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there will be no penalty for that question.

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NDA Previous Year Question Paper 2009 – Shift 1 |

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NDA 2009-I Mathematics Test Booklet (Series A)
T.B.C. : O-OEBA-J-N
Serial No. 187329
Time Allowed: Two Hours and Thirty Minutes
Maximum Marks: 300

(Full instructions as printed on the cover page are omitted here for brevity; the paper contains 120 objective questions, each with four options, equal marks, and negative marking of 1/3 for wrong answers.)

1. If A and B are subsets of a set X, then what is (A ∩ (X − B)) ∪ B equal to?
(a) A ∪ B (b) A ∩ B (c) A (d) B

2. The total number of subsets of a finite set A has 56 more elements than the total number of subsets of another finite set B. What is the number of elements in the set A?
(a) 5 (b) 6 (c) 7 (d) 8

3. What is the smallest natural number n such that n! is divisible by 990?
(a) 9 (b) 11 (c) 33 (d) 99

4. Which one of the following is correct?
(a) A × (B − C) = (A − B) × (A − C)
(b) A × (B − C) = (A × B) − (A × C)
(c) A ∩ (B ∪ C) = (A ∩ B) ∪ C
(d) A ∪ (B ∩ C) = (A ∪ B) ∩ C

5. In an examination out of 100 students, 75 passed in English, 60 passed in Mathematics and 45 passed in both English and Mathematics. What is the number of students passed in exactly one of the two subjects?
(a) 45 (b) 60 (c) 75 (d) 90

6. Let
R = {x | x ∈ N, x is a multiple of 3 and x ≤ 100}
S = {x | x ∈ N, x is a multiple of 5 and x ≤ 100}
What is the number of elements in (R × S) ∩ (S × R)?
(a) 36 (b) 33 (c) 20 (d) 6

7. If A = {a, b, c} and R = {(a, a), (a, b), (b, c), (b, b), (c, c), (c, a)} is a binary relation on A, then which one of the following is correct?
(a) R is reflexive and symmetric, but not transitive
(b) R is reflexive and transitive, but not symmetric
(c) R is reflexive, but neither symmetric nor transitive
(d) R is reflexive, symmetric and transitive

8. If log₁₀(x + 1) + log₁₀ 5 = 3, then what is the value of x?
(a) 199 (b) 200 (c) 299 (d) 300

9. What is the value of 2 log₈ 2 − (log₃ 9)/3?
(a) 0 (b) 1 (c) 8/3 (d) 16/3

10. Which one of the following is one of the roots of the equation (b − c)x² + (c − a)x + (a − b) = 0?
(a) (c − a)/(b − c) (b) (a − b)/(b − c) (c) (b − c)/(a − b) (d) (c − a)/(a − b)

11. What is the value of x satisfying the equation 16[(a − x)/(a + x)]³ = (a + x)/(a − x)?
(a) a/2 (b) a/3 (c) a/4 (d) 0

12. If α, β are the roots of the equation 2x² − 2(1 + n²)x + (1 + n² + n⁴) = 0, then what is the value of α² + β²?
(a) 2n² (b) 2n⁴ (c) 2 (d) n²

13. The roots of Ax² + Bx + C = 0 are r and s. For the roots of x² + px + q = 0 to be r² and s², what must be the value of p?
(a) (B² − 4AC)/A² (b) (B² − 2AC)/A² (c) (2AC − B²)/A² (d) B² − 2C

14. What is the value of r if P(5, r) = P(6, r − 1)?
(a) 9 (b) 5 (c) 4 (d) 2

15. What is the number of words formed from the letters of the word ‘JOKE’ so that the vowels and consonants alternate?
(a) 4 (b) 8 (c) 12 (d) None of the above

16. If the sum of the first two terms and the sum of the first four terms of a geometric progression with positive common ratio are 8 and 80 respectively, then what is the 6th term?
(a) 88 (b) 243 (c) 486 (d) 1458

17. What is the decimal equivalent of (101·101)₂?
(a) (5·225)₁₀ (b) (5·525)₁₀ (c) (5·625)₁₀ (d) (5·65)₁₀

18. If x > 1 and log₂ x, log₃ x, logₓ 16 are in GP, then what is x equal to?
(a) 9 (b) 8 (c) 4 (d) 2

19. What is the term independent of x in the expansion of (1 + x + 2x³)(3x⁻²/2 − 1/(3x))⁹?
(a) 1/3 (b) 19/54 (c) 1/4 (d) No such term exists in the expansion

