ANSWERS 1.3 EXERCISE 1 3,1, (i) {2} (ii) {0, 1} (iii) {1, p} 2. (i) {0, 1, 1} (ii) (iii) { 3, 2, 2, 3}

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1 ANSWERS. EXERCISE. (i) {} (ii) {0, } (iii) {, p}. (i) {0,, } (ii) (iii) {,,, }. {,,,,... P,( p } 4. (i) True (ii) False (iii) True (iv) True 7. (i) {, 4, 6, 8,..., 98} (ii) (,4, 9, 6, 5, 6, 49, 64, 8,} 8. (i) {4, 8, } (ii) {7, 8, 9} (iii),, (iv) {0,, } 9. (i) {4, 5, 6,...0} (ii) {5} (iii) {,,, 4, 5} 0... True 4. False 5. True 6. True 7. True. T = {0} 4. (i) (ii) (iii) (iv) (a) 00 (b) (i) 6, (ii), (iii) 9, (iv), (v), (vi) 6, (vii) 0, (viii) 0 9. C 0. B. B. D. C 4. D 5. B 6. B 7. C

2 04 EXEMPLAR PROBLEMS MATHEMATICS 8. C 9. C 40. A 4. B 4. B 4. C 44. [,] n (B) 47. A B 48. {φ, {}, {}, {, } 49. {0,,,, 4, 5, 6, 8} 50.(i) {,5, 9, 0 } (ii) {,,, 5, 6, 7, 9, 0 } 5. A Β 5. (i) (b) (ii) (c) (iii) (a) (iv) (f) (v) (d) (vi) (e) 5. True 54. False 55. False 56. True 57. True 58. False. EXERCISE. (i) {(, ), (, ), (, ), (, ), (, ), (, )} (ii) {(, ), (, ), (, ), (, ), (, ), (, )} (iii) {(, ), (, ), (, ), (, )} (iv) {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )}. {(0, ), (0, ), (, ), (, ), (, ), (, )}. (i) {(0, ), (, )} (ii) {(0, ), (0, ), (0, 4), (0, 5), (, ), (, ), (, 4), (,5)} 4. (i) a = and b = (ii) a = 0 and b = 5. (i) {(, 4), (, ), (, ), (4, ) } (ii) {(, ), (, ), (,),(, ), (, ), (, )} (iii) { (4, 5), (5, 4), (5, 5)} 6. Domain of R = {0,, 4, 5} = Range of R 7. Domain of R = [ 5, 5 ] and Range of R = [, 7 ] 8. R = {(0, 8), (8, 0) (0, 8), ( 8, 0)} 9. Domain of R = R and range of R = R + {0} 0. (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. (a) 6 (b) 64 4 (c) (d) t _ 4 (e) t + 5. (a) x = 4 (b) x > 4. (i) (f + g) x = x + x + (ii) (f g) x = x x

3 ANSWERS 05 f (iii) ( f g) x = x + x + x + (iv) g x = x + x + 4. (i) f = {(, 0), (0, ), (, 8), (7, 44), (9, 70)} 5. x =, 4 6. Yes, α =, β = 7. (i) R {nπ : n Z} (ii) R + (iii) R (iv)r {, } (v) R {4} 8. (i) [, ) (ii) (, ] (iii) [ 0, ) (iv) [, 4] 9. x, x< f ( x) = 4, x <, x. (i) (f + g) x = x + x (ii) (f g ) x = x x (iii) (fg) x = x (iv). Domain of f = (5, ) and Range of f = R + f x = g 4. D 5. D 6. B 7. C 8. B 9. B 0. A. C. C. A 4. B 5. A 6. {,, 4, 5} 7. (a) (iii) (b) (iv) (c) (ii) (d) (i) 8. False 9. False 40. True 4. False 4. True.. EXERCISE x cos x cos x

