Nda (Sep 2010) Mathematics Orc Academy

NDA (SEP 2010) MATHEMATICS ORC ACADEMY

1. The geometric mean of three numbers was computed as 6. It was subsequently found that, in this computation, a number 8 was wrongly read as 12. What is the correct geometric mean ?

(A) 4 (B) 35

2. Let A = 1234 = | aij |, where i, j = 1, 2. If its inverse matrix is |bij|, what is h22 ?

(A) – 2 (B) 1

3. The angle A lies in the third quadrant and it satisfies the equation 4 (sin2 x + cos x) = 1. What is the measure of the angle A ?

(A) 225° (B) 240°

4. What is the area enclosed between the curves y2 = I2x and the lines x = 0 and y = 6 ?

(A) 2 square units (B) 4 square units

5. In a triangle ABC, BC =39, AC = 5 and AB = 7. What is the measure of the angle A ?

(A) p4 (B) p3 (C) p2 (D) p6

6. What is the modulus of 1+2i1-(1-i)2 ?

(A) 1 (B) 5 (C) 3 (D) 5

7. If the line through the points A (k, 1, –1) and
B (2k, 0, 2) is perpendicular to the line
through the points B and C (2 + 2k, k, 1), then
what is the value of k ?

(A) – 1 (B) 1 (C) – 3 (D) 3

8. What is 11+ e2 dx equal to ?

(A) x – In x + c (B) x – In (tan x) + c

where c is a constant of integration.

9. The function f(x) = x cosec x is—

(A) Continuous for all values of x

(B) Discontinuous everywhere

(D) Continuous for all x except at x = np/2, where n is an integer

10. What is the solution of the differential equation a xdydx+2y= xydydx ?

(A) x2 = kyey/a (B) y2 = kyey/a

11. A vector b is collinear with the vector a = (2, 1, –1) and satisfies the condition a, b = 3. What is b equal to ? ,

(A) (1, 1/2, –1/2) (B) (2/3, 1/3, –1/3)

12. What is the least positive integer n for which 1 + i1- in= 1 ?

(A) 16 (B) 12

13. The vector a = xi + yj+zk, b = k, c are such that they form a right-handed system.

What is c equal to ?

(A) j (B) yj – xk

14. If x = t2,y = t3, then what is equal to ?

(A) 1 (B) 32t

15. What is -p/4p/4tan3 x dx equal to ?

(A) 3 (B) 13

16. What is the number of ways of arranging the letters of the word 'BANANA' so that no two N's appear together ?

(A) 40 (B) 60

17. Consider the equation (x – p) (x – 6) + 1=0 having integral coefficients. If the equation has integral roots, then what values can p have ?

(A) 4 or 8 (B) 5 or 10

18. What is the equivalent binary number of the decimal number 13-625 ?

(A) 1101111 (B) 1111-101

19. What is the value of

i + 3-i + 3200+ i - 3i + 3200 + 1 ?

(A) – 1 (B) 0 (C) 1 (D) 2

20. The order of a set A is 3 and that of a set B is2. What is the number of relations from A to B ?

(A) 4 (B) 6 (C) 32 (D) 64

21. What is the value of log αβ (H)log αβ (H) ?

(A) logαβ (α) (B) (αβ)

22. The 59th term of an AP is 449 and the 449th term is 59. Which term is equal to 0 (zero) ?

(A) 501st term (B) 502nd term

23. For a set A, consider the following statements—

1. A È P(A) = P(A)

2. {A} Ç P(A) = A

3. P(A) – {A} = P(A)

where P denotes power set.

Which of the statements given above is/are correct?

(A) 1 only (B) 2 only

24. If the AM and HM of two numbers are 27 and 12 respectively, then what is their GM equal to ?

(A) 12 (B) 18

25. If tan A = 12 and tan B = 13, then what is the value of (A + B) ?

(A) 0 (B) p4

26. If (4, 0) and (–4, 0) are the foci of an ellipse and the semi-minor axis is 3, then the ellipse passes through which one of the following points ?

