346 NDA Mathematics Practice Set 1. Let A = { 1, 2, 5, 8}, B = {0, 1, 3, 6, 7} and R be the relation is one less than from A to B, then how many elements will R contain? 2 3 5 9 7. 1 only 2 only 1 and 2 1, 2, 3 and 4 2. If a set X contains n (n > 5) elements, then what is the number of subsets of X containing less than 5 elements? 3. If A and B are two sets satisfying A B = B A, then which one of the following is correct? 4. 2 only - 2 only 2 or 2 1 or 1 5. If A = {1, 2, 3, 4} and R = {(1, 1), (1, 3), (2, 2), (3, 1), (3, 4), (4, 3), (4, 4)} is a relation on A A, then which one of the following is correct? R is reflexive R is symmetric and transitive R is transitive but not reflexive R is neither reflexive nor transitive {1, 2} {1, 1} { 1, 1, 2} {1} 8. 9. In binary system the decimal number 0.3 can be expressed as 10. The relation has the same father as over the set of children is: only reflexive only symmetric only transitive an equivalence relation 11. 6. 1 and 2 only
347 1, 2 and 3 2 only 1 only 12. 13. 14. 10 8 6 4 18. p + r p + q q + r p q 19. 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. Assertion (A): 1/8, loga 2 a, loga a 2 are in GP but not in AP. Reason: (R): x, y, z are in AP as well as in GP if x = y =z. 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 20. 15. no roots one roots two equal roots infinite roots 16. 17. 21. If the number of terms of an A.P. is (2n + 1), then what is the ratio of the sum of the odd terms to the sum of even terms?
348 22. If the AM and HM of two numbers are 27 and 12 respectively, then what is their GM equal to? 12 18 24 27 23. What is the 10th common term between the series 2 + 6 + 10 +... and 1 + 6 + 11 +...? 180 186 196 206 24. What is the sum of first eight terms of the series 1 1/2 + 1/4 1/8 +...? 89/128 57/384 85/128 None of the above 25. 26. 27. 2-1 -2 1 28. real part of z is zero. The imaginary part of z is zero. The real part of z is equal to imaginary The sum of real and imaginary parts of z is z. 29. 2 7 8 None of these 30. 9 18 48 84 31. If 7 points out of 12 are in the same straight line, then what is the number of triangles formed? 84 175 185 201 32. What is the number of signals that can be sent by 6 flags of different colours taking one or more at a time? 21 63 720 1956 33. What is the angle between the two straight lines y = (2-3)x + 5 and y = (2 + 3)x 7? 60 45 30 15 34. The numerical value of the perimeter of a square exceeds that of its area by 4. What is the side of the square? 1 unit 2 unit 3 unit 4 unit 35.
349 If the straight lines x 2y = 0 and kx + y = 1 intersect at the point (1, 1/2), then what is the value of k? 1 2 1/2 1/2 36. If (p, q) is the point on the x-axis equidistant from the points (1, 2) and (2, 3), then which one of the following is correct? p = 0, q = 4 p = 4, q = 0 p = 3/2, q = 0 p = 1, q = 0 37. Let p, q, r, s be the distances from origin of the points (2, 6), (3, 4), (4, 5) and ( 2, 5) respectively. Which one of the following is a whole number? p q r s 38. What does the equation x 3 y + xy 3 xy = 0 represent? A pair of straight lines only A pair of straight lines and a circle A rectangular hyperbola only A rectangular hyperbola and a circle 39. ABC is an equilateral triangle inscribed in a circle of centre O and radius 5 cm. Let the diameter through C meet the circle again at D. Assertion (A): AD. BD < OB. OC Reason (R): 2(AD 2 + BD 2 ) = CD 2 = 100 sq cm 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 40. If the eccentricity and length of latus rectum of a hyperbola are 13/3 and 10/3 units respectively, then what is the length of the transverse axis? 7/2 unit 12 unit 15/2 unit 15/4 unit 41. What are the equations of the directrices of the ellipse 25x 2 + 16y 2 = 400? 