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Padasalai.Net’s CENTUM COACHING TEAM- SPECIAL QUESTION PAPER – 2017

Class: XII STD Subject: MATHEMATICS Marks: 200 Time: 3hrs.

PART – A (40 x 1 = 40)

(i) Answer all the questions. (ii) Choose and write the correct answer.

1. A discrete random variable X has probability mass function p(x), then

(a) 0 p(x) 1 (b) p(x) 1 (c) p(x) 1 (d) 0 p(x) 1

2. In 16 throws of a die getting an even number is considered a success, then the variance of

the successes is (a) 4 (b) 6 (c) 2 (d) 256

3. The value of [ + , + , + ] is equal to

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

4. If x ( x ) + x ( x ) + x ( x ) = x then…..

(a) = (b) = (c) and are parallel (d) =0 or =0 or and are parallel

5. The following two lines are

=

=

and

=

=

(a) parallel (b) intersecting (c) skew (d) perpendicular

6. If ‘t1’ , ‘t2’ are the extremities of any focal chord of a parabola y2 = 4ax then,

t1 , t2 is (a) -1 (b) 0 (c) 1 (d)

7. The angle between the asymptotes to the hyperbola

-

= 1 is

(a) – 2 tan-1

(b) – 2 tan

-1

(c) 2 tan

-1

(d) 2 tan

-1

8. In the group (G, .), G = {1, -1, i, -i}, the order of –i is (a) 2 (b) 0 (c) 4 (d) 3

9. Let p be “Anbu is going to tuition” and q be “There are twenty students in

the class”.“Anbu is not going to tuition or there are twenty students in the class” stands

for (a) p q (b) p∧q (c) ∼p (d) ∼p q

10. In the multiplicative group of nth roots of unity, the inverse of is (k n)

(a) (b) (c) (d)

11. If ‘f’ has a local extreme (maximum and minimum) at c and if f’(c) exists, then f’(c) = 0 is

(a) the extreme value theorem (b) Fermat’s theorem

(c) mean value theorem (d) Rolle’s theorem

12. The curve y = ax3 + bx

2 + cx + d has a point of inflexion at x = 1 then

(a) a+ b = 0 (b) a + 3b = 0 (c) 3a +b = 0 (d) 3a + b = 1

13. The value of

dx is

(a)

(b)

(c)

(d)

14. In which of the following function is increasing in (0, )

(a) (b)

(c) - (d)

15. The integrating factor of the differential equation

+ Py = Q is

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(a) (b) (c) (d)

16. The order and degree of the differential equation are sinx[dx+dy] = cosx [dx-dy] are

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

17. The differential equation formed by eliminating A and B from the relation

y= ex[Acosx + Bsinx] is

(a) y2+y1 = 0 (b) y2-y1 = 0 (c) y2 - 2y1 +2y= 0 (d) y2 - 2y1 -2y= 0

18. A monoid becomes a group if it also satisfies the

(a) closure axiom (b) associative axiom (c) identity axiom (d) inverse axiom

19. Given E(X+ c) = 8 and E(X- c) = 12 then the value of c is

(a) -2 (b) 4 (c) -4 (d)2

20. In the homogenous system with three unknowns, p(A) = number of unknowns, then the

system

a) Has only trivial solution

b) reduces to 2 equations and has infinitely many solutions

c) reduces to a single equation and has infinitely many solutions

d) has no solution (inconsistent)

21. If A is a square matrix of order n then | adjA | is

(a) | A |2 (b) | A |

n (c) | A |

n-1 (d) | A |

22. If the matrix

has an inverse then the values of k ………

(a) k is any real number (b) k = -4 (c) k (d) k

23. If - lies in the third quadrant then z lies in the

(a) first quadrant (b) second quadrant (c) third quadrant (d) fourth quadrant

24. If Z represents a non-zero complex number then arg (z) + arg ( ) is

(a)

(b)

(c) 0 (d) -

25. Which of the following statements is correct?

(a) Negative complex numbers exist (b) order relation does not exist in real numbers

(c) order relation exists in complex numbers (d) (1 + i) (3 -2i) is meaningless

26. If is non-zero vector and m is a non-zero scalar then m is a unit vector if

(a) m = 1 (b) a = (c) a =

(d) a = 1

27. If u = f (x, y) then with usual notations, = if

(a) u is continuous (b) is continuous (c) is continuous

(d)u, are continuous

28. The curve y2(x-2) = x

2(1+x)has

(a) an asymptote parallel to x-axis (b) an asymptote parallel to x-axis

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(c) an asymptote parallel to both axes (d)no asymptotes

29. The Rolle’s constant for the function y = x2 on [-2,2] is

(a)

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

30. The curve y2 (1+x)= x

2(1-x) is defined for

(a) -1 x 1 (b) -1 x 1 (c) -1 x 1 (d) -1 x 1

31. Which of the following are correct?

(a) Re (Z) (b) Im (Z) (c) = (d) =

32. The volume generated when the region bounded by y = x, y = 1, x = 0 is rotated by y –

axis (a)

(b)

(c)

(d)

