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1 373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an infinite set? The set of human beings on the earth The set of water drops in a glass of water The set of trees in a forest The set of all primes 3. an integer a rational number but not an integer an irrational number none of the above a negative number a transcendental number an irrational number a rational number only 2 only Both 1 and 2 Neither 1 nor 2 5. If X and Y are any two non-empty sets, then what is (X - Y ' ) equal to? 6. Natural numbers are divided into groups as (1), (2, 3), (4, 5, 6), (7, 8, 9, 10) and so on. What is the sum of the numbers in the 11th group? What is the equivalent binary number of the decimal number ? /2 12. a/(bc) b/ac -b/(ac) -a/(bc) 13.

2 real imaginary positive negative /49 49/47-47/49-49/ /8 1/4 20. What is the geometric mean of the ratio of corresponding terms of two series where G1 and G2 are geometric means of the two series? log G1 log G2 log G1 + log G2 G1/G2 G1G2 21. If the sum of n terms of an arithmetic progression is n 2 2n, then what is the n th term? 3n n 2 2n 3 2n + 3 2n What is the sum of all natural numbers between 200 and 400 which are divisible by 7? If the 10th term of a GP is 9 and 4th term is 4, then what is its 7th term? /14 56/ The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively. Q. What is its first term? p = 25/12 p < 25/12 p > 25/12 p 25/ If x + 1, 4x + 1, and 8x + 1 are in geometric progression, then what is the non-trivial value of x?

3 only 2 only Both 1 and 2 Neither 1 or x/19 83x 19x 83x/ What is the sum of the coefficients in the expansion of (1 + x) n? 2 n 2 n 1 2 n + 1 n In how many ways can 3 books on Hindi and 3 books on English be arranged in a row on a shelf, so that not all the Hindi books are together? In how many ways can a committee consisting of 3 men and 2 women be formed from 7 men and 5 women? What is the image of the point (2, 3) in the line y = x? ( 3, 2) ( 3, 2) ( 2, 3) (3, 2) 34. If (a, b), (c, d) and (a c, b d) are collinear, then which one of the following is correct? bc ad = 0 ab cd = 0 bc + ad = 0 ab + cd = What is the maximum number of straight lines that can be drawn with any four points in a plane such that each line contains at least two of these points? If p is the length of the perpendicular drawn from the origin to the line x/a + y/b = 1, then which one of the following is correct? 37. From the point (4, 3) a perpendicular is dropped on the x- axis as well as on the y-axis. If the lengths of perpendiculars are p, q respectively, then which one of the following is correct? p = q 3p = 4q 4p = 3q p + q = What is the value of λ if the straight line (2x + 3y + 4) + λ (6x y + 12) = 0 is parallel to y-axis?

4 376 If x-axis is tangent to the circle x 2 + y 2 + 2gx + 2fy + k = 0, then which one of the following is correct? g 2 = k g 2 = f f 2 = k f 2 = g 40. In how many points do the ellipse x 2 /4 + y 2 /8 = 1 and the circle x 2 +y 2 = 9 intersect? One Two Four None of the above 41. Let E be the ellipse x 2 /9 + y 2 /4 = 1 and C be the circle x 2 + y 2 = 9. Let P = (1, 2) and Q = (2, 1). Which one of the following is correct? Q lies inside C but outside E Q lies outside both C and E P lies inside both C and E P lies inside C but outside E. 42. If an angle B is complement of an angle A, what are the greatest and least values of cos A cos B respectively? 0, - 1/2 1/2, , 0 1/2, - 1/2 43. What is the value of sin(a + B) sin (A B) + sin (B + C) sin (B C) + sin (C + A) sin (C A)? 0 sina + sin B + sin C cosa + cos B + cos C If α = π/8, what is the value of cos α cos 2 α cos 4 α? 0 1/4 8 4 What is the value of cos10 + cos110 + cos130? None of these tan x 2 cosec x 2 cos x 2 sin x 50. What is sin 2 A sin 2 B equal to? 0 1/ sin A cos A tan A The angles A, B, C of a triangle are in the ratio 2: 5: 5. What is the value of tan B tan C? 4 + 3

5 Which one of the following is not correct? sin 1 { sin (5π/4)} = π /4 sec 1 { sec (5π/4)} = 3π /4 tan 1 { tan (5π/4)} = π /4 cosec 1 { cosec (7π/4)} = π /4 54. π/4 π/3 π/2 π/6 55. The sides a, b, c of a triangle ABC are in arithmetic progression and a is the smallest side. What is cos A equal to? 56. The foot of a tower of height h m is in a direct line between two observers A and B. If the angles of elevation of the top of the tower as seen from A and B are α and β respectively and if AB = d m, then what is h/d equal to? h tanθ h cosθ h sinθ 58. Consider the following statements: 1. The function f (x) = greatest integer x, x R is a continuous function. 2. All trigonometric functions are continuous on R. Which of the statements given above is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor 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): The function f : (1, 2, 3) (a, b, c, d) defined by f = {(1, a), (2, b), (3, c)} has no inverse. Reason (R): f is not one-one. 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 The limit does not exist From the top of a building of height h meter, the angle of depression of an object on the ground is θ. What is the distance (in meter) of the object from the foot of the building? h cotθ 1/ Let A = {x R x 0}. A function f: A A is defined by f (x) = x 2. Which one of the following is correct?

