 # UPSC NDA & NA (I) 2016: Maths Question Paper & Answer Key

For the benefit of NDA aspirants, Jagran Josh is providing the NDA & NA (I) 2016 Maths Question Paper and the official Answer Key as given by the UPSC.

The Union Public Service Commission (UPSC) conducted the National Defence Academy & Naval Academy (NDA & NA) (I) Exam on 17 April 2017. The exam consisted of two papers – General Ability Test (GAT) and Mathematics. For the benefit of NDA aspirants, the Jagran Josh is providing the Mathematics Question Paper (Series A) and the official Answer Key as provided by the UPSC, which is as given below.    (a) [0, 1)

(b) [0, 1]

(c) (0, 1)

(d) (0, 1]

Ans. (a) [0, 1)

7. A straight line intersects x and y axes at P and Q respectively. If (3, 5) is the middle point of PQ, then what is the area of the triangle OPQ?

(a) 12 square units

(b) 15 square units

(c) 20 square units

(d) 30 square units

Ans. (d) 30 square units 9. What is the maximum value of the function f (θ) ?

(a) 1

(b) 2

(c) 3

(d) 4

Ans. (d) 4

10. What is the minimum value of the function f (θ) ?

(a) 0

(b) 1

(c) 2

(d) 3

Ans. (d) 3 Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (c) Both 1 and 2

For the next two (2) items that follow:

Consider the curves 12. What do the curves intersect?

(a) At (2, 3) only

(b) At (-1, -2) only

(c) At (2, 3) and (-1, -2)

(d) Neither at (2, 3 nor at (-1, -2)

Ans. (c) At (2, 3) and (-1, -2) 14. How many solutions does the function f(x) = 1 have?

(a) One

(b) Two

(c) Three

(d) Four

Ans. (b) Two

15. How many solutions does the function f(x) = 1 have?

(a) One

(b) Two

(c) Three

(d) Four

Ans. (a) One   18. Which one of the following statements is correct?

(a) f(x) is continuous but not differentiable at x = 0

(b) f(x) is continuous but not differentiable at x = 1

(c) f(x) is differentiable at x = 1

(d) f(x) is not differentiable at x = 0 and x = 1

Ans. (b) f(x) is continuous but not differentiable at x = 1   23. Consider the following statements:

1. The sequence {a2n} is in AP with common difference zero.

2. The sequence {a2n+1} is in AP with common difference zero.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (c) Both 1 and 2 25. Which of the following statements are not correct?

1. y as a function of x is not defined for all real x.

2. y as a function of x is not continuous at x = 0.

3. y as a function of x is differentiable for all x.

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

Ans. (d) 1, 2 and 3   29. What is the number of points of local minima of the function f(x)?

(a) None

(b) One

(c) Two

(d) Three

Ans. (c) Two

30. What is the number of points of local maxima of the function f(x)?

(a) None

(b) One

(c) Two

(d) Three

Ans. (b) One

31. Let f(x) and g(x) be twice differentiable functions on [0, 2] satisfying f’’(x) = g’’(x), f’(1) = 4, g’(1) = 6, f(2) = 3 and g(2) = 9. Then what is f(x) – g(x) at x = 4 equal to?

(a) – 10

(b) – 6

(c) – 4

(d) 2

Ans. (a) – 10

For the next two (2) items that follow:

Consider the curves

y = |x – 1| and |x| = 2

32. What is/are the point(s) of intersection of the curves?

(a) (-2, 3) only

(b) (2, 1) only

(c) (-2, 3) and (2, 1)

(d) Neither (-2, 3) nor (2, 1)

Ans. (c) (-2, 3) and (2, 1)

33. What is the area of the region bounded by the curves and x-axis?

(a) 3 square units

(b) 4 square units

(c) 5 square units

(d) 6 square units

Ans. (c) 5 square units 34. What is the value of f’(0)?

(a) p3

(b) 3p3

(c) 6p3

(d) -6p3

Ans. (d) -6p3  37. What is the value of

(a)

(b)

(c)

(d) None of the above

Ans. (d) None of the above

For the next two (2) items that follow: 40.  What is the distance between the centres of the two circles?

