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Question Paper 2017-2018 XII - STD CENTUM SPECIAL QUESTION PAPER TIME : 3 Hrs.

SUBJECT: MATHS MAX.MARKS: 200 Instructions : Use Blue or Black ink to writes and pencil to draw diagrams.

PART – A 40 x 1 = 40

Note : (i) All questions are compulsory

(ii) Choose the most suitable answer from the given four alternatives and write the option code and the

corresponding answer.

1. If A is a matrix of order 3 , then det

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

2. If A and B are any two matrices such that AB = O and A is a non singular , then

(a) B = O (b) B is singular (c) B is non – singular (d) B = A

3. If the equation – such that

, then the system is

a non – zero unique solution (b) trivial solution

(c) Infinitely many solution (d) No solution

4. If then the system is

(a) Consistent and has unique solution (b) Consistent and has infinitely many solution

(c) Inconsistent

(d) Either consistent or inconsistent

5. If = 0 , | | = 3 , | | = 4 , | | = 5 then the angle between is

(a)

(b)

(c)

(d)

6. is the equation of

(a) a straight line joining the points (b) plane (c) plane (d) plane

7. If the projection of on and projection of are equal then the angel between and is

(a)

(b)

(c)

(d)

8. The vector equation of a sphere whose centre is origin and radius ‘a’ is

(a) r = a

(b) acr

(c) ar

(d) r

= a

9. The marks secured by 400 students in mathematical test were normally distributed with mean 65. If 120 students

got more marks above 85, the number of students securing marks between 45 and 65 is

(a) 120 (b)20 (c)80 (d)160

10. If is a p.d.f of a normal distribution with mean μ then

is

(a)1 (b)0.5 (c)0 (d)0.25

11. The random variable X follows normal distribution

. Then the value of c is

(a) (b)

(c) 5 (d)

12. For a standard normal distribution the mean and variance are

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

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

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

14. The P.I of is

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

15. The integrating factor of the differential equation

is

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

16. The order and degree of the differential equation are 3

2

3 )y(y y

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

17. If x2 + y

2 = 1 then the value of

iyx

iyx

1

1 is

a) x – iy b) 2x c) -2iy d) x + iy

18. If z represents a complex number then arg (z) + arg ( z ) is

a) 4

b)

2

c) 0 d)

4

19.

i

i

e

e

1

1=

a) cos + isin b) cos - isin c) sin + icos d) sin + icos

20. The complex conjugate of –4 – i9 is

a) –4 + i9 b) 4 + i9 c) 4 – i9 d) –4 – i9

21. The length of the latus rectum of the parabola whose vertex is (2,-3) and the directix x = 4 is

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

22. The eccentricity of the conic 9x2 + 5y

2 – 54x – 40y + 116 = 0 is

a) 3

1 b)

3

2 c)

9

4 d)

5

2

23. The straight line 2x-y+c =0 is a tangent to the ellipse 4x2 + 8y

2 = 32 if c is

a) 23 b) 6 c) 36 d) 4

24. The eccentricity of the hyperbola 12y2-4x

2-24x+48y-127 = 0 is

a) 4 b) 3 c) 2 d) 6

25. The angle between the asymptotes to the hyperbola 1916

22

yx

is

a) - 2 tan-1

4

3 b) - 2 tan

-1

3

4 c) 2 tan

-1

4

3 d) 2 tan

-1

3

4

26. If the centre of the ellipse is (2, 3) one of the foci is (3, 3)then the other focus is

a) (1, 3) b) (–1, 3) c) (1, –3) d) (–1, –3)

27. The slope of the tangent to the curve y = 3x2 + 3 sin x at x = 0 is

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

28. The equation of the normal to the curve = t

1at the point (-3, -

3

1)is

a) 3 = 27 t-80 b) 5 = 27 t -80 c) 3 = 27 t + 80 d) = t

1

29. The radius of a cylinder is increasing at the rate of 2cm/sec and its altitude is decreasing at the rate of 3cm/sec.

The rate of change of volume when the radius is 3cm and the altitude is 5cm is

a) 23 b) 33 c) 43 d) 53

30. The function (x) = x3 has

a) absolute maximum b) absolute minimum c) local maximum d) no extrema

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

2 (1 +x) has

a) an asymptote parallel to x – axis b) an asymptote parallel to y – axis

c) asymptotes parallel to both axes d) no asymptotes

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32. If u(x, y) = x4 + y

3 + 3x

2y

2 + 3x

2y then

yx

u

2

is

a) 12xy + 6x b) 12xy – 6x c) 12x2y – 6x d) 12xy

2 – 6x

33. The value of dxxx

1

0

4)1( is a) 12

1 b)

30

1 c)

24

1 d)

20

1

34. The area bounded by the line y = x, the x – axis, the ordinates x = 1, x = 2 is

a) 2

3 b)

2

5 c)

2

1 d)

2

7

35. The curved surface area a sphere of radius 5, intercepted between two parallel planes of distances 2 and 4 from

the centre is

a) 20 b) 40 c) 10 d) 30

36. If n is even then 2

0sin

xdxnis

a) 2

....5

4.

