The UPSC conducted the NDA & NA (II) 2017 Exam on 10 September 2017. The exam consisted of two papers – Mathematics and General Ability Test (GAT). The Mathematics Question paper consisted of questions from various topics such as – Algebra, Matrices, Trigonometry, Calculus, etc. For the benefit of NDA Exam aspirants, we are providing the Mathematics Qn. Paper below.
1. If x + log_{10}(1 + 2^{x}) = xlog_{10} 5 + log_{10} 6 then x is equal to
(a) 2, –3
(b) 2 only
(c) 1
(d) 3
2. The remainder and the quotient of the binary division (101110)_{2} ÷ (110)_{2} are respectively
(a) (111)_{2} and (100)_{2}
(b) (100)_{2} and (111)_{2}
(c) (101)_{2} and (111)_{2}
(d) (100)_{2} and (100)_{2}
3. The matrix A has x rows and x + 5 columns. The matrix B has y rows and 11 – y columns. Both AB and BA exist.
What are the values of x and y respectively?
(a) 8 and 3
(b) 3 and 4
(c) 3 and 8
(d) 8 and 8
4. If S_{n} = nP + n(n – 1)Q/2, where S_{n} denotes the sum of the first n terms of an AP, then the common difference is
(a) P + Q
(b) 2P + 3Q
(c) 2Q
(d) Q
5. The roots of the equation (q – r)x^{2} + (r – p)x + (p – q) = 0 are
(a) (r – p) / (q – r), 1/2
(b) (p – q) / (q – r), 1
(c) (q – r) / (p – q), 1
(d) (r – p) / (p – q), 1/2
6. If E is the universal set and A = B ∪ C, then the set
E – (E – (E – (E – (E – A)))) is same as the set
(a) B' ∪ C'
(b) B ∪ C
(c) B' ∪ C'
(d) B ∪ C
7. If A = {x : x is a multiple of 2}, B = {x : x is a multiple of 5} and C = {x : x is a multiple of 10}, then A ∩ (B ∩ C) is equal to
(a) A
(b) B
(c) C
(d) {x : x is a multiple of 100}
9. If |a| denotes the absolute value of an integer, then which of the following are correct?
I. |ab| = |a| |b|
II. |a + b| ≤ |a| + |b|
III. |a – b| ≥ ||a| - |b||
Select the correct answer using the code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2, and 3
10. How many different permutations can be made out of the letters of the word ‘PERMUTATION’?
(a) 19958400
(b) 19954800
(c) 19952400
(d) 39916800
12. The sum of all real roots of the equation
|x – 3|^{2} + |x – 3| – 2 = 0 is
(a) 2
(b) 3
(c) 4
(d) 6
13. If is given that the roots of the equation x^{2} – 4x – log_{3} P = 0 are real. For this, the minimum value of P is
(a) 1/27
(b) 1/64
(c) 1/81
(d) 1
14. If A is a square matrix, then the value of adj A^{T} – (adj A)^{T} is equal to
(a) A
(b) 2 |A| I, where I is the identity matrix
(c) null matrix whose order is same as that of A
(d) unit matrix whose order is same as that of A
15. The value of the product
(a) 6
(b) 36
(c) 216
(d) 512
16. The value of the determinant
(a) 1
(b) cos θ
(c) sin θ
(d) cos 2θ
17. The number of terms in the expansion of (x + a)^{100} + (x – a)^{100} after simplification is
(a) 202
(b) 101
(c) 51
(d) 50
18. In the expansion of (1 + )^{50}, the sum of the coefficients of odd powers of x is
(a) 226
(b) 249
(c) 250
(d) 251
20. A person is to count 4500 notes. Let an denote the number of notes he counts in the nth minute. If a_{1} = a_{2} = a_{3} = .. = a_{10} = 150, and a_{10}, a_{11}, a_{12}, … are in AP with the common difference –2, then the time taken by him to count all the notes is
(a) 24 minutes
(b) 34 minutes
(c) 125 minutes
(d) 135 minutes
(a) 1
(b) 4
(c) 8
(d) 16
22. If we define a relation R on the set N x N as (a, b) R (c, d) a + d = b + c for all (a, b), (c, d) ∈ N x N, then the relation is
