NDA/NA Mathematics Sample Question Paper 1 PDF

NDA / NA MATHEMATICS Sample Question National Defence Academy / Naval Academy Paper 1 Time : 2 :30 Hour...

NDA / NA MATHEMATICS Sample Question National Defence Academy / Naval Academy Paper 1 Time : 2 :30 Hour Total Marks : 300 Important Instructions : 1. This ‘test Booklet contains 120 items (questions). Each item is printed in English. Each item comprises four responses (answer,s). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item. 2. You have to mark all your responses ONLY on the separate Answer Sheet provided. 3. All items carry equal marks. 4. Before you proceed to mark in the Answer Sheet the response to various items in the Test Booklet, you have to fill in some particulars in the Answer Sheet as per instructions. 5. Penalty for wrong answers : THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE MOCK PAPERS. (i) There are four alternatives for the answer to every question. For each question for which a wrong answer has been given by the candidate, one·third of the marks assigned to that question will be deducted as penalty. (ii) If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to that question. (iii) If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question. Q. 1. If set A and B has respectively 4 and 2 Q. 4. If α and β are the roots of the equation elements. What is the ratio of number of x2 + 3x + 1 = 0, then the equation whose proper subset of A to the number of relation roots are α3 + β3 and α2 + β2 will be ? from set A to set B. (a) x2 – 11x – 126 = 0 (a) 15 : 256 (b) 8 : 256 (b) x2 + 11x – 126 = 0 (c) 1 : 2 (d) None of these (c) x2 + 11x + 126 = 0 Q. 2. If (x + 3)3 + 27 = 0 has three roots x1, x2 and (d) x2 – 11x + 126 = 0 x3, than the value of x13 + x23 + x33 will be. Q. 5. If C(30, p + 2) = C(30, p + 6) and C(50, q + 5) (a) –122 (b) –142 = C(50, q + 7) the value of p2 + q2 = ? (c) –162 (d) –182 (a) 480 (b) 481 Q. 3. Let tn = sinn θ + cosn θ (c) 482 (d) 483 t100 − t98 Q. 6. If Sn denotes the sum of n terms of A.P. where then the value of equal to? t98 − t96 t9 + t7 Sn = 2n2 + 3n, then the value of , where t5 − t3 t100 t98 (a) (b) tn denotes the nth term of the A.P. t96 t96 33 31 (a) (b) t96 t100 4 4 (c) (d) t94 t94 29 27 (c) (d) 4 4 306 Oswaal NDA/NA Year-wise Solved Papers Q. 14. If one root of the equation x2 + ax + b = 0 is. tan 2 7 x − tan 2 3x Q. 7. The value of the limit lim Square of the other root, then which one is x →0 tan 2 5x − tan 2 x true? 7 a 1+b a 1+b (a) 1 (b) (a) = 2 (b) = 2 5 b a + 3b b a − 3b 21 5 (c) (d) a 1−b a 1−b 5 3 (c) = 2 (d) = 2 b a + 3b b a − 3b Q. 8. Solve the differential equation by suitable Q. 15. A binary number is represented by (abaacdcc)2, method. where a > b and c > d. What is it’s decimal ydx – xdy = xy2 dy + y3dx equivalent to? y x (a) (185)10 (b) (187)10 (a) = xy + c (b) = xy + c x y (c) (189)100 (d) (191)10 (c) x2 = y2 + c (d) None of these Q. 16. If A = 3 + cos4 θ – cos2 θ, then which one of Q. 9. If 4x + 5y – 30 = 0 and 5x – 4y – 17 = 0 are the following is correct. the two equations of diagonals of a square. 