UPKAR PRAKASHAN, AGRA–2

Editorial BoardPratiyogita Darpan

© Publishers

Publishers

UPKAR PRAKASHAN(An ISO 9001 : 2000 Company)

2/11A, Swadeshi Bima Nagar, AGRA–282 002Phone : 4053333, 2530966, 2531101Fax : (0562) 4053330, 4031570E-mail : care@upkar.in, Website : www.upkar.in

Branch Offices :4845, Ansari Road, Daryaganj,New Delhi—110 002Phone : 011–23251844/66

Pirmohani Chowk,Kadamkuan,Patna—800 003Phone : 0612–2673340

1-8-1/B, R.R. Complex (Near SundaraiahPark, Adjacent to Manasa Enclave Gate),Bagh Lingampally,Hyderabad—500 044 (A.P.)Phone : 040–66753330

28, Chowdhury Lane, ShyamBazar, Near Metro Station,Gate No. 4Kolkata—700004 (W.B.)Phone : 033–25551510

B-33, Blunt Square, KanpurTaxi Stand Lane, Mawaiya,Lucknow—226 004 (U.P.)Phone : 0522–4109080

● The publishers have taken all possible precautions in publishing this book, yet ifany mistake has crept in, the publishers shall not be responsible for the same.

● This book or any part thereof may not be reproduced in any form byPhotographic, Mechanical, or any other method, for any use, without writtenpermission from the Publishers.

● Only the courts at Agra shall have the jurisdiction for any legal dispute.

ISBN : 978-81-7482-229-1

Price : Rs. 210·00(Rs. Two Hundred Ten Only)

Code No. 432

Printed at : Upkar Prakashan (Printing Unit) Bye-pass, Agra

National Defence AcademyExamination

Solved Paper

2013(Held on 11 August, 2013)

Mathematics1. What is the seventh term of the sequence 0, 3,

8, 15, 24, …… ?(A) 63 (B) 48(C) 35 (D) 33

2. What is tan 15° equal to ?

(A) 2 – √⎯ 3 (B) 2 + √⎯ 3

(C) 1 – √⎯ 3 (D) 1 + √⎯ 3

3. Consider the following—

(1) tan ( )π6

(2) tan ( )3π4

(3) tan ( )5π4

(4) tan ( )2π3

What is the correct order ?(A) 1 < 4 < 2 < 3 (B) 4 < 2 < 1 < 3(C) 4 < 2 < 3 < 1 (D) 1 < 4 < 3 < 2

4. If cosx = 13 , then what is

sin x · cot x · cosec x · tan xequal to ?

(A)23

(B)32

(C) 2 (D) 1

5. What is log81 243 equal to ?(A) 0·75 (B) 1·25(C) 1·5 (D) 3

6. If the three vertices of the parallelogramABCD are A(1, a), B(3, a), C(2, b), then D isequal to—(A) (3, b) (B) (6, b)(C) (4, b) (D) (5, b)

7. The value of k for which the lines 2x + 3y + a= 0 and 5x + ky + a = 0, represent family ofparallel lines is—(A) 3 (B) 4·5(C) 7·5 (D) 15

8. What is the equation of the line which passesthrough (4, –5) and is perpendicular to 3x +4y + 5 = 0 ?(A) 4x – 3y – 31 = 0(B) 3x – 4y – 41 = 0(C) 4x + 3y – 1 = 0(D) 3x + 4y + 8 = 0

9. The foci of the hyperbola4x2 – 9y2 – 1 = 0 are—

(A) ( )±√⎯⎯13‚ 0 (B) ⎝⎜⎛

⎠⎟⎞

± √⎯⎯13

6‚ 0

(C) ⎝⎜⎛

⎠⎟⎞

0‚ ± √⎯⎯13

6(D) None of the above

10. For what value of k are the two straight lines3x + 4y = 1 and 4x + 3y + 2k = 0 equidistantfrom the point (1, 1) ?

(A)12

(B) 2

(C) – 2 (D) – 12

11. A point P moves such that its distances from(1, 2) and (– 2, 3) are equal. Then the locus ofP is—(A) Straight Line (B) Parabola(C) Ellipse (D) Hyperbola

12. The equation of the locus of a point which isequidistant from the axes is—(A) y = 2x (B) x = 2y(C) y = ±x (D) 2y + x = 0

