# UPSC NDA 2017 (1) Mathematics Paper

1. Let S be the set of all persons living in Delhi. We say that x, y in S are related if they were born in Delhi on the same day. Which one of the following is correct?

• (a) The relation is an equivalent relation
• (b) The relation is not reflexive but it is symmetric and transitive
• (c) The relation is not symmetric but it is reflexive and transitive
• (d) The relation is not transitive but it is reflexive and symmetric

2. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10). Then the number of subsets of A containing two or three elements is

• (a) 45
• (b) 120
• (c) 165
• (d) 330

7. Three-digit numbers are formed from the digits 1, 2 and 3 in such a way that the digits are not repeated. What is the sum of such three-digit numbers?

• (a) 1233
• (b) 1322
• (c) 1323
• (d) 1332

10. If the sum of m terms of an AP is n and the sum of n terms is m, then the sum of (m + n) terms is

• (a) mn
• (b) m+n
• (c) 2(m + n)
• (d) -(m+n)

13. If |z + 4| ≤ 3, then the maximum value of |z+ 1| is

• (a) 0
• (b) 4
• (c) 6
• (d) 10

16. The sum of the roots of the equation x2 + bx + c = 0 (where b and c are non-zero) is equal to the sum of the reciprocals of their squares. Then 1/c, b, c/b are in

• (a) AP
• (b) GP
• (c) HP
• (d) None of the above

19. The number of different words (eight-letter words) ending and beginning with a consonant which can be made out of the letters of the word 'EQUATION' is

• (a) 5200
• (b) 4320
• (c) 3000
• (d) 2160

20. The fifth term of an AP of n terms, whose sum is n2 - 2n, is

• (a) 5
• (b) 7
• (c) 8
• (d) 15

21. The sum of all the two-digit odd numbers is

• (a) 2475
• (b) 2530
• (c) 4905
• (d) 5049

28. If B is a non-singular matrix and A is a square matrix, then the value of det (B-1AB) is equal to

• (a) det (B)
• (b) det (A)
• (c) det (B-1)
• (d) det (A-1)

31. The equations

x +2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4

• (a) have the unique solution
• (b) have infinitely many solutions
• (c) are inconsistent
• (d) None of the above

35. Consider the set A of all matrices of order 3x3 with entries 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is -1. Then which one of the following is correct?

• (a) C is empty
• (b) B has as many elements as C
• (c) A = B ∪ C
• (d) B has thrice as many elements as C

41. From the top of a lighthouse, 100 m high, the angle of depression of a boat is tan-1(5/12). What is the distance between the boat and the lighthouse?

• (a) 120 m
• (b) 180 m
• (c) 240 m
• (d) 360 m

53. If the three consecutive vertices of a parallelogram are (-2, -1), (1, 0) and (4, 3), then what are the coordinates of the fourth vertex?

• (a) (1, 2)
• (b) (1, 0)
• (c) (0, 0)
• (d) (1, -1)

54. The two circles x2 + y2 = r2 and x2 + y2 - 10x + 16 = 0 intersect at two distinct points. Then which one of the following is correct?

• (a) 2 < r < 8
• (b) r = 2 or r = 8
• (c) r < 2
• (d) r > 2