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

May 10, 2017 18:20 IST

    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.

     

    UPSC NDA (I) 2016 Maths Question Paper Analysis Answer Key

                                                     Image Source: National Defence Academy

    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.

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    (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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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)

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    34. What is the value of f’(0)?

    (a) p3

    (b) 3p3

    (c) 6p3

    (d) -6p3

    Ans. (d) -6p3

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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:

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    Ans. (b) In a

    NDA NA-I 2016 Mathematics

     

    (a) 0

    (b) In a

    (c) 1-a-1

    (d) Limit does not exist

    NDA NA-I 2016 Mathematics

    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:

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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:

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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.

    NDA NA-I 2016 Mathematics

    (a) – 1

    (b) 0

    (c) 1

    (d) 2

    Ans. (c) 1

    NDA NA-I 2016 Mathematics

    (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?

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    (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

    NDA NA-I 2016 Mathematics

    (a) 0

    (b) 1

    (c) 2

    (d) 4

    Ans. (c) 2

    NDA NA-I 2016 Mathematics

    (a) – 3

    (b) 2

    (c) 3

    (d) 5

    Ans. (b) 2

    NDA NA-I 2016 Mathematics

    (a) a2

    (b) a-2

    (c) – a2

    (d) – a

    Ans. (a) a2

    https://www.jagranjosh.com/imported/images/E/Articles/nda-na-i-2016-mathematics46.jpg

    (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

    NDA NA-I 2016 Mathematics

    (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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    (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?

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    (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?

    NDA NA-I 2016 Mathematics

    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?

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    (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?

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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?

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    (a) 30°

    (b) 45°

    (c) 60°

    (d) 75°

    Ans. (a) 30°

    NDA NA-I 2016 Mathematics

    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

    NDA NA-I 2016 Mathematics

    (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

    For Previous NDA Exam Question Papers (2005-17) & Answer Keys, Click here

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