NDA-II-2018-Mathematics.pdf - TejasX

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NDA-II 2018 Mathematics �estion Paper

NDA II 2018 Maths�estions

1. What is the value of log7log7√7√7√7 equal to?

a) 3log27b) 1− 3log27c) 1− 3log72

d) 78

Ans(c)

2. If an in�nite GP has the �rst term x and the sum 5, then which one of the fol-lowing is correct?a) x < −10b) −10 < x < 0c) 0 < x < 10d) x > 10

Ans(c)

3. Consider the following expressions

1. x+ x2 − 1x

2.√ax2 + bx+ x− c+

d

x−e

x2

3. 3x2 − 5x+ ab

4. 2x2 − ax+ b3

5. 1x−

22+ 5

Which one of the above are rational expressions?

a) 1, 4 and 5b) 1, 3, 4 and 5c) 2, 4 and 5d) 1 and 2

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Ans(a)

4. A square matrix A is called orthogonal if

a) A = A2

b) A ′ = A−1

c) A = A−1

d) A = A ′

Ans(b)

5. If A, B and C are subsets of a universal set, then which one of the following isnot correct?

a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)b) A ′ ∪ (A ∪ B) = (B ′ ∩A) ′ ∪Ac) A ′ ∪ (B ∪ C) = (C ′ ∩ B) ′ ∩A ′

d) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)

Ans(c)

6. Let x be the number of integers lying between 2999 and 8001 which have at leasttwo digits equal. �en x is equal toa) 2480b) 2481c) 2482d) 2483

Ans(b)

7. �e sum of the series 3− 1+ 13 −

19 +….. is equal to

a) 209

b) 920

c) 94

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d) 49

Ans(c)DirectionConsider the information given below and answer the two items that follow.A survey was conducted among 300 students. It was found that 125 studentslike to play cricket, 145 students like to play football and 90 students like to playtennis. 32 students like to play exactly two games out of the three games.

8. How many students like to play all the three games?

a) 14b) 21c) 28d) 35

Ans(a)

9. How many students like to play exactly only one game?

a) 196b) 228c) 254d) 268

Ans(c)

10. If α and β 6= (0) are the roots of the quadratic equation x2 + αx− β = 0, thenthe quadratic expression −x2 + αx+ β, where x ∈ R has

a) Least value -1/4b) Least value -9/4c) Least value 1/4d) Least value 9/4

Ans(d)

11. What is the coe�cient of themiddle term in the binomial expansion of (2+3x)4?

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a) 6b) 12c) 108d) 216

Ans(d)

12. For a square matrix A, which of the following hold?1. (A−1)−1 = A

2. det(A−1) =1

detA

3. (λA)−1 = λA−1, where λ is a scalarSelect the correct answer using the code given below.

a) 1 and 2b) 2 and 3c) 1 and 3d) 1, 2 and 3

Ans(d)

13. Which of the following factors does the expansion of the determinant

∣∣∣∣∣∣x y 3x2 5y3 9x3 10y5 27

∣∣∣∣∣∣contains?

a) x-3b) x-yc) y-3d) x-3y

Ans(a)

14. What is the adjoint of the matrix(Cos(−θ) −sin(−θ)−sin(−θ) cos(−θ)

)?

a)(Cos(θ) −sin(θ)−sin(θ) cos(θ)

)

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b)(Cos(θ) sin(θ)sin(θ) cos(θ)

)c)(Cos(θ) sin(θ)−sin(θ) cos(θ)

)d)(Cos(θ) −sin(θ)sin(θ) cos(θ)

)Ans (a)

15. What is the value of (−1+ i√3

2

3n

) + (−1− i

√3

2

3n

), where i =√−1 ?

a) 3b) 2c) 1d) 0

Ans(b)

16. �ere are 17 cricket players, out of which 5 players can bowl. In howmanywayscan a team of 11 players be selected so as to include 3 bowlers?a) C(17, 11)b) C(12, 8)c) C(17, 5)× C(5, 3)d) C(5, 3)× C(12, 8)

Ans (d)

17. What is the value of log927+ log832?

a) 72

b) 196

c) 4d) 7

Ans(b)

