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 4. 1111.101 1101.101 1111.111 8. 2 4 6 8 9. a negative number a transcendental number an irrational number a rational number 10. 5 7 9 11 11. 1 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? 605 615 671 693 7. What is the equivalent binary number of the decimal number 13.625? 1101.111 0-1 1 3/2 12. a/(bc) b/ac -b/(ac) -a/(bc) 13.
374 14. 15. real imaginary positive negative 16. 47/49 49/47-47/49-49/47 17. 18. 1/8 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 5 22. What is the sum of all natural numbers between 200 and 400 which are divisible by 7? 6729 8712 8729 9276 23. If the 10th term of a GP is 9 and 4th term is 4, then what is its 7th term? 6 14 27/14 56/15 24. The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively. Q. What is its first term? 2 3 4 5 25. p = 25/12 p < 25/12 p > 25/12 p 25/12 19. If x + 1, 4x + 1, and 8x + 1 are in geometric progression, then what is the non-trivial value of x? 1 1 1-1 2 0 26. 4-4 8
-8 27. 1 2 2 24 2 48 28. 375 1 only 2 only Both 1 and 2 Neither 1 or 2 29. 20x/19 83x 19x 83x/19 30. What is the sum of the coefficients in the expansion of (1 + x) n? 2 n 2 n 1 2 n + 1 n + 1 31. 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? 144 360 576 720 32. In how many ways can a committee consisting of 3 men and 2 women be formed from 7 men and 5 women? 45 350 700 4200 33. 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 = 0 35. 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? 2 4 6 12 36. 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 = 5 38. What is the value of λ if the straight line (2x + 3y + 4) + λ (6x y + 12) = 0 is parallel to y-axis? 3 6 4 3 39.
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, -1 2-1, 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 1 44. 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? 1 0 1 2 47. 48. 0 1 1 None of these 49. 2 tan x 2 cosec x 2 cos x 2 sin x 50. What is sin 2 A sin 2 B equal to? 0 1/2 1 2 51. 45. 46. sin A cos A tan A 0 52. 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
377 4 + 2 3 7 + 4 3 3 +3 3 53. 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 2 59. 1 0 1 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): 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 61. 0 1 1 The limit does not exist 62. 57. 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 1/2 1 2 0 63. 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?
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 70. 64. 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 8/663 65. 1/6 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 7/12 68. 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 2 71. 1 only 2 only 3 only 2 and 3 72. 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 = 6 73. 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 = 0 74. 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.
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 77. 120 108 100 90 80. If the values of a set are measured in cm, what will be the unit of variance? cm cm 2 cm 3 No unit 81. 78. 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? 12.2 12.8 13.2 13.8 79. 1995 1996 1997 1998 82.
380 The algebraic sum of the deviations of 20 observations measured from 30 is 2. What would be the mean of the observations? 30 32 30.2 30.1 83. The standard deviation of the observations 5, 5, 5, 5, 5 is 0 5 20 25 84. 88. What is the maximum slope of the curve y = x 3 + 3x 2 + 2x 27? 1 2 5 23 89. 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 90. 85. 2 3 4 4 3 1/3 2 1/2 91. The function f (x) = x 3 3x 2 + 6 is an increasing function for: 0 < x < 2 x < 2 x > 2 or x < 0 all x 92. 86. 0 1 ½ 1 87.
381 93. 1 only 2 only Both 1 and 2 Neither 1 nor 2 94. 1/110 1/111 1/112 1/119 95. 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 96. 99. 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 101. 97. 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.
382 S = {2, 3} 0 1 2 2 2xyz 105. 108. 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 = 1 110. 106. 111. 107. 112. S = {0, 2, 3} S = { 1, 2, 3} S = {1, 2, 3}
383 116. 1 and 2 only 2 and 3 only 1 and 3 only 1, 2 and 3 113. 2 2 4 4 117. 118. 114. Only 1 2 and 3 1 and 2 1 and 3 1 only 2 only Both 1 and 2 Neither 1 nor 2 115. 119. 100 200 300 400 120. 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
384 4. 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 6. 7. 8. 9. 1. 2. 3. 4. 5. 10. 11.
385 12. 16. 13. 14. 17. 15.
386 18. 21. 22. 19. 20. 23.
387 24. 27. 25. 28. 26. 29.
388 35. 36. 37. 30. 31. 32. 33. 38. 34. 39. 40.
389 41. 44. 42. 45. 43. 46.
390 51. 52. 47. 48. 53. 49. 54. 50.
391 55. 57. 56. 58.
392 59. 63. 60. 64. 61. 65. 62.
393 69. 70. 71. 72. 66. 67. 73. 74. 68. 75.
394 76. 77. 82. 78. 79. 83. 80. 84. 81.
395 87. 85. 88. 86. 89.
396 90. 92. 93. 91. 94.
397 95. 98. 96. 97. 99.
398 103. 104. 100. 105. 101. 106. 102.
399 107. 110. 108. 109. 111.
400 115. 112. 116. 113. 114. 117. 118.
401 119. 120.