(A) (2, 0) (B) (0, 5)

27. Under what condition do the planes bx – ay = n, cy – bz = l, az – cx = m intersect in a line ?

(A) a + b + c = 0

(B) a =b =c

(D) l + m + n = 0

28. What is the maximum point on the curve x - exy ?

(A) (1, e) (B) (l, e–1)

29. The function f(x) =ex,xe R is—

(A) Onto but not one-one

(B) One-one onto

(D) Neither one-one not onto

30. If y = sin-1 4x1+ 4x2 , then what is dydx equal to ?

(A) 11+ 4x2 (B) – 11+ 4x2

31. If abclmnpqr= 2, then what is the value of

the determinant 6a3b15c2lm5n2pq5r ?

(A)10 (B) 20 (C) 40 (D) 60

32. Let X be the set of all graduates in India.
Elements x and y in X are said to be related if
they are graduates of the same university.
Which one of the following statements is
correct ?

(A) Relation is symmetric and transitive only

(B) Relation is reflexive and transitive only

(D) Relation is reflexive, symmetric and transitive

33. If x2 + y2 = 1, then what is 1+x+iy1+x-iy equal to ?

(A) x – iy (B) x + iy

34. Consider the following statements—

1. For any three vectors a, b, c ;

a . {( b + c ) ´ ( a + b + c)} = 0

2. For any three coplanar unit vectors

d, e, f ; ( d ´ e) . d, f = 1

Which of the statements given above is/are correct?

(A) 1 only (B) 2 only

Directions—(Q. 35 to 37) Consider the following lists :

Each item under List I is associated with one or more items under List II.

List I List II

(Function) (Property)

A. sin x 1. Periodic function

B. cos x 2. Non-periodic function

C. tan x 3. Continuous at every point

on (– ¥, ¥)

4. Discontinuous function

5. Differentiable at every

point on (– ¥, ¥)

6. Not differentiable at every

point on (– ¥, ¥)

7. Has period p

8. Has period 2n

9. Increases on (0, p/2)

10. Decreases on (0, p/2)

11. Increases on (p/2, p)

12. Decreases on (p/2, p)

35. A is associated with—

(A) 1, 3, 5, 8, 9, 12

(B) 2, 4, 6, 8, 10, 11

(D) None of the above

36. B is associated with—

(A) 2, 3, 5, 8, 9, 12

(B) 1, 3, 5, 8, 10, 12

(D) None of the above

37. C is associated with—

(A) 1, 4, 6, 7, 9, 11

(B) 2, 4, 6, 8, 9

(D) None of the above

38. If p and q are positive integers, then which one of the following equations has p –q as one of its roots ?

(A) x2 – 2px - (p2 – q) = 0

(B) x2 – 2px+(p2 – q) = 0

(D) x2 + 2px + (p2 – q) = 0

Given two squares of sides x and y such that y = x + x2. What is the rate of change of area of the second square with respect to the area of the first square ?

(A) 1 + 3x + 2x2 (B) 1 + 2x+ 3x2

39. The planes px + 2y+2z – 3 = 0 and 2x – y + z + 2 + 2 = 0 intersect at an angle p/4. What is the value of p2 ?

(A) 24 (B) 12 (C) 6 (D) 3

40. The growth of a quantity N(r) at any instant t is given by = dN(t)dt = α N(r). Given that N(r) = cekt,c is a constant. What is the value of α ?

(A) c (B) k (C) c + k (D) c – k

41. A circle is drawn with the two foci of an

x2a2 + y2b2 = 1 ellipse at the end of the diameter. What is the equation to the circle ?

1. x2 + y2 = a2 + b2

2. x2 + y2 = a2 – b2

3. x2 + y2 — 2 (a2 + b2)

4. x2 + y2 = 2 (a2 – b2)

42. What is the image of the point (1, 2) on the line 3x + 4y – 1 = 0 ?

(A) -75, -65 (B) 78, 12

43. If the product of the roots of the equation

x2 – 5x + k = 15 is – 3, then what is the value of k ?

(A) 12 (B) 15 (C) 16 (D) 18

45. Consider the following statements—

1. Every function has a primitive.

2. A primitive of a function is unique.

Which of the statements given above is/are correct?

(A) 1 only (B) 2 only

46. Let O (0, 0, 0), P (3, 4, 5), Q (m, n, r) and R (1, 1, 1) be the vertices of a parallelogram taken in order. What is the value of m + n + r ?

(A) 6 (B) 12(C) 15 (D) More than 15

47. What is the solution of the differential equation 3 ex tan y dx + (1 + ex) sec2 y dy = 0 ?

(A) (1 + ex))tan y = c

(B) (1+ ex)3 tan y = c

(D) (1 + ex) sec2 y = c

where c is a constant of integration.