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18. If A and B are two invertible square matrices of same order, then what is (AB)−1

equal to?a) B−1(A)−1

b) A−1B−1

c) B−1A

d) A−1B

And(a)

19. If a+b+c=0, then one of the solutions of

∣∣∣∣∣∣a− x c bc b− x ab a c− x

∣∣∣∣∣∣ isa) x=a

b) x =√

3(a2 + b2 + c2)2

c) x =√

2(a2 + b2 + c2)3

d) x=0

Ans(d)

20. What should be the value of x, so that the matrix(

2 4−8 x

)does not have an

inverse?

a) 16b) -16c) 8d) -8

Ans(-b)

21. �e system of equations 2x+y-3z=5, 3x-2y+2z=5 and 5x-3y-z=16

a) is inconsistentb) is consistent, with a unique solutionc) is consistent, with in�nity many solutions

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d) has its solution lying along X-axis in three dimensional space.

Ans(b)

22. Which one of the following is correct in respect of the cube roots of unity?a) �ey are collinearb) �ey lie om a circle of radius

√3

c) �ey form an equilateral triangle.d) None of the above

Ans(c)

23. If u, v and w(all positive) are the pth, qth and nth terms of a GP, the the deter-

minant of the matrix

lnu p 1ln v q 1lnw r 1

is

a) 0b) 1c) (p-q)(q-r)(r-p)d) lnu× ln v× lnw

Ans(a)

24. Let the coe�cient of the middle term of the binomial expansion of (1+ x)2n beα and those of two middle terms of the binomial expansion of (1+ x)2n−1 be βand γ. Which one of the following relations is correct?a) α > β+ γ

b) α < β+ γ

c) α = β+ γ

d) α = βγ

Ans(c)

25. Let A = [x ∈ R : −1 6 x 6 1], B = [y ∈ R : −1 6 y 6 1], and S be the subsetof A × B, de�ned by S = [(x,y) ∈ A × B : x2 + y2 = 1] .Which one of thefollowing is correct?a) S is a one-one function from A into B

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b) S is a many-one function from A into Bc) S is a bijective mapping from A into B.d) S is not a function

Ans(d)

26. Let Tr be the rth term of an AP for r=1,2,3….. If for some distinct positive inte-gers m and n we have tm =

1n

and tn =1m

, then what is tmn equal to?

a) (mn)−1

b) m−1 + n−1

c) 1d) 0

Ans(c)

27. Suppose f(x) is such a quadratic expression that it is positive for all real x. ifg(x) = f(x) + f ′(x) + f ′′(x), then for any real x.a) g(x) < 0b) g(x) > 0c) g(x) = 0d) g(x) > 0

Ans(b)

28. Consider the following in respect of matrices A, B and C of same order. 1. (A+B+ C)

′= A ′ + B ′ + C ′

2. (AB) ′ = A ′B ′

3. (ABC) ′ = C ′B ′A ′

Where A ′ is the transpose of the matrix A. Which of the above are correct?

a) 1 and 2b) 2 and 3c) 1 and 3d) 1, 2 and 3

Ans(c)

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29. �e sum of the binary numbers (11011)2, (10110110)2 and (10011x0y)2 is thebinary numbers (101101101)2. What are the values of x and y?

a) x=1, y=1b) x=1, y=0c) x=0, y=1d) x=0, y=0

Ans(b)

30. Let matrix B be the adjoint of a square matrix A, I be the identity matrix of sameorder as A. If k 6= (0) is the determinant of the matrix A, then what is AB equalto?a) Ib) kIc) k2I

d) 1kI

Ans(b)

31. If (0.2)x = 2 and log10 2 = 0.3010, then what is the value of x to the nearesttenth?a) -10.0b) -0.5c) -0.4d) -0.2

Ans(c)

32. �e total number of 5-digit numbers that can be composed of distinct digits from0 to 9 isa) 45360b) 30240c) 27216d) 15120

Ans(c)

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33. What is the determinant of the matrix

x y y+ zz x z+ xy z x+ y

?

a) (x-y)(y-z)(z-x)b) (x-y)(y-z)c) (y-z)(z-x)d) (z− x)2(x+ y+ z)

Ans(d)

34. If A, B and C are the angles of a triangle and∣∣∣∣∣∣1 1 1

1+ sinA 1+ sinB 1+ sinCSinA+ Sin2A sinB+ sin2B sinC+ sin2C

∣∣∣∣∣∣ = 0,

then which of the following is correct?

a) �e triangle ABC is isoscelesb) �e triangle ABC is equilateralc) �e triangle ABC is scalened) No conclusion can be drawn with regard to the nature of triangle.

