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1 212 NDA Mathematics Practice Set 1. Let X be any non-empty set containing n elements. Then what is the number of relations on X? 2 n 2 2n 2 2n n 2 2. Only 1 2 and 3 1 and 2 1 and 3 3. Consider the following statements: 1. Parallelism of lines is an equivalence relation. 2. x R y, if x is a father of y, is an equivalence relation. Which of the statements given above is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor only 2 only 1, 2 5. a natural number an irrational number a rational number a whole number 6. 9/ Let M be the set of men and R is a relation is son of defined on M. Then, R is an equivalence relation a symmetric relation only a transitive relation only None of the above 10. The set A = {x : x + 4 = 4} can also be represented by: 11. Which one of the following is a null set? R is reflexive and symmetric, but not transitive R is reflexive and transitive, but not symmetric R is reflexive, but neither symmetric nor transitive R is reflexive, symmetric and transitive

2 /2 3/4 3/2 2/ AP HP GP None of these 22. If x 2, y 2, z 2 are in AP, then y + z, z + x, x + y are in AP HP GP None of these 23. In a GP of positive terms, any term is equal to one-third of the sum of next two terms. What is the common ratio of the GP? a = c b = c a = -c b = : 1 1 : 2 1 : 3 3 : It the sum of first 10 terms of an arithmetic progression with first term p and common difference q, is 4 times the sum of the first 5 terms, then what is the ratio p: q? 1: 2 1: 4 2: 1 4: The geometric mean and harmonic mean of two non negative observations are 10 and 8 respectively. Then what is the arithmetic mean of the observations equal to? , i i i/2 - i/2 26.

3 214 None of these 1 only 2 only Both 1 and 2 Neither 1 or x iy x + iy 2x 2iy C(12, 7) x 3 y 3 C(12, 6) x 3 y 3 C(12, 7) x 3 y 3 C(12, 6) x 3 y Always an odd integer A perfect square None of the above 32. How many times does the digit 3 appear while writing the integers from 1 to 1000? What is the number of ways that 4 boys and 3 girls can be seated so that boys and girls alternate? The coordinates of P and Q are ( 3, 4) and (2, 1), respectively. If PQ is extended to R such that PR = 2QR, then what are the coordinates of R? (3, 7) (2, 4) (- 1/2, 5/2) (7, 2) 35. What is the locus of a point which is equidistant from the point (m + n, n m) and the point (m n, n + m)? mx = ny nx = my nx = my mx = ny 36. What is the locus of the point which is at a distance 8 units to the left of Y-axis? X = 8 Y = 8 X = 8 Y = The line making an angle ( 120 ) with x-axis is situated in the: first quadrant second quadrant third quadrant fourth quadrant 38. Two straight line paths are represented by the equation 2x y = 2 and 4x + 2y = 6. Then the paths will cross each other at one point not cross each other cross each other at two points cross each other at infinitely many points 39. What is the equation of circle which touches the lines x = 0, y = 0 and x = 2? x 2 + y 2 + 2x + 2y + 1 = 0 x 2 + y 2 4x 4y + 1 = 0 x 2 +y 2 2x 2y + 1 = 0 None of these

4 P (2, 2) is a point on the parabola y 2 = 2x and A is its vertex. Q is another point on the parabola such that PQ is perpendicular to AP. What is the length of PQ? The curve y 2 = 4ax (a > 0) lies in First and fourth quadrants First and second quadrants Second and third quadrants Third and fourth quadrants 42. The sum of the focal distances of a point on the ellipse x 2 /4 + y 2 /9 = 1 is: 4 units 6 units 8 units 10 units 43. π/2 π/3 π/4 π/6 44. If α and β are such that tan α = 2 tan β, then what is sin (α + β) equal to? 1 2 sin (α β) sin (α β) 3 sin (α β) 45. Which one of the following pairs is not correctly matched? sin 2π : sin ( 2π) tan 45 : tan ( 315 ) cot (tan 1 0.5) : tan (cos 1 0.5) tan 420 : tan ( 60 ) 46. One radian is approximately equal to which one of the following? Consider the following statements I. If θ = 1200, then (sec θ + tan θ) 1 is positive. II. If θ = 1200, then (cosec θ cot θ) is negative. Which of the statements given above is/are correct? I only II only Both I and II Neither I nor II 48. If α and β are positive angles such that α + β = π/4, then what is (1 + tan α) (1 + tan β) equal to? If sin A + sin B + sin C = 3 then what is cos A + cos B + cos C equal to? What is the maximum value of sin3θcos2θ + cos3θsin2θ? What is tan ( 585 ) equal to? What is the value of x that satisfies the equation cos 1 x = 2 sin 1 x? 1/ /2 53. π π/2 π/4 π/3 54. If sin 1 x cot 1 (1/2) = π/2 then what is the value of x? 0 1/ 5