(a) symmetric only
(b) symmetric and transitive only
(c) equivalence relation
(d) reflexive only
23. If y = x + x^{2} + x^{3} + … up to infinite terms, where x < 1, then which one of the following is correct?
(a) x = y / 1+y
(b) x = y / 1 – y
(c) x = 1 + y / y
(d) x = 1 – y /y
24. If α and β are the roots of the equation 3x^{2} + 2x + 1 = 0, then the equation whose roots are α + β^{–1} and β + α^{–1} is
(a) 3x^{2} + 8x + 16 = 0
(b) 3x^{2} – 8x – 16 = 0
(c) 3x^{2} + 8x – 16 = 0
(d) x^{2} + 8x + 16 = 0
(a) log_{e} 9
(b) 0
(c) 1
(d) log_{e} 3
NDA & NA (II) 2017 Exam – GAT Question Paper & Solution
26. A tea party is arranged for 16 people along two sides of a long table with eight chairs on each side. Four particular men wish to sit on one particular side and two particular men on the other side. The number of ways they can be seated is
(a) 24 X 8! X 8!
(b) (8!)^{3}
(c) 210 X 8! X 8!
(d) 16!
27. The system of equation kx + y + z = 1, x + ky + z = k and x = y + kz = k^{2} has no solution if k equals
(a) 0
(b) 1
(c) –1
(d) –2
28. If 1.3 + 2.32 + 3.33 + … + n.3^{n} = (2n – 1)3^{a} + b / 4
then a and b are respectively
(a) n, 2
(b) n, 3
(c) n + 1, 2
(d) n + 1, 3
29. In ΔPQR, ∠R = π/2, If tan (P/2) and tan (Q/2) of the equation ax^{2} + bx + c = 0, then which one of the following is correct?
(a) a = b + c
(b) b = c + a
(c) c = a + b
(d) b = c
30. If |z – 4/z| = 2, then the maximum value of |z| is equal to
(a) 1 + √3
(b) 1 + √5
(c) 1 – √5
(d) √5 – 1
31. The angle of elevation of a stationary cloud from a point 25 m above a lake is 150 and the angle of depression of its image in the lake is 450. The height of the cloud above the lake level is
(a) 25 m
(b) 25√3 m
(c) 50 m
(d) 50√3 m
32. The value of
tan9^{} – tan27^{} – tan63^{} + tan81^{} is equal to
(a) –1
(b) 0
(c) 1
(d) 4
33. The value of √3 cosec 20^{} – sec 20^{} is equal to
(a) 4
(b) 2
(c) 1
(d) –4
34. Angle α is divided into two parts A and B such that A – B = x and tan A : tan B = p : q. The value of sin x is equal to
(a) (p + q) sin α / p – q
(b) p sin α / p + q
(c) p sin α / p – q
(d) (p – q) sin α / p + q
35. The value of
sin^{-1} (3/5) + tan^{-1} (1/7) is equal to
(a) 0
(b) π/4
(c) π/3
(d) π/2
36. The angles of elevation of the top of a tower from the top and foot of a pole are respectively 300 and 450. If h_{T} is the height of the tower and hp is the height of the pole, then which of the following are correct?
Select the correct answer using the code given below.