6 378 The function does not have inverse f is its own inverse The function has an inverse but f is not its own inverse None of the above What is the probability of having a knave and a queen when two cards are drawn from a pack of 52 cards? 16/663 2/663 4/663 8/ /6 1/3 2/3 1/2 66. Two dice are thrown. What is the probability that the sum of the faces equals or exceeds 10? 1/12 1/4 1/3 1/6 67. Two balls are selected from a box containing 2 blue and 7 red balls. What is the probability that at least one ball is blue? 2/9 7/9 5/12 7/ There is a point inside a circle. What is the probability that this point is close to the circumference than to the centre? 3/4 1/2 1/4 1/3 69. Two dice each numbered from 1 to 6 are thrown together. Let A and B be two events given by A: even number on the first die. B: number on the second die is greater than 4. Q. What is P(A B) equal to? 1/2 1/4 2/3 1/6 1 only 2 only Both 1 and 2 Neither 1 nor only 2 only 3 only 2 and Which one of the following planes is normal to the plane 3x + y + z = 5? x + 2y + z = 6 x 2y + z = 6 x + 2y z = 6 x 2y z = Under what condition do the planes bx ay = n, cy bz = l, az cx = m intersect in a line? a + b + c = 0 a = b = c al + bm + cn = 0 l + m + n = What is the angle between two planes 2x y + z = 4 and x + y + 2z = 6? π/2 π/3 π/4 π/6 75.

7 379 What is the equation to the straight line passing through (a, b, c) and parallel to z-axis? 76. If from the point of intersection of two ogives, a perpendicular is drawn on the x-axis, what does the xcoordinate give? Arithmetic Mean Mode Median Geometric Mean If the values of a set are measured in cm, what will be the unit of variance? cm cm 2 cm 3 No unit The arithmetic mean of 4 numbers is 15. The arithmetic mean of another 6 numbers is 12. What is the arithmetic mean of the combined 10 numbers?

8 380 The algebraic sum of the deviations of 20 observations measured from 30 is 2. What would be the mean of the observations? The standard deviation of the observations 5, 5, 5, 5, 5 is What is the maximum slope of the curve y = x 3 + 3x 2 + 2x 27? If x cosθ + y sinθ = 2 is perpendicular to the line x y = 3, then what is one of the value of θ? p/6 p/4 p/2 p/ /3 2 1/2 91. The function f (x) = x 3 3x is an increasing function for: 0 < x < 2 x < 2 x > 2 or x < 0 all x ½ 1 87.

9 only 2 only Both 1 and 2 Neither 1 nor /110 1/111 1/112 1/ What is the area enclosed between the curves y 2 = 12x and the lines x = 0 and y = 6? 2 sq unit 4 sq unit 6 sq unit 8 sq unit What is the solution of the differential equation 3e x tan y dx + (1 + e x ) sec 2 y dy = 0? (1 + e x ) tan y = c (1 + e x ) 3 tan y = c (1 + e x ) 2 tan y = c (1 + e x ) sec 2 y = c 100. A set of circles having centre on the Y-axis A set of circles having centre on the X-axis A set of ellipses A pair of straight lines What does the solution of the differential equation xdy ydx = 0 represent? Rectangular hyperbola Straight line passing through (0, 0) Parabola with vertex at (0, 0) Circle with centre at (0, 0) [2006-I] 98. What is the differential equation of the curve y = ax 2 + bx? y = e x + c y = e x + c y = e x + c y = e x + c 102. If A is a square matrix such that A A T = 0, then which one of the following is correct? A must be a null matrix A must be a unit matrix A must be a scalar matrix None of the above 103. Let A be a square matrix of order n n where n 2. Let B be a matrix obtained from A with first and second rows interchanged. Then which one of the following is correct? det A = det B det A = det B A = B A = B 104.

10 382 S = {2, 3} xyz If a matrix A has inverses B and C, then which one of the following is correct? B may not be equal to C B should be equal to C B and C should be unit matrices None of the above 109. There exist more than one but finite number of B s such that AB = BA There exists exactly one B such that AB = BA There exist infinitely many B s such that AB = BA There cannot exist any B such that AB = BA a = 5 a = 5 [2012-II] a = 0 a = S = {0, 2, 3} S = { 1, 2, 3} S = {1, 2, 3}

11 and 2 only 2 and 3 only 1 and 3 only 1, 2 and Only 1 2 and 3 1 and 2 1 and 3 1 only 2 only Both 1 and 2 Neither 1 nor A = B only B = C only A = C only A = B = C 1. b 31. c 61. a 91. c 2. d 32. b 62. a 92. b 3. b 33. a 63. c 93. d

12 d 34. a 64. c 94. a 5. c 35. c 65. b 95. c 6. c 36. a 66. d 96. d 7. c 37. c 67. a 97. b 8. a 38. a 68. b 98. a 9. a 39. a 69. d 99. b 10. d 40. d 70. c 100. b 11. c 41. d 71. b 101. b 12. b 42. d 72. d 102. d 13. d 43. a 73. c 103. b 14. d 44. a 74. b 104. c 15. a 45. c 75. b 105. c 16. c 46. b 76. c 106. c 17. c 47. c 77. b 107. a 18. b 48. b 78. c 108. b 19. c 49. b 79. a 109. a 20. c 50. b 80. d 110. c 21. b 51. b 81. a 111. b 22. c 52. c 82. d 112. d 23. a 53. d 83. a 113. a 24. b 54. b 84. a 114. a 25. b 55. c 85. d 115. c 26. a 56. b 86. d 116. c 27. d 57. a 87. b 117. d 28. b 58. d 88. c 118. b 29. d 59. b 89. b 119. c 30. a 60. c 90. c 120. b

(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