4 06 EXEMPLAR PROBLEMS MATHEMATICS 8. + π π 5. θ = nπ + ( ) n θ = nπ + 7 π 4 7. θ = nπ ± π 8. θ = π 5π, π π π 9. x =,, nπ ± 4 π 8. nπ π π π ± 9. θ = nπ ± C. D. D. C 4. B 5. C 6. B 7. C 8. A 9. B 40. D 4. D 4. A 4. D 44. C 45. B 46. C 47. C 48. C 49. B 50. C 5. B 5. C 5. C 54. A 55. B 56. A 57. B 58. B 59. D tan β 6. 4 [4 (a ) ], a 64. x x [, ] 67. sin A 68. True 69. False 70. False 7. True 7. False 7. True 74. True 75. True 76. (a) (iv) (b) (i) (c) (ii) (d) (iii)

5 ANSWERS EXERCISE. P(n) : n < n. P(n) : n = 6. A 7. B 8. A False 5. EXERCISE n ( n + ). n. + i. (0, ) (, 0) 6. icot θ. i. i. : 4. 0,0, ± i, ± i. i. 5π 5π cos + isin 5. (i) ( a b )( z z ) + + (ii) 5 (iii) (iv) 0 (v) i (vi) z (vii) 0 (viii) 6 and 0 (ix) a circle (x) + i 6. (i) F (ii) F (iii) T (iv) T (v) T (vi) T (vii) F (viii) F 7. (a) (v), (b) (iii), (c) (i), (d) (iv), (e) (ii), (f) (vi), (g) (viii) and (h) (vii)

6 08 EXEMPLAR PROBLEMS MATHEMATICS 8. i 9. No ( a + ) 4a +. + i.. π 4. Real axis 5. D 6. C 7. B 8. A 9. B 40. A 4. A 4. B 4. D 44. D 45. B 46. B 47. C 48. C 49. C 50. A 6. EXERCISE. x. [0,] [,4]. (, 5 ) (, ) [5, ) 4. [ 4, ] [,6] More than Between 7.77 and , 9. More than 0 litres but less than 90 litres. 0. Between 04 F and F. 4 cm.. Between 8 km and 0 km. No Solution 4. x + y 0, x + y 48, x 0, y 0 6. No Solution 5. x + y 8, x + y 4, x 5, y 5, x 0, y 0 7. No Solution. 9. C 0. C. A. B. D 4. C 5. B 6. A 7. D 8. B 9. A 0. B. (i) F (ii) F (iii) T (iv) F (v) T (vi) F (vii) T (viii) F

7 ANSWERS 09 (ix) T (x) F (xi) T (xii) F (xiii)f (xiv) T (xv) T.. (i) (ii) (iii) > (iv) > (v) > (vi) > (vii) <, > (viii). 7. EXERCISE , n C ( r )!! r r = !. (6!). (a) C 4 (b) 6C 5C (c) 6C 4 + 5C 4. (i) 4C 9 (ii) 4C 4. (0C 5 0C 6 ) 5. (i), (ii) 44 (iii) 9 6. A 7. B 8. C 9. B 0. C. A. B. D 4. B 5. C 6. D 7. A 8. C 9. B 40. B 4. n = n r 44.,5, False 5. False 5. False 54. True 55. True 56. True 57. True 58. False 59. False 60. (a) (ii) (b) (iii) and (c) (i) 6. (a) (iii) (b) (i) (c) (iv), (d) (ii) 6. (a) (iv) (b) (iii) (c) (ii), (d) (i) 6. (a) (i) (b) (iii) (c) (iv), (d) (ii) 64. (a) (iii) (b) (i) (c) (ii) 8. EXERCISE. 5 C k = ± ( 0 ) ( 5 )