33. The area between the ellipse

+

= 1and its auxillary circle is

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

34. If x = r cosθ, y = r sin θ, then

is equal to

(a) secθ (b) sin θ (c) cosθ (d) cosec θ

35. The length of the arc of the curve + = 4 is

(a) 48 (b) 24 (c) 12 (d)96

36. The projection of on a unit vector equal thrice the area of parallelogram OPRQ

is (a) tan-1

(

) b) cos

-1(

) c) sin

-1(

) d) sin

-1(

)

37. The centre and radius of the sphere – = 5 are

(a) (2, -1, 4) and 5 (b) (2, 1, 4) and 5 (c) (-2, 1, 4) and 6 (d) (2, 1,- 4) and 5

38. The vector equation of a plane passing through a point where P.V is and perpendicular

to a vector is

(a) . = . (b) x = x (c) + = + (d) - = -

39. If + + = 0, = 3, = 4, = 5 then the angle between and is

(a)

(b)

(c)

(d)

40. The modulus and amplitude of the complex number [ ]3 are respectively

(a) ,

(b) ,-

(c) ,

(d) ,

PART - B 10x6=60

(i)Answer any 10 questions. (ii) Question no.55 is compulsory and chooses any 9 questions

from the remaining. (iii) Each question carries 6 marks.

41. Solve by matrix determinant method x+y+3z = 4, 2x+2y +6z =7, 2x+y+z =10

42. For A=

, show that A = A-1

43. (a) For any vector , prove that x ( x ) + x ( x ) x ( x ) = 2

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(b) A force given by 3 +2 -4 is applied at the point (1,-1, 2). Find the moment of the

force about the point (2,-1,3)

44. Prove that (1+i)n + (1-i)

n = 2

cos

(n N)

45. Find the square root of (-7+24i)

46. Find the equation of the two tangents that can be drawn from the point (2, -3) to the

parabola y2 = 4x

47. Evaluate

48. Find the intervals of concavity and the points of inflexion of the function

f(x) = 2x3 + 5x

2 -4x

49. Verify Euler’s theorem forf(x,y) =

50. Evaluate

51. Show that (( p)∨q) ∨(p∧( q))is a tautology.

52. State and prove the Reversal law of a group.

53. (a) In a Binomial distribution if n = 5 and P(X = 3) = 2P(X = 2) find p

(b) In a Poisson distribution if P(X = 3) = P(X = 2) find P(X = 5) [ = 0.050]

54. A game is played with asingle pair die, A player wins Rs. 20 if a 2 turns up, Rs. 40 if a 4

turns up, loses Rs. 30 if a 6 turns up. While he neither wins nor loses if any other faces

turns up. Find the expected sum of money he can win.

55. (a) Find the coordinates of the centre and the radius of the sphere whose vector

equation is - . (8 – 6 + 10 )– 50 = 0[OR]

(b) Solve

+ y cot x= 2 cosx

PART-C (10 x 10=100)

(i)Answer any 10 questions. (ii)Question no.70 is compulsory and chooses any9

questions from the remaining. (iii) Each question carries 10 marks.

56. For what values of µ the equations x+y+3z = 0, 4x+3y+µz = 0, 2x+y+2z =0, have a

(i) trivial solution (ii) non-trivial solution (use rank method).

57. Prove that cos (A −B) = cosA cosB + sin A sin B

58. Find the vector and Cartesian equations of the plane through the point (2, -1, -3) and

parallel to the lines

=

=

and

=

=

59. If a = cos2 + i sin 2 , b = cos2 + i sin 2 , c = cos2 +i sin2 prove that

(a) +

= 2cos ( ) (b)

= 2cos 2 ( )

60. Find the eccentricity, centre, vertices and foci of the ellipse 16x2 + 9y

2 -32x +36y-92 = 0

and sketch of the graph.

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61. A comet is moving in a parabolic orbit around the sun which, is at the focus of the

parabola, when the comet is 80 million km from the sun, the line segment from the sun to

the comet makes an angle of

radians with the axis of orbit. Find (i) the equation of

comet’s orbit (ii) how close the comet comes nearer to the sun.(Takes the orbit as open

rightward)

62. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line

x + 2y −5 = 0 and passes through the points (6, 0) and (−3, 0)

63. The top and bottom margins of a poster are each 6 cms and the side margins are each

4 cms. If the area of the printed material on the poster is fixed at 384 cms2, find the

dimension of the poster with the smallest area.

64. Trace the curve y2 = 2x

3.

65. Find the length of the curve 4y2 = x

3 between x = 0 and x = 1

66. The rate at which the population of a city increases at any time is proportional to the

population at that time. If there were 1, 30,000 people in the city in 1960 and 1,60,00 in

1990 what population may be anticipated in 2020.

67. Solve(D2 - 1)y = cos 2x – 2sin2x

68. Show that G be the set of all rational numbers expect 1 forms an abelian group with

respect to the operation given by a * b = a + b – ab for all a, b G

69. The probability density function of a random variable x is

f(x) =

. Find (i) k (ii) P(X 10)

70. (a) Prove that the sum of the intercepts on the co-ordinate axes of any tangent to the curve

x = a cos4 , y = a sin

4 , 0

is equal to a[OR]

(b) Compute the area between the curve y = sin x and y = cos x and the lines x = 0 and

x=

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