Ans(a)

35. Consider the following in respect of matrices A and B of same order1.A2 − B2 = (A+ B)(A− B)

2.(A− I)(A+ I) = O⇔ A2 = 1Where I is the identity matrix and O is the null matrix.Which of the above is/are correct?a) 1 onlyb) 2 onlyc) Both 1 and 2d) Neither 1 nor 2

Ans(b)

36. What is 2tanθ1+ tan2θ

equal to?

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a) cos2θb) tan2θc) sin2θd) cosec2θ

Ans(c)

37. If sec(θ + α), secθ and sec(θ + α) are in AP, where cosα 6= 1, then what isthe value of sin2θ+ cosα?

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

Ans(a)

38. If A+ B+ C = 180°, then what is sin2A-sin2B-sin2C equal to?a) -4sinAsinBsinCb) -4cosAsinBcosCc) -4cosAcosBsinCd) -4sinAcosBcosC

Ans(d)

39. A balloon is directly above one end of a bridge. �e angle of depression of theother end of the bridge from the balloon is 48 °. If the height of the balloon abovethe bridge is 122 m, then what is length of the bridge?a) 122 sin48°mb) 122 tan42°mc) 122 cos48°md) 122 tan48°m

Ans(b)

40. A is an angle in the fourth quadrant it satis�es the trigonometric equation 3(3−tan2A− cotA2 = 1). Which one of the following is a value of A?

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a) 300 °b) 315 °c) 330 °d) 345 °

Ans(a)

41. �e top of a hill observed from the top and bo�om of a building of height h isat angles of elevation π6 and π3 respectively. What is the height of the hill?

a) 2hb) 3h/2c) hd) h/2

Ans(b)

42. What is/are the solutions(s) of the trigonometric equation cosecx+cotx =√3

where 0 < x < 2π ?

a) 5π3 only

b) π3 only

c) π only

d) π, π3 ,5π3 , only

Ans(b)

43. If θ =π

8 , then what is the value of (2cosθ+ 1)10(2cos4θ− 1)10(2cosθ− 1)10 ?

a) 0b) 1c) 2d) 4

Ans(b)

44. If cosα and cosβ(0 < α < β < π) are the roots of quadratic equation 4x2−3 =0, then what is the value of secα × secβ ?

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a) -4/3b) 4/3c) 3/4d) -3/4

Ans(a)

45. Consider the following values of x:1. 82. -43. 1/64. -1/4Which of the above values of x is/are the solution(s) of the equation tan−1(2x)+tan−1(3x) = π

4 ?

a) 3 onlyb) 2 and 3c) 1 and 4d) 4 only

Ans (a)

46. If the second term of a GP is 2 and the sum of its in�nite terms is 8, then the GPis

a) 8, 2, 12 ,18 , ……

b) 10, 2, 25 ,225 , ……..

c) 4, 2, 1, 12 ,122 , ….

d) 6, 3, 32 ,34 , ………

Ans (c)

47. If a,b,c are in AP or GP or HP, then a− b

b− cis equal to

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a) baor 1 or b

c

b) caor bcor 1

c) 1 or abor ac

d) 1 or abor ca

Ans (c)

48. What is the sum of all three-digit numbers that can be formed using all the digits3, 4 and 5, when repetition of digits is not allowed?a) 2664b) 3882c) 4044d) 4444

Ans(a)

49. �e ratio of roots of the equations ax2 + bx+ c = 0 and px2 + qx+ r = 0 areequal. If D1 and D2 are respective discriminant, then what is D1

D2equal to?