20. In a geometric progression with first term a and common ratio r, what is the arithmetic mean of first five terms?
(a) a + 2r (b) ar² (c) a(r⁵ − 1)/(r − 1) (d) a(r⁵ − 1)/[5(r − 1)]

21. If 2x = 3 + 5i, then what is the value of 2x³ + 2x² − 7x + 72?
(a) 4 (b) −4 (c) 8 (d) −8

22. If A = [[3 2] [1 4]], then what is A(Adj A) equal to?
(a) [[0 10] [10 0]] (b) [[10 0] [0 10]] (c) [[1 10] [10 1]] (d) [[10 1] [1 10]]

23. What is the inverse of
[[0 0 1]
[0 1 0]
[1 0 0]] ?
(a) Identity matrix (b) The same matrix (c) −I with off-diagonals (d) Negative of the given matrix

24. Consider the following statements in respect of symmetric matrices A and B:

1. AB is symmetric.
2. A² + B² is symmetric.
   Which of the above statements is/are correct?
   (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

25. If A = I, B = [[0 1] [−1 0]], C = [[cos α sin α] [−sin α cos α]], then which one of the following is correct?
(a) C = A cos α − B sin α
(b) C = A sin α + B cos α
(c) C = A sin α − B cos α
(d) C = A cos α + B sin α

26. What is the value of [(-1 + i√3)/2]⁹⁰⁰ + [(-1 − i√3)/2]³⁰¹?
(a) (−1 + i√3)/2 (b) (1 − i√3)/2 (c) (−1 − i√3)/2 (d) (1 + i√3)/2

27. If ω is an imaginary cube root of unity, then what is the value of the determinant of the matrix
[[1 + ω ω² ω]
[1 + ω² ω ω²]
[ω + ω² ω ω²]] ?
(a) 0 (b) −2ω (c) −2ω² (d) 3ω²

28. In how many ways can the letters of the word ‘GARDEN’ be arranged so that in each word the vowels should appear in alphabetical order?
(a) 120 (b) 240 (c) 360 (d) 480

29. If V = {x : x + 2 = 0}, R = {x : x² + 2x = 0}, S = {x : x² + x − 2 = 0}, then for what value of x, V = R = S?
(a) 0 (b) −1 (c) −2 (d) 1

30. Out of a group of 20 teachers in a school, 10 teach Mathematics, 9 teach Physics and 7 teach Chemistry. 4 teach both Mathematics and Physics, but none teach both Mathematics and Chemistry. What is the number of teachers who teach both Chemistry and Physics?
(a) 1 (b) 2 (c) 3 (d) 4

31. Let α, γ be the roots of Ax² − 4x + 1 = 0 and β, δ be the roots of Bx² − 6x + 1 = 0. If α, β, γ, δ are in HP, then what are the values of A and B respectively?
(a) 3, 8 (b) −3, −8 (c) 3, −8 (d) −3, 8

32. Under which one of the following conditions does the system of equations
kx + y + z = k − 1
x + ky + z = k − 1
x + y + kz = k − 1
have no solution?
(a) k = 1 (b) k ≠ −2 (c) k = 1 or k = −2 (d) k = −2

33. If the sides of a triangle are 6 cm, 10 cm and 14 cm, then what is the largest angle included by the sides?
(a) 90° (b) 120° (c) 135° (d) 150°

34. If X and Y are any two non-empty sets, then what is (X − Y)′ equal to?
(a) X′ − Y′ (b) X′ ∩ Y (c) X′ ∪ Y (d) X − Y′

35. For finding the area of a triangle ABC, which of the following entities are required?
(a) Angles A, B and side a
(b) Angles A, B and side b
(c) Angles A, B and side c
(d) Either (a) or (b) or (c)

36. Let A = [[1 2] [3 4]] and B = [[a 0] [0 b]], where a, b are natural numbers, then which one of the following is correct?
(a) There exist more than one but finite number of Bs such that AB = BA
(b) There exists exactly one B such that AB = BA
(c) There exist infinitely many Bs such that AB = BA
(d) There cannot exist any B such that AB = BA

37. Consider a matrix M = [[3 4 0] [2 1 0] [3 1 k]] and the following statements:
Statement A: Inverse of M exists.
Statement B: k ≠ 0
Which one of the following in respect of the above matrix and statements is correct?
(a) A implies B, but B does not imply A
(b) B implies A, but A does not imply B
(c) Neither A implies B nor B implies A
(d) A implies B as well as B implies A

