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

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1 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,} 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 cosx 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 nn+ ( ). 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 ± 8 0. 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 ( ) a n = m n acosa k = 8 9. x + x+ 0. x x 4 x x +. xsec x+ 5sec x+ tan x+. tanxsec x x 5x. ( 5 x 7x+ 9) x cosx+ 5sec sinx+ sin x x 5. cosec x( xcot x) 6. ( ax + cot x)( q sin x) + ( p + qcos x)( ax cosec x) 7. bccos x + ad sin x + db ( c+ dcos x) 8. cosx

13 ANSWERS 5 9. ( )( )( ) x x x+ 40. x cos x+ xsin x sin x 4. sin cos 4 x x 4. ( + b) ax ( ax + bx + c) 4. sin ( ) x x ad bc ( cx + d ) 45. x 46. cos x xsin x α x x x+ 48. 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 : p : 4 is multiple of 4 and q : 4 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 : is an number ; q : is a prime number 4. (i) The number 7 is not prime (ii) (iii) Violet are not blue (iv) 5 is not a rational number (v) 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 66 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 : 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 nn 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, BW, B B, B W, WB, WB BW, BW, W B, WW, 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)

19 DESIGN OF THE QUESTION PAPER MATHEMATICS - CLASS XI Time : Hours Max. Marks : 00 The weightage of marks over different dimensions of the question paper shall be as follows:. Weigtage of Type of Questions Marks (i) Objective Type Questions : (0) 0 = 0 (ii) Short Answer Type questions : () 4 = 48 (viii) Long Answer Type Questions : (7) 7 6 = 4 Total Questions : (9) 00. Weightage to Different Topics S.No. Topic Objective Type S.A. Type L.A. Type Total Questions Questions Questions. Sets - (4) - 4(). Relations and Functions - - (6) 6(). Trigonometric Functions () (4) (6) (4) 4. Principle of Mathematical - (4) - 4() Induction 5. Complex Numbers and () (4) - 6() Quadratic Equations - 6. Linear Inequalities () (4) - 5() 7. Permutations and Combinations - (4) - 4() 8. Binomial Theorem - - (6) 6() 9. Sequences and Series - (4) - 4() 0. Straight Lines () (4) (6) (4). Conic Section - - (6) 6(). Introduction to three - (4) - 4() dimensional geometry. Limits and Derivatives () (4) - 5() 4. Mathematical Reasoning () (4) - 5() 5. Statistics - (4) (6) 0() 6. Probability () - (6) 7() Total 0(0) 48() 4(7) 00(9)

20 EXEMPLAR PROBLEMS MATHEMATICS SAMPLE QUESTION PAPER Mathematics Class XI General Instructions (i) (ii) (iii) (iv) The question paper consists of three parts A, B and C. Each question of each part is compulsory. Part A (Objective Type) consists of 0 questions of mark each. Part B (Short Answer Type) consists of questions of 4 marks each. Part C (Long Answer Type) consists of 7 questions of 6 marks each. PART - A. If tan θ = and tan φ =, then what is the value of (θ + φ)?. For a complex number z, what is the value of arg. z + arg. z, z 0?. Three identical dice are rolled. What is the probability that the same number will appear an each of them? Fill in the blanks in questions number 4 and The intercept of the line x + y 6 = 0 on the x-axis is.... cosx 5. lim x 0 is equal to.... x In Questions 6 and 7, state whether the given statements are True or False: 6. x+, x> 0 x 7. The lines x + 4y + 7 = 0 and 4x + y + 5 = 0 are perpendicular to each other. In Question 8 to 9, choose the correct option from the given 4 options, out of which only one is correct. 8. The solution of the equation cos θ + sinθ + = 0, lies in the interval (A) π π, 4 4 (B) π π, 4 4 (C) π 5π, 4 4 (D) 5π 7π, 4 4

