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Padasalai.Net’s Special - Centum Coaching Team Question Paper 2017-2018

Mathematics

Marks:200 Time: 3 hours

Class : XII

Part I Choose the Best Answer 40 x 1 = 40

1. If 102A , then the rank of TAA is, a) 1 b) 2 c) 3 d) 0

2. If A is a square matrix of order n then Aadj is a) 2A b) nA c) 1nA

d) A

3. The system of equations ax+y+z = 0 ; x+by+z = 0; x+y+cz = 0 has a non – trivial solution then

c1

1

b1

1

a1

1 . a) 1 b) 2 c) –1 d) 0

4. If a

and b

include an angle 120 and their magnitude are 2 and 3 then ba is equal to

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

3

5. If

PR = kji2

,

QS = k2j3i

then the area of the quadrilateral PQRS is

a) 35 b) 310 c) 2

35 d) 2

3

6. The shortest distance of the point (2, 10, 1) from the plane 262)43( kjir

is

a) 262 b) 26 c) 2 d) 26

1

7. The point of intersection of the line k7j2i3tkir

and the plane 8)kji(r

is

a) (8, 6, 22) b) (-8, - 6, -22) c) (4, 3, 11) d) (-4, -3, -11)

8. The centre and radius of the sphere given by 01z10y8x6zyx 222

a) (-3, 4, -5) , 49 b) (-6, 8, -10) , 1 c) (3, -4, 5) , 7 d) (6, -8, 10) , 7

9. The modulus and amplitude of the complex number 33 4i

e are respectively

a) 2,9

e b) 2

,9 e

c) 4

3,6

e d) 4

3,9

e

10. If ))()(( 332211 ibaibaibaiBA then 22 BA is

a) 2

32

32

22

22

12

1 bababa b) 23212

321 bbbaaa

c) 23

23

22

22

21

21 bababa d) 2

32

22

12

32

22

1 bbbaaa

11. If the amplitude of a complex number is 2

then the number is

a) purely imaginary b) purely real c) 0 d) neither real nor imaginary

12. If 2i is one is one root equation 02 cbxax , then the other root is

a) 2i b) 2i c) i2 d) ii 2

13. The angle between the two tangents drawn from the point (-4, 4) to x16y2 is.

a) 45 b)

30 c) 60 d)

90

14. The straight line 0cyx2 is a tangent to the ellipse 32y8x4 22 if c is.

a) 32 b) 6 c) 36 d) 4

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15. The line 0k4y2x5 is a tangent to 36yx4 22 then k is.

a) 9

4

b) 3

2

c) 4

9

d) 16

81

16. The parametric equations of the curve 323232 ayx are.

a) 33 cos;sin ayax b) 33 sin;cos ayax

c) cos;sin 33 ayax d) sin;cos 33 ayax

17. A spherical snowball is melting in such a way that its volume is decreasing at a rate of .min/1 3cm The

rate at which the diameter is decreasing when the diameter is 10 cms is

a) min/

50

1cm

b) min/

50

1cm

c) min/

75

11cm

d) min/

75

2cm

18. If y = 6x –x³ and x increases at the rate of 5 units per second, the rate of change of slope when x = 3 is.

a) – 90 units / sec. b) 90 units / sec. c) 180 units / sec. d) – 180 units / sec

19. For the curve x = et cos t ; y = et sin t the tangent line is parallel to the x axis when t is equal to a) –

4 b) 4 c) 0 d) 2

20. The curve )1()2( 22 xxxy has

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

c) asymptotes parallel to both axes d) no asymptotes

21. In which region the curve y² (a+x) = x² (3a-x) does not lie ?

a) x > 0 b) 0 < x < 3a. c) x < -a and x > 3a d) –a < x < 3a

22. The value of

0

32 xdxcosxsin

is a) b) 2 c) 4 d) 0.

23. The surface area of the solid of revolution of the region bounded by y = 2x, x = 0 and x = 2 about x axis

is. a) 58 b) 52 c) 5 d) 54

24. The area between the ellipse 1b

y

a

x

2

2

2

2

and its auxillary circle is

a) bab b) baa 2 c) baa d) bab 2

25. If cos x is an integrating factor of the differential equation QPydx

dy then P =

a) – cot x. b) cot x c) tan x d) – tan x.

26. The amount present in a radio active element disintegrates at a rate proportional to its amount . The

differential equation corresponding to the above statement is ( k is negative)

a) p

k

dt

dp b) kt

dt

dp c) kp

dt

dp d) kt

dt

dp

27. The P.I of (3D²+D-14) y = 13e2x is..

a) 26 x e2x. b) 13 x e2x. c) x e2x. d) x22

e2

x

28. The conditional statement qp is equivalent to

a) p v q. b) p v ~ q c) ~p v q. d) p ^ q.

29. The order of –i in the multiplicative group of 4th roots of unity

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

30. The value of [ 3 ] + 11( [5] + 11[6]) is.

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

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

a) Closure axiom. b) Associative axiom c) identity axiom d) inverse axiom.