13. What angle does the line segment joining (5,2) and (6, – 15) subtend at (0, 0) ?

(A)π6

(B)π4

(C)π2

(D)3π4

4 | NDA (II) 2013

14. The length of latus rectum of the ellipse 4x2 +9y2 = 36 is—

(A)43

(B)83

(C) 6 (D) 12

15. What is the equation to the straight linepassing through (5, – 2) and (– 4, 7) ?(A) 5x – 2y = 4 (B) – 4x + 7y = 9(C) x + y = 3 (D) x – y = – 1

16. What is sin2 20° + sin2 70° equal to—(A) 1 (B) 0

(C) – 1 (D)12

17. What is the value of the determinant

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪1 bc a(b + c)

1 ca b(c + a)1 ab c(a + b)

?

(A) 0 (B) abc(C) ab + bc + ca (D) abc (a + b + c)

18. Consider the following—(1)A ∪ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)(2)A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)Which of the above is/are correct ?(A) 1 only (B) 2 only(C) Both 1 and 2 (D) Neither 1 nor 2

19. Let n be a positive integer and(1 + x)n = a0 + a1x + a2x2 + …… + anxn

What is a0 + a1 + a2 + …… + an equal to ?

(A) 1 (B) 2n

(C) 2n – 1 (D) 2n + 1

20. If α, β are the roots of the equation ax 2 + bx +

b = 0, then what is the value of

αβ

+ βα

+ ba ?

(A) – 1 (B) 0(C) 1 (D) 2

21. A number in binary system is 110001. It isequal to which one of the following numbersin decimal system ?(A) 45 (B) 46(C) 48 (D) 49

22. The roots of the equation x2 – 8x + 16 = 0(A) are imaginary

(B) are distinct and real(C) are equal and real(D) cannot be ascertained

23. What is tan–1 ( )12

+ tan–1 ( )13

equal to ?

(A)π2

(B)π3

(C)π4

(D)π6

24. What is (1 – sin2 θ) (1 + tan2 θ) equal to ?(A) sin2 θ (B) cos2 θ(C) tan2 θ (D) 1

25. How many terms are there in the expansion of(1 + 2x + x2)10 ?(A) 11 (B) 20(C) 21 (D) 30

26. The sum of the first five terms and the sum ofthe first ten terms of an AP are same. Whichone of the following is the correct statement ?(A) The first term must be negative(B) The common difference must be negative(C) Either the first term or the common

difference is negative but not both(D) Both the first term and the common

difference are negative

27. If A = {1, 3, 5, 7} then what is the cardinalityof the power set P(A) ?(A) 8 (B) 15(C) 16 (D) 17

28. What is the difference in the roots of theequation x2 – 10x + 9 = 0 ?(A) 2 (B) 3(C) 5 (D) 8

29. If 8x – 9y = 20 and 7x – 10y = 9 then what is2x – y equal to ?(A) 10 (B) 11(C) 12 (D) 13

30. The quadratic equation x2 + bx + 4 = 0 willhave real roots if—(A) b ≤ – 4 only (B) b ≥ 4 only(C) – 4 < b < 4 (D) b ≤ – 4, b ≥ 4

31. What is the positive square root of 7 + 4√⎯ 3 ?

(A) √⎯ 3 – 1 (B) √⎯ 3 + 1

(C) √⎯ 3 – 2 (D) √⎯ 3 + 2

NDA (II) 2013 | 5

32. If A = {1, 2}, B = {2, 3} and C = {3, 4} thenwhat is the cardinality of (A × B) ∩ (A × C) ?(A) 8 (B) 6(C) 2 (D) 1

33. If α, β are the roots of the equation x2 + x + 2

= 0, then what is α10 + β10

α–10 + β–10 equal to ?

