318 NDA Mathematics Practice Set 1. (1001)2 (101)2 (110)2 (100)2 2. z 1/z 2z z/2 3. The multiplication of the number (10101)2 by (1101)2 yields which one of the following? (100011001)2 (100010001)2 (110010011)2 (100111001)2 4. R is reflexive only R is symmetric only R is transitive only R is reflexive and transitive n n 2 8. If A and B are two disjoint sets, then which one of the following is correct? 9. Null set {P} {P, Q, R} {Q, R} 10. Complement of A Complement of B B A 11. 5. {(2, 4), (3, 5), (1, 5)} {(1, 4), (2, 4), (3, 4)} {(1, 4), (2, 5), (3, 4)} {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} 6. 12 21 27 30 7. Sets A and B have n elements in common. How many elements will (A B) and (B A) have in common? 0 1 2k 7 2 K + 1 12. 1 3 3xyz 27xyz 13. Signs of a and c should be like Signs of b and c should be like Signs of a and b should be like None of the ablve 14.
319 A quadratic polynomial with two distinct roots has one real root. Then, the other root is Not necessarily real, if the coefficients are real Always imaginary Always real Real, if the coefficients are real 15. 20. If, for positive real numbers x, y, z, the numbers x + y, 2y and y + z are in harmonic progression, then which one of the following is correct? x, y, z are in geometric progression x, y, z are in arithmetic progression x, y, z are in harmonic progression None of the above 21. always complex always real always purely imaginary None of these 16. 17. 18. If the roots of a quadratic equation are m + n and m n, then the quadratic equation will : 19. What is the product of first 2n + 1 terms of a geometric progression? The (n + 1)th power of the nth term of the GP The (2n + 1)th power of the nth term of the GP The (2n + 1)th power of the (n + 1)th term of the GP The nth power of the (n + 1)th terms of the GP 22. The 59th term of an AP is 449 and the 449th term is 59. Which term is equal to 0 (zero)? 501 st term 502 nd term 508 th term 509 th term 23. Which one of the following options is correct? sin 2 30, sin 2 45, sin 2 60 are in GP cos 2 30, cos 2 45, cos 2 60 are in GP cot 2 30, cot 2 45, cot 2 60 are in GP tan 2 30, tan 2 45, tan 2 60 are in GP 24. Consider the following statements: 1. The sum of cubes of first 20 natural numbers is 44400. 2. The sum of squares of first 20 natural numbers is 2870. Which of the above statements is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor 2 25.
320 26. 1 0 ω ω 2 27. 32. What is the number of words that can be formed from the letters of the word UNIVERSAL, the vowels remaining always together? 720 1440 17280 21540 33. The lines (p + 2q) x + (p 3q) y = p q for different values of p and q pass through the fixed point given by which one of the following? -1 0 1 2 28. -8 0 4 8 29. What is the approximate value of (1.02) 8? 1.171 1.175 1.177 1.179 30. In the expansion of [1 + x] n, what is the sum of even binomial coefficients? 2 n 2 n 1 2 n + 1 None of the above 31. A group consists of 5 men and 5 women. If the number of different five-person committees containing k men and (5 k) women is 100, what is the value of k? 2 only 3 only 2 or 3 4 34. If O be the origin and A (x1, y1), B (x2, y2) are two points, then what is (OA) (OB) cos AOB? 35. What is the equation to the straight line joining the origin to the point of intersection of the lines x/a + y/b = 1 and x/b + y/a = 1? x + y = 0 x + y +1 = 0 x y = 0 x + y + 2 = 0 36. How many diagonal will be there in an n-sided regular polygon?