3x ± 25 = 0 3y ± 25 = 0 x ± 15 = 0 y ± 25 = 0 42. A positive acute angle is divided into two parts whose tangents are 1/8 and 7/9. What is the value of this angle? π/3 π/4 π/6 π/12 43. 0 1 1 44. If sin A = sin B and cos A = cos B, then which one of the following is correct? B = nπ + A A = 2nπ B A = 2nπ + B B = nπ A (n is an integer) 45. If sin x + sin y = a, cos x + cos y = b, then what is the value of cos (x y)? 46. If sin 2 x + sin 2 y = 1, then what is the value of cot(x + y)? 1 3 0 1/ 3 47. If tan A = 1/2 and tan B = 1/3, then what is the value of (A+B)? 0 π/4 π/2 π 48. What is the value of sin 15 sin 75? 1/4
350 1/8 1/16 1 49. What is the value of tan 15. tan195? 7 4 3 7 + 4 3 7 + 2 3 7 + 6 3 50. What is tan(a + 2B). tan(2a + B) equal to? 1 0 1 2 51. If sinθ + 2 cosθ = 1, then what is 2sinθ cosθ equal to? 0 1 2 4 52. In Δ ABC, if A: B: C = 1: 2: 3, then what is BC: CA: AB? 1: 2: 3 1: 3: 2 2: 3 :1 3 :1: 2 53. The equation sin 1 (3x 4x 3 ) = 3 sin 1 (x) is true for all values of x lying in which one of the following intervals? 55. In a triangle ABC, a = 8, b = 10 and c = 12. What is the angle C equal to? A/2 2A 3A 3A/2 56. PT, a tower of height 2 x metre, P being the foot, T being the top of the tower. A, B are points on the same line with P. If AP = 2 x + 1 m, BP = 192 m and if the angle of elevation of the tower as seen from b is double the angle of the elevation of the tower as seen from A, then what is the value of x? 6 7 8 9 57. The angle of elevation of a tower at a level ground is 30. The angle of elevation becomes θ when 10 m moved towards the tower. If the height of tower is 5 3 m, then what is θ equal to? 45 60 75 None of the above 58. If f (x) = (1 + x) 5/x is continuous at x = 0, then what is the value of f (0)? 0 1 e 5 59. 54. Both statements I and II are independently correct and statement II is the correct explanation of statement I Both statements I and II are independently correct but statement II is not the correct explanation of statement I Statement I is correct but statement II is false. Statement I is false but statement II is correct. p = 0 p > 0 p < 0 No value of p 60. 2 3 1/(3 3) 2/(3 3) 1/ 3
351 61. If f : R R, g: R R and g(x) = x + 3 and (fog) (x) = (x + 3) 2, then what is the value of f ( 3)? 9 0 9 3 62. 63. Consider the following statements: 1. If f (x) = x 3 and g(y) = y 3 then f = g. 2. Identity function is not always a bijection. Which of the above statements is/are correct? 1 only 2 only Both 1 and 2 Neither 1 nor 2 64. From a pack of 52 cards, two cards are drawn, the first being replaced before the second is drawn. What is the probability that the first is a diamond and the second is a king? ¼ 4/13 1/52 4/15 65. In a school there are 40% science students and the remaining 60% are arts students. It is known that 5% of the science students are girls and 10% of the arts students are girls. One student selected at random is a girl. What is the probability that she is an arts student? 1/3 3/4 1/5 3/5 66. 3/4 1/4 1/3 2/3 67. In tossing three coins at a time, what is the probability of getting at most one head? 3/8 7/8 1/2 1/8 68. A husband and wife appear in an interview for two vacancies in the same post. The probability of husband s selection is 1/5 and that of wife s selection is 1/3. What is the probability that only one of them will be selected? 1/5 2/5 3/5 4/5 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 70. What is the angle between the two lines whose direction numbers are ( 3 1, 3 1, 4) and ( 3 1, 3 1, 4)? π/6 π/4 π/3 π/2 71. 1 0 1 a 2 + b 2 72. The equation by + cz + d = 0 represents a plane parallel to which one of the following?