(a) 1 and 3 only
(b) 2 and 3 only
(c) 1 and 2 only
(d) 1, 2 and 3
37. In a triangle ABC, a – 2b + c = 0. The value of cot (A/2) cot (C/2) is
(a) 9/2
(b) 3
(c) 3/2
(d) 1
39. In triangle ABC, if sin^{2} A sin^{2} B sin^{2} C/cos^{2} A + cos^{2} B + cos^{2} C = 2 then the triangle is
(a) right-angled
(b) equilateral
(c) isosceles
(d) obtuse-angled
40. The principal value of sin^{–1} x lies in the interval
41. The points (a, b), (0, 0), (–a, –b) and (ab, b^{2}) are
(a) the vertices of a parallelogram
(b) the vertices of a rectangle
(c) the vertices of a square
(d) collinear
42. The length of the normal from origin to the plane x = 2y – 2z = 9 is equal to
(a) 2 units
(b) 3 units
(c) 4 units
(d) 5 units
43. If α, β and γ are the angles which the vector OP (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct?
1. cos^{2}α + cos^{2}β = sin^{2}γ
2. sin^{2}α + sin^{2}β = cos^{2}γ
3. sin^{2}α + sin^{2}β + sin^{2}γ = 2
Select the correct answer using the code given below.
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
44. The angle between the lines x + y – 3 = 0 and x – y + 3 = 0 is α and the acute angle between the lines x – √3y + 2√3 = 0 and √3x – y + 1 = 0 is β. Which one of the following is correct ?
(a) α = β
(b) α > β
(c) α < β
(d) α = 2
(a) √3 units
(b) 2√3 units
(c) √3/2 units
(d) 1/√3 unit
(a) 5 units
(b) 7 units
(c) 9 units
(d) 10 units
49. A man running round a racecourse notes that the sum of the distances of two flag-posts from him is always 10 m and the distance between the flag-posts is 8 m. The area of the path he encloses is
(a) 18π square metres
(b) 15π square metres
(c) 12π square metres
(d) 8π square metres
50. The distance of the point (1, 3) from the line 2x + 3y = 6, measured parallel to the line 4x + y = 4, is
(a) 5/√13 units
(b) 3/√17 units
(c) √17 units
(d) √17/2 units
(a) 0
(b) 1
(c) a + b + c
(d) abc
52. The point of intersection of the line joining the points (–3, 4, –8) and (5, –6, 4) with the XY -plane is
53. If the angle between the lines whose direction ratios are (2, -1, 2) and (x, 3, 5) is π/4, then the smaller value of x is
(a) 52
(b) 4
(c) 2
(d) 1
54. The position of the point (1, 2) relative to the ellipse
2x^{2} + 7y^{2} = 20 is
(a) outside the ellipse
(b) inside the ellipse but not at the focus
(c) on the ellipse
(d) at the focus
55. The equation of a straight line which cuts off an intercept of 5 units on negative direction of y-axis and makes an angle 120^{} with positive direction of x-axis is
(a) y + √3x + 5 = 0
(b) y – √3x + 5 = 0
(c) y + √3x – 5 = 0
(d) y – √3x – 5 = 0
56. The equation of the line passing through the point (2, 3) and the point of intersection of lines 2x – 3y + 7 = 0 and 7x + 4y + 2 = 0 is
(a) 21 + 46y – 180 = 0
(b) 21x – 46y + 96 = 0
(c) 46x + 21y – 155 = 0
(d) 46x – 21y – 29 = 0
57. The equation of the ellipse whose centre is at origin, major axis is along x-axis with eccentricity 3/4 and latus rectum 4 units is
58. The equation of the circle which passes through the points (1, 0), (0, –6) and (3, 4) is
(a) 4x^{2} + 4y^{2} + 142x + 47y + 140 = 0
(b) 4x^{2} + 4y^{2} – 142x + 47y + 138 = 0
(c) 4x^{2} + 4y^{2} – 142x + 47y + 138 = 0
(d) 4x^{2} + 4y^{2} + 150x – 49y + 138 = 0
59. A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B and C respectively. The locus of the centre of the sphere OABC, O being the origin, is
(a) x/2 + y/b + z/c = 1
(b) a/x + b/y + c/z = 1
(c) a/x + b/y + c/z = 2
(d) x/a + y/b + z/c = 2
60. The equation of the plane passing through the line of intersection of the planes x + y + z = 1, 2x + 3y + 4z = 7, and perpendicular to the plane x – 5y + 3z = 5 is given by
(a) x + 2y + 3z – 6 = 0
(b) x + 2y + 3z + 6 = 0
(c) 3x + 4y + 5z – 8 = 0
(d) 3x + 4y + 5z + 8 = 0
61. The inverse of the function y = 5^{ln x} is
62. A function is defined as follows:
Which one of the following is correct in respect of the above function?