321 37. What is the perpendicular distance between the parallel lines 3x + 4y = 9 and 9x + 12y + 28 = 0? 7/3 units 8/3 units 10/3 units 11/3 units 38. What is the locus of the point of intersection of the straight lines (x/a) + (y/b) = m and (x/a) (y/b) = 1/m? Circle Parabola Ellipse Hyperbola 39. The equation of the circle which touches the axes at a distance 5 from the origin is y 2 + x 2 2ax 2ay + a 2 =0. What is the value of a? 4 5 6 7 40. What is the equation of the parabola, whose vertex and focus are on the x-axis at distance a and b from the origin respectively? (b > a > 0) y 2 = 8 (b a) (x a) y 2 = 4(b + a)( x a) y 2 = 4(b a)( x +a) y 2 = 4(b a)(x a) 41. A circle is drawn with the two foci of an ellipse x 2 /a 2 + y 2 /b 2 = 1 at the end of the diameter. What is the equation of the circle? x 2 + y 2 = a 2 + b 2 x 2 + y 2 = a 2 b 2 x 2 + y 2 = 2(a 2 + b 2 ) x 2 + y 2 = 2(a 2 b 2 ) 42. The difference of two angles is 1 ; the circular measure of their sum is 1. What is the smaller angle in circular measure? 43. If sin 3 θ + cos 3 θ = 0, then what is the value of θ? -π/4 0 π/4 π/3 44. Let 0 < θ < 45. Which one of the following is correct? sin 2 θ + cos 6 θ = sin 6 θ cos 2 θ cosec 2 θ + cot 6 θ = cosec 6 θ cot 2 θ sin 2 θ cos 4 θ = sin 4 θ cos 2 θ cosec 2 θ cot 4 θ = cosec 4 θ cot 2 θ 45. How many values of θ between 0 and 360 satisfy tan θ = k 0, where k is a given number? 1 2 4 Many 46. If cos θ < sin θ and θ lies in the first quadrant, then which one of the following is correct? 0 < θ < π/4 π/4 < θ < π/2 0 < θ < π/3 π/3 < θ < π/2 47. What is the value of 3 cosec 20 sec 20? 4 3 2 1 48. If tan A = 3/4 and tan B = 12/5, then how many values can cot (A B) have depending on the actual values of A and B? 1 2 3 4 49. If sinθ = cos 2 θ, then what is cos 2 θ (1 + cos 2 θ) equal to? 1 0 cos 2 θ 2 sin θ 50.
322 π/6 π/3 π/4 π/8 51. If cosecθ + cotθ = c, then what is cosθ equal to? 52. If the perimeter of a triangle ABC is 30 cm, then what is the value of a cos 2 (C/2) + c cos 2 (A/2)? 15 cm 10 cm 15/2 cm 13 cm 53. Let 1 x 1. If cos (sin 1 x) = 1/2, then how many value does tan (cos 1 x) assume? One Two Four Infinite 54. If angles A, B and C are in AP, then what is sin A + 2 sin B + sin C equal to? elevation of the top of the flag at the foot of the ladder is 60 )? 20 feet 30 feet 40 feet 20 2 feet 57. Two poles are 10 m and 20 m high. The line joining their tips makes an angle of 15 with the horizontal. What is the distance between the poles? 10( 3 1) m 5(4 + 2 3) m 20( 3 + 1) m 10 ( 3 + 1) m 58. 0 5/4 5/16 25/4 59. 0 1 Both 0 and 1 None of these 60. 55. If the sides of a triangle are in the ratio 2: 6 : 1 + 3, then what is the smallest angle of the triangle? 75 60 45 30 56. What should be the height of a flag where a 20 feet long ladder reaches 20 feet below the flag (The angle of 0 1 1 61. e e 2 e 4 e 5 62. At how many points is the function f(x) = [x] discontinuous? 1 2 3