(a) 5 units

(b) 6 units

(c) 8 units

(d) 10 units

Ans. (a) 5 units

41. If the circles intersect at two distinct points, then which one of the following is correct?

(a) r = 1

(b) 1 < r < 2

(c) r = 2

(d) 2 < r < 8

Ans. (d) 2 < r < 8

For the next two (2) items that follow:

Consider the two lines

x + y + 1 = 0 and 3x + 2y + 1 = 0

42. What is the equation of the line passing through the point of intersection of the given lines and parallel to x-axis?

(a) y + 1 = 0

(b) y – 1 = 0

(c) y – 2 = 0

(d) y + 2 = 0

Ans. (d) y + 2 = 0

43. What is the equation of the line passing through the point of intersection of the given lines and parallel to y-axis?

(a) x + 1 = 0

(b) x – 1 = 0

(c) x – 2 = 0

(d) x + 2 = 0

Ans. (b) x – 1 = 0

For the next two (2) items that follow:

Consider the equation

K sin x + cos 2x = 2k – 7

44. If the equation possesses solution, then what is the minimum value of k?

(a) 1

(b) 2

(c) 4

(d) 6

Ans. (b) 2

45. If the equation possesses solution, then what is the maximum value of k?

(a) 1

(b) 2

(c) 4

(d) 6

Ans. (d) 6 Ans. (b) In a (a) 0

(b) In a

(c) 1-a-1

(d) Limit does not exist 49. Consider the following statements:

1. z1z2z3 is purely imaginary.

2. z1z2 + z2z3 + z3z1 is purely real.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (c) Both 1 and 2

For the next two (2) items that follow:

Given that logx y, log z x, logy z are in GP, xyz = 64 and x3, y3, z3 are in AP.

50. Which one of the following is correct?

x, y and z are

(a) in AP only

(b) in GP only

(c) in both AP and GP

(d) neither in AP nor in GP

Ans. (c) in both AP and GP

51. Which one of the following is correct?

xy, yz and zx are

(a) in AP only

(b) in GP only

(c) in both AP and GP

(d) neither in AP nor in GP

Ans. (c) in both AP and GP

For the next two (2) items that follow:   Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (c) Both 1 and 2 Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (d) Neither 1 nor 2

For the next two (2) items that follow: Which of the above is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (b) 2 only

For the next three (3) items that follow:

Consider a parallelogram whose vertices are A (1, 2), B (4, y), C(x, 6) and D(3, 5) taken in order.

58. What is the value of AC2 – BD2 ?

(a) 25

(b) 30

(c) 36

(d) 40

Ans. (c) 36  61. What is f(1) equal to?

(a) – 2

(b) – 1

(c) 0

(d) 4

Ans. (d) 4

62. What is f’ (1) equal to?

(a) – 6

(b) – 5

(c) 1

(d) 0

Ans. (b) – 5

63. What is f’’’(10) equal to?

(a) 1

(b) 5

(c) 6

(d) 8

Ans. (c) 6

64. Consider the following:

1.  f(2) = f(1) – f(0)

2. f’’(2) – 2f’(1) = 12

Which of the above is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (c) Both 1 and 2

For the next three (3) items that follow:

A plane P passes through the line of intersection of the planes 2x – y + 3z = 2, x + y – z = 1 and the point (1, 0, 1).

65. What are the direction ratios of the line of intersection of the given planes?

(a) (2, -5, -3)

(b) (1, -5, -3)

(c) (2, 5, 3)

(d) (1, 3, 5)

Ans. (a) (2, -5, -3)

66. What is the equation of the plane P?

(a) 2x + 5y – 2 = 0

(b) 5x + 2y – 5 = 0

(c) x + y – 2 = 0

(d) 2x – y – 2z = 0

Ans. (b) 5x + 2y – 5 = 0 68. What is f’(4) equal to?

(a) – 4

(b) – 3

(c) 3

(d) 2

Ans. (c) 3

69. What is f”(2.5) equal to?

(a) -3

(b) -2

(c) 0

(d) 2

Ans. (b) -2

For the next two (2) items that follow:

Let f(x) be the greatest integer function and g(x) be the modulus function. (a) – 1

(b) 0

(c) 1

(d) 2

Ans. (c) 1 (a) -1

(b) 0

(c) 1

(d) 2

Ans. (b) 0

For the next two (2) items that follow:

Consider a circle passing through the origin and the points (a, b) and (-b, -a).