11 13 Square has one vertex (–10, 2). Find the Area (a) ≤A≤3 (b) ≤A≤3 4 4 of the square. (a) 400 (b) 425 11 13 (c) ≤A≤4 (d) ≤A≤4 (c) 450 (d) 475 4 4 Q. 17. The value of cot 3A – cosec 3A will be equals Q. 10. In a lottery of 20 tickets numbered from 11 to? to 30, three tickets are drawn simultaneously what is the probability that all the three  3A   3A  (a) tan   (b) − tan   number are prime number?  2   2  1 2  3A   3A  (a) (b) (c) cot   (d) − cot   57 57  2   2  3 3 Q. 18. If a, b and c all are positive numbers, then (a + (c) (d) 57 20 1 1 1 b + c)  + +  will be. Q. 11. The function f(x) = |x| + x + x will be? 2 4 a b c (a) even 1 (b) odd (a) ≥ 9 (b) ≥ 9 2 (c) both even and odd 1 (d) Neither even nor odd (c) ≥10 (d) 10 2 Q. 12. Total number of terms in the expansion of Q. 19. If the ratio of G.M. to A.M. between two (x3 + 6x2 + 12x + 8)5 + (y3 + 9y2 + 27y + 27)6 numbers a and b is 2 : 3. Then the value of will be a:b=? (a) 35 (b) 30 (c) 28 (d) 27 7+3 5 7−3 5 (a) (b) Q. 13. A man has 7 friends. In how many ways he 2 2 can invite two or more than two friends for a 7+3 5 7−3 5 (c) (d) party? 4 4 (a) 125 (b) 123 (c) 121 (d) 120 Sample Question Paper – 1 307  2 −5  Q. 29. In a ∆ABC, if ∠A = 60°, a = 2 cm and a + b + Q. 20. If the inverse of A =   is equal to c = 10 cm, then the value of inradius will be,  −1 3  where a, b, c are sides of triangle? p q  r s  , then the value of pq + rs will be. 1   (a) 3 cm (b) cm (a) 11 (b) 13 3 (c) 15 (d) 17 1 (c) 2 3 cm (d) cm Q. 21. If p = sin θ + tan θ, then the value of 2 3 p3 + (p – 1)3 + (p – 2)2 + 4 will be equals to? Q. 30. The shadow of a tower standing on a level (a) 1 (b) 3 ground is found to be 120 m longer, when the (c) 5 (d) 7 sun’s altitude is 30° than when it is 60°. The Q. 22. A car is moving towards a building after height of the tower will be? 10 seconds, it’s angle of elevation with the building becomes 60° from 45°. In how many (a) 120 ( ) 3 +1 m (b) 60 ( ) 3 +1 m seconds it will reach the foot of building from (c) 60 3 m (d) 120 3 m the initial position? (a) 21.66 sec (b) 22.65 π Q. 31. If sin–1(1 – x) – 2sin–1 x = , then the solutions (c) 23.65 sec (d) 24.65 sec. 2 of x will be? Q. 23. If b > a, then the equation (x – a)(x – b) = 1 has (a) only one (b) Two values root in which one of the following interval? (c) Three values (d) None of these (a) (–∞, a) (b) (b, ∞) (c) (–∞, a), (b, ∞) (d) (a, b) sin 2 A cot Α 1 Q. 24. What is the number of diagonals of a Q. 32. For any ∆ABC, the value of sin 2 B cot B 1 decagon? will be? sin 2 C cot C 1 (a) 28 (b) 35 (a) –1 (b) 0 (c) 38 (d) 42 (c) 1 (d) None of these Q. 25. If Sn = 5n + 27n2 for an A.P, then the value of Q. 33. What is the value of log25 125 + log8 256 common difference will be? + log36 216 will be. (a) 6 (b) 12 (c) 18 (d) 9 20 19 (a) (b) a+b 3 3 Q. 26. If a2 + b2 = 14ab, then the value of log    4  17 16 will be equals to? (c) (d) 3 3 1 (a) log (ab) (b) log (ab) Q. 34. If A a square matrix of order 4 × 4, and |A| = 2 k, then the value of |adj A| + |adj(adj A)| will 1 1 be equals to? (c) log (ab) (d) log (ab) 3 4 (a) k2(1 + k6) (b) k3(1 + k6) Q. 27. What is the least value of 4cosec2 x + 9sec2 x ? (c) k (1 + k ) 2 3 (d) k(k2 + k3) (a) 25 (b) 27 Q. 35. If x < 0, then the value of sin −1 1 − x 2 will (c) 36 (d) 72 be. sin x 1 cos x 3 (a) π + cos–1 x (b) π – cos–1 x Q. 28. If = and = , then the value sin y 2 cos y 2 π (c) + cos–1 x (d) cos–1 x of tan (x + y) will be ? 