323 Infinite 63. Let f: R R be a function whose inverse is x + 5/3. What is f (x) equal to? f (x) = 3x + 5 f (x) = 3x 5 f (x) = 5x 3 f (x) does not exist 64. The probability that a student passes in mathematics is 4/9 and that he passes in physics is 2/5. Assuming that passing in mathematics and physics are independent of each other, what is the probability that he passes in mathematics but fails in physics? 4/15 8/45 26/45 19/45 65. What is the probability that in a family of 4 children there will be at least one boy? 15/16 3/8 1/16 7/8 66. Given that P(A) = 1/3, P(B) = 1/4, P(A/B) = 1/6, then what is P(B/ A) equal to? 1/4 1/8 3/4 1/2 67. Consider the following statements: 1. The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year. 2. The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor 2 68. Two letters are drawn at random from the word HOME. What is the probability that both the letters are vowels? 1/6 5/6 1/ 2 1/3 69. An urn contains one black ball and one green ball. A second urn contains one white and one green ball. One ball is drawn at random from each urn. Q. What is the probability of getting at least one green ball? 1/2 1/3 2/3 ¾ 70. A line makes 45 with positive x-axis and makes equal angles with positive y, z axes, respectively. What is the sum of the three angles which the line makes with positive x, y and z axes? 180 165 150 135 71. 72. The line passing through (1, 2, 3) and having direction ratios given by < 1, 2, 3 > cuts the x-axis at a distance k from origin. What is the value of k? 0 1 2 3 73. What is the equation of the sphere which has its centre at (6, 1, 2) and touches the plane 2x y + 2z 2 = 0? x 2 + y 2 + z 2 + 12x 2y + 4z + 16 = 0 x 2 + y 2 + z 2 + 12x 2y + 4z 16 = 0 x 2 + y 2 + z 2 12x + 2y 4z + 16 = 0 x 2 + y 2 + z 2 12x + 2y 4z + 25 = 0 74. What is the angle between the lines whose direction cosines are proportional to (2, 3, 4) and (1, 2, 1) respectively?
324 90 60 45 30 75. What is the equation to the plane through (1, 2, 3) parallel to 3x + 4y 5z = 0? 3x + 4y + 5z + 4 = 0 3x + 4y 5z + 14 = 0 3x + 4y 5z + 4 = 0 3x + 4y 5z 4 = 0 76. 4 3 2 1 80. (28, 13) (23, 18) (31, 10) (25, 16) 77. If AM of numbers x1, x2... xn is μ, then what is the AM of the numbers which are increased by 1, 2, 3,...n respectively? 25 26 27 28 81. 78. For a set of discrete numbers, three measures of central tendency are given below 1. Arithmetic mean 2. Median 3. Geometric mean Which of the above measures may not have a meaningful definition? 1 only 2 only 3 only All of them are meaningfully defined 79. 90 100 110 120 82. What is the mean of first n odd natural numbers?
325 83. Variance is always independent of the change of origin but not scale scale only both origin and scale None of the above 84. 87. 88. The motion of a particle is described as s = 2 3t + 4t 3. What is the acceleration of the particle at the point where its velocity is zero? 0 4 unit 8 unit 12 unit 89. A stone thrown vertically upward satisfies the equation s = 64t 16t 2, where s is in meter and t is in second. What is the time required to reach the maximum height? 1s 2s 3s 4s 90. 85. 86. 0 1 1 2 91. Which one of the following statement is correct? e x is an increasing function e x is a decreasing function e x is neither increasing nor decreasing function e x is a constant function 92. cosx log tanx + log tan (x/2) + c cosx log tanx + log tan (x/2) + c cosx log tanx + log cot (x/2) + c cosx log tanx + log cot (x/2) + c 93.