a) a2

p2

b) b2

q2

c) c2

r2

d) None of the above

Ans (b)

50. IfA = Sin2θ+ cos4θ, then for all real θ, which one of the following is correct?

a) 1 6 A 6 2

b) 34 6 A 6 1

c) 1316 6 A 6 1

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d) 34 6 A 6

1316

Ans(b)

51. �e equation of a circle whose end points of a diameter are (x1,y1) and (x2,y2)is

a) (x− x1)(x− x2) + (y− y1)(y− y2) = x2 + y2

b) (x− x1)2 + (y− y1)

2 = x2y2

c) x2 + y2 + 2x1x2 + 2y1y2 = 0d) (x− x1)(x− x2) + (y− y1)(y− y2) = 0

Ans(d)

52. �e second degree equation x2 + 4y2 − 2x− 4y+ 2 = 0 represents

a) A pointb) An ellipse of semi-major axis 1

c) An ellipse with eccentricity√32

d) None of the above.

Ans(d)

53. �e angle between the two line lx +my + n = 0 and l ′x +m ′y + n ′ = 0 isgiven by tan−1θ. What is θ equal to?

a)∣∣∣∣ lm ′ − l ′m

ll ′ −mm ′

∣∣∣∣b)∣∣∣∣ lm ′ + l ′m

ll ′ +mm ′

∣∣∣∣c)∣∣∣∣ lm ′ − l ′m

ll ′ +mm ′

∣∣∣∣d)∣∣∣∣ lm ′ + l ′m

ll ′ −mm ′

∣∣∣∣Ans(c)

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54. Consider the following statements :

1. �e distance between the lines y = mx+ c1 and y = mx+ c2 is|c1 − c2|√1+m2

2. �e distance between the lines ax + by + c1 = 0 and ax + by + c2 = 0 is|c1 − c2|√a2 + b2

3. �e distance between the line x = c1 and x = c2 is |c1 − c2|.Which of the above statements are correct?a) 1 and 2 onlyb) 2 and 3 onlyc) 1 and 3 onlyd) 1, 2 and 3

Ans(b)

55. What is the equation of straight line passing through the point of intersectionof the line x2 +

y

3 = 1 and x3 +y

2 = 1, and parallel to the line 4x+5y-6=0?

a) 20x+25y-54=0b) 25x+20y-54=0c) 4x+5y-54=0d) 4x+5y-45=0

Ans(a)

56. What is the distance of the point (2,3,4) from the plane 3x-6y+2z+11=0?a) 1 unitb) 2 unitsc) 3 unitsd) 4 units

(a)

57. Coordinates of the points O,P,Q and R are respectively (0,0,0), (4,6,2m) (2,0,2n)and (2,4,6). Let L,M,N and K be points on the sides OR, OP, PQ and QR respec-tively such that LMNK is a parallelogram whose two adjacent sides LK and LMare each length

√2. What are the values of m and n respectively?

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a) 6,2b) 1,3c) 3,1d) None of the above

Ans(c)

58. �e line x− 12 =

y− 23 =

z− 34 is given by

a) x+y+z=6, x+2y-3z=-4b) x+2y-2z=-1, 4x+4y-5z-3=0c) 3x+2y-3z=0, 3x-6y+3z=-2d) 3x+2y-3z=-2, 3x-6y+3z=0

Ans(d)

59. Consider the following statements:1. �e angle between the planes 2x-y+z=1 and x+y+2z=3 is π/32. �e distance between the planes 6x-3y+6z+2=0 and 2x-y+2z+4=0 is 10/9.Which one of the above statements is/are correct?a) 1 onlyb) 2 onlyc) Both 1 and 2d) Neither 1 nor 2

An(c)

60. Consider the following statementsStatement I: If the line segment joining the point P(m,n) ans Q(r,s) subtents anangle α at the origin, then cosα =

ms− nr√(m2 + n2)(r2 + s2)

.

Statement II: In any triangle ABC, it is true that a2 = b2 + c2 − 2bc cosAWhich of the following is correct in respect of the above two statements?

a) Both the statements are individually true and statement II is the correctexplanation of statement I.