8 0 EXEMPLAR PROBLEMS MATHEMATICS 5. (i) 5 (ii) x ; 9 6 x y x 9. r = p = ± 4. n = (C) 9. (A) 0. (C). (D). (B). (B) 4. (C) 5. 0 C ( n ) ( n + ) 7. 6 C 8. n = a 6 a C 4 a 56 b 4.. Third term. 4. F 5. T 6. F 7. F 8. T 9. F 40. F 9. EXERCISE. Rs 400. Rs 8080, Rs days cm m 9. Rs 75. (i) 4n + 9n + 6n (ii) T r = 6r 7. D 8. C 9. A 0. B. C. B. B 4. A 5. D 6. A 7. a b or b c 8 8. First term + last term F. T. T. F 4. F 5. (a) (iii) (b) (i) (c) (ii) 6. (a) (iii) (b) (i) (c) (ii)(d) (iv) 0. EXERCISE. x + y + = 0. x 4y + = or 0 5

9 ANSWERS x y 4. x + y = 7 or + = 5. (, ), ( 7, ) y x + = 0 8. x + 4y + = a =, b = x 5y + 60 = 0. x + y = 8. x 7y = (, ) 5. 5 or x 0y + 96 = 0 8. x 4y + 6 = 0 and 4x y + = 0 0. (0, + 5 ). A. A 4. B 5. B 6. C 7. D 8. A 9. A 0. A. B. B. A 4. C 5. A 6. B 7. B 8. C 9. D 40. B 4. B 4. (, ) 4. x + y + = x y 7 = 0, x + y 9 = opposite sides 46. (x + y ) 8 x + 64 y + 8 = x y = p (x + y ) 48. True 49. False 50. False 5. True 5. True 5. 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). EXERCISE a b. x + y ax ay + a = 0.,

10 EXEMPLAR PROBLEMS MATHEMATICS 4. x + y x 4y + = x + y + 4x + 4y + 4 = 0 7. (, ) 8. x + y x + 4y 0 = 0 9. k ± x + y 6x + y 5 = 0.. ecentricity = 4 5 and foci (4, 0) and ( 4, 0) x 4y + = (, 4), (, 4) a cosθ 7. sin θ 8. x + 8y = 9. m = 0. x y =.. x y 4 =.. x + y x + y = x + y 4x 0y + 5 = 0 5. (x ) + (y + ) = 8 6. x + y 8x 6y + 0 = 0 7. x + y 8x 6y + 6 = 0 8. (a) y = x 6, (b) x = 8y, (c)4x + 4xy + y + 4x + y + 6 = 0 9. x + 4y 6x = x + 5y = 80. (a) 5x y = 5 (b) 9x 7y + 4 = 0, (c) y x = 5. False 4. False 5. True 6. False 7. True 8. False 9. True 40. True 4. (x ) + (y + 4) = x + y 46x + y = , x 4y + = x + 4xy + y + 4x + y + 6 = y x = and (0, ± 0) (C) 48. (C) 49. (C) 50. (C)

11 ANSWERS 5. A 5. B 5. A 54. A 55. D 56. B 57. C 58. A 59. A. EXERCISE. (i) st octant (ii) 4 th octant (iii) viii th octant (iv) v th octant (v) nd octant (vi) rd octant (vii) viii th octant (viii) vi th octant. (i) (,0,0), (0,4,0), (0,0,) (ii) ( 5, 0, 0), (0,,0), (0,0,7) (iii) (4,0,0), (0,, 0), (0,0,5) 4. (i) (,4,0), (0,4,5), (,0,5) (ii) ( 5,, 0),(0,,7), ( 5, 0, 7) (iii) (4,, 0), (0,, 5), (4, 0, 5) (, 4, 6). (,, ). (,, ). (, 4, 7), (7,, 5) and (,, 7) 4. (4, 7, 6) 5. (4, 5, ), (,, ) 6. a =, b = 8, c = 7. 7,,9 8. : externally 9. vertices are (,4,5), (,6, 7), (,,) and centroid is (,4, ) 0. : externally. (,0,0), (,,0), (0,,0), (0,,) (0,0,) (,0,), (0,0,0), (,,). A. B 4. A 5. B 6. A 7. B 8. B 9. A 0. A. B. A. D 4. A 5. Three cordinates planes 6. Three pairs 7. given point 8. Eight 9. (0, y, z) 40. x = 0 4. (0, 0, z) 4. x = 0, y = 0 4. z- cordinates 44. (y, z cordinates) 45. yz-plane 46. x-axis a = 5 or 49. (,, ) 50. (a) (iii) (b) (i) (c) (ii) (d) (vi) (e) (iv) (f) (v) (g) (viii) (h) (vii) (i) (x) (j) (ix)