38. If 2ˣ + 3ʸ = 17 and 2^{x+2} − 3^{y+1} = 5, then what is the value of x?
(a) 3 (b) 2 (c) 1 (d) 0

39. If P(32, 6) = k C(32, 6), then what is the value of k?
(a) 6 (b) 32 (c) 120 (d) 720

40. What is (√3 + i)/(1 + √3 i) equal to?
(a) 1 + i (b) 1 − i (c) √3(1 − i)/2 (d) (√3 − i)/2

41. What is the binary equivalent of decimal number (0.8125)₁₀?
(a) (0.1101)₂ (b) (0.1001)₂ (c) (0.1111)₂ (d) (0.1011)₂

42. If |y x y+z ; z y x+y ; x z z+x| = 0, then which one of the following is correct?
(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

43. What is the value of k if
|k b+c b²+c² ;
k c+a c²+a² ;
k a+b a²+b²| = (a − b)(b − c)(c − a)?
(a) 1 (b) −1 (c) 2 (d) 0

44. What is the number of proper subsets of a given finite set with n elements?
(a) 2n − 1 (b) 2n − 2 (c) 2ⁿ − 1 (d) 2ⁿ − 2

45. If (x + a) is a factor of both the quadratic polynomials x² + px + q and x² + lx + m, where p, q, l and m are constants, then which one of the following is correct?
(a) a = (m − q)/(l − p) (l ≠ p)
(b) a = (m + q)/(l + p) (l ≠ −p)
(c) l = (m − q)/(a − p) (a ≠ p)
(d) p = (m − q)/(a − l) (a ≠ l)

46. If A, B and C are three finite sets, then what is [(A ∪ B) ∩ C]′ equal to?
(a) A′ ∪ B′ ∩ C′ (b) A′ ∩ B′ ∩ C′ (c) A′ ∩ B′ ∪ C′ (d) A ∩ B ∩ C

47. What is the value of tan(−1575°)?
(a) 1 (b) 1/2 (c) 0 (d) −1

48. For which acute angle θ, cosec² θ = 3√3 cot θ − 5?
(a) 5π/12 (b) π/3 (c) π/6 (d) π/4

49. If tan² θ = 2 tan² φ + 1, then which one of the following is correct?
(a) cos 2θ = cos 2φ − 1
(b) cos 2θ = cos 2φ + 1
(c) cos 2θ = [cos 2φ − 1]/2
(d) cos 2θ = [cos 2φ + 1]/2

50. Which one of the following is correct in respect of the matrix
A = [[0 0 −1]
[0 −1 0]
[−1 0 0]]?
(a) A⁻¹ does not exist
(b) A = (−1)I
(c) A is a unit matrix
(d) A² = I

51. The formula sin⁻¹{2x(1 − x²)} = 2 sin⁻¹ x is true for all values of x lying in the interval
(a) [−1, 1] (b) [0, 1] (c) [−1, 0] (d) [−1/√2, 1/√2]

52. What is the value of 1 − sin 10° sin 50° sin 70°?
(a) 1/8 (b) 3/8 (c) 5/8 (d) 7/8

53. The sines of two angles of a triangle are equal to 5/13 and 99/101. What is the cosine of the third angle?
(a) 255/1313 (b) 265/1313 (c) 275/1313 (d) 770/1313

54. After subtending an angle of 1000° from its initial position, the revolving line will be situated in which one of the following quadrants?
(a) First (b) Second (c) Third (d) Fourth

55. One radian is approximately equal to which one of the following?
(a) 90° (b) 180° (c) 57° (d) 47°

56. If cot(x + y) = 1/√3 and cot(x − y) = √3, then what are the smallest positive values of x and y respectively?
(a) 45°, 30° (b) 30°, 45° (c) 15°, 60° (d) 45°, 15°

57. If sin A = 1/√5, cos B = 3/√10; A, B being positive acute angles, then what is (A + B) equal to?
(a) π/6 (b) π/4 (c) π/3 (d) π/2

58. If sin⁻¹(2a/(1 + a²)) − cos⁻¹((1 − b²)/(1 + b²)) = tan⁻¹(2x/(1 − x²)), then what is the value of x?
(a) a/b (b) ab (c) b/a (d) (a − b)/(1 + ab)

59. x = sin θ cos θ and y = sin θ + cos θ are satisfied by which one of the following equations?
(a) y² − 2x = 1 (b) y² + 2x = 1 (c) y² − 2x = −1 (d) y² + 2x = −1