5 216 2/ 5 3/ /5 1/5 56. A vertical pole with height more than 100 m consists of two parts, the lower being one-third of the whole. At a point on a horizontal plane through the foot and 40 m from it, the upper part subtends an angle whose tangent is 1/2. What is the height of the pole? 110 m 200 m 120 m 150 m 57. A tower of height 15 m stands vertically on the ground. From a point on the ground the angle of elevation of the top of the tower is found to be 30. What is the distance of the point from the foot of the tower? 15 3 m 10 3 m 5 3 m 30 m 58. f (x) 3[f (x)] [f (x)] 3 3[f (x)] 59. Let f:r R be defined as f(x) = ax 2 + bx + c, a,b, c being fixed non-zero real numbers. Which one of the following statements is correct, in general? If b 2 4ac > 0, then f 1 (0) does not contain 0 If b 2 4ac < 0, then f 1 (0) must contain 0 If b 2 4ac > 0, then f 1 (0) may contain 0 If b 2 4ac < 0, then f 1 (0) may contain e e 3 e 9 e /2 1/ Five coins whose faces are marked 2, 3 are thrown. What is the probability of obtainining a total of 12? 1/16 3/16 5/16 7/ Each of A and B tosses two coins. What is the probability that they get equal number of heads? 3/16 5/16 4/16 6/ Consider the following statements related to the nature of Bayes theorem 1. Bayes theorem is a formula for computation of a conditional probability.

217 2. Bayes theorem modifies an assumed probability of an event in the light of a related event which is observed. Which of the statements given above is/are correct?

6 Bayes theorem modifies an assumed probability of an event in the light of a related event which is observed. Which of the statements given above is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor Given that P(A) = 1/3, P(B) = 3/4 and P(A B) = 11/12 then what is P(B/A)? 1/6 4/9 1/2 1/ Three coins are tossed simultaneously. What is the probability that they will fall two heads and one tail? 1/3 1/2 1/4 3/8 70. Two numbers are successively drawn from the set U = {1, 2, 3, 4, 5, 6, 7, 8}, the second being drawn without replacing the first. The number of elementary events in the sample is: Which one of following is correct? The three planes 2x + 3y z 2 = 0, 3x + 3y + z 4 = 0, x y + 2z 5 = 0 intersect at a point at two points at three points in a line 72. Which one of the following planes contains the z-axis? x z = 0 z + y = 0 3x + 2y = 0 3x + 2z = What is the value of n so that the angle between the lines having direction ratios (1, 1, 1) and (1, 1, n) is 60? None of these 74. What is the acute angle between the planes x + y + 2z = 3 and 2x + y z = 11? π/ 5 π/ 4 π/ 6 π/3 75. If a line OP of length r (where O is the origin) makes an angle α with x-axis and lies in the xz-plane, then what are the coordinates of P? (r cos α, 0, r sin α) (0, 0, r sin α) (r cos α, 0, 0) (0, 0, r cos α) 76. In a manufacture of ready-made garments, which average is used to find the most frequent size? Arithmetic mean Geometric mean Mode Harmonic mean 77. If you want to measure the intelligence of a group of students,which one of the following measures will be more suitable? Arithmetic mean Mode Median Geometric mean 78. What is the value of n for which the numbers 1, 2, 3,..., n have variance 2? Product A Product B Product C

218 Product D 80. What is the missing frequency y? 20 16 15 12 81. 85. 810 830 970 1030 82. Consider the following statements: 1. A continuous random variable can take all values in an interval. 2. A random variable which takes a finite number of values is necessarily discrete.