(A) 4096 (B) 2048(C) 1024 (D) 512

34. If a and b are rational and b is not perfectsquare, then the quadratic equation withrational coefficients whose one root is 3a +

√⎯ b is—(A) x2 – 6ax + 9a2 – b = 0

(B) 3ax2 + x – √⎯ b = 0

(C) x2 + 3ax + √⎯ b = 0

(D) √⎯ bx2 + x – 3a = 0

35. If A is a finite set having n elements, then thenumber of relations which can be defined inA is—

(A) 2n (B) n2

(C) 2n2 (D) nn

36. If the positive integers a, b, c and d are in AP,then the numbers abc, abd, acd, bcd are in—

(A) HP (B) AP

(C) GP (D) None of the above

37. Which one of the following is an example ofnon-empty set ?

(A) Set of all even prime numbers

(B) (x : x2 – 2 = 0 and x is rational)

(C) {x : x is a natural number, x < 8 andsimultaneously x > 12}

(D) {x : x is a point common to any twoparallel lines)

38. What is one of the square roots of 3 + 4i,

where i = √⎯⎯– 1 ?

(A) 2 + i (B) 2 – i

(C) – 2 + i (D) – 3 – i

39. The number 83 is written in the binary systemas—

(A) 100110 (B) 101101

(C) 1010011 (D) 110110

40. The relation R in the set Z of integers givenby R = {(a, b) : a – b, is divisible by 5} is—(A) reflexive(B) reflexive but not symmetric(C) symmetric and transitive(D) an equivalence relation

41. What is n

Σr = 0

C (n, r) equal to ?

(A) 2n – 1 (B) n(C) n! (D) 2n

42. What is 0·9 + 0·09 + 0·009 + …… equal to ?(A) 1 (B) 1·01(C) 1·001 (D) 1·1

43. How many real roots does the quadraticequation f (x) = x2 + 3 | x | + 2 = 0 have ?(A) One (B) Two(C) Four (D) No real root

44. In a group of 50 people, two tests wereconducted, one for diabetes and one for bloodpressure. 30 people were diagnosed withdiabetes and 40 people were diagnosed withhigh blood pressure. What is the minimumnumber of people who were having diabetesand high blood pressure ?(A) 0 (B) 10(C) 20 (D) 30

45. Consider the following statements—(1) The product of two non-zero matrices

can never be identity matrix.(2) The product of two non-zero matrices

can never be zero matrix.Which of the above statements is/are correct ?(A) 1 only (B) 2 only(C) Both 1 and 2 (D) Neither 1 nor 2

46. What is the area of the parabola x 2 = y

bounded by the line y = 1 ?

(A)13 square unit (B)

23 square unit

(C)43 square units (D) 2 square units

47. What is the area bounded by y = tan x, y = 0

and x = π4 ?

(A) ln 2 square units

(B)ln 22

square units

6 | NDA (II) 2013

(C) 2(ln 2) square units(D) None of the above

48. What is ∫ (x cos x + sin x) dx equal to ?

(A) x sin x + c (B) x cos x + c(C) – x sin x + c (D) – x cos x + c

where, c is an arbitrary constant.

49. What is ∫

2 0 e

ln x dx equal to ?

(A) 1 (B) 2(C) 4 (D) None of the above

50. If the angle between the vectors i – mj and

j + k is π3 , then what is the value of m ?

(A) 0 (B) 2(C) – 2 (D) None of the above

51. What is the vector perpendicular to both the

vectors i – j and i ?

(A) i (B) – j

(C) j (D) k

52. The position vectors of the points A and B are

respectively 3i – 5j + 2k and i + j – k. Whatis the length of AB ?(A) 11 (B) 9(C) 7 (D) 6

53. The vectors i – 2xj – 3yk and i + 3xj + 2ykare orthogonal to each other. Then the locusof the point (x, y) is—(A) hyperbola (B) ellipse(C) parabola (D) circle

54. What is the value of p for which the vector

p (2i – j + 2k) is of 3 units length ?(A) 1 (B) 2(C) 3 (D) 6

55. If →a = 2i + 2j + 3k,

→b = –i + 2j + k and

→c =

3i + j are three vectors such that →a + t

→b is

perpendicular to →c , then what is equal to ?

(A) 8 (B) 6(C) 4 (D) 2

56. The mean of 20 observations is 15. Onchecking, it was found that two observationswere wrongly copied as 3 and 6. If wrong

observations are replaced by correct values8 and 4, then the correct mean is—(A) 15 (B) 15·15(C) 15·35 (D) 16

57. The Arithmetic Mean of the squares of thefirst n natural numbers is—

(A)n(n + 1) (2n + 1)

6(B)

n(n + 1) (2n + 1)2

(C)(n + 1) (2n + 1)

6(D)

(n + 1) (2n + 1)3

58. Consider the following statements :(1) Both the regressin coefficients have same

sign.(2) If one of the regression coefficients is

greater than unity, the other must be lessthan unity.