3

2.

1

n

n

n

n

n

n b)

22

1....

4

5.

2

3.

1

n

n

n

n

n

n

c) 1.2

3....

5

4.

3

2.

1

n

n

n

n

n

n d) 1.

3

2....

4

5.

2

3.

1

n

n

n

n

n

n

37. Which of the following is a tautology?

a) p q b) p q c) p ~ p d) p ~ p

38. Which of the following is not a group?

a) (Zn, +n) b) (Z, +) c) (Z, .) d) (R,+)

39. In the set of integers under the operation * defined by a * b = a + b – 1, the identify element is

a) 0 b) 1 c) a d) b

40. Which of the following are not statements?

i) Three plus four is eight ii) The sun is a planet.

iii) Switch on the light iv) Where are you going?

a) i), ii) b) ii), iii) c) iii) and iv) d) iv) only

PART – B 10 x 6 = 60

Note : (i) Answer any ten questions.

(ii) Question No. 55 is compulsory and choose any nine questions from the remaining

41. Show that the adjoint of A=

is A itself

42. A =

verify the result ( A-1

)T = ( A

T )

-1

43. For any three vectors prove that [ ] =

44. Find the vector and Cartesian equations of the sphere whose center is (1 , 2, 3) and which passes through the

point ( 5, 5, 3)

45. Prove that the triangle formed by the points representing the complex numbers (10 + 8i), (-2 + 4i) and

(-11 + 31i) on the Argand plane is right angled triangle

46. Find the real values of x and y for which

47. Find the angle between the asymptotes to the hyperbola

48. Obtain the Maclaurin’s series for

.

49. Find the approximate value of using differentials

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50. Evaluate (i)

(ii)

51. Solve

52. Show that

53. State and prove reversal law on inverses of a group.

54. Alpha particles are emitted by a radioactive source at an average rate of 5 in a 20 minutes interval. Using Poisson

distribution, find the probability that there will be

(i) 2 emissions (ii) at least 2 emissions

In a particular 20 minute interval. ( )

55. (a) Let x be the binomially distributed variable with mean 2 and standard deviation

. Find the corresponding

probability function. (OR)

(b) (i)Find the critical numbers of

(ii) Determine the domain of concavity (convexity) of the curve

PART – C 10 x 10 = 100

Note : (i) Answer any ten questions.

(ii) Question No. 70 is compulsory and choose any nine questions from the remaining

56. Investigate for what values of λ, μ the simultaneous equations

x+ y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ have

(i) no solution (ii) a unique solution and (iii) an infinite number of solutions

57. + + , =2 + , + + , + +2 Verify

( ) x ( ) = [ - [ 58. The girder of a railway bridge is in the parabolic form with span 100 ft. and the highest point on the arch is 10 ft.

above the bridge. Find the height of the arch of the bridge at 10 ft. to the left or right from the mid- point of the

bridge .

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

passes through the points (6, 0) and (-3, 0).

60. Find the eccentricity , centre, foci, vertices of the ellipse and hence draw the

diagram.

61. If and are the roots of , prove that

and deduce ,

62. Show that the equation of the normal to the curve at is

63. Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is

x

(volume of the sphere)

64. Verify Euler’s theorem for

65. Show that the surface area of the solid obtained by revolving the arc of the curve from x = 0 to

about x – axis is

66. Find the area of the region common to the circle and the parabola

67. Solve

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68. Radium disintegrates at a rate proportional to the amount present. If 5% of the original amount disintegrates in 50

years, how much will remain at the end of 100 years ? [Take Ao as the initial amount ]

69. Show that the set is an abelian group under multiplication.

70. (a) Find and of the normal distribution whose probability function is given by

(OR)

(b) Find the vector and Cartesian equations of the plane, through the point (1, 2, - 2) and parallel to the line

and the perpendicular to the plane .

R.RAJASEKARAN.,M.Sc.,B.Ed.,M.Phil.

DAYANANDA VIDYALAYA MATRIC HR SEC SCHOOL

KURUSILAPATTU

VANIYAMBADI TK , VELLROE DT– 635710

CELL : 9047879191

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