(a) f(x) is continuous at x = 0 but not differentiable at x = 0
(b) f(x) is continuous as well as differentiable at x = 0
(c) f(x) is discontinuous at x = 0
(d) None of the above
64. Consider the following:
1. x + x^{2} is continuous at x = 0
2. x + cos 1/x is discontinuous at x = 0
3. x^{2} + cos 1/x is continuous at x = 0
Which of the above are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
65. Consider the following statements:
1. dy/dx at a point on the curve gives slope of the tangent at that point.
3. If s(t) gives displacement of a particle at time t, then ds/dt gives its acceleration at that instant.
Which of the above statements is/are correct?
(a) 1 and 2 only
(b) 2 only
(c) 1 only
(d) 1, 2 and 3
(a) 0
(b) 1
(c) x-1 / x+1
(d) x+1 / x-1
68. A function is defined in (0, ∞) by
Which one of the following is correct in respect of the derivative of the function, i.e., f '(x)?
(a) f '(x) = 2x for 0 < x ≤ 1
(b) f '(x) = –2x for 0 < x ≤ 1
(c) f '(x) = –2x for 0 < x < 1
(d) f '(x) = 0 for 0 < x < ∞
69. Which one of the following is correct in respect of the function f(x) = x(x – 1)(x + 1)?
(a) The local maximum value is larges than local minimum value
(b) The local maximum value is smaller than local minimum value
(c) The function has no local maximum
(d) The function has no local minimum
70. Consider the following statements:
1. Derivative of f(x) may not exist at some point.
2. Derivative of f(x) may exist finitely at some point
3. Derivative of f(x) may be infinite (geometrically) at some point.
Which of the above statements are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
71. The maximum value of In x / x is
(a) e
(b) 1/e
(c) 2/e
(d) 1
72. The function f(x) = |x| = |x| – x^{3} is
(a) odd
(b) even
(c) both even and odd
(d) neither even nor odd
74. The general solution of dy/dx = ax + h / by + k represents a circle only when
(a) a = b = 0
(b) a = –b ≠ 0
(c) a = b ≠ 0, h = k
(d) a = b ≠ 0
(a) l = 1, m = 1
(b) l = 2/π, m = ∞
(c) l = 2/π, m = 0
(d) l = 1, m = ∞
(a) 8
(b) 4
(c) 2
(d) 0
77. The area bounded by the curve |x| + |y| = 1 is
(a) 1 square unit
(b) 2√2 square units
(c) 2 square units
(d) 2√3 square units
79. The left-hand derivative of f(x) = [x] sin(πx) at x = k where k is an integer and [x] is the greatest integer function, is