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b) Both the statements are individually are individually true but statement IIis not the correct explanation of statement I.

c) Statement I is true but statement II is falsed) Statement I is false but statement two is correct.

Ans(d)

61. What is the area of the triangle with vertices(x1,

1x1

),(x2,

1x2

) (x3,

1x3

)?

a) |(x1 − x2)(x2 − x3)(x3 − x1)|

b) 0

c)∣∣∣∣ (x1 − x2)(x2 − x3)(x3 − x1)x1x2x3

∣∣∣∣d)∣∣∣∣ (x1 − x2)(x2 − x3)(x3 − x1)2x1x2x3

∣∣∣∣Ans(d)

62. If y-axis touches the circle x2 + y2 + gx+ fy+c

4 = 0, then the normal at thispoint intersects the circle at the point.

a)(−g

2 ,−f

2

)b)(−g,− f2

)c)(−g

2 , f)

d) (−g,−f)Ans(b)

63. Let∣∣−→a ∣∣ 6= 0,

∣∣∣−→b ∣∣∣ 6= 0.

(−→a +−→b ).(−→a +

−→b ) =

∣∣−→a ∣∣2 + ∣∣∣−→b ∣∣∣2 holds if and only if

a) −→a and−→b are perpendicular

b) −→a and−→b are parallel

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c) −→a and −→b are inclined at an angle of 45°

d) −→a and−→b are anti-parallel.

Ans(a)

64. If −→r = xi+ yj+ zk, then what is −→r .(i+ j+ k) equal to?a) xb) x+yc) -(x+y+z)d) (x+y+z)

Ans(d)

65. A unit vector perpendicular to each of the vectors 2i− j+ k and 3i− 4j− k is

a) 1√3i+

1√3j−

1√3k

b) 1√2i+

12 j+

12 k

c) 1√3i−

1√3j−

1√3k

d) 1√3i+

1√3j+

1√3k

Ans(a)

66. If∣∣−→a ∣∣ = 3,

∣∣∣−→b ∣∣∣ = 4 and∣∣∣−→a −

−→b∣∣∣ = 5, then what is the value of value of∣∣∣−→a +

−→b∣∣∣ ?

a) 8b) 6c) 5√3

d) 5Ans(d)

67. Let−→a ,−→b and−→c be the three mutually perpendicular vectors each of unit mag-nitude. If −→A = −→a +

−→b +−→c , −→B = −→a −

−→b +−→c and −→C = −→a −

−→b −−→c , then

which one of the following is correct?

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a)∣∣∣−→A ∣∣∣ > ∣∣∣−→B ∣∣∣ > ∣∣∣−→C ∣∣∣

b)∣∣∣−→A ∣∣∣ = ∣∣∣−→B ∣∣∣ 6= ∣∣∣−→C ∣∣∣

c)∣∣∣−→A ∣∣∣ = ∣∣∣−→B ∣∣∣ = ∣∣∣−→C ∣∣∣

d)∣∣∣−→A ∣∣∣ 6= ∣∣∣−→B ∣∣∣ 6= ∣∣∣−→C ∣∣∣

Ans(c)

68. What is (−→a −−→b )× (−→a +

−→b ) equal to?

a) −→0

b) −→a ×−→b

c) 2(−→a ×−→b )

d)∣∣−→a ∣∣2 − ∣∣∣−→b ∣∣∣2

Ans(c)

69. A spacecra� located at i + 2j + 3k is subjected to a force λk by �ring a rocket.�e spacecra� is subjected to a moment of magnitude.

a) λb)√3λ

c)√5λ

d) None of the above

Ans(c)

70. In a triangle ABC, if taken in order, consider the following statements:

1. −→AB+−→BC+

−→CA =

−→02. −→AB+

−→BC−

−→CA =

−→03.−→AB−

−→BC+

−→CA =

−→04.−→BA−

−→BC+

−→CA =

−→0How many of the above statements are correct?

a) One

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b) Twoc) �reed) Four

Ans(a)

71. Let the slope of the curve y = cos−1(sinx) be the tanθ. �en the value of θ inthe interval (0,π) is

a) π6

b) 3π4

c) π4d) π2

Ans(b)

