185 NDA Mathematics Practice Set 1. Which of the following statements is not correct for the relation R defined by arb if and only if b lives within one kilometer from a? R is reflexive R is symmetric R is not anti-symmetric None of the above 2. X is well defined but Y is not a well defined set Y is well defined but X is not a well defined set Both X and Y are well defined sets Neither X nor Y is a well defined set 3. 4. 36 33 20 6 7. In a town 35.4% of the people are not literates, 27% have education up to primary school, 18.6% have education up to middle school. The people with education up to high school are twice the number of people with education up to Pre-University. Of the remaining, 660 are graduates. If the population of the town is 15000, then what is the number of people with education up to high school? 3120 1560 1460 None of these 8. Out of 32 persons, 30 invest in National Savings Certificates and 17 invest in shares. What is the number of persons who invest in both? 13 15 17 19 9. 1 2 4 5 Both A and R are individually true, and R is the correct explanation of A. Both A and R are individually true but R is not the correct explanation of A. A is true but R is false. A is false but R is true. 5. A = B only B = C only A = C only A = B = C 6. 10. The number 292 in decimal system is expressed in binary system by 100001010 100010001 100100100 101010000 11. If d is the number of degrees contained in an angle, m is the number of minutes and s is the number of seconds, then the value of (s m)/(m d) is: 1 60 1/60 None of these 12.
186 1 b -b 0 18. 13. 14. 15. 2n 2 2n 4 2 n 2 16. 1, 0 0, 1-2, 0-2, 1 17. 1 r q r 1 + r q + r 19. (x + 1) 2 1 = 0 has one real root two real roots two imaginary roots four real roots 20. a, b, c are in G.P. with 1 < a < b < n, and n > 1 is an integer. loga n, logb n, logc n form a sequence. This sequence is which one of the following? Harmonic progression Arithmetic progression Geometric progression None of these 21. The difference between the nth term and (n 1)th term of a sequence is independent of n. Then the sequence follows which one of the following? AP GP HP None of these 22. If (1 + 3 + 5 +... + p) + (1 + 3 + 5 +... + q) = (1 + 3 + 5 +...+ r) where each set of parentheses contains the sum of consecutive odd integers as shown, what is the smallest possible value of (p + q + r) where p > 6? 12 21 45 54 23. If A, B and C are in AP and b: c = 3: 2, then what is the value of sin C? 1 1/ 3 3 1/ 2
187 24. If p, q, r are in AP as well as G.P., then which one of the following is correct? p = q r p q r p q = r p = q = r 25. 0 1/2 1 2 26. 30. What is the sum of all the coefficients in the expansion of (1 + x) n? 2 n 2 n 1 2 n 1 2(n 1) 31. In how many ways can be letters of the word CABLE be arranged so that the vowels should always occupy odd positions? 12 18 24 36 32. In a football championship 153 matches were played. Every team played one match with each other team. How many teams participated in the championship? 21 18 17 15 33. 27. -2-1 0 1 28. 1 3 4 5 29. What is the coefficient of x 5 in the expansion (1 2x + 3x 2 4x 3 +... ) 5? (10!)/(5!) 2 5 5 5 5 10!/{6!)(4!)} 2 n 1 2 n 2 n 1 2 n + 1 34. The straight line ax + by + c = 0 and the coordinate axes form an isosceles triangle under which one of the following conditions? a = b a = c b = c none of these 35. The middle point of the segment of the straight line joining the points (p, q) and (q, p) is (r/2, s/2). What is the length of the segment? [(s 2 + r 2 ) 1/2 ]/2 [(s 2 + r 2 ) 1/2 ]/4 (s 2 + r 2 ) 1/2 s + r 36. What is the equation of the line passing through (2, 3) and parallel to Y-axis? Y = 3 Y = 2 X = 2
188 X = 3 37. What is the perpendicular distance of the point (x, y) from x-axis? x y x y 38. What is the inclination of the line 3x y 1 = 0? 30 60 135 150 39. Consider a circle of radius R. What is the length of a chord which subtends an angle θ at the centre? 40. If the latus rectum of an ellipse is equal to one half its minor axis, what is the eccentricity of the ellipse? 1/2 3/2 3/4 15/4 41. What does an equation of the first degree containing one arbitrary parameter passing through a fixed point represent? Circle Straight line Parabola Ellipse 42. What is the eccentricity of the conic 4x 2 + 9y 2 = 144? 5/3 5/4 3 5 2/3 43. Let A and B be obtuse angles such that sin A = 4/5 and cos B = 12/13. What is the value of sin (A + B)? -63/65-33/65 33/65 63/65 44. 1/ 3 1/ 3 3 3 45. For what value of x does the equation 4 sin x + 3 sin 2x 2 sin 3x + sin 4x = 2 3 hold? π/6 π/4 π/3 π/2 46. After subtending an angle of 1000 from its inital position, the revolving line will be situated in which one of the following quadrants? First quadrant Second quadrant Third quadrant Fourth quadrant 47. 1 1 1/3 3 48. sec 18 cosec 18 sec 18 cosec 18 49. 1 0 1 2 50. If cosecθ cotθ = 1/ 3 where θ 0, then what is the value of cos θ?