(a) (–1)^{k}(k – 1)π
(b) (–1)^{k–1}(k – 1)π
(c) (–1)^{k}kπ
(d) (–1)^{k–1}kπ
80. If f(x) = x/2 – 1 , then on the interval [0, π] which one of the following is correct?
(a) tan [f(x)], where [.] is the greatest integer function, and 1/f(x) are both continuous
(b) tan [f(x)], where [.] is the greatest integer function, and f^{–1}(x) are both continuous
(c) tan [f(x)], where [.] is the greatest integer function, and 1/f(x) are both discontinuous
(d) tan [f(x)], where [.] is the greatest integer function, is discontinuous but 1/f(x) is continuous
(a) 3 and 2
(b) 2 and 2
(c) 2 and 3
(d) 1 and 3
82. If y = cos^{-1}(2x/1+x^{2}), then dy/dx is equal to
(a) – 2/1+x^{2} for all |x| < 1
(b) – 2/1+x^{2} for all |x| > 1
(c) 2/1+x^{2} for all |x| < 1
(d) None of the above
83. The set of all points, where the function f(x) = √1 – e^{-x2} differentiable, is
84. Match List-I with List-II and select the correct answer using the code given below the lists:
List-I (Function) |
List-II (Maximum value) |
||
A. |
sin x + cos x |
1. |
√10 |
B. |
3sin x + 4cos x |
2. |
√2 |
C. |
2sin x + cos x |
3. |
5 |
D. |
sin x + 3cos x |
4. |
√5 |
Code:
(a) (A-2), (B-3), (C-1), (D-4)
(b) (A-2), (B-3), (C-4), (D-1)
(c) (A-3), (B-2), (C-1), (D-4)
(d) (A-3), (B-2), (C-4), (D-1)
85. If f(x) = x(√x - √x + 1), then f(x) is
(a) continuous but not differentiable at x = 0
(b) differentiable at x = 0
(c) not continuous at x = 0
(d) None of the above
(a) 251
(b) 250
(c) 1
(d) 0
(a) x(ln x)^{–1} + c
(b) x(ln x)^{–2} + c
(c) x(ln x) + c
(d) x(ln x)^{2} + c
89. A cylindrical jar without a lid has to be constructed using a given a given surface area of a metal sheet. If the capacity of the jar is to be maximum then the diameter of the jar must be k times the height of the jar. The value of k is
(a) 1
(b) 2
(c) 3
(d) 4
(a) π/4
(b) π/2
(c) π/2√2
(d) π/√2
91. Let g be the greatest integer function. Then the function f(x) = (g(x))^{2} – g(x^{2}) is discontinuous at
(a) all integers
(b) all integers except 0 and 1
(c) all integers except 0
(d) all integers except 1
92. The differential equation of minimum order by eliminating the arbitrary constants A and C in the equation y = A [sin (x + C) + cos(x + C)] is
(a) y" + (sin x + cos x)y' = 1
(b) y" = (sin x + cos x)y'
(c) y" = (y')^{2} + sin x cos x
(d) y" + y = 0
93. Consider the following statements:
Statement I: x > sin x for all x > 0
Statement II: f(x) = x – sin x is an increasing function for all x > 0
Which one of the following is correct in respect of the above statements?
(a) Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I
(b) Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I
(c) Statement I is true but Statement II is false
(d) Statement I is false but Statement II is true
95. If f(x) 4x + x^{4} / 1 + 4x^{3} and g(x) = In (1 + x / 1 – x), then what is the value of f ^{o }g(e – 1 / e + 1) equal to?
(a) 2
(b) 1
(c) 0
(d) 1/2
96. The value of the determinant is equal to
(a) (α – β) (β – γ) (α – γ)
(b) (α – β) (β – γ) (γ – α)
(c) (α – β) (β – γ) (γ – α) (α + β + γ)
(d) 0
(a) A^{2} = –2A
(b) A^{2} = –4A
(c) A^{2} = –3A
(d) A^{2} = 4A
99. Geometrically Re(z^{2} – i) = 2, where i = √-1 and Re is the real part, represents
(a) circle
(b) ellipse
(c) rectangular hyperbola
(d) parabola
(a) 0
(b) 1
(c) pa + qb + rc
(d) pq + qb + rc + a + b + c
101. A committee of two persons is selected from two men and two women. The probability that the committee will have exactly one woman is
(a) 1/6
(b) 2/3
(c) 1/3
(d) ½
102. Let a die be loaded in such a way that even faces are twice likely to occur as the odd faces. What is the probability that a prime number will show up when the die is tossed?