326 99. What is the solution of the differential equation xdy y dx = xy 2 dx? y + x 2 = c y 2 + 2x 1 = c y + x 1 = c x 2 + 2xy 1 = c 100. 94. 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 A A is true but R is false A is false but R is true 95. What is the area bounded by the curve y = 4x x 2 3 and the x-axis? 2/3 sq unit 4/3 sq unit 5/3 sq unit 4/5 sq unit 96. What is the area enclosed by the equation x 2 + y 2 = 2? 4π square units 2π square units 4π 2 square units 4 square units 97. What is the area of the region enclosed by y = 2 x and y = 4? 2 square unit 4 square unit 8 square unit 16 square unit 98. What is the solution of the differential equation (x + y) (dx dy) = dx + dy? 2 1 2 1 101. 5 4 3 2 102. Let A and B be two matrices such that AB is defined. If AB = 0, then which one of the following can be definitely concluded? A = 0 or B = 0 A = 0 and B = 0 A and B are non-zero square matrices A and B cannot both be non-singular 103. Let A be an m n matrix.under which one of the following conditions does A 1 exist? m = n only m = n and det A 0 m = n and det A= 0 m n 104. If the matrix B is the adjoint of the square matrix A and a is the value of the determinant of A, then what is AB equal to?
327 105. 5 0 5 10 110. 106. A matrix X has (a + b) rows and (a + 2) columns; and a matrix Y has (b + 1) rows and (a + 3) columns. If both XY and YX exist, then what are the values of a, b respectively? 3, 2 2, 3 2, 4 4, 3 107. If A is a square matrix, then what is adj A T (adj A) T equal to? 2 A 2 A I Null Matrix Unit Matrix 111. m takes 1 value, n takes 1 value m takes 1 value, n takes 2 values m takes 2 value, n takes 1 value m takes 2 value, n takes 2 values 112. 108. 1 1 9/8 9/8 109. 113.
328 114. What is the area of the triangle with vertices (0, 2, 2), (2, 0, 1) and (3, 4, 0)? 15/2 sq unit 15 sq unit 7/2 sq unit 7 sq unit 115. r > s > q > p s > r > p > q r > s > p > q s > r > q > p 116. 117. 119. an integer a rational number but not an integer an irrational number none of the above 120. 1. d 2. b 3. b 4. d 5. d 6. d 7. d 8. a 9. b 10. d 11. b 12. d 13. a 14. c 15. b 16. d 17. b 18. c 19. c 20. a 21. b 22. c 23. d 24. b 25. a 26. b 27. c 28. b 29. a 30. b 31. c 32. c 33. d 34. c 35. c 36. b 37. d 38. d 39. b 40. d 41. b 42. c 43. a 44. d 45. b 46. b 47. a 48. d 49. a 50. b 51. c 52. a 53. b 54. b 55. c 56. b 57. b 58. c 59. b 60. c 61. d 62. d 63. b 64. a 65. a 66. b 67. a 68. c 69. d 70. b 71. c 72. a 73. c 74. a 75. c 76. c 77. a 78. d 79. a 80. b 81. a 82. a 83. a 84. b 85. b 86. a 87. a 88. c 89. b 90. d 91. a 92. b 93. a 94. b 95. b 96. b 97. c 98. b 99. d 100. c 101. c 102. c 103. b 104. d 105. d 106. b 107. c 108. d 109. b 110. d 111. d 112. c 113. c 114. a 115. c 116. c 117. b 118. b 119. b 120. b 1. 118.
329 2. 5. 6. 3. 4. 7. 8. 9.
330 13. 10. 14. 15. 11. 16. 17. 12.
331 22. 23. 18. 19. 24. 20. 25. 21. 26.
332 27. 32. 28. 33. 29. 34. 35. 30. 31.
333 39. 36. 37. 40. 38.
334 44. 41. 45. 42. 46. 47. 43.
335 52. 48. 49. 53. 50. 51. 54.
336 55. 57. 56. 58.
337 59. 62. 60. 63. 61. 64.
338 70. 65. 66. 71. 67. 72. 68. 69.
339 73. 74. 78. 75. 79. 76. 80. 81. 77.
340 82. 88. 83. 84. 89. 85. 86. 87.
341 90. 93. 94. 91. 92. 95.
342 96. 97. 99. 98. 100.
343 106. 101. 107. 108. 102. 103. 104. 105. 109.
344 110. 112. 113. 114. 111.
345 118. 119. 115. 120. 116. 117.