60. 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

61. If sin⁴ x − cos⁴ x = p, then which one of the following is correct?
(a) p = 1 (b) p = 0 (c) |p| > 1 (d) |p| ≤ 1

62. If cos θ < sin θ and θ lies in the first quadrant, then which one of the following is correct?
(a) 0 < θ < π/4 only (b) π/4 < θ < π/2 (c) 0 < θ < π/3 (d) π/3 < θ < π/2

63. The angle of elevation from a point on the bank of a river of the top of a temple on the other bank is 45°. Retreating 50 m, the observer finds the new angle of elevation as 30°. What is the width of the river?
(a) 50 m (b) 50√3 m (c) 50/(√3 − 1) m (d) 100 m

64. If sin² x + sin² y = 1, then what is the value of cot(x + y)?
(a) 1 (b) √3 (c) 0 (d) 1/√3

65. What is the value of cos 10° + cos 110° + cos 130°?
(a) −1 (b) 0 (c) 1 (d) 2

66. If dy/dx = 1 + x + y + xy and y(−1) = 0, then what is y(x) equal to?
(a) e^{(1+x)²/2} − 1 (b) e^{(1−x)²/2} (c) ln(1 + x) − 1 (d) ln(1 − x)

67. What is the value of ∫₀^{π/2} ln(tan x) dx?
(a) 0 (b) 1 (c) −1 (d) π/4

68. What is ∫ tan² x sec⁴ x dx equal to?
(a) (sec⁵ x)/5 + (sec³ x)/3 + c
(b) (tan⁵ x)/5 + (tan³ x)/3 + c
(c) (tan⁵ x)/5 + (sec³ x)/3 + c
(d) (sec⁵ x)/5 + (tan³ x)/3 + c

69. What is lim_{x→0} (sin² a x)/(b x) (a, b constants) equal to?
(a) 0 (b) a/b (c) a²/b (d) Does not exist

70. If f(x) = tan x + e^{−2x} − 7x³, then what is the value of f′(0)?
(a) −2 (b) −1 (c) 0 (d) 3

71. What are the degree and order respectively of the differential equation of the family of rectangular hyperbolas whose axes of symmetry are the coordinate axes?
(a) 1, 1 (b) 1, 2 (c) 2, 1 (d) 2, 2

72. The function f(x) = x² − 2x increases for all
(a) x > −1 (b) x < −1 only (c) x > 1 only (d) x < 1

73. What is ∫₀¹ x(1 − x)ⁿ dx equal to?
(a) 1/[n(n + 1)] (b) 1/[(n + 1)(n + 2)] (c) 1 (d) 0

74. Let a and b be two distinct roots of a polynomial equation f(x) = 0. Then there exists at least one root lying between a and b of the polynomial equation
(a) f(x) = 0 (b) f′(x) = 0 (c) f″(x) = 0 (d) None of the above

75. If 3ˣ + 3ʸ = 3^{x+y}, then what is dy/dx equal to?
(a) (3^{x+y} − 3ˣ)/3ʸ (b) 3^{x−y}(3ʸ − 1)/(1 − 3ˣ) (c) (3ˣ + 3ʸ)/(3ˣ − 3ʸ) (d) (3ˣ + 3ʸ)/(1 + 3^{x+y})

76. What is ∫ sec x° dx equal to?
(a) ln(sec x° + tan x°) + c
(b) (π/180) ln tan(π/4 + x/2) + c
(c) (180/π) ln tan(π/4 + x/2) + c
(d) (180/π) ln tan(π/4 + πx/360) + c

77. The profit function, in rupees, of a firm selling x items (x ≥ 0) per week is given by P(x) = −3500 + (400 − x)x. How many items should the firm sell so that the firm has maximum profit?
(a) 400 (b) 300 (c) 200 (d) 100

78. A stone thrown vertically upward satisfies the equation s = 64t − 16t², where s is in metre and t is in second. What is the time required to reach the maximum height?
(a) 1 s (b) 2 s (c) 3 s (d) 4 s

79. If f(x) = 3x² + 6x − 9, then
(a) f(x) is increasing in (−1, 3)
(b) f(x) is decreasing in (3, ∞)
(c) f(x) is increasing in (−∞, −1)
(d) f(x) is decreasing in (−∞, −1)

80. If f(x) = sin²(x²), then what is f′(x) equal to?
(a) 4x sin(x²) cos(x²) (b) 2 sin(x²) cos(x²) (c) 4 sin(x²) sin² x (d) 2x cos²(x²)