189 0 3/2 1/2 1/ 2 51. If sec α = 13/5 where 270 < α < 360 then what is sin α equal to? 5/13 12/13 12/13 13/12 52. What is the value of tan (tan 1 x + tan 1 y + tan 1 z) cot (cot 1 x + cot 1 y + cot 1 z)? 0 2 (x + y + z) 3π/2 3π/2 + x + y + z 53. 54. If in a Δ ABC, cos B = (sin A)/ (2 sin C), then the triangle is Isosceles triangle Equilateral triangle Right angled triangle Scalene triangle 55. Only I Only II Both I and II Neither I nor II 56. 57. At a point 15 m away from the base of a 15 m high house, the angle of elevation of the top is 90 60 45 30 58. Let f (x + y) = f (x). f (y) and f (1) = 2 for all x, y R, where f (x) is continuous function. What is f ' (1) equal to? 2 ln 2 ln 2 1 0 59. If f(x) is differentiable everywhere, then which one of the following is correct? f is differentiable everywhere f 2 is differentiable everywhere f f is not differentiable at some points None of the above 60. DIRECTIONS : The following questions consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answers. Q. Assertion (A): f (x) = x sin 1 (1/x) is differentiable at x = 0. Reason (R): f (x) is continuous at x = 0. Both A and R are individually true, and R is the correct explanation of A. Both A and R are individually true but R is not the correct explanation of A. A is true but R is false. A is false but R is true. 61. A mapping f : R R which is defined as f (x) = cos x; x R is One-one only
190 Onto only One-one onto Neither one-one nor onto 62. The function is one-one into The function is many-one into The function is one-one onto The function is many-one onto 63. 0 1/2 1 ½ 64. From past experience it is known that an investor will invest in security A with a probability of 0.6, will invest in security B with a probability 0.3 and will invest in both A and B with a probability of 0.2. What is the probability that an investor will invest neither in A nor in B? 0.7 0.28 0.3 0.4 65. A box contains 10 identical electronic components of which 4 are defective. If 3 components are selected at random from the box in succession, without replacing the units already drawn, what is the probability that two of the selected components are defective? 1/5 5/24 3/10 1/40 66. A coin is tossed. If a head is observed, a number is randomly selected from the set {1, 2, 3} and if a tail is observed, a number is randomly selected from the set {2, 3, 4, 5}. If the selected number be denoted by X, what is the probability that X = 3? 2/7 1/5 1/6 7/24 67. Two numbers X and Y are simultaneously drawn from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. What is the conditional probability of exactly one of the two numbers X and Y being even, given (X + Y) = 15? 1 3/4 1/2 1/4 68. A box contains 6 distinct dolls. From this box, 3 dolls are randomly selected one by one with replacement. What is the probability of selecting 3 distinct dolls? 5/54 12/25 1/20 5/9 69. 1/3 2/3 1/9 2/9 70. If X follows a binomial distribution with parameters n = 100 and p = 1/3, then P(X = r) is maximum when r = 16 r = 32 r = 33 r = 34 71. If the sum of the squares of the distance of the point (x, y, z) from the points (a, 0, 0) and ( a, 0, 0) is 2c 2, then which one of the following is correct? x 2 + a 2 = 2c 2 y 2 z 2 x 2 + a 2 = c 2 y 2 z 2 x 2 a 2 = c 2 y 2 z 2 x 2 + a 2 = c 2 + y 2 + z 2 72. If θ is the acute angle between the diagonals of a cube, then which one of the following is correct? θ < 30 θ = 60 30 < θ < 60 θ > 60 73. What is the equation of a plane through the x-axis and passing through the point (1, 2, 3)? x + y + z = 6
191 x = 1 y + z = 5 z + y = 1 74. What is the distance of the origin from the plane 2x + 6y 3z + 7 = 0? 1 2 3 6 75. If the distance between the points (7, 1, 3) and (4, 5, λ) is 13 units, then what is one of the values of λ? 20 10 9 8 76. The production of food grains in Maharashtra is given for the 12 years from 1992 to 2003. Which one of the following representations is most suitable to depict the data? A simple bar diagram A pie diagram A component bar diagram with the components arranged in chronological order A broken line graph 77. If the slopes of the line of regression of Y and X and of X and Y are 30 and 60 respectively, then r(x, Y) is: 1 1 1/ 3 1/ 3 78. 6 5 4 2 80. What is the lower limit of the median class? 10 15 20 25 81. A variate X takes values 2, 3, 4, 2, 5, 4, 3, 2, 1. What is the mode? 2 3 4 5 82. What is the mode for the data 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 25? 7 21 22 25 83. The marks obtained by 13 students in a test are 10, 3, 10, 12, 9, 7, 9, 6, 7, 10, 8, 6, 7. The median of this data is? 7 8 9 10 84. Mean < Mode < Median Mode < Median < Mean Mode < Mean < Median Median < Mean < Mode 79.