Which of the above statements is/are correct ?(A) 1 only (B) 2 only(C) Both 1 and 2 (D) Neither 1 nor 2

59. Which one of the following measures isdetermined only after the construction ofcumulative frequency distribution ?(A) Arithmetic Mean(B) Mode(C) Median(D) Geometric Mean

60. Coefficient of correlation is the measure of—(A) central Tendency(B) dispersion(C) both central tendency and dispersion(D) neither central tendency nor dispersion

61. What is x → 0lim

l – cos xx

equal to ?

(A) 0 (B)12

(C) 1 (D) 2

62. What is x → 0lim

cos xπ – x

equal to ?

(A) 0 (B) π

(C)1π

(D) 1

63. What is x → 0lim

sin 2x + 4x2x + sin 4x

equal to ?

(A) 0 (B)12

(C) 1 (D) 2

NDA (II) 2013 | 7

64. The maximum value of the function f (x) = x3

+ 2x2 – 4x + 6 exists at—(A) x = – 2 (B) x = 1(C) x = 2 (D) x = – 1

65. The minimum value of the funtion

f(x) = | x – 4 | exists at—

(A) x = 0 (B) x = 2

(C) x = 4 (D) x = – 4

66. What is ∫

2 1 ln x dx equal to ?

(A) ln 2 (B) 1

(C) ln ( )4e

(D) ln ( )e4

67. What is ∫ dx

√⎯⎯⎯⎯4 + x2 equal to ?

(A) ln | |√⎯⎯⎯⎯4 + x2 + x + c

(B) ln | |√⎯⎯⎯⎯4 + x2 – x + c

(C) sin –1 ( )x2

+ c

(D) None of the abovewhere, c is an arbitrary constant.

68. What is the order of the differential equation

( )dydx

2 +

dydx

– sin2 y = 0 ?

(A) 1 (B) 2(C) 3 (D) Undefined

69. y = 2cos x + 3sin x, satisfies which of thefollowing differential equations ?

(1)d2ydx2 + y = 0

(2) ( )dydx

2 +

dydx

= 0

Select the correct answer using the code givenbelow—(A) 1 only (B) 2 only(C) Both 1 and 2 (D) Neither 1 nor 2

70. The differential equation of all circles whosecentres are at the origin is—

(A)dydx

= yx

(B)dydx

= xy

(C)dydx

= – xy

(D) None of the above

71. What is ∫ sin2 x dx + ∫ cos2 x dx equal to ?

(A) x + c (B)x2

2 + c

(C) x2 + c (D) None of the abovewhere, c is an arbitrary constant.

72. What is ∫ eex ex dx equal to ?

(A) eex + c (B) 2eex + c

(C) eex ex + c (D) 2eex ex + cwhere, c is an arbitrary constant.

73. What is the area bounded by the lines x = 0,y = 0 and x + y + 2 = 0 ?

(A)12 square unit (B) 1 square unit

(C) 2 square units (D) 4 square units

74. What is the derivative of sin (sin x) ?(A) cos (cos x)(B) cos (sin x)(C) cos (sin x) cos x(D) cos (cos x) cos x

75. What is the derivative of | x – 1 | at x = 2 ?(A) – 1(B) 0(C) 1(D) Derivative does not exist

76. If three events A, B, C are mutually exclusive,then which one of the following is correct ?

(A) P (A ∪ B ∪ C) = 0

(B) P (A ∪ B ∪ C) = 1(C) P (A ∩ B ∩ C) = 0(D) P (A ∩ B ∩ C) = 1

77. What is the variance of the first 11 naturalnumbers ?

(A) 10 (B) 11

(C) 12 (D) 13

78. If A and B are independent events such that

P(A) = 15

, P(A ∪ B) = 710

, then what is P(–B)

equal to ?

(A)27

(B)37

(C)38

(D)79

N.D.A. Solved Papers

Publisher : Upkar Prakashan ISBN : 9788174822291 Author : Pratiyogita Darpan

Type the URL : http://www.kopykitab.com/product/4216

Get this eBook

30%OFF