81. If f(x) = cos x, g(x) = ln x and y = (g ∘ f)(x), then what is the value of dy/dx at x = 0?
(a) 0 (b) 1 (c) −1 (d) 2

82. If f(x) = {3x − 4, 0 ≤ x ≤ 2; 2x + λ, 2 < x ≤ 3} is continuous at x = 2, then what is the value of λ?
(a) 1 (b) −1 (c) 2 (d) −2

83. If x cos θ + y sin θ = 2 is perpendicular to the line x − y = 3, then what is one of the values of θ?
(a) π/6 (b) π/4 (c) π/2 (d) π/3

84. If x-axis is tangent to the circle x² + y² + 2gx + 2fy + k = 0, then which one of the following is correct?
(a) g² = k (b) g² = f (c) f² = k (d) f² = g

85. What is the sum of focal radii of any point on an ellipse equal to?
(a) Length of latus rectum (b) Length of major axis (c) Length of minor axis (d) Length of semi-latus rectum

86. What does an equation of the first degree containing one arbitrary parameter passing through a fixed point represent?
(a) Circle (b) Straight line (c) Parabola (d) Ellipse

87. What is the foot of the perpendicular from the point (2, 3) on the line x + y − 11 = 0?
(a) (1, 10) (b) (5, 6) (c) (6, 5) (d) (7, 4)

88. Consider the following statements:

1. The equation to a straight line parallel to the axis of x is y = d, where d is a constant.
2. The equation to the axis of x is x = 0.
   Which of the statements given above is/are correct?
   (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2

89. What is the angle between the planes 2x − y + z = 6 and x + y + 2z = 3?
(a) π/2 (b) π/3 (c) π/4 (d) π/6

90. What is the equation of a plane through the x-axis and passing through the point (1, 2, 3)?
(a) x + y + z = 6 (b) x = 1 (c) y + z = 5 (d) z + y = 1

91. The points (1, 3, 4), (−1, 6, 10), (−7, 4, 7) and (−5, 1, 1) are the vertices of a
(a) rhombus (b) rectangle (c) parallelogram (d) square

92. What is the number of planes passing through three non-collinear points?
(a) 3 (b) 2 (c) 1 (d) 0

93. What is the angle between the lines x + z = 0, y = 0 and 20x = 15y = 12z?
(a) cos⁻¹(1/5) (b) cos⁻¹(1/7) (c) sin⁻¹(1/5) (d) sin⁻¹(1/7)

94. What is the eccentricity of an ellipse if its latus rectum is equal to one-half of its minor axis?
(a) 1/4 (b) 1/2 (c) √3/4 (d) √3/2

95. Under what condition does the equation x² + y² + z² + 2ux + 2vy + 2wz + d = 0 represent a real sphere?
(a) u² + v² + w² = d² (b) u² + v² + w² > d (c) u² + v² + w² < d (d) u² + v² + w² < d²

96. Which one of the following vectors of magnitude √51 makes equal angles with the three vectors â = (î − 2ĵ + 2k̂)/3, b̂ = (−4î − 3k̂)/5 and ĉ = ĵ?
(a) 5î − ĵ − 5k̂ (b) 5î + ĵ + 5k̂ (c) −5î − ĵ + 5k̂ (d) 5î + 5ĵ − k̂

97. If |a⃗| = 2, |b⃗| = 5 and |a⃗ × b⃗| = 8, then what is the value of a⃗ · b⃗?
(a) 4 (b) 6 (c) 8 (d) 10

98. If |a⃗ + b⃗| = |a⃗ − b⃗|, then which one of the following is correct?
(a) a⃗ is parallel to b⃗ (b) a⃗ is perpendicular to b⃗ (c) a⃗ = b⃗ (d) Both a⃗ and b⃗ are unit vectors

99. If a⃗ = î − 2ĵ + 5k̂, b⃗ = 2î + ĵ − 3k̂, then what is (b⃗ − a⃗) · (3a⃗ + b⃗) equal to?
(a) 106 (b) −106 (c) 53 (d) −53

100. Let a⃗, b⃗, c⃗ be the position vectors of points A, B, C respectively. Under which one of the following conditions are the points A, B, C collinear?
(a) a⃗ × b⃗ = 0
(b) b⃗ × c⃗ is parallel to a⃗ × b⃗
(c) a⃗ × b⃗ is perpendicular to b⃗ × c⃗
(d) (a⃗ × b⃗) + (b⃗ × c⃗) + (c⃗ × a⃗) = 0

