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NDA Mathematics Practice Set 1.
(a) {1, 3} (b) {1, 2, 3}
(c) {0, 1, 3}
(d) {0, 1, 2, 3} 2.
(a) 2 (b) 4 (c) 6 (d) 8 3.
(a)
(b)
(c)
(d) 4.
(a)
(b)
(c)
(d) 5.
(a) (b)
(c)
(d) 6.
(a) (0.1101)2 (b) (0.1001)2 (c) (0.1111)2 (d) (0.1011)2 7.
(a) 1 (b) 2 (c) 4 (d) 8 8. The order of a set A is 3 and that of a set B is 2. What is the number of relations from A to B? (a) 4 (b) 6 (c) 32 (d) 64 9.
(a) I and II only
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(b) I and III only (c) II and III only (d) I, II and III 10. If A = {a, b, c}, then what is the number of proper subsets of A? (a) 5 (b) 6 (c) 7 (d) 8 11.
(a)
(b)
(c)
(d) 12.
(a) 4 (b) 3 (c) 2 (d) 1 13.
(a) –2 (b) 0 (c) 30 (d) 34 14.
(a) x > 0 (b) y > 0 (c) y < 0 (d) x <0
15.
(a) 2 (b) 3 (c) 4 (d) 9 16.
(a) -2 only (b) 1 onlay (c) -2 and 1 (d) -2 and -1 17.
(a)
(b)
(c)
(d) 18.
(a)
(b)
(c)
(d) 19. If 4x – 6.2x + 8 = 0, then the values of x are (a) 1, 2 (b) 1, 1 (c) 1, 0 (d) 2, 2 20. The equation (a2 + b2) x2 – 2b (a + c) x + (b2 + c2) = 0 has equal roots. Which one of the following is correct about a, b, and c? (a) They are in AP (b) They are in GP (c) They are in HP (d) They are neither in AP, nor in GP, nor in HP 21. If the points with the coordinates (a, ma), {b, (m + 1) b}, {c, (m + 2) c} are collinear, then which one of the following is correct?
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(a) a, b, c are in arithmetic progression for all m (b) a, b, c are in geometric progression for all m (c) a, b, c are in harmonic progression for all m (d) a, b, c are in arithmetic progression only for m = 1 22. If a, 2a + 2, 3a + 3 are in GP, then what is the fourth term of the GP? (a) – 13.5 (b) 13.5 (c) –27 (d) 27 23.
(a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 24. If log10 2, log10 (2x – 1), log10 (2x + 3) are three consecutive terms of an AP, then which one of the following is correct? (a) x = 0 (b) x = 1 (c) x = log2 5 (d) x = log5 2 25. The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively. What is the common difference? (a) 1 (b) 2 (c) 3 (d) 4 26.
(a) 3ω (b) 3ω2 (c) -3ω (d) 3ω3 27.
(a)
(b)
(c)
(d) 28.
(a) 0 (b) 1 (c) -1 (d) None of these 29.
(a) i (b) -i (c) 0 (d) i – 1 30.
(a) 0 (b) 1 (c) 2 (d) 28 31. How many 3-digit numbers, each less than 600, can be formed from {1, 2, 3, 4, 7, 9} if repetition of digits is allowed? (a) 216 (b) 180 (c) 144 (d) 120 32. How many words, with or without meaning can be formed by using all the letters of the word ‘MACHINE’, so that the vowels occurs only the odd positions? (a) 1440 (b) 720 (c) 640
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(d) 576 33. What is the total number of combination of n different things taken 1, 2, 3, ...., n at a time? (a) 2n + 1 (b) 22n + 1 (c) 2n – 1 (d) 2n – 1 34. The middle point of A (1, 2) and B (x, y) is C (2, 4). If BD is perpendicular to AB such that CD = 3 unit, then what is the length BD? (a) 2 √2 unit (b) 2 unit (c) 3 unit (d) 3 √2 unit 35. The point of intersection of the two lines 2x + 3y + 4 = 0 and 4x + 3y + 2 = 0 is at a distance d from origin. What is the value of d? (a) √2 (b) √3 (c) √5 (d) √7 36. A square is drawn by joining mid points of the sides of a square. Another square is drawn inside the second square in the same way and the process is continued in definitely. If the side of the first square is 16 cm, then what is the sum of the areas of all the squares? (a) 256 sq cm (b) 512 sq cm (c) 1024 sq cm (d) 512/ 3 sq cm 37. For what value of k, are the lines x + 2y – 9 = 0 and kx + 4y + 5 = 0 parallel? (a) 2 (b) – 1 (c) 1 (d) 0 38. The line y = 0 divides the line joining the points (3, –5) and (–4, 7) in the ratio: (a) 3: 4 (b) 4: 5 (c) 5: 7 (d) 7: 9 39. An equilateral triangle is inscribed in the circle x2 + y2 = a2 with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex?