72. If f(x) =√x− 1x− 4 de�nes a function on R, then what is its domain?

a) (−∞, 4) ∪ (4,∞)

b) [4,∞)

c) (1, 4) ∪ (4,∞)

d) [1, 4) ∪ (4,∞)

Ans(d)

73. Consider the function

f(x) =sin2x5x if x 6= 0 and

f(x) =215 if x=0

Which one of the following is correct in respect of the function?

a) It is not continuous at x=0b) It is continuous at every xc) It is not continuous at x = πd) It is continuous at x=0

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Ans(a)

74. For the function f(x) = |x− 3|, which one of the following in not correct?

a) �e function is not continuous at x=-3b) �e function is continuous at x=3c) �e function is di�erentiable at x=0d) �e function is di�erentiable at x=-3

Ans(a)

75. If the function f(x) = 2x− sin−1x

2x+ tan−1xis continuous at each point in its domain,

then what is the vale of f(0)?

a) -1/3b) 1/3c) 2/3d) 2

Ans(b)

76. If f(x) =√25− x2, then what is lim

x→1

f(x) − f(1)x− 1 equal to?

a) −1√24

b) 1√24

c) −1

4√3

d) 14√3

Ans(a)

77. If y = tan−1(5− 2tan

√x

2+ 5tan√x

), then what is dy

dxequal to?

a) −1

2√x

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b) 1c) -1

d) 12√x

Ans(a)

78. Which one of the following is correct in respect of the function f(x) = xsinx+cosx+

12cos

2x

a) It is increasing in the interval(0, π2

)b) It remains constant in the interval

(0, π2

)c) It is decreasing in the interval

(0, π2

)d) It is decreasing in the interval

(π4 ,π

2

)Ans(a)

79. What is limθ→0

√1− cosθθ

equal to?

a)√2

b) 2√2

c) 1√2

d) −1

2√2

Ans(c)

80. A function f : A → R is de�ned by the equation f(x) = x2 − 4x + 5 whereA=(1,4). What is the range of the function?a) (2,5)b) (1,5)c) [1, 5)d) [1, 5]

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Ans(c)

81. Whats is∫ba

[x]dx+

∫ab

[−x]dx equal to, where [,] is the greatest integer function?

a) b-ab) a-bc) 0d) 2(b-a)

Ans(b)

82. What is∫ 82|x− 5|dx equal to?

a) 2b) 3c) 4d) 9

Ans(d)

83. What is∫sin3 x cos xdx equal to?

(Where c is the constant of integration.)a) cos4x+ cb) sin4x+ c

c) (1+ sin2x)2

4 + c

d) (1− sin2x)2

4 + c

Ans(d)

84. What is∫eln(tanx)dx equal to?

(Where c is the constant of integration.)

a) ln|tanx|+ cb) ln|secx|+ c

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c) tanx +cd) etanx + c

Ans(b)

85. What is∫ 1−1

[d

dx

(tan−1 1

x

)]dx equal to?

a) 0

b) −π

4c) −

π

2d) π2

Ans(d)

86. In which of the following intervals is the function f(x) = x2−5x+6 decreasing?a) (−∞, 2]b) [3,∞)

c) (−∞,∞)

d) (2,3)Ans(a)

87. �e di�erential equation of the family of curves y = pcos(ax) + qsin(ax),where p, q are arbitrary constants, is

a) d2y

dx2− a2y = 0

b) d2y

dx2− ay = 0

c) d2y

dx2+ ay = 0

d) d2y

dx2+ a2y = 0

Ans(d)

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88. �e equation of the curve passing through the point (-1, -2) which satis�es dydx

=

−x2 −1x3

, is

a) 17x2y− 6x2 + 3x5 − 2 = 0b) 6x2 + 17x2 + 2x5 − 3 = 0c) 6xy− 2x2 + 17x5 + 3 = 0d) 17x2y+ 6xy− 3x5 + 5 = 0

Ans(b)

89. What is the order of the di�erential equation whose solution is y = acosx +bsinx+ ce−x + d, where a, b, c and d are arbitrary constants?a) 1b) 2c) 3d) 4

Ans(d)