12 4 EXEMPLAR PROBLEMS MATHEMATICS. EXERCISE. 6.. x ( ) 5 a n = m n a cos a k = 8 9. x + x + 0. x x 4 x x +. xsec x + 5sec x + tan x +. tan xsec x x 5x. ( 5 x 7x + 9) x cos x + 5sec sin x + sin x x 5. cosec x( x cot x) 6. ( ax + cot x)( q sinx) + ( p + q cosx)( ax cosec x) 7. bc cos x + ad sin x + db ( c + d cos x) 8. cos x

13 ANSWERS 5 9. ( )( )( ) x x x x cos x + xsin x sin x sin cos 4 x x 4. ( + ) ax b ( ax + bx + c) 4. sin ( ) x x ad bc ( cx + d ) 45. x 46. cos x xsin x α x x x α β 47. sec ( tan ) k = 6 5. c = 54. C 55. A 56. A 57. B 58. A 59. C 60. C 6. D 6. B 6. D 64. C 65. D 66. B 67. B 68. D 69. A 70. A 7. A 7. A 7. B 74. C 75. A 76. D m = 79. y EXERCISE. (i) to (v) and (viii) to (x) are statements.. (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 : 00 is divisble by (iv) p : Chandigarh is capital of Haryana q : 00 is divisible by q : Chandigarh is the capital of U.P r : 00 is divisible by 5

14 6 EXEMPLAR PROBLEMS MATHEMATICS (v) p : (vii) (viii) (ix) 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 p : plants use sunlight for photosynthesis q : plants use water for photosynthesis r : plants use carbondioxide for photosynthesis p : two lines in a plane intersect at one point q : two lines in a plane are parallel p : a rectangle is a quadrilateral q : a rectangle is a 5- sided polygons.. (i) Compound statement is true and its component statements are : p : 57 is divisible by and q : 57 is divisble by (ii) component statement is true and its component statements are : (iii) p : 4 is multiple of 4 and q : 4 is multiple of 6 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 : is an number ; q : is a prime number 4. (i) The number 7 is not prime (ii) (iii) Violet are not blue (iv) (vi) (vii) (ix) (x) 5 is not a rational number (v) is a prime number There exists a real number which is not an irrational number Cow has not four legs (viii) A leap year has not 66 days There exist similar triangles which are not congruent 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 : is factor of ; q : is factor of ; r : 6 is factor of (iv) p q where p : x is an odd integer ; q : x + is an odd integer (v) p q where p : a number is divisible by, q : it is divisibe by (vi) p q where p : x = is a root of x x 0 = 0, q : x = is a root of x x 0 = 0

15 ANSWERS 7 (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) (iii) (iv) (v) (vi) It is false that all real numbers are rational or irrational x = is not a root of the quadratic equation x 5x + 6 = 0 or x = is not a root of the quadratic equation x 5x + 6 = 0 A triangle has neither -sides nor 4-sides 5 is not a prime number and it is not a complex number 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 or it is not divisible by. 7. (i) If the number is odd number then its square is odd number (ii) (iii) If you take the dinner then you will get sweet dish 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 b = 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. (iii) A triangle is an equilateral triangle if and only if all three sides of triangle are equal. 9. (i) If x then x y or y (ii) (iii) (iv) If n is not an integer then it is not a natural number. If the triangle is not equilateral then all three sides of the triangle are not equal If xy is not positive integer then either x or y is not negative integer. (v) If natural number n is not divisible by and then n is not divisible by 6. (vi) The weather will not be cold if it does not snow. 0. (i) If the rectangle R is rhombus then it is square. (ii) (iii) If tomorrow is Tuesday then today is Monday. If you must visit Taj Mahal you go to Agra.