72. On which line does the centre of the circle lie?

(a) x + y = 0

(b) x – y = 0

(c) x + y = a + b

(d)  x – y = a2 – b2

Ans. (a) x + y = 0

73. What is the sum of the squares of the intercepts cut off by the circle on the axes?    Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (b) 2 only Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (d) Neither 1 nor 2 (a) 1, 2

(b) 2, 2

(c) 1, 3

(d) 1, 4

Ans. (d) 1, 4

(a) 0

(b) 1

(c) 2

(d) 4

Ans. (c) 2 (a) 0

(b) 1

(c) 2

(d) 4

Ans. (c) 2 (a) – 3

(b) 2

(c) 3

(d) 5

Ans. (b) 2 (a) a2

(b) a-2

(c) – a2

(d) – a

Ans. (a) a2 (a) 0

(b) 1

(c) – 1

(d) Limit does not exist

Ans. (a) 0

83. If A is a square matrix, then what is adj (A– 1) – (adj A) –1 equal to?

(a) 2 |A|

(b) Null matrix

(c) Unit matrix

(d) None of the above

Ans. (b) Null matrix

84. What is the binary equivalent of the decimal number 0.3125?

(a) 0.0111

(b) 0.1010

(c) 0.0101

(d) 0.1101

Ans. (c) 0.0101

85. Let R be a relation on the set N of natural numbers defined by ‘nRm n is a factor of m’. Then which one of the following is correct?

(a) R is reflexive, symmetric but not transitive

(b) R is transitive, symmetric but not reflexive

(c) R is reflexive, transitive but not symmetric

(d) R is an equivalence relation

Ans. (c) R is reflexive, transitive but not symmetric (a) 0

(b) 2

(c) 4

(d) 8

Ans. (d) 8

87. What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?

(a) 860

(b) 854

(c) 840

(d) 824

Ans. (b) 854

88. (a, 2b) is the mid-point of the line segment joining the points (10, -6) and (k, 4). If a – 2b = 7, then what is the value of k?

(a) 2

(b) 3

(c) 4

(d) 5

Ans. (a) 2 Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) 1 and 2

(d) 2 and 3

Ans. (b) 2 only (a) – 1

(b) 0

(c) 1

(d) 2

Ans. (c) 1

91. What is the mean deviation from the mean of the numbers 10, 9, 21, 16, 24?

(a) 5.2

(b) 5.0

(c) 4.5

(d) 4.0

Ans. (a) 5.2

92. Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?   (a) 1500

(b) 1600

(c) 1700

(d) 1800

Ans. (c) 1700

96. A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is queen of spade? 97. If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4? 98. A certain type of missile hits the target with probability p = 0.3. What is the least number of missiles should be fired so that there is at least an 80% probability that the target is hit?

(a)  5

(b) 6

(c) 7

(d) None of the above

Ans. (a)  5  (a) 1

(b) 3

(c) 6

(d) 9

Ans. (a) 1

102. A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?  104. For the data 3, 5, 1, 6, 5, 9, 5, 2, 8, 6 the mean, median and mode are x, y and z respectively. Which one of the following is correct?

(a) x = y ≠ z

(b) x ≠ y = z

(c) x ≠ y ≠ z

(d) x = y = z

Ans. (d) x = y = z

105. Consider the following statements in respect of a histogram:

1. The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and the x-axis.

2. When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (c) Both 1 and 2

106. A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times? 107. What is the number of ways in which 3 holiday travel tickets are to be given to 10 employees of an organization, if each employee is eligible for any one or more of the tickets?

(a) 60

(b) 120

(c) 500

(d) 1000

Ans. (d) 1000 109. What is the number of four-digit decimal numbers (< 1) in which no digit is repeated?

(a) 3024

(b) 4536

(c) 5040

(d) None of the above

Ans. (b) 4536   1. A2 = -A

2. A3 = 4A

Which of the above is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Ans. (b) 2 only 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

Ans. (d) 1, 2 and 3 (a) 30°

(b) 45°

(c) 60°

(d) 75°

Ans. (a) 30° 117. What is the number of different messages that can be represented by three 0’s and two 1’s?

(a) 10

(b) 9

(c) 8

(d) 7

Ans. (a) 10 (a) 10

(b) 2

(c) 1

(d) 0

Ans. (d) 0

120. Suppose ω1 and ω2 are two distinct cube roots of unity different from 1. Then what is (ω1 ─ ω2)2 equal to?

(a) 3

(b) 1

(c) ─1

(d) ─3

Ans. (d) ─3