48. What is the locus of points, the difference of whose distances from two points being constant ?

(A) Pair of straight lines

(B) An ellipse

(D) A parabola

49. What is the differential equation for

y2 = 4a (x – a)?

(A) yy' – 2xyy' + y2 = 0

(B) yy’ (yy' + 2x) + y2 = 0

(D) yy' – 2xyy' + y = 0

50. If the angle between the vectors a and b is π3,what is the angle between –5 a and 6 b —

(A) π6 (B) 2π3 (C) 2π5 (D) 3π7

51. What is the degree of the differential equation d2ydx2 – 1+dydx3= 0 ?

(A) 1 (B) 2 (C) 3 (D) 6

52. If x2 ln x dx = x3m ln x x3m + c the values of m and n respectively ?

(A) 1/3, –1/9 (B) 3, –9

where c is a constant of integration.

53. What is the principal value of cosec–1 -2 ?

(A) p4 (B) p2 (C) – p4 (D) 0

54. If f : R → R, g : R → R and g(x) = x + 3 and (fog) (x) = (x + 3)2, then what is the value of f(– 3) ?

(A) – 9 (B) 0 (C) 9 (D)

55. What is the valuelimx →1 x-1|x-1|2?
(A) 0 (B) 0

56. A balloon is pumped at the rate of 4 cm3 per
second. What is the rate at which its surface
area increases when its radius is 4 cm ?

(A) 1 cm2/sec (B) 2 cm2/sec

57. What is the value of 1+tan15°1-tan15°

(A) 1 (B) 1`2 (C) 13 (D) 3

58. If f(x) = kx2 – 9x2 + 9x + 3 is monotonically increasing in every interval, then which one of the following is correct ?

(A) k < 3 (B) k < 3

59. If sin–1 5x + sin–1 p2, then what is the value of x ?

(A) 1 (B) 7

60. If a, P are the roots of the quadratic equation
x2 – x + 1 = 0, then which one of the
following is correct ?

(A) (α4 – β4) is real

(B) 2(α5 + β5) = (α β)5

(D) (α 8 + β8) = (αβ)8

61. If A = {a, b, c, d}, then what is the number of proper subsets of A ?

(A) 16 (B) 15

62. What is the number of three-digit odd
numbers formed by using the digits 1, 2, 3, 4,
5, 6 if repetition of digits is allowed ?

(A) 60 (B) 108

63. Let A = 5612-15 Let there exist a matrix B such that AB = 35492913 What is B equal

(A) 5612-15 (B) 263514

64. Consider the followig statements—

1. The probability that there are 53 Sundays
in a leap year is twice the probability that
there are 53 Sundays in a non-leap year.

2. The probability that there are 5 Mondays
in the month of March is thrice the
probability that there are 5 Mondays in

the month of April.

Which of the statements given above is/are
correct ?

(A) 1 only

(B) 2 only

(D) Neither 1 nor 2

65. Consider the following statements—

1. If A' = A, then A is a singular matrix,
where A' is the transpose of A.

2. If A is a square matrix such that A3 = I, then A is non-singular.

Which of the statements given above is/are
correct ?

(A) 1 only (B) 2 only

66. If p times the pth term of an AP is q times the qth term, then what is the (p + q)th term equal
to?

(A) p+q (B) pq (C) 1 (D) 0

67. A team of 8 players is to be chosen from a group of 12 players. Out of the eight players
one is to be elected as captain and another as
vice-captain. In how many ways can this be
done ?

(A) 27720 (B) 13860 (C) 6930 (D) 495

68. In tossing three coins at a time, what is the probability of getting at most one head ?

(A) 38 (B) 78 (C) 12 (D) 18

69. What is the sum of the coefficients of all the
terms in the expansion of (45x - 49)4 ?

(A) –256 (B) – 100

70. Two balls are selected from a box containing
2 blue and 7 red balls. What is the probability that at least one ball is blue ?

(A) 29 (B) 79 (C) 512 (D) 712

71. If the equation x2 – b x + 1 = 0 does not
possess real roots, then which one of the
following is correct ?

(A) – 3 < b < 3 (B) –2<b <2

72. The probability of guessing a correct answer is x12. If the probability of not guessing the
correct answer is 23, then what is x equal to ?