2 (a) 10 (b) 13 (c) 15 (d) 17 308 Oswaal NDA/NA Year-wise Solved Papers −1  3 sin 2 α  Q. 44. The equation whose roots are 2x1 and 2x2 will Q. 36. The value of tan   be?  5 + 3 cos 2α  (a) x2 – 4x+ 16 = 0 (b) x2 + 4x + 16 = 0  tan α  + tan −1   will be equal to? (c) x2 – 2x + 16 = 0 (d) x2 – 2x + 16 = 0  4  (a) α (b) 2α Q. 45. The value of x12 + x22 : x14 + x24 will be (a) 1 : 2 (b) 1 : 4 α (c) α2 (d) (c) 1 : 8 (d) 1 : 16 4 Q. 46. The value of (1 + i)5 (1 + i)7 (1 + i)8 will be. Q. 37. If f(x3) = 4x4, then the value of f′(8) will be (a) 512 (b) –512 4 10 (c) 1024 (d) –1024 (a) (b) 3 3  i 2 n +1 + i 2 n + 2 + i 2 n + 3  16 32   (c) (d)  + i 2 n + 4 + i 2 n + 5 + i 2 n + 6  3 3 Q. 47. The value of  will Q. 38. If A and B are two sets, the A ∩ (A ∪ B)′ will be (i + 1) be equals to? 2 n +1 (a) ( −1 ) (b) ( −1 ) 2n (a) φ (b) A ⋅i ⋅i (c) B (d) A′ n +1 (c) ( −1 ) ⋅ i (d) ( −1 ) n ⋅i Q. 39. In a class of 100 students, 65 students have passed in mathematics and 77 students have sin x + log ( 1 − x ) passed in physics. Then how many students Q. 48. The value of lim will be? x →0 x2 have passed only in physics? (a) 0 (b) –1 (a) 10 (b) 23 (c) 42 (d) 35 1 (c) − (d) 1 Q. 40. The number of reflexive relation of a set with 2 four elements will be equals to? Q. 49. In how many ways number greater than (a) 216 (b) 212 24000 can be formed by using the digits 1, 2, (c) 2 10 (d) 24 3, 4, 5 when no digit is repeated? Q. 41. 10 Coins are tossed simultaneously. The (a) 84 (b) 92 probability of getting at least 8 head is ? (c) 98 (d) 120 Q. 50. If f(x) = –4x2 + 4x + 5, then the maximum 7 9 (a) (b) value of f(x) will be? 128 256 45 93 6 (c) (d) (a) 6 (b) 512 1024 5 Q. 42. The function f(x) = sin |x| is ? 7 (c) 7 (d) (a) Differentiable everywhere 11 (b) Not differentiable at x = 0 6 Q. 51. The fifth H.M. between 3 and will be? (c) Discontinuous everywhere 13 (d) None of these 25 36 Information (Q. 43–45): If x2 + 2x + 4 = 0 has roots x1 (a) (b) 47 67 and x2 then answer the following questions. 39 109 Q. 43. The value of x12 + x22 + 5x1x2 will be (c) (d) 73 201 (a) 16 (b) 12 Q. 52. The value of log3 4⋅log4 5⋅log5 6⋅log6 7⋅log7 (c) 8 (d) 7 8⋅log8 9 will be? (a) 2 (b) 4 (c) 6 (d) 8 Sample Question Paper – 1 309 Q. 53. If A = {x : x2 – 5x + 6 = 0}, B = {2, 4} and C Q. 62. The direction cosines of the line 2x – 1 = 3y = {4, 5}, then A × (B ∩ C) will be equals to? – 2 = 4z – 3 will be? (a) {(2, 4), (4, 3)} (b) {(4, 2), (4, 3)} 2 3 4 1 2 3 (a) , , (b) , , (c) {(4, 2), (3, 4)} (d) {(2, 4), (3, 4)} 29 29 29 14 14 14 Q. 54. In a binomial probability distribution, mean 6 4 3 (c) , , (d) None of these 3 61 61 61 is 3 and standard deviation is , then the 2 1 probability distribution will be? Q. 63. ∫ dx will be equals to ? ( ) 2 12 9 e x + e −x 3 1 3 1 (a)  +  (b)  +  1 1 4 4 4 4 (a) +c (b) − +c 3 1 10 3 1 6 ( 2 1+ e 2x ) ( 2 1 + e2x ) (c)  +  (d)  +  4 4 4 4 1 1 (c) +c (d) − +c Q. 55. The number of solution of the equation 2 2 ( 2 1−e 2x ) ( 2 1 − e2x ) 16 sin x + 16 cos x = 10, will be, when 0 ≤ x ≤ π.  (a) two (b) four Q. 64. The projection of a = ˆi + 2 ˆj + 3kˆ on the (c) three (d) six  vector b = ˆi + ˆj + kˆ will be ? cos2 x + cos x − 2 Q. 56. The value of lim will be (a) 3 units (b) 2 3 units π cos 2 x + 4 cos x − 5 equals to? x → 3 1 8 (c) units (d) units 5 6 3 3 (a) (b) 11 11 Q. 65. The line joining the points (2, –3) and (–5, 6) is 7 8 divided by x–axis in the ratio ? (c) (d) 11 11 (a) 2 : 5 (b) 5 : 2 Q. 57. For the function f(x) = |x + a|, then which (c) 3 : 5 (d) 5 : 3 one of the following is not correct. Q. 66. Area Bounded by the curve y = sin x between (a) f(x) is continuous at x = a x = 0 and x = 3π will be? (b) f(x) is continuous at x = –a (a) 3 units2 (b) 6 units2 (c) f(x) is differentiable at x = a (c) 9 units2 (d) 2 units2 (d) f(x) is differentiable at x = –a Q. 67. If the two opposite vertex of a rectangle Q. 58. If f(x) = x3 – 6x2 – 36x + 7 is increasing are (–3, –2) and (5, 7), then the area of the function, then rectangle will be ? (a) (–∝, –2) ∪ (6, ∝) (b) (–∝, 2) ∪ (6, ∝) (a) 36 sq. units (b) 54 sq. units (c) (–∝, 6) (d) (6, ∝) (c) 72 sq. units. (d) 90 sq. units Q. 59. The minimum value f(x) = 4 ⋅ e 2 x + 9 ⋅ e −2 x is ? Q. 68. The equation of the circle in the first quadrant (a) 9 (b) 10 touching each co-ordinate axis at a distance (c) 11 (d) 12 of two unit from the origin? Q. 60. Plane XOZ divides the line joining by the (a) x2 + y2 – 4x – 4y + 4 = 0 points (2, 3, 1) and (6, –7, 1) in the ratio of ? (a) 3 : 7 (b) 7 : 3 (b) x2 + y2 + – 4x – 4y + 16 = 0 (c) 1 : 2 (d) 2 : 1 (c) x2 + y2 – 4x – 4y – 4 = 0 Q. 61. The equation of a sphere touching all the (d) x2 + y2 – 4x – 4y – 16 = 0 coordinate axis will be, if p is the radius of the Q. 69. Two consecutive sides of a parallelogram are sphere? 4x + 5y = 0 and 7x + 2y = 0. If the equation (a) x2 + y2 + z2 + 2p(x + y + z) + 2p2 = 0 of one diagonal is 11x + 7y = 9, then the (b) x2 + y2 + z2 – 2p(x + y + z) – 2p2 = 0 equation of the other diagonal will be. (c) x2 + y2 + z2 – 2p(x + y + z) + 2p2 = 0 (a) x + y = 0 (b) x – y = 0 (d) x2 + y2 + z2 + 2p(x + y + z) – 2p2 = 0 (c) x + 2y = 0 (d) x – 2y = 0 310 Oswaal NDA/NA Year-wise Solved Papers Q. 70. The solution of the differential equation x2 dy Q. 78. It is given that f′(a) exist, then = –2xy dx is x ⋅ f (a) − a ⋅ f (x ) (a) xy = c (b) x2y = c lim will be equals to? x →a x−a (c) xy2 = c (d) x = yc (a) f′(a) + a⋅f(a) (b) f′(a) – a⋅f(a) 1 Q. 71. The equation of tangent with slope to the (c) f(a) + a⋅f′(a) (d) f(a) – a⋅f′(a) 2 parabola 2x2 = 7y is ? Q. 79. A bag contains 6 white and 2 black balls (a) 2x – 4y + 7 = 0 (b) 2x – 4y – 7 = 0 and 4 balls are successively drawn out and (c) 2x + 4y – 7 = 0 (d) 2x + 4y + 7 = 0 not replaced. The probability that they are Q. 72. If 4P(A) = 6P(B) = 10P(A ∩ B) = 3, then alternately of different colours, is? B P   will be ? 