352 x-axis y-axis z-axis None of these 73. What are the direction ratios of the line determined by the planes x y + 2z = 1 and x + y z = 3? ( 1, 3, 2) ( 1, 3, 2) (2, 1, 3) (2, 3, 2) (2) collecting data from public libraries (3) collecting data by telephonic interview Select the correct answer using the code given below All the three methods give secondary data 1 and 2 give secondary and 3 gives primary data 1 and 3 give secondary and 2 gives primary data 2 and 3 give secondary and 1 gives primary data 79. 74. What is the locus of points of intersection of a sphere and a plane? Circle Ellipse Parabola Hyperbola 75. What are the direction ratios of the line of intersection of the planes x = 3z + 4 and y = 2z 3? 1, 2, 3 2, 1, 3 3, 2, 1 1, 3, 2 76. 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. Assertion (A): We cannot find out the regression of x on y from that of y on x. Reason (R): In one equation x is dependent variable and y is independent whereas in other equation y is dependent variable and x is independent. 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. 5/2 5/3 4/3 16/9 80. Consider the following statements with regard to correlation coefficient r between random variables x and y. I. r = + 1 or 1 means there is a linear relationship between the variables. II. 1 r 1and r 2 is a measure of the linear relationship between the variables. Which of the statements given above is/are correct? Only I Only II Both I and II Neither I nor II 81. 77. In computing a measure of the central tendency for any set of 51 numbers, which one of the following measures is welldefined but uses only very few of the numbers of the set? Arithmetic mean Geometric mean Median Mode 78. Consider the following three methods of collecting data (1) collecting data from government offices
353 85. 1995 1996 1997 1998 82. The arithmetic mean of numbers a, b, c, d, e is M. What is the value of (a M) + (b M) + (c M) + (d M) + (e M)? M a + b + c + d + e 0 5 M 83. If two lines of regression are perpendicular, then the correlation coefficient r is 2 1/2 0 None of the above 84. 86. 87. Under what conditions is the tangent to a given curve at a point perpendicular to x-axis? 88. What is the product of two parts of 20, such that the product of one part and the cube of the other is maximum? 75 91 84
96 354 89. If f (x) = 3x 2 + 6x 9, then f (x) is increasing in ( 1, 3) f (x) is decreasing in (3, ) f (x) is increasing in (, 1) f (x) is decreasing in (, 1) 90. If at any instant t, for a sphere, r denotes the radius, S denotes the surface area and V denotes the volume, then what is dv/dt equal to? 94. What is the area enclosed by the curve 2x 2 + y 2 = 1? 95. π π/2 π/4 0 96. 91. The radius of a circle is uniformly increasing at the rate of 3 cm/s. What is the rate of increase in area, when the radius is 10 cm? 6π cm 2 /s 10π cm 2 /s 30π cm 2 /s 60π cm 2 /s 92. K = 3 0 3 < K K 4 K = 0 97. What is the area of the parabola y 2 = x bounded by its latus rectum? 1/12 square unit 1/6 square unit 1/3 square unit None of the above 98. What is the only solution of the initial value problem y' = t (1+ y), y (0) = 0? Both A and R are true and R is the correct explanation of A 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 93. tan x + cot x + c tan x cot x + c (tan x + cot x) 2 + c (tan x cot x) 2 + c 99. When a and b are eliminated from the equation xy = ae x + be x, the resulting differential equation is of first order and first degree first order and second degree second order and first degree second order and second degree 100.