32. If f(x) =

x,x16

1A

2 is a p.d.f of a continuous random variable X, then the value of A is.

a) 16 b) 8 c) 4 d) 1

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

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

34. If in a Poisson distribution P(X=0) = k then the variance is.

a) log 1/ k b) log k. c) e d) 1 / k

35. If f(x) is a p.d.f. of a normal variate X and 2,~ NX then

µ

dx)x(f is

a) undefined. b) 1 c) . 5 d) - .5

36. If 0 then the system is

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

c) Inconsistent d) Either consistent or inconsistent

37. If,60 ba and

40 baand

46b then

ais

a) 22 b) 21 c) 18 d) 11

38. The equation of the L.R. of yx 42 is a) x = -1 b) y = -1 c) x =1 d) y = 1

39. dxex x 2

0

6

a)

72

!6

b)62

!6

c) !626 d) !627

40. In finding the differential equation corresponding to mxey where m is the arbitrary constant, then m

is a) 'y

y

b) y

y'

c) y’ d) y

Part II Answer Any 10 Of the Following Questions 10 x 6 = 60

Q.No 55 is compulsory

41. State and prove reversal law for inverses of matrices.

42. Solve the following non-homogeneous equations of three unknowns.

423;1;522 zyxzyxzyx 43. If the diagonals of a parallelogram are equal then it is a rectangle

44. Find the vector and Cartesian equation of the sphere on the join of the points A and B having position

vectors kji 762 and kji 342 respectively as a diameter. Find also the centre

and radius of the sphere.

45. For what values of x and y, the numbers yix 23 and iyx 42 are complex conjugate of each other?

46. Solve : 010332256 234 xxxx given that one of the root is i2

47. Newton’s law of cooling is given by kte 0 , where the excess of temperature at zero time is

C0 and at time t seconds is C . Determine the rate of change of temperature after 40 s, given that

C160 and .03.0k 3201.32.1 e

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48. The current at time t in a coil with resistance R, inductance L and subjected to a constant

electromotive force E is given by

L

RT

eR

Ei 1

Obtain a suitable formula to be used when R is

very small.

49. If byaxzeV and z is a homogenous function of degree n in x and y prove that

.Vnbyaxy

Vy

x

Vx

50. Find the volume of the solid generated when the region enclosed by 2, yxy and 0x is

revolved about the y – axis.

51. Show that the set of all non-zero complex numbers is an abelian group under the usual multiplication

of complex numbers.

52. Show that pqqpqp

53. Solve: 22 14 xyDD

54. In a continuous distribution the p.d.f of X is

otherwise

xxxxf

,0

20,24

3

Find the mean and the variance of the distribution.

55. Alpha particles are emitted by a radio active source at an average rate of 5 in a 20 minutes interval.

Using Poisson distribution find the probability that there will be (i) 2 emission (ii) at least 2 emission

in a particular 20 minutes interval. .0067.05 e (Or)

Points A and B are 10km apart and it is determined from the sound of an explosion heard at those

points at different times that the location of the explosion is 6km closer to A than B. Show that the

location of the explosion is restricted to a particular curve and find an equation of it.

Part III Answer any 10 of the following questions 10 x 10 = 100

Q.no 70 is compulsory

56. Investigate for what values of , the simultaneous equations

zyxzyxzyx 2,1032,6 have (i) no solution (ii) a unique solution and

(iii) an infinite number of solutions.

57. Altitudes of a triangle are concurrent – prove by vector method

58. Find the vector and Cartesian equation of the plane containing the line3

1

3

2

2

2

zyxand parallel

to the line .1

1

2

1

3

1

zyx

59. If 2sin2cos,2sin2cos ibia and 2sin2cos ic Prove that

(i) cos2

1

abcabc

(ii)

2cos2222

abc

cba

60. A cable of a suspension bridge is in the form of a parabola whose span is 40 mts. The road way is 5 mts

below the lowest point of the cable. If an extra support is provided across the cable 30 mts above

the ground level find the length of the support if the height of the pillars are 55 mts.

61. Prove that the line 9125 yx touches the hyperbola 99 22 yx and find its point of contact.

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62. Find the eccentricity, centre, foci, vertices of the following ellipses: 0443272436 22 yxyx

63. A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 km

downstream on the opposite bank, as quickly as possible. He could row his boat directly across the

river to point R and then run to Q, or he could row directly to Q, or he could row to some point to

between Q and R and then run to Q. If he can row at 6 km/h and run at 8 km/h where should he land

to reach Q as soon as possible?

64. Use differentials to find the an approximate value for the given number 43 02.102.1 y

65. Find the area of the region bounded by the ellipse 12

2

2

2

b

y

a

x

66. Find the length of the curve tayttax cos1,sin between 0t and .

67. Solve :

xeydx

dy

dx

yd 3

2

2

223

when 0,2log yx and 0,0 yx

68. A bank pays interest by continuous compounding, that is by treating the interest rate as the

instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year

compounded continuously. Calculate the percentage increase in such an account over one year. [ Take

0833.108 e ].

69. Let G be the set of all rational numbers except 1 and * be defined on G by abbaba * for all

., Gba Show that ( G , * )is an infinite abelian group.

70. The mean score of 1000 students for an examination is 34 and S.D. is 16. (i) How many candidates can

be expected to obtain marks between 30 and 60 assuming the normally of the distribution and (ii)

determine the limit of the marks of the central 70% of the candidates. (Or)

Gravel is being dumped from a conveyor belt at a rate of 30 ft3 / min and its coarsened such that it

forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the

height of the pile increasing when the pile is 10 ft high?

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PG TEACHER

MARY ANN MATRIC HIGHER SECONDARY SCHOOL

K.PUDUR, MADURAI -625007

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S.SARAVANA KUMAR

2/291, KULANTHAI YESU STREET BACK STREET, ANKAYAR KANNI COLONY

AATHIKULAM, K.PUDUR

MADURAI -625007

PHONE: 7092681321, 8608033637

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