1 1 A (a) (b) 7 14 1 1 (a) (b) 2 3 5 3 (c) (d) 7 14 2 2 (c) (d) Q. 80. The arithmetic mean of 7 consecutive integers 3 5 starting with a is m. Then the arithmetic mean Q. 73. In a throw of a dice the probability of getting of 11 consecutive integers starting with a + 2 one in even number of throw is ? is 3 4 (a) (b) (a) m + 1 (b) m + 2 11 11 (c) m + 3 (d) m + 4 5 6 (c) (d) Q. 81. The correlation coefficient between x and y 11 11 Q. 74. The weighted mean of first n natural number for the given data will be ? whose weights are equal to the squares of Σx = 40, Σy = 50, Σxy = 220 corresponding numbers is ? Σx2 = 200, Σy2 = 262, n = 10 n ( n + 1) 3n ( n + 1 ) (a) 0.90 (b) 0.91 (a) (b) ( 2n + 1) ( 2n + 1) (c) 0.92 (d) 0.93 Q. 82. If bxy and byx are positive regression coefficients, n ( n + 1) 3n ( n + 1 ) (c) (d) then which one is correct? 2 ( 2n + 1) 2 ( 2n + 1) 1 1 1 1 1 2 Q. 75. In a binomial distribution, the mean is 4 and (a) + > (b) + > bxy byx r bxy byx r variance is 3. Then it’s mode is ? (a) 4 (b) 5 1 1 1 1 1 2 (c) + < (d) + < (c) 6 (d) 7 bxy byx r bxy byx r Q. 76. The orthocenter of the triangle formed by the Q. 83. If A and B are two events such that P(A ∪ B) lines x + y = 1, xy = 0 is? (a) (0, 0)  1 (b) (1, 2) 1 1 = 3 4 1 , P(A ∩ B) = , P A = 4 2 3 then the( ) (c)  1,  (d)  ,   2 2 3 ( ) value of P B will be Q. 77. What is the value of 1 2 ( 2 n + 1 ) 2 ( 4 n 2 + 5n + 6 ) (a) 3 (b) 3 lim ? n→ ∝ ( 3n + 5 )3 ⋅ ( 2 n + 3 ) 1 5 (c) (d) 8 4 6 6 (a) (b) 27 3 8 (c) (d) 27 51 Sample Question Paper – 1 311 Q. 84. The area of the region y = |x – 1| and y = 1 Q. 91. The probability of A, B, C solving a problem will be 1 2 3 1 are , and respectively. If all the three (a) 1 Sq. units (b) Sq. units 3 7 8 2 try to solve the problem simultaneously, the 1 1 probability that exactly one of them will solve (c) Sq. units (d) Sq. units 3 4 the problem is ? π 17 19 2 cos x (a) (b) Q. 85. The value of I = ∫ (1 + sin x )( 2 + sin x ) dx 56 56 will be equals to? 0 23 25 (c) (d) 56 56 2 4 (a) log   (b) log   Q. 92. If x dy = y(dx + y dy), y > 0, and y(4) = 2, then 3 3 y(3) will be 5 1 (a) 2 (b) 3 (c) log   (d) log   3 3 (c) 4 (d) 5 Q. 86. In a frequency distribution, if mean is 20 and Q. 93. The integrated factor of the equation (x2 + 1) median is 26, then the mode will be? (a) 27 (b) 29 dy + 2xy = x2 – 1 is (c) 36 (d) 38 dx (a) (1 + x2)2 (b) log (1 + x2) Q. 87. In a cricket league there are 10 teams, each team plays every other team twice, if they 1 + x2 (c) (d) 1 + x2 can play 6 match every week, then how 2 many weeks the league will end? 1 2 (a) 10 weeks (b) 12 weeks  d3 y   dy  2 Q. 94.  