(a) 2x – a = 0 (b) x + a = 0 (c) 2x + a = 0 (d) 3x – 2a = 0 40. The circle x2 + y2 + 4x –4y + 4 = 0 touches (a) Only the x-axis (b) Only the y-axis (c) Both the axes (d) Neither of the axes 41.
(a) 2 (b) 3 (c) 4 (d) 16 42. A point P moves such that the difference of its distances from two given points (c, 0) and (–c, 0) is constant. What is the locus of the point P? (a) Circle (b) Ellipse (c) Hyperbola (d) Parabola 43. Three expressions are given below: Q1 = sin (A + B) + sin (B + C) + sin (C + A) Q2 = cos (A – B) + cos (B – C) + cos (C – A) Q3 = sin A (cos B + cos C) + sin B (cos C + cos A) + sin C (cos A + cos B) Which one of the following is correct? (a) Q1 = Q2 (b) Q2 = Q3 (c) Q1 = Q3 (d) All the expressions are different 44. Given that tan α = m/(m + 1), tan β = 1/(2m+ 1), then what is the value of α + β? (a) 0 (b) π/4 (c) π/6 (d) π/3 45. What is the value of cot (– 870°)? (a) √3 (b) 1/√3 (c) - √3 (d) -1/√3 46.
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(a)
(b)
(c)
(d) 47. What is the length of arc of a circle of radius 5 cm subtending a central angle measuring 15°? (a) 5π/12cm (b) 7π/12cm (c) π/12cm (d) π/5cm 48.
(a) 1 (b) 1/√2 (c) 1/√3 (d) √3 49.
(a)
(b)
(c)
(d) 50. If sin 3A = 1, then how many distinct values can sin A assume? (a) 1 (b) 2 (c) 3 (d) 4 51. What is the value of sin420°.cos390° + cos(– 300°).sin(–330°)? (a) 0 (b) 1 (c) 2 (d) – 1 52. What is tan4 A – sec4 A + tan2 A + sec2 A equal to? (a) 0 (b) 1
(c) 2 (d) –1 53. If A, B and C are angles of a triangle such that tan A = 1, tan B = 2, then what is the value of tan C? (a) 0 (b) 1 (c) 2 (d) 3 54. If sin–1x + sin–1y = π/2 and cos–1 x – cos–1 y = 0, then values x and y are respectively
(a)
(b)
(c)
(d) 55.
(a) π/3 (b) π/2 (c) π/4 (d) π/6 56.
(a) 63/65 (b) 33/65 (c) 22/65 (d) 11/65 57. A man observes the elevation of a balloon to be 30°. He, then walks 1 km towards the balloon and finds that the elevation is 60°. What is the height of the balloon? (a) 1/ 2 km (b) √3/2 km (c) 1/ 3 km (d) 1 km 58. The top of a hill observed from the top and bottom of a building of height h is at angles of elevation a and b respectively. The height of the hill is:
(a)
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(b)
(c)
(d) 59.
(a)
(b)
(c)
(d) 60.
(a)
(b) (c) ab (d) log (ab) 61. 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): y = 2x + 3 is a one to one real valued function. Reason (R): x1 ≠ x2 y1 ≠ y2, y1 = 2x1 + 3, y2 = 2x2 + 3, for any two real x1 and x2 (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A (c) A is true but R is false (d) A is false but R is true 62. A is associated with (a) 1, 3, 5, 8, 9, 12 (b) 2, 4, 6, 8, 10, 11 (c) 1, 3, 5, 7, 10, 11 (d) None of these 63.
(a) 3 (b) 2 (c) 1 (d) – 1 64.
(a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2 65. Consider the following statement: "The mean of a binomial distribution is 3 and variance is 4." Which of the following is correct regarding this statement? (a) It is always true (b) It is sometimes true (c) It is never true (d) No conclusion can be drawn 66. The outcomes of 5 tosses of a coin are recorded in a single sequence as H (head) and T (tail) for each toss. What is the number of elementary events in the sample space? (a) 5 (b) 10 (c) 25 (d) 32 67. For a binomial distribution B(n, p), np = 4 and variance npq = 4/3. What is the probability P(x ≥ 5) equal to? (a) (2/3)6 (b) (1/3)6 (c) (1/3)6 (d) (28 /36) 68.