355 1 2 3 4 101. xy = c x = cy x + y = c x 2 + y 2 = c 102. If A is a matrix of order p q and B is a matrix of order s t, under which one of the following conditions does AB exist? p = t p = s q = t q = s 103. Let Aand B be two matrices of order n n. Let A be nonsingular and B be singular. Consider the following: 1. AB is singular 2. AB is non-singular 3. A 1 B is singular 4. A 1 B is non singular Which of the above is/ are correct? 1 and 3 2 and 4 only 1 only 3 only 104. 105. Under which one of the following condition does the system of equations kx + y + z = k 1 x + ky + z = k 1 x + y + kz = k 1 have no solution? k = 1 k 2 k = 1 or k = 2 k = 2 106. 10 20 40 60 107. 0 1 2 3
356 108. Consider the following statements: [2012-I] 1. every zero matrix is a square matrix. 2. A matrix has a numerical value. 3. A unit matrix is a diagonal matrix. Which of the above statements is/are correct? 2 only 3 only 2 and 3 1 and 3 109. If the matrix AB is a zero matrix, then which one of the following is correct? A must be equal to zero matrix or B must be equal to zero matrix. A must be equal to zero matrix and B must be equal to zero matrix. It is not necessary that either A is zero matrix or B is zero matrix. None of the above 110. A, B, C are not collinear 113. What is the length of the vector (1, 1)? 0 1 2 1/2 114. 115. 3/16 3 ¾ 2 116. 111. 1 ½ 2/3 2 P, Q, R are the vertices of an equilateral triangle P, Q, R are the vertices of an isosceles triangle P, Q, R are collinear None of the above 117. 112. C is in between A and B A is in between C and B B is in between A and C 1 2 3 4 118.
357 X is well defined but Y is not a well defined set Y is well defined but X is not a well defined set Both X and Y are well defined sets Neither X nor Y is a well defined set 119. 4. 120. 1 only 2 only Both 1 and 2 Neither 1 nor 2 5. 1. d 2. d 3. c 4. c 5. d 6. b 7. c 8. a 9. a 10. a 11. d 12. b 13. c 14. b 15. a 16. a 17. d 18. c 19. a 20. a 21. c 22. b 23. b 24. c 25. a 26. d 27. b 28. a 29. d 30. d 31. c 32. b 33. a 34. b 35. c 36. b 37. b 38. b 39. d 40. c 41. b 42. b 43. b 44. a 45. c 46. c 47. b 48. a 49. a 50. c 51. c 52. b 53. d 54. d 55. b 56. c 57. b 58. d 59. b 60. d 61. c 62. b 63. a 64. c 65. b 66. b 67. c 68. b 69. c 70. c 71. a 72. a 73. a 74. a 75. c 76. a 77. d 78. b 79. c 80. c 81. c 82. c 83. c 84. b 85. d 86. b 87. c 88. a 89. d 90. b 91. d 92. b 93. b 94. d 95. c 96. a 97. b 98. a 99. c 100. b 101. a 102. d 103. b 104. b 105. c 106. d 107. a 108. b 109. c 110. b 111. b 112. b 113. c 114. b 115. c 116. c 117. b 118. c 119. c 120. c 6. 7. 1. 2. 3. 8.
358 9. 10. 13. 11. 12. 14.
359 18. 19. 15. 20. 21. 16. 17.
360 26. 22. 27. 23. 28. 24. 29. 25.
361 33. 34. 30. 35. 36. 31. 32. 37.
362 43. 38. 39. 44. 40. 41. 45. 42. 46.
363 50. 51. 47. 52. 53. 54. 48. 55. 56. 57. 58. 49. 59.
364 63. 64. 60. 65. 66. 61. 62. 67.
365 68. 71. 69. 72. 73. 70.
366 74. 80. 75. 81. 76. 82. 77. 78. 83. 79. 84. 85.
367 86. 89. 90. 87. 88. 91.
368 92. 96. 93. 97. 94. 95.
369 100. 98. 101. 99. 102. 103. 104.
370 105. 107. 108. 106. 109. 110.
371 111. 115. 112. 116. 113. 117. 114.
372 118. 119. 120.