE 12.4 1. Yelllow : 484 m2; Red : 242 m2; Green : 373.04 m2 2. 20 30 cm 2 3. 23 cm, 27 cm 5. Rs 19200 6. 3 cm 8. 1632 cm2, 1868 cm2 1. (D) 5. (B)9. (A) 1. True, EXERCISE 13.1 2. (C) 3. 6. (B) 7. 10. (A) EXERCISE 13.2 4 32 2π=π 3 r 3 (2 ) r r 1 r 2 1 4. 7. EXEMPLAR PROBLEMS 374 cm2 45 cm, 40 cm (B) 4. (C) (A) 8. (B) ⎛⎞ h2. False, since new volume = π⎜⎟⋅2 = (Original volume)3 ⎝⎠22 3. True, since r2 + h2 = l2 h4. True, 2πrh= 2π(2r) . 2 12 r 2 r35. True, since volume of cone = π⋅r (2 ) =π= volume of hemisphere33 6. True, since V1 = volume of cylinder = πr2h 12since V2 = volume of cone = πrhTherefore, V1 = 3V23 12 237. True, V = πrr,V=πr, V = πr2 r13 2338. False, 3a=63 == a 6 Therefore, edge = 6 cm 9. True, V1 (volume of cube) = a3 a 4 a3 Radius of sphere = .V2 (Volume of sphere) = π 2 38 V1 : V2 = 6 : π 2 ⎛⎞ 2π(2 ) ⋅h 2[] h10. True, new volume = r ⎜⎟= π r .Therefore, volume is doubled.2⎝⎠ EXERCISE 13.3 1. 488 cm3 2. 7.5 cm3 3. 14.8 cm3 4. 471.42 m2 5. 5 cm 6. 739.2 litres 7. 200 revolutions 8. 40 days 9. 8 laddoos 10. 304 cm3, 188.5 cm2 EXERCISE 13.4 1. 8800 cm3 2. 677.6 cm3 3. 110, 241.7 cm3 4. 668.66 m3 5. 16 : 9 6. 30.48 cm3 7. 50% 8. (i) 9152 cm2 (ii) 55440 cm3 EXERCISE 14.1 1. (B) 2. (D) 3. (B) 4. (C) 5. (B) 6. (B) 7. (B) 8. (C) 9. (B) 10. (D) 11. (D) 12. (C) 13. (B) 14. (D) 15. (B) 16. (B) 17. (C) 18. (B) 19. (D) 20. (B) 21. (C) 22. (C) 23. (C) 24. (B) 25. (D) 26. (C) 27. (C) 28. (C) 29. (C) 30. (D) EXERCISE 14.2 1. Not correct. The classes are of varying widths, not of uniform widths. 2. Median will be a good representative of the data, because (i) each value occcurs once, (ii) The data is influenced by extreme values. 3. Data has to be arranged in ascending (or descending ) order before finding the median. 4. No, the data have first to be arranged in ascending (or descending) order before finding the median. 5. It is not correct. In a histogram, the area of each rectangle is propotional to the frequency of its class. 6. It is not correct. Reason is that differnce between two consecutive marks should be equal to the class size. 7. No. Infact the number of children who watch TV for 10 or more hours a week is 4 + 2, i.e., 6. 8. No, since the number of trials in which the event can happen cannot be negative, and the total number of trials is always positive. 9. No, since the number of trials in which the event can happen cannot be greater than the total number of trials. 10. No. As the number of tosses of a coin increases, the ratio of the number of 11heads to the total number of tosses will be nearer to , not exactly .22 EX ERCISE 14.3 1. 2. 3. 4. Blood Group A B AB O Total Number of Students (frequency) 12 8 4 6 30 Digit 0 1 2 3 4 5 6 7 8 9 Frequency 1 2 5 6 3 4 3 2 5 4 Scores 48 58 64 66 69 71 73 81 83 84 Frequency 3 3 4 7 6 3 2 1 2 2 Class 0 -10 10 - 20 20 - 30 30 - 40 40 - 50 Frequency 4 8 13 12 6 Class size = 10 5. Class intervals Frequency 149.5 - 153.5 7 153.5 - 157.5 7 157.5 - 161.5 15 161.5 - 165.5 10 165.5 - 169.5 5 169.5 - 173.5 6 153.5 is included in the class interval 153.5-157.5 and 157.5 in 157.5 - 161.5. 9. 20 10. 8.05 11. 72.2 12. 80.94 13. 20 14. Median = 12, mode = 10 15. 16. (i) 0.06 (ii) 0.19 (iii) 17. (i) 0.06 (ii) 0.086 (iii) 0.282 (iv) 0.254 4 59 669 18. (i) (ii) (iii)19. (i) 0.25 (ii) 0.75 (iii) 0.73 (iv) 0 20. (i) 0.675 (ii) 0.325 (iii) 0.135 (iv) 0.66 Class intervals Frequency 150 - 200 50 200 - 250 30 250 - 300 35 300 - 350 20 350 - 400 10 Total 145 3 400 7350700 EXERCISE 14.4 1. 2. Class 0 -9 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99 Frequency 1 2 5 6 3 4 3 2 5 4 Class intervals Frequency 0 - 10 4 10 - 20 7 20 - 30 5 30 - 40 10 40 - 50 5 50 - 60 8 60 - 70 5 70 - 80 8 80 - 90 5 90 - 100 3 10. a = 5, frequency of 30 is 28 and that of 70 is 24. 11. 2 : 1 12. Mean = 75.64, Median = 77, Mode = 85