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The probability of guessing a correct answer is X/12. If the probability of not guessing the correct answer is 2/3, then what is x equal to? (a) 2 (b) 3 (c) 4 (d) 6 69. In a random arrangement of the letters of the word ‘UNIVERSITY’, what is the probability that two I’s do not come together? (a) 4/5 (b) 1/5 (c) 1/10 (d) 9/10 70. Consider a random experiment of throwing together a die and two coins. The associated sample space has (a) 8 points (b) 12 points (c) 24 points (d) 36 points 71. If the centre of the sphere ax2 + by2 + cz2 – 2x + 4y + 2z – 3 = 0 is (1/2, – 1, – 1/2), what is the value of b? (a) 1 (b) – 1 (c) 2 (d) – 2 72. What is the equation of the plane passing through (x1, y1, z1) and normal to the line with < a, b, c > as direction ratios? (a) ax + by + cz = ax1 + by1 + cz1 (b) a (x + x1) + b (y + y1) + c (z + z1) = 0 (c) ax + by + cz = 0 (d) ax + by + cz = x1 + y1 + z1 = 0 73. If the radius of the sphere x2 + y2 + z2 – 6x – 8y + 10z + λ = 0 is unity, what is the value of λ? (a) 49 (b) 7 (c) – 49 (d) – 7 74. The planes px + 2y + 2z – 3 = 0 and 2x – y + z + 2 = 0 intersect at an angle 4/π. What is the value of p2? (a) 24 (b) 12 (c) 6 (d) 3
75. What is the equation of the plane passing through the point (1, – 1, – 1) and perpendicular to each of the planes x – 2y – 8z = 0 and 2x + 5y – z = 0? (a) 7x – 3y + 2z = 14 (b) 2x + 5y – 3z = 12 (c) x – 7y + 3z = 4 (d) 14x – 5y + 3z = 16 76. The sum of the direction cosines of z-axis is (a) 0 (b) 1/3 (c) 1 (d) 3 77. The marks scored by two students A and B in six subjects are given below: A 71 56 45 89 54 44 B 55 74 83 54 38 52 Which one of the following statements is correct? (a) The average scores of A and B are same but A is consistent (b) The average scores of A and B are not same but A is consistent (c) The average scores of A and B are same but B is consistent (d) The average scores of A and B are not same but B is consistent 78. The following question 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 answer. While constructing the cumulative frequency column of a frequency distribution, it is noticed that these cumulative frequencies are in arithmetic progression. Assertion (A): All the class frequencies are equal. Reason (R): When all the class frequencies are equal, the cumulative frequencies are in arithmetic progression. (a) Both A and R are individually true, and R is the correct explanation of A. (b) Both A and R are individually true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true. 79.
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(a) Month 1 (b) Month 2 (c) Month 3 (d) Month 4 80. What is the mean deviation of the data 2, 9, 9, 3, 6, 9, 4? (a) 2.23 (b) 2.57 (c) 3.23 (d) 3.57 81. What is the cumulative frequency curve of statistical data commonly called? (a) Cartogram (b) Histogram (c) Ogive (d) Pictogram 82.
(a)
(b)
(c)
(d) 83. The median of 27 observations of a variable is 18. Three more observations are made and the values of these observations are 16, 18 and 50. What is the median of these 30 observations? (a) 18 (b) 19 (c) 25.5 (d) Can not be determined due to insufficient data 84.
(a)
(b)
(c)
(d)
85.
(a)
(b)
(c)
(d) 86.
(a)
(b)
(c)
(d) 87.
(a) 2.5 (b) 2.8 (c) 3 (d) 3.5 88. The sum of two numbers is 20. What are the numbers if the product of the square of one and the cube of the other is maximum? (a) 6, 14 (b) 15, 5 (c) 12, 8
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(d) 10, 10 89. What is the area of the largest rectangular field which can be enclosed with 200 m of fencing? (a) 1600 m2 (b) 2100 m2 (c) 2400 m2 (d) 2500 m2 90. The function y = tan–1 x – x (a) is always decreasing (b) is always increasing (c) first increases and then decreases (d) first decreases and then increases 91.
(a) Only I (b) Only II (c) Both I and II (d) Neither I nor II 92. What is the minimum value of | x |? (a) – 1 (b) 0 (c) 1 (d) 2 93.
(a)
(b)
(c)
(d) 94.
(a)
(b)
(c)
(d) 95.