7 218 Product D 80. What is the missing frequency y? Consider the following statements: 1. A continuous random variable can take all values in an interval. 2. A random variable which takes a finite number of values is necessarily discrete. 3. Construction of a frequency distribution is based on data which are discrete. Which of the above statements are correct? 1 and 2 only 2 and 3 only 1 and 3 only 1, 2 and Consider the following statements: 1. Both variance and standard deviation are measures of variability in the population. 2. Standard deviation is the square of the variance. Which of the above statements is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor The derivative of x at x = 0 is 1 is 1 is 0 does not exist Assertion (A): The tangent to the curve y = x 3 x 2 x + 2 at (1, 1) is parallel to the x-axis. Reason (R): The slope of the tangent to the curve at (1, 1) is zero. Both A and R are true and R is the correct explanation of A

8 219 Both A and R are true but R is not the correct explanation of A A is true but R is false A is false but R is true 90. What is the least vlaue of f(x) = 2x 3 3x 2 12x + 1 on [ 2, 2.5]? Which one of the following statement is correct? The derivative of a function f(x) at a point will exist if there is one tangent to the curve y = f(x) at that point and the tangent is parallel to y-axis The derivative of a function f(x) at a point will exist if there is one tangent to the curve y = f(x) at that point and the tangent must be parallel to x-axis The derivative of a function f(x) at a point will exist if there is one and only one tangent to the curve y= f[x] at that point and the tangent is not parallel to y-axis None of the above 92. 1/2 1 3/ π/ What is the area under the curve y = x + x 1 between x = 0 and x = 1? π/2 π/4 π/8 None of the above 98. What is the solution of the differential equation (x + y) (dx dy) = dx + dy? x + y + ln (x + y) = c x y + ln (x + y) = c y x + ln (x + y) = c y x ln (x y) = c 99. Rate of growth of bacteria is proportional to the number of bacteria present at that time. If x is the number of bacteria present at any instant t, then which one of the following is correct? (Take proportional constant equal to 1) x = log t x = ce t e x = t x = t 100.

9 /p+1/q A family of circles having centre on the y-axis. A family of circles having centre on the x-axis. A family of circles touching the x-axis A family of ellipses A set of straight lines A set of ellipses A set of circles None of the above If A is a 3 3 matrix such that A =4, then what is A(adj A) equal to? A is symmetric matrix A is anti- symmetric matrix A is singular matrix A is non-singular matrix 107. If A = 8, where A is square matrix of order 3, then what is adj A equal to?

10 None of the above Consider the following statements: 1. A matrix is not a number 2. Two determinants of different order may have the same value. Which of the above statements is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor /5 1/ 5 1/29 1/ /3 1 8/ in the plane of XY in the plane of YZ in the plane of XZ along the X-axis 118.

11 only 2 only 1 and 2 1, 2, 3 and The set of isosceles triangles; the set of non- isosceles right angled triangles The set of isosceles triangles; the set of right angled triangles The set of equilateral triangles; the set of right angled triangles The set of isosceles triangles; the set of equilateral triangles R is reflexive only R is symmetric only R is transitive only R is reflexive and transitive 1. a 2. b 3. a 4. d 5. b 6. c 7. b 8. c 9. d 10. d 11. d 12. d 13. d 14. a 15. c 16. d 17. a 18. a 19. a 20. d 21. b 22. a 23. b 24. c 25. d 26. d 27. b 28. c 29. d 30. b 31. c 32. c 33. d 34. d 35. c 36. c 37. c 38. b 39. c 40. d 41. c 42. a 43. a 44. d 45. d 46. c 47. b 48. c 49. b 50. a 51. b 52. a 53. c 54. b 55. d 56. c 57. a 58. b 59. a 60. a 61. c 62. c 63. d 64. c 65. b 66. c 67. c 68. a 69. d 70. b 71. d 72. c 73. b 74. d 75. a 76. c 77. b 78. b 79. d 80. a 81. a 82. b 83. d 84. a 85. b 86. c 87. d 88. a 89. a 90. c 91. c 92. b 93. a 94. b 95. a 96. b 97. c 98. c 99. b 100. b 101. c 102. c 103. b 104. a 105. c 106. d 107. c 108. a 109. b 110. c 111. d 112. d 113. b 114. d 115. a 116. d 117. b 118. d 119. a 120. d

(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz

(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz 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