3  =  2 +  has degree ? (c) 15 weeks (d) 18 weeks  dx   dx    4 2  (a) 2 (b) 4 Q. 88. If A =   , then the value of A⋅(adj A) will (c) 6 (d) 3 be? 3 4  Q. 95. The equation x2 – 2xy + y2 + 3x + 2 = 0 10 0  16 0  represents (a)   (b)    0 10   0 16  (a) A circle (b) A parabola (c) An ellipse (d) A hyperbola  20 0  5 0  (c)   (d)    0 20  0 5  x2 y2 Q. 96. If the equation + + 1 = 0 represents 2−r r −5 Q. 89. The ratio of the sum of n terms of two A.P.’s an ellipse, then the value of r will be? is 3n – 13 : 5n + 21, then the ratio of their 20th (a) 2 < r < 5 (b) r < 2 term will be equals to ? (c) r > 5 (d) 5 < r < 2 13 14 (a) (b) 4 x + 3, 1 ≤ x ≤ 2 27 27 Q. 97. If f(x) =  , then the value of  3x + 4, 2 < x ≤ 4 17 19 (c) (d) 4 27 27 ∫ f ( x ) dx will be equals to? Q. 90. What is the coefficient of x5 in the expansion 1 1+ x  2 (a) 35 (b) 36 of   ? (c) 37 (d) 38 1−x  (a) 10 (b) 20 (c) 30 (d) 40 312 Oswaal NDA/NA Year-wise Solved Papers π Q. 104. If x = a(t cos t – sin t), y = a(t sin t + cos t), ∫ tan n Q. 98. If In = x dx, then the value of I + I 7 5 dy 0 then the value of will be equals to? dx will be equals to ? (a) –cot t (b) cot t 1 1 (c) –tan t (d) tan t (a) (b) 3 7 x 1 1 Q. 105. If p = tan −1   , then px : py will be, where (c) (d) y 5 6 px and py are partial differential coefficients? Q. 99. If the lines 2x – 3y = 5 and 3x – 4y = 7 are (a) x : y (b) –x : y the diameters of a circle of area 154 sq. units. (c) y : x (d) –y : x Then the equation of the circle will be? (a) (x – 1)2 + (y + 1)2 = 49 Q. 106. The value of 22 + 42 + 62 + 82 + …………. + (b) (x – 1)2 + (y – 1)2 = 49 302 will be equals to? (a) 4680 (b) 4690 (c) (x – 1)2 + (y + 1)2 = 64 (c) 4860 (d) 4960 (d) (x – 1)2 + (y – 1)2 = 64 Q. 100. The equation of the line perpendicular to dy Q. 107. If y = sin −1 x 3 , then will be equals to? x y dx the line − = 1 and passes the point a b 3 x 3 1 − x3 (a) (b) (b, a) will be? 2 1 − x3 2 x x y x y (a) + =1 (b) + =2 3 x 2 1 − x3 b a b a (c) (d) 2 1 − x3 3 x x y x y (c) + =3 (d) + =4 b a b a dy Q. 108. If ln(x + y) = 2xy, then the value of at    dx Q. 101. If a , b and  c are three non-zero vectors (1, 2) will be?    such that b = 9c and a = −3c , then the  11 11 angle between a and b will be? (a) (b) − (a) 0° (b) 90° 5 5 (c) 180° (d) 270° 5 5 (c) (d) − 11 11 Q. 102. The unit vector perpendicular to 2ˆi + 3ˆj − kˆ Q. 109. If the roots of the equation p(q – r)x2 + q(r – and 3ˆi − ˆj + 5kˆ will be ? p)x + r(p – q) = 0 are equal, than p, q, r will 14ˆi − 13ˆj − 11kˆ 14ˆi + 13ˆj + 11kˆ be in ? (a) (b) 9 6 9 6 (a) A.P. (b) G.P. (c) H.P. (d) None 14ˆi − 13ˆj + 11kˆ 14ˆi + 13ˆj − 11kˆ (c) (d) Q. 110. If f(x) = x2 + 2 and g(x) = x + 3, then the 9 6 9 6 value of fog(3) will be ? x −1 y−2 z −3 x −1 (a) 37 (b) 38 Q. 103. If lines = = , = −3 2k 2 3k (c) 39 (d) 40 y−5 z−6 Q. 111. Let P ≡ (–1, 0), Q ≡ (0, 0) and R ≡ (3, 3 3 ) = are at right angles, then the be the three points, then the equation of the 1 −5 value of k will be ? bisector of the angle PQR is ? 10 10 (a) x – 3y = 0 (b) x + 3y = 0 (a) (b) − 7 7 (c) 3x + y = 0 (d) 3x – y = 0 7 7 (c) (d) − 10 10 Sample Question Paper – 1 313 Q. 116. The area of the region enclosed by the Q. 112. If f(x) = 2 log 3 x , then the value of f–1(16) will be? 1 curves y = x, x = e, y = , and positive 1 x (a) 27 (b) x-axis will be? 27 1 (a) 1 unit2 (b) unit2 1 2 (c) 81 (d) 81 3 (c) unit2 (d) 2 unit2 Q. 113. Area of the rhombus formed by the lines 2 3x ± 4y ± 5 = 0 will be.    