(a) 2 (b) 1 (c) ¼ (d) 0 96.
(a) √3 (b) 1/3 (c) 1/2s (d) 0 97.
(a) 2 (b) 1 (c) π (d) 0 98. Which one of the following differential equations represents the system of circles touching y-axis at the origin?
(a)
(b)
(c)
(d) 99.
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(a) 4 (b) 3 (c) 2 (d) 1 100. What is the differential equation for y2 = 4a(x – a)? (a) yy’–2xy’+y2= 0 (b) yy’(yy’+2x)+y2= 0 (c) yy’(yy’–2x)+y2 = 0 (d) yy’2xyy’) +y = 0 101.
(a) 3 (b) 2 (c) 2/3 (d) Not defined 102. The differential equation of the curve y = sin x is
(a)
(b)
(c)
(d) 103. What is the largest value of a third order determinant whose elements are 0 or 1? (a) 0 (b) 1 (c) 2 (d) 3 104. What should be the value of k so that the system of linear equations x – y + 2z = 0, kx – y + z = 0, 3x + y – 3z = 0 does not possess a unique solution? (a) 0 (b) 3 (c) 4 (d) 5
105. If | An× n | = 3 and | adj A | = 243, what is the value of n? (a) 4 (b) 5 (c) 6 (d) 7 106.
(a) Either x + y = z or x = y (b) Either x + y = – z or x = z (c) Either x + z = y or z = y (d) Either z + y = x or x = y 107. If the system of equations 2x + 3y = 7 and 2ax + (a+ b) y = 28 has infinitely many solutions, then which one of the following is correct? (a) a = 2b (b) b = 2a (c) a = –2b (d) b = –2a 108. Consider the following statements. I. The inverse of a square matrix, if it exists, is unique. II. If A and B are singular matrices of order n, then AB is also a singular matrix of order n. Which of the statements given above is/are correct? (a) Only I (b) Only II (c) Both I and II (d) Neither I nor II 109.
(a) 0 (b) 1 (c) 10 (d) 20 110.
(a) 0 (b) x2 + y2 + z2
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(c) x2 + y2 + z2 + 1 (d) None of the above 111.
(a)
(b)
(c)
(d) 112.
(a) ¾ (b) 4/3 (c) 9/16 (d) 3/5 113.
(a) 1 (b) 2 (c) 3 (d) 4 114.
(a) 4 (b) 6 (c) 8 (d) 10 115.
(a)
(b)
(c)
(d) 116.
(a) –8 (b) 4 (c) 8 (d) 12 117.
(a) 12 (b) 16 (c) 20 (d) 24 118.
(a) 0 (b) ½ (c) 1 (d) 2 119.
(a)
(b)
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(c)
(d) 120.
(a)
(b)
(c)
(d) 1. c 2. c 3. b 4. c 5. b 6. a 7. c 8. b 9. b 10. c 11. a 12. b 13. d 14. a 15. a 16. c 17. c 18. c 19. a 20. b 21. c 22. a 23. c 24. c 25. b 26. d 27. c 28. b 29. d 30. a 31. c 32. d 33. d 34. b 35. c 36. b 37. a 38. c 39. c 40. c 41. c 42. c 43. c 44. b 45. a 46. b 47. a 48. d 49. c 50. b 51. b 52. a 53. c 54. d 55. b 56. b 57. b 58. b 59. a 60. a 61. a 62. a 63. a 64. d 65. c 66. b 67. d 68. c 69. a 70. c 71. c 72. a 73. a 74. a 75. d 76. c 77. d 78. a 79. c 80. b 81. c 82. a 83. b 84. b 85. d 86. a 87. c 88. c 89. d 90. a 91. a 92. b 93. a 94. c 95. d 96. d 97. a 98. b 99. c 100. c 101. b 102. b 103. c 104. d 105. c 106. b 107. b 108. a 109. a 110. a 111. d 112. a 113. b 114. b 115. c 116. c 117. b 118. c 119. d 120. C 1.
2.
3.
4.
5.
6.
7.
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8.
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15.
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21.
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22.
23. 24.
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27.
28.
29.
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30.
31.
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34.
35.
36. 37.
38.
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39.
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41. 42.
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49.
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50.
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55.
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56.
57. 58. 59.
60.
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63.
64.
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65.
66.
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70.
71.
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73.
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74.
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79.
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82.
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83.
84.
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88.
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89.
90.
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92.
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96.
97.
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98.
99.
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102.
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103.
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110.
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111.
112.
113.
114.
115.
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116.
117.
118.
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120.