101. If a⃗ = î + ĵ + k̂, b⃗ = î − ĵ + k̂ and c⃗ = î + ĵ − k̂, then what is a⃗ × (b⃗ + c⃗) + b⃗ × (c⃗ + a⃗) + c⃗ × (a⃗ + b⃗) equal to?
(a) 2î + 3ĵ − k̂ (b) 2î − 3ĵ − k̂ (c) 3î + ĵ + k̂ (d) 0

102. If A, B, C are any three arbitrary events, then which one of the following expressions shows that both A and B occur but not C?
(a) A ∩ B̅ ∩ C̅ (b) A ∩ B ∩ C̅ (c) A ∩ B ∩ C (d) A ∩ B̅ ∩ C

103. The average sales and standard deviation of sales for four months for a company are as follows:

Month 1 2 3 4
Average sales 30 57 82 28
S.D. of sales 2 3 4 2

During which month are the sales most consistent?
(a) Month 1 (b) Month 2 (c) Month 3 (d) Month 4

104. By Bayes’ theorem, which one of the following probabilities is calculated?
(a) Prior probability (b) Likelihood probability (c) Posterior probability (d) Conditional probability

105. Given that P(A) = 1/3, P(B) = 1/4, P(A|B) = 1/6, then what is P(B|A) equal to?
(a) 1/4 (b) 1/8 (c) 3/4 (d) 1/2

106. If A and B are two mutually exclusive and exhaustive events with P(B) = 3P(A), then what is the value of P(B̅)?
(a) 3/4 (b) 1/4 (c) 1/3 (d) 2/3

107. Two dice are thrown. What is the probability that the sum of the faces equals or exceeds 10?
(a) 1/12 (b) 1/4 (c) 1/3 (d) 1/6

108. For a binomial distribution b(n, p), np = 4 and variance = 4/3. What is the probability P(x ≥ 5) equal to?
(a) (2/3)⁶ (b) 2⁵/3⁶ (c) (1/3)⁶ (d) 2⁸/3⁶

109. The harmonic mean of two numbers is 21.6. If one of the numbers is 27, then what is the other number?
(a) 16.2 (b) 17.3 (c) 18 (d) 20

110. The marks scored by two students A and B in six subjects are given below:
A: 71 56 55 75 54 49
B: 55 74 83 54 38 52
Which one of the following statements is most appropriate?
(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

111. In a factory, there are 30 men and 20 women employees. If the average salary of men is Rs 4,050 and the average salary of all the employees is Rs 3,550, then what is the average salary of women?
(a) Rs 3,800 (b) Rs 3,300 (c) Rs 3,000 (d) Rs 2,800

112. What is the standard deviation of numbers 7, 9, 11, 13, 15?
(a) 2.2 (b) 2.4 (c) 2.6 (d) 2.8

113. When a card is drawn from a well-shuffled pack of cards, what is the probability of getting a Queen?
(a) 2/13 (b) 1/13 (c) 1/26 (d) 1/52

114. If the monthly expenditure pattern of a person who earns a monthly salary of Rs 15,000 is represented in a pie diagram, then the sector angle of an item on transport expenses measures 15°. What is his monthly expenditure on transport?
(a) Rs 450 (b) Rs 625 (c) Rs 675 (d) Cannot be computed from the given data

115. If ∑{i=1}^n (x_i − 2) = 110, ∑{i=1}^n (x_i − 5) = 20, then what is the mean?
(a) 11/2 (b) 2/11 (c) 17/3 (d) 17/9

116. Let f : R → R be a function defined as f(x) = x|x|; for each x ∈ R. Which one of the following is correct?
(a) f is one-one but not onto
(b) f is onto but not one-one
(c) f is both one-one and onto
(d) f is neither one-one nor onto

117. What is the set of all points where the function f(x) = x/(1 + |x|) is differentiable?
(a) (−∞, ∞) (b) (0, ∞) only (c) (−∞, 0) ∪ (0, ∞) only (d) (−∞, 0) only

118. Let y(x) = a xⁿ and δy denote small change in y. What is lim_{δx→0} (δy/δx)?
(a) 0 (b) 1 (c) a n x^{n−1} (d) a xⁿ log(a x)

119. What is the solution of the differential equation dy/dx = e^{x−y}(e^{y−x} − e^y)?
(a) y = x − e^x + c (b) y = x + e^x + c (c) y = e^{x−y} − e^y + c (d) None of the above

120. What is the area of the triangle formed by the lines y − x = 0, y + x = 0, x = c?
(a) c/2 (b) c² (c) 2c² (d) c²/2

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