UPSC NDA 2017 (2) Mathematics Paper

1. If x + log₁₀(1+2^x) = x log₁₀ 5 + log₁₀ 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 (101)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

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. It is given that the roots of the equation x² - 4x - log₃ P = 0 are real. For this, the minimum value of P is

(a) 1/27
(b) 1/64
(c) 1/81
(d) 1

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+ x)⁵⁰, the sum of the coefficients of odd powers of x is

(a) 2²⁶
(b) 2⁴⁹
(c) 2⁵⁰
(d) 2⁵¹

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 equations kx + y + z = 1, x + ky + z = k and x + y + kz= k² has no solution if k equals

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

31. The angle of elevation of a stationary cloud from a point 25 m above a lake is 15° and the angle of depression of its image in the lake is 45°. 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 tan 9° - tan 27º – tan 63° + tan 81° is equal to

(a) -1
(b) 0
(c) 1
(d) 4

41. The points (a b), (0, 0), (-a, -b) and (ab, b²) 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

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

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 2x² + 7y² = 20 is

(a) outside the ellipse
(b) inside the ellipse but not at the focus
(c) on the ellipse
(d) at the focus