16 8 EXEMPLAR PROBLEMS MATHEMATICS (iv) (v) If the triangle is right angle then sum of squares of two sides of a triangle is equal to the square of third side. If the triangle is equilateral then all three anlges of triangle are equal. (vi) If x = y then x:y = : (vii) If the opposite angles of a quadrilaterals are supplementary then S is cyclic. (viii) If x is neither positive nor negative than x is 0. (ix) If the ratio of corresponding sides of two triangles are equal then trianges are similar.. (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 7.. C 8. D 9. B 0. D. C. B. A 4. B 5. C 6. A 7. C 8. B 9. A 0. C. B. A. C 4. A 5. C 6. D 7. (i), (ii) and (iv) are statement; (iii) and (v) are not statements. 5. EXERCISE n 4n 4. n 4 5. n ( ) + ( ) ( ) + n + n ( n + n ) n s n s n n x x Mean =.8, SD = , Mean = 5.7, SD =.5 5. Mean = 5.5, Var. = Mean = 9 40, SD =.85

17 ANSWERS 9 9. Var. =.6gm, S.D =.08 gm 0. Mean = d ( n ) a +, n S.D = d. Hashina is more intelligent and consistent Mean = 4., Var B 5. B 6. B 7. C 8. A 9. C 0. C. A. C. A 4. D 5. D 6. A 7. D 8. A 9. A 40. SD 4. 0, less Independent 44. Minimum 45. Least 46. greater than or equal 6. EXERCISE (a) 5 k elements (b) 5 k (a) 0.65 (b) 0.55 (c) 0.8 (d) 0 (e) 0.5 (f) (a) 0.5 (b) 0.77 (c) 0.5 (d) (a) 9 (b) (a)p(john promoted) = 8, p(rita promoted) = 4, p(aslam promoted) =, p(gurpreet promoted) = 8 (b) P(A) = 4. (a) 0.0 (b) 0.7 (c) 0.45 (d) 0. (e) 0.5 (f) 0.5. (a) S = { B B, B W, B B, B W, WB, WB BW, BW, W B, W W, W B, W W } (b) 6 (c)

18 0 EXEMPLAR PROBLEMS MATHEMATICS. (a) 5 4 (b) 8 4 (c) (a) 4 (b) 4 (c) 5 6 (d) (a) p(a) =.5, p(b) =., p(a Β) =.7 (b) p(a B) =.40 (c).40 (d) (a) (b) 4 (c) 6 (d) A 9. B 0. C. C. D. A 4. A 5. C 6. B 7. C 8. C 9. B 0. False. False. False. True 4. True 5. False 6. True E = {, 4,6} (a) (iv) (b) (v) (c) (i) (d) (iii) (e) (ii) 4. (a) (iv) (b) (iii) (c) (ii) (d) (i)

ANSWERS 1.3 EXERCISE. 1. (i) {2} (ii) {0, 1} (iii) {1, p}

ANSWERS 1.3 EXERCISE. 1. (i) {2} (ii) {0, 1} (iii) {1, p} ANSWERS. EXERCISE. (i) {} (ii) {0, } (iii) {, p}. (i) {0,, } (ii). {,,,,... P,( p } (iii) {,,, } 4. (i) True (ii) False (iii) True (iv) True 7. (i) {, 4, 6, 8,..., 98} (ii) (,4, 9, 6, 5, 6, 49, 64, 8,}