192 6/7 1/2 85. 86. 89. What is the minimum value of 2x 2 3x + 5? 0 3/4 31/4 31/8 90. Statement I: y = tan 1 (x 1 ) + 1 is an increasing function of x. Statement II: dy/dx is positive for all values of x. Which one of the following is correct in respect of the above statements? Both statements I and II are independently correct and statement II is the correct explanation of statement I Both statements I and II are independently correct but statement II is not the correct explanation of statement I Statement I is correct but statement II is false. Statement I is false but statement II is correct. 91. What are the points on the curve x 2 + y 2 2x 3 = 0 where the tangents are parallel to x-axis? (1, 2) and (1, 2) (0, 3) and (0, 3 ) (3, 0) and ( 3, 0) (2, 1) and (2, 1) 92. 87. e x cot x + c cos 2 (x e x ) + c log sin (x e x ) + c cot (x e x ) + c 93. 88. What is the x-coordinate of the point on the curve f (x) = x (7x 6), where the tangent is parallel to x-axis? 1/3 2/7 94.
193 1 0 1 2 100. 95. 1/2 and 2 1/2 and 2 1/2 and 2 1/2 and 2 96. What is the area bounded by the curves y = e x, y = e x and the straight line x = 1? Family of straight lines through the origin Family of circles with their centres at the origin Family of parabolas with their vertices at the origin Family of straight lines having slope 1 and not passing through the origin 101. 97. What is the area of the region bounded by the lines y = x, y = 0 and x = 4? 4 square units 8 square units 12 square units 16 square units 98. 102. If a matrix B is obtained from a square matrix A by interchanging any two of its rows, then what is A + B equal to 2 A 2 B 0 A B 103. 5 0 0.01 (10) 1 99.
194 104. If the least number of zeroes in a lower triangular matrix is 10, then what is the order of the matrix? 3 3 4 4 5 5 10 10 105. If l + m + n = 0, then the system of equations 2x + y + z = l x 2y + z = m x + y 2z = n has a trivial solution no solution a unique solution infinitely many solutions 106. If a matrix A is symmetric as well as anti-symmetric, then which one of the following is correct? A is a diagonal matrix A is a null matrix A is a unit matrix A is a triangular matrix 107. 3 1 1 1 1 3 3 3 110. If D is determinant of order 3 and D is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct? D = D 2 D = D 3 D = 2D 2 D = 3D 3 111. 112. 3 + i 1 + 3i 3i 0 108. The simultaneous equations 3x + 5y = 7 and 6x + 10y = 18 have no solution infinitely many solutions unique solution any finite number of solutions 109. 0 1 2 2 113. 0 2 1 1/2 114.
195 120. 115. 1 1 2 2 116. 8 4 8 12 117. Both A and R are individually true, and R is the correct explanation of A. Both A and R are individually true but R is not the correct explanation of A. A is true but R is false A is false but R is true. 1. c 2. c 3. c 4. a 5. b 6. a 7. c 8. b 9. b 10. c 11. c 12. c 13. c 14. a 15. d 16. a 17. d 18. a 19. b 20. a 21. a 22. b 23. d 24. d 25. c 26. a 27. c 28. b 29. a 30. a 31. b 32. b 33. a 34. a 35. c 36. c 37. d 38. b 39. a 40. b 41. b 42. a 43. a 44. c 45. a 46. d 47. b 48. a 49. b 50. c 51. c 52. a 53. b 54. a 55. c 56. d 57. c 58. a 59. c 60. d 61. d 62. c 63. b 64. c 65. c 66. d 67. a 68. c 69. b 70. c 71. b 72. d 73. b 74. a 75. c 76. a 77. c 78. d 79. d 80. b 81. a 82. c 83. b 84. d 85. a 86. c 87. c 88. b 89. d 90. a 91. a 92. d 93. a 94. a 95. a 96. c 97. b 98. a 99. c 100. a 101. a 102. c 103. c 104. b 105. d 106. b 107. d 108. a 109. b 110. a 111. c 112. c 113. c 114. d 115. a 116. c 117. d 118. d 119. a 120. a 1. 1/8 1/64 182 1/ 182 2. 118. (1001)2 (101)2 (110)2 (100)2 3. 119. 4.
196 5. 6. 9. 7. 10. 11. 8.
197 15. 12. 16. 17. 13. 18. 14. 19.
198 20. 21. 23. 22. 24.
199 25. 27. 26. 28.
200 29. 34. 30. 31. 32. 35. 33.
201 41. 36. 37. 42. 38. 39. 43. 40.
202 46. 47. 44. 48. 45.
203 52. 49. 53. 50. 54. 51. 55.
204 56. 57. 62. 63. 58. 64. 59. 65. 60. 61. 66. 67.
205 72. 68. 69. 70. 73. 74. 71. 75.
206 76. 77. 81. 82. 83. 78. 84. 79. 80.
207 85. 86. 91. 87. 88. 92. 93. 89. 90.
208 96. 94. 97. 95. 98.
209 99. 103. 104. 100. 105. 101. 102.
210 106. 107. 111. 108. 112. 109. 113. 114. 110. 115.
211 118. 116. 119. 120. 117.