21 DESIGN OF THE QUESTION PAPER 9. If z = + i, the value of z z is (A) 7 (B) 8 (C) i (D) 0. What is the contrapositive of the statement? If a number is divisible by 6, then it is divisible by. PART - B. If A B = U, show by using laws of algebra of sets that A B, where A denotes the complement of A and U is the universal set.. If cos x = and cos y =, x, y being acute angles, prove that x y = Using the principle of mathematical induction, show that n is divisible by 7 for all n N. 4. Write z = 4 + i 4 in the polar form. 5. Solve the system of linear inequations and represent the solution on the number line: x 7 > (x 6) and 6 x > x b+ c c+ a a + b 6. If a + b + c 0 and,, a b c also in A.P. are in A.P., prove that,, a b c are 7. A mathematics question paper consists of 0 questions divided into two parts I and II, each containing 5 questions. A student is required to attempt 6 questions in all, taking at least questions from each part. In how many ways can the student select the questions? 8. Find the equation of the line which passes through the point (, ) and cuts off intercepts on x and y axes which are in the ratio 4 :. 9. Find the coordinates of the point R which divides the join of the points P(0, 0, 0) and Q(4,, ) in the ratio : externally and verify that P is the mid point of RQ. x 0. Differentiate f(x) = with respect to x, by first principle. + 4x

22 4 EXEMPLAR PROBLEMS MATHEMATICS. Verify by method of contradiction that p = is irrational.. Find the mean deviation about the mean for the following data: x i f i PART C. Let f(x) = x and g(x) = x be two functions defined over the set of nonnegative real numbers. Find: f (i) (f + g) (4) (ii) (f g) (9) (iii) (fg) (4) (iv) (9) g 4. Prove that: (sin 7x+ sin 5 x) + (sin 9x+ sin x) = tan 6x (cos7x+ cos5 x) + (cos9x+ cos x) 5. Find the fourth term from the beginning and the 5th term from the end in the x expansion of x A line is such that its segment between the lines 5x y + 4 = 0 and x + 4y 4 = 0 is bisected at the point (, 5). Find the equation of this line. 7. Find the lengths of the major and minor axes, the coordinates of foci, the vertices, the ecentricity and the length of the latus rectum of the ellipse + = x y 8. Find the mean, variance and standard deviation for the following data: Class interval: Frequency: What is the probability that (i) a non-leap year have 5 Sundays. (ii) a leap year have 5 Fridays (iii) a leap year have 5 Sundays and 5 Mondays.

23 DESIGN OF THE QUESTION PAPER 5 MARKING SCHEME MATHEMATICS CLASS XI PART - A Q. No. Answer Marks. π 4. Zero True 7. False 8. D 9. A 0. If a number is not divisible by, then it is not divisible by 6. PART - B. B = B φ = B (A A ) = (B A) (B A ) = (B A) (A B) = (B A) U (Given) = B A A B.

24 6 EXEMPLAR PROBLEMS MATHEMATICS. cos x = 7 sin x = 4 cos x = = 49 7 cos y = 4 sin y = 69 = 96 4 cos(x y) = cosx cosy + sinx siny = 4 + = x y = π. Let P(n) : n is divisble by 7 P() = = 8 = 7 is divisible by 7 P() is true. Let P(k) be true, i.e, k is divisible by 7, k = 7a, a Z We have : (k + ) = k. =( k ) = 7a = 7(8a + ) P(k + ) is true, hence P(n) is true n N 4. Let 4 + i 4 = r (cosθ + i sinθ) r cosθ = 4, r sinθ = 4 r = = 64 r = 8.

25 DESIGN OF THE QUESTION PAPER 7 tanθ = θ = π z = 4 + i 4 = 8 5. The given in equations are : π π = π π cos + i sin x 7 > (x 6)... (i) and 6 x > x... (ii) (i) x x > + 7 or x > 5... (A) (ii) x + x > 6 or x > 5... (B) From A and B, the solutions of the given system are x > 5 Graphical representation is as under: b+ c c+ a a + b 6. Given,, a b c are in A.P. + b+ c c a a b,, a b + c will also be in A.P. a + b+ c a + b+ c a + b+ c,, will be in A.P. a b c Since,a + b + c 0,, a b c will also be in A.P. 7. Following are possible choices: Choice Part I Part II } (i) 4 (ii) (iii) 4