Q. 117. If a , b and c are three vector such    6 12 that [ a b c ] = 3, then the value of (a) Sq. units (b) Sq. units 25 25    a × b b × c c × a  will be ? 25 24   (c) Sq. units (d) Sq. units (a) 0 (b) 3 24 25 (c) 9 (d) 27  1   2   1  Q. 118. The common chord of the circle x2 + y2 + 4x Q. 114. The points  , 1 ,  , 2 ,  , 3  are  3   3   3  + 1 = 0 and x2 + y2 + 6x + 2y + 3 = 0 is. the vertices of ? (a) x + y – 1 = 0 (b) x + y – 2 = 0 (a) An equilateral triangle (c) x + y + 1 = 0 (d) x + y + 2 = 0 (b) An isosceles triangle Q. 119. If mean deviation is 36, then the value of (c) A right angled triangle standard deviation will be ? (d) None of these (a) 45 (b) 35 Q. 115. If f(x) = |x – 3| + |x + 2| + |x – 7|, then the (c) 25 (d) 15 value of f′(–10) will be? Q. 120. If a, b, c are in G.P., then the value of (a) –3 (b) –2 a c (c) –1 (d) 0 2 2 − will be equals to ? b −c a − b2 2 1 1 1 1 (a) + (b) + a b b c 1 1 1 1 1 (c) + (d) + + c a a b c Finished Solving the Paper ?  Time to evaluate yourself !  SCAN THE CODE SCAN For elaborated Solutions 314 Oswaal NDA/NA Year-wise Solved Papers Answers Mathematics Q. No. Answer Key Topic Name Chapter Name 1 (a) Proper Set Sets 2 (c) Cube Root Complex Number 3 (c) Trigonometric Identities Trigonometry 4 (b) Formation of Equation Quadratic Equation 5 (c) Combination Permutation and Combination 6 (a) Sum of A.P. Sequence and Series 7 (d) Values Limits 8 (b) Differential Equation Differential Equation 9 (c) Area Co-ordinate Geometry 10 (a) Probability Probability 11 (a) Function Function 12 (a) Number of Terms Binomial Theorem 13 (d) Permutation and Combination Permutation and Combination 14 (a) Values Quadratic Equation 15 (b) Binary Number Binary Number 16 (a) Maximum and Minimum Value Trigonometry 17 (b) Values Trigonometry 18 (a) Properties of A.M & H.M Sequence and Series 19 (a) Properties of A.M & G.M Sequence and Series 20 (d) Properties of Matrix Matrix and Determinant 21 (d) Values Trigonometry 22 (c) Height and Distance Trigonometry 23 (c) Properties of Roots Quadratic Equation 24 (b) Number of Diagonals Permutation and Combination 25 (a) Properties of A.P. Sequence and Series 26 (b) Logarithm Logarithm 27 (a) Maximum and Minimum Value Trigonometry 28 (c) Values Trigonometry 29 (a) Properties of Triangle Trigonometry 30 (c) Height and Distance Trigonometry 31 (a) Solutions of Equation Trigonometry 32 (b) Values Matrix and Determinant 33 (a) Logarithm Logarithm 34 (b) Properties of Matrix Matrix and Determinant 35 (b) Inverse Trigonometry Trigonometry 36 (a) Inverse Trigonometry Trigonometry 37 (d) Values Function Sample Question Paper – 1 315 Q. No. Answer Key Topic Name Chapter Name 38 (a) Properties of Sets Sets 39 (d) Properties of Sets Sets 40 (b) Reflexive Relation Relation and Function 41 (a) Probability Probability 42 (b) Continuity and Differentiability Differentiation 43 (a) Finding Value