(a) 1/3
(b) 2/3
(c) 4/9
(d) 5/9
103. Let the sample space consist of non-negative integers up to 50, X denote the numbers which are multiples of 3 and Y denote the odd numbers. Which of the following is/are correct?
1. P(X) = 8/25
2. P(Y) = ½
Select the correct answer using the code given below.
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
(a) 1/6
(b) 1/4
(c) 1/3
(d) 1/2
105. Consider the following statements:
1. Coefficient of variation depends on the unit of measurement of the variable.
2. Range is a measure of dispersion.
3. Mean deviation is least when measured about median.
Which of the above statements are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
106. Given that the arithmetic mean and standard deviation of a sample of 15 observations are 24 and 0 respectively. Then which one of the following is the arithmetic mean of the smallest five observations in the data?
(a) 0
(b) 8
(c) 16
(d) 24
107. Which one of the following can be considered as appropriate pair of values of regression coefficient of y on x and regression coefficient of x on y?
(a) (1, 1)
(b) (-1, 1)
(c) (- ½, 2)
(d) (1/3, 10/3)
(a) 1/5
(b) 1/7
(c) 1/8
(d) 1/10
109. In a binomial distribution, the mean is 2/3 and the variance is 5/9. What is the probability that X = 2?
(a) 5/36
(b) 25/36
(c) 25/216
(d) 25/54
110. The probability that a ship safely reaches a port is 1/3. The probability that out of 5 ships, at least 4 ships would arrive safely is
(a) 1/243
(b) 10/243
(c) 11/243
(d) 13/243
111. What is the probability that at least two persons out of a group of three persons were born in the same month (disregard year)?
(a) 33/144
(b) 17/72
(c) 1/144
(d) 2/9
(a) Y = 3.2X + 58
(b) X = 3.2Y + 58
(c) X = –8 + 0.2Y
(d) Y = –8 + 0.2X
(a) 1/12
(b) ¾
(c) 1/5
(d) 1/9
114. The following table gives the monthly expenditure of two families:
Expenditure (in Rs.) |
||
Items |
Family A |
Family B |
Food |
3,500 |
2,700 |
Clothing |
500 |
800 |
Rent |
1,500 |
1,000 |
Education |
2,000 |
1,800 |
Miscellaneous |
2,500 |
1,800 |
In constructing a pie diagram to the above data, the radii of the circles are to be chosen by which one of the following ratios?
(a) 1 : 1
(b) 10 : 9
(c) 100 : 91
(d) 5 : 4
115. If a variable takes values 0, 1, 2, 3, …, n with frequencies 1, C(n, 1), C(n, 2), C(n, 3), …, C(n, n) Respectively, then the arithmetic mean is
(a) 2n
(b) n + 1
(c) n
(d) n/2
116. In a multiple-choice test, an examinee either knows the correct answer with probability p, or guesses with probability 1 – p. The probability of answering a question correctly is 1/m, if he or she merely guesses. If the examinee answers a question correctly, the probability that he or she really knows the answer is
(a) mp/1+mp
(b) mp/1+(m-1)p
(c) (m-1)p/1+(m-1)p
(d) (m-1)p/1+mp
117. If x_{1} and x_{2} are positive quantities, then the condition for the difference between the arithmetic mean and the geometric mean to be greater than 1 is
118. Consider the following statements:
1. Variance is unaffected by change of origin and change of scale.
2. Coefficient of variance is independent of the unit of observations.
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
119. Five sticks of length 1, 3, 5, 7 and 9 feet are given. Three of these sticks are selected at random. What is the probability that the selected sticks can form a triangle?
(a) 0.5
(b) 0.4
(c) 0.3
(d) 0
120. The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is
(a) 0.36
(b) 0.2
(c) 0.6
(d) 0.9
NDA Exam Question Papers (2005-17) with Answer Keys
NDA & NA (II) 2017 Exam & SSB: All you need to know to clear the Exam