26 8 EXEMPLAR PROBLEMS MATHEMATICS Total number of ways of selecting the questions are: = ( 5 C 5 C 5 C 5 C 5 C 5 C ) = = Let the intercepts on x-axis and y-axis be 4a, a respectively x y Equation of line is : + = 4a a or x + 4y = a (, ) lies on it a = 7 Hence, the equation of the line is x + 4y + 7 = 0 9. Let the coordinates of R be (x, y, z) x = (4) (0) = 4 y = ( ) (0) = z = ( ) (0) = R is ( 4,, ) 4 4 Mid point of QR is,, i.e., (0, 0, 0) Hence verified. 0. f (x) = x f (x + Δx) = ( x + Δx ) + 4x + 4( x + Δx) f (x) = lim Δ x 0 lim x Δx x f( x+δx) f( x) Δ x 0 + 4x + 4Δ x + 4x Δx = Δx

27 DESIGN OF THE QUESTION PAPER 9 = lim Δ x 0 ( x Δ x)(+ 4 x) (+ 4x+ 4 Δx)( x) ( Δ x)(+ 4x+ 4 Δ x)(+ 4 x) = Δ x x x 4x Δx 4x Δx 9 + x x+ 4x Δ x+ 4xΔx lim = ( Δ x)(+ 4x+ 4 Δ x)(+ 4 x) 5Δx 5 lim = = ( Δ x )( + 4 x+ 4 Δ x )( + 4 x ) ( + 4 x ) = Δ x 0. Assume that p is false, i.e., ~p is true i.e., is rational There exist two positive integers a and b such that a =, a and b are coprime b a = b divides a divides a a = c, c is a positive integer, 9c = b b = c divides b also is a common factor of a and b which is a contradiction as a, b are coprimes. Hence p : is irrational is true.. x i : f i : f i = 80 f i x i : fx i i = 4000 d = x x : Mean = 50 i i f i d i : fi d i = 80 Mean deviation = =

28 0 EXEMPLAR PROBLEMS MATHEMATICS PART C. (f + g) (4) = f(4) + g(4) = (4) + 4 = 6 + = 8 (f g) (9) = f(9) g(9) = (9) 9 = 8 = 78 (f. g) (4) = f(4). g(4) = (4). (4) = (6) () = f g (9) = f (9) (9) 8 = = = 7 g(9) 9 4. sin 7x + sin 5x = sin 6x cosx sin 9x + sin x = sin 6x cos x cos 7x + cos 5x = cos 6x cosx cos 9x + cos x = cos 6x cos x sin 6x cos x+ sin 6xcosx L.H.S = cos 6x cos x+ cos 6xcosx sin 6 x (cosx+ cos x) sin 6x = = cos 6 x (cos x+ cos x) cos 6x = tan 6x 5. Using T r + = C n n r r r x y we have T 4 = 0C 7 x x 5 5 = x = 40 x th term from end = ( 5 + ) = 7 th term from beginning

29 DESIGN OF THE QUESTION PAPER T 7 = 0C 4 6 x 6 x = = Let the required line intersects the line 5x y + 4 = 0 at (x, y ) and the line x + 4y 4 = 0 at (x, y ). 5x y + 4 = 0 y = 5x + 4 x + 4y 4 = 0 y = 4 x 4 4 x Points of inter section are (x, 5x + 4), x, 4 x + x = and 4 x + 5x = 5 x + x = and0x x = Solving to get x =, x = y =, y = 8 Equation of line is y 5 = or 07x y 9 = 0 5 ( x ) 6

30 EXEMPLAR PROBLEMS MATHEMATICS 7. Here a = 69 and b = 44 a =, b = Length of major axis = 6 Length of minor axis = 4 Since e = b = = e = a foci are (± ae, 0) = ±,0 = (± 5, 0) vertices are (± a, 0) = (±, 0) latus rectum = b (44) 88 a = = 8. Classes: f: f = 50 x i : d i : = xi f i d i : fd i i = 5 i i f d : , fd i i = 05 5 Mean x = 65 0 = 65 = 6 50 Variance σ = = S.D. σ = 0 = (i) Total number of days in a non leap year = 65 = 5 weeks + day

31 DESIGN OF THE QUESTION PAPER P(5 sun days) = 7 (ii) Total number of days in a leap year = 66 = 5 weeks + days These two days can be Monday and Tuesday, Tuesday and Wednesday, Wednesday and Thursday, Thursday and Friday, Friday and Saturday, Saturday and Sunday, Sunday and Monday P(5 Fridays) = 7 (iii) P(5 Sunday and 5 Mondays) = 7 (from ii)

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}

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} 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,}