90. What is the solution of the di�erential equation ln(dy

dx

)= ax+ by? (where

c is an arbitrary constant.)

a) aeax + beby = c

b) 1aeax +

1beby = c

c) aeax + be−by = c

d) 1aeax +

1be−by = c

Ans(d)

91. If u = eaxsinbx and v = eaxcosbx, then what is ududx

+ vdv

dxequal to?

a) ae2ax

b) (a2 + b2)eax

c) abe2ax

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d) (a+ b)eax

Ans(a)

92. If y = sin(lnx), then which one of the following is correct?

a) d2y

dx2+ y = 0

b) d2y

dx2= 0

c) x2d2y

dx2+ x

dy

dx+ y = 0

d) x2d2y

dx2− x

dy

dx+ y = 0

Ans(c)

93. A �ower-bed in the form of a sector has been fenced by a wire of 40m length.If the �ower-bed has the greatest possible area, then what is the radius of thesector?a) 25mb) 20mc) 10md) 5m

Ans(c)

94. What is the minimum value of [x(x− 1) + 1]13 , where 0 6 x 6 1 ?

a)(34

) 13

b) 1

c) 12

d)(38

) 13

Ans(a)

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95. If |sinx||x|, then what is value of dydx

at x = −π

6 ?

a) 2−π

6 (6ln2−√3π)

6

b) 2π

6 (6ln2+√3π)

6

c) 2−π

6 (6ln2+√3π)

6

d) 2π

6 (6ln2−√3π)

6Ans(a)

96. What is d√1− sin2xdx

equal to, where π4 < x <π

2 ?

a) cosx+ sinxb) −(cos+ sinx)

c) ±(cosx+ sinx)d) None of the above

Ans(a)

97. What is∫

dx

a2sinn2x+ b2cos2xequal to?

a) c+ 1abtan−1

(atanx

b

)b) c− 1

abtan−1

(btanx

a

)c) c+ 1

abtan−1

(btanx

a

)d) None of the above

Ans(a)

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98. Let f(x + y) = f(x)f(y) and f(x) = 1 + xg(x)φ(x), where limx→0

g(x) = a andlimx→0

φ(x) = b. �en what is f’(x) equal to?

a) 1+abf(x)b) 1+abc) abd) abf(x)

Ans(d)

99. What is the solution of the di�erential equation dxdy

=x+ y+ 1x+ y− 1? (where c is

an arbitrary constant).

a) y− x+ 4ln(x+ y) = cb) y+ x+ 2ln(x+ y) = cc) y− x+ ln(x+ y) = c

d) y+ x+ 4ln(x+ y) = c

Ans(c)

100. What is limx→π

6

2sin2x+ six− 12sin2x− 3sinx+ 1 equal to?

a) -1/2b) -1/3c) -2d) -3

Ans(d)

101. If two dice are thrown and at least one of the dice shows 5, then the probabilitythat sum is 10 or more isa) 1/6b) 4/11c) 3/11d) 2/11

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Ans(c)

102. �e correlation coe�cient computed from a set of 30 observations is 0.8. �enthe percentage of variation not explained by linear regression isa) 80%b) 20%c) 64%d) 36%

Ans(b)

103. �e average age of a combined group of men and women is 25 years. If theaverage age of the group of men is 26 years and that of the group of women is21 years, then the percentage of men and women in the group is respectivelya) 20, 80b) 40, 60c) 60, 40d) 80, 20

Ans(d)

104. If sinβ is the harmonic mean of sinα and cosα and sin θ is the arithmetic meanof sinα and cosα, then which of the following is/are correct?