Quadratic Equation 44 (b) Roots Quadratic Equation 45 (b) Finding Value Quadratic Equation 46 (d) Finding Value Quadratic Equation 47 (c) Finding Value Complex Number 48 (c) Values Limits 49 (a) Permutation and Combination Permutation and Combination 50 (a) Maximum and Minimum Value Quadratic Equation 51 (b) Harmonic Mean Sequence and Series 52 (a) Logarithm Logarithm 53 (d) Properties of Sets Relation and Functions 54 (a) Binomial Distribution Probability 55 (a) Number of Solutions Trigonometry 56 (a) Values Limits 57 (d) Continuity and Differentiability Differentiation 58 (a) Increasing and Decreasing Function Differentiation 59 (d) Maximum and Minimum Value Differentiation 60 (a) Planes Three Dimensional Geometry 61 (c) Equation of Sphere Three Dimensional Geometry 62 (c) Direction Cosine Three Dimensional Geometry 63 (b) Indefinite Integration Integration 64 (b) Projection Vectors 65 (a) Section Formula 2D 66 (b) Area Under Curves Integration 67 (c) Area Co-ordinate Geometry 68 (a) Equation of Circle Co-ordinate Geometry 69 (b) Equation of Line Co-ordinate Geometry 70 (b) Differential Equation Differentiation 71 (a) Parabola 2D 72 (d) Probability Probability 73 (c) Probability Probability 74 (d) Mean Statistics 75 (a) Mode Binomial Distribution 76 (a) Orthocentre 2D 77 (a) Values Limits 78 (d) Values Limits 79 (b) Probability Probability 316 Oswaal NDA/NA Year-wise Solved Papers Q. No. Answer Key Topic Name Chapter Name 80 (d) Mean Statistics 81 (b) Correlation Coefficient Statistics 82 (b) Regression Coefficient Statistics 83 (a) Probability Probability 84 (a) Area under Curves Integration 85 (b) Definite Integration Integration 86 (d) Mean Median and Mode Integration 87 (c) Permutation and Combination Permutation and Combination 88 (a) Properties of Matrix Matrix and Determinant 89 (a) Sum of A.P. Sequence and Series 90 (b) Coefficients Binomial Theorem 91 (d) Probability Probability 92 (b) Differential Equation Differentiation 93 (d) Integrated Factor Differential Equation 94 (b) Order and Degree Differential Equation 95 (b) Locus Conic Section 96 (a) Properties of Ellipse 2D 97 (c) Definite Integration Integration 98 (d) Definite Integration Integration 99 (a) Equation of Circle Circle 100 (b) Equation of Line 2D 101 (c) Angle between Two Vector Vector 102 (a) Unit Vector Vector 103 (b) Angle between Two Lines Three Dimensional Geometry 104 (a) Differential Coefficient Differentiation 105 (d) Partial Differentiation Differentiation 106 (d) Sum of Series Sequence and Series 107 (a) Differential Coefficient Differentiation 108 (b) Differential Coefficient Differentiation 109 (c) Properties of Roots Quadratic Equation 110 (b) Compostion of Function Function 111 (c) Equation of Line 2D 112 (c) Inverse Function Function 113 (c) Area 2D 114 (b) Type of Triangle 2D 115 (a) Differential Coefficient Differentiation 116 (c) Area under Curves Integration 117 (c) Values Vectors 118 (c) Common Chord Circle 119 (a) Standard Deviation Statistics 120 (c) Properties of A.P. and G.P. Sequence and Series