1.√2sin

(α+

π

4

)sinβ = sin2α

2.√2sinθ = cos

(α−

π

4

)Select the correct answer using the code given below:a) 1 onlyb) 2 onlyc) Both 1 and 2d) Neither 1 nor 2

Ans(c)

105. Let A, B and C be three mutually exclusive and exhaustive events associatedwith a random experiment. If P(B) = 1.5P(A) and P(C) = .5P(B), then P(A) isequal to

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a) 3/4b) 4/13c) 2/3d) 1/2

Ans(b)

106. In a bolt factory, machines X, Y, Z manufacture bolts that are respectively 25%,35%, and 40% of the factory’s output. �e machine X,Y,Z respectively produce2%, 4% and 5% defective bolts. A bolt is drawn at random from the product andis found to be defective. What is the probability that it was manufactured bymachine X?

a) 5/39b) 14/39c) 20/39d) 34/39

Ans(a)

107. 8 coins are tossed simultaneously. �e probability of ge�ing at least 6 heads isa) 7/64b) 57/64c) 37/256d) 299/256

Ans(c)

108. �ree groups of children contain 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and3 boys. One child is selected at random from each group. �e probability thatthe three selected consist of 1 girl and 2 boys.

a) 13/32b) 9/32c) 3/32d) 1/32

Ans(a)

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109. Consider the following statements:1. If 10 is added to each entry on a list, then the average increases by 10.2. If 10 is added to each entry on a list, then the standard deviation increases by10.3. If each entry on a list is doubled, then the average doubled.Which of the above statements are correct?

a) 1, 2 and 3b) 1 and 2 onlyc) 1 and 3 onlyd) 2 and 3 only

Ans(c)

110. �e variance of 25 observations is 4. If 2 is added to each observation, then thenew variance of the resulting observation is

a) 2b) 4c) 6d) 8

Ans(b)

111. If xi > 0,yi > 0(i = 1, 2, 3….n) are the values of two variable X and Y withgeometric means P and Q respectively, then the geometric mean of X

Yis

a) PQ

b) antilog(P

Q

)c) n(logP − logQ)

d) n(logP + logQ)

Ans(a)

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112. If the probability of simultaneous occurrence of two events A and B is p andthe probability that exactly one of A, B occurs is q, then which of the followingis/are correct?1. P(A) + P(B) = 2− 2p− q2. P(A ∩ B) = 1− p− qSelect the correct answer using the code given below:

a) 1 onlyb) 2 onlyc) Both 1 and 2d) Neither 1 nor 2

Ans(c)

113. If the regression coe�cient of Y on X is -6, and the correlation coe�cient be-tween X and Y is -1/2, then the regression coe�cient of X and Y would bea) 1/24b) -1/24c) -1/6d) 1/6

Ans(b)

114. �e set of bivariate observations (x1,y1), (x2,y2), ….(xn,yn) are such that allthe values are distinct and all the observations fall on a straight line with line-zero slope. �en the possible values of the correlation coe�cient between x andy are

a) 0 and 1 onlyb) 0 and -1 onlyc) 0,1 and -1d) -1 and 1 only

Ans(d)

115. Two integers x and y are chosen with replacement from set (0,1,2,…,10). �eprobability that |x− y| > 5 is

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a) 6/11b) 35/121c) 30/121d) 25/121

Ans(c)

116. An analysis of monthly wages paid to the workers in two �rms A and B belong-ing to the same industry gives the following result:

Firm A Firm BNumber of workers 500 600

Average monthly wage Rs 1860 Rs 1750Variance of distribution of wages 81 100

�e average of monthly wage and variance of distribution of wages of all theworkers in the �rms A and B taken together are

a) Rs 1860, 100b) Rs 1750, 100c) Rs 1800, 81d) None of the above

Ans(d)

117. �ree dice having digits 1,2,3,4,5 and 6 on their faces are marked I, II and III androlled. Let x,y and z represent the number on die-I, die-II and die-III respectively.What is the number of possible outcomes such that x > y > z?

a) 14b) 16c) 18d) 20

Ans(d)

118. Which one of the following can be obtained from an ogive?a) Meanb) Median

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c) Geometric meand) Mode

Ans(b)

119. In any discrete series(when all values are not same), if x represents mean devi-ation about mean and y represents standard deviation, then which one of thefollowing is correct?

a) y > x

b) y 6 x

c) x = yd) x < y

Ans(d)

120. In which one of the following cases would you expect to get a negative correla-tion?

a) �e ages of husbands and wivesb) Show size and intelligencec) Insurance companies pro�ts and the number of claims they have to payd) Amount of rainfall and yield of crop.

Ans(c)

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