xii std mathematics part a · pdf filexii std mathematics ... f0, 48 determine where the ......

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XII Std MATHEMATICS Time Allowed 3.00 Hrs Total Marks : 200

Instructions :

1. Check the question paper for fairness of printing. If there is any lack of

fairness, inform the Hall Supervisor immediately.

2. Use Blue or Black ink to write and pencil to draw diagrams

Part A

Note 1 All questions are compulsory

2 Choose the most suitable answer from the given four alternatives

and write the option code and the correspoinding answer.

1x40 =40

1 The random variable X follows normal distribution f(x) = 25

)100x(2/1 2

ce

. Then the value of c is.

a) 2 b) 2

1 c) 25 d) 25

1

2. The mean of binomial distribution is 5 and its standard deviation is 2. Then the value of n and p are.

a)

25,

5

4 b)

5

4,25 c)

25,

5

1 d)

5

1,25

3. If f(x) =

elsewhere,0

3x0,kx2is a probability density function then the value of k is.

a) 1/ 3 b) 1/ 6 c) 1/ 9 d) 1/12.

4. Which of the following is a contradiction ?

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

5. If a compound statement is made up of three simple statements, then the number of rows in the truth

table is.

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

6. Which of the following is correct ?

a. an element of a group can have more than one inverse.

b if every element of a group is its own inverse, then the group is abelian.

c the set of all 2 x 2 real matrices forms a group under matrix multiplication.

d. (a*b)-1 = a-1 * b-1for all a, b G

7. If xxf and f(1) = 2 then f(x) is..

a) )2xx(3

2 b) )2xx(

2

3 c) )2xx(

3

2 d) )2x(x

3

2

8 A particular integral of (D² -4D+4) y = e2x is.

a) x22

e2

x b) xe2x. c) xe-2x. d) x2e2

x

9. The integrating factor of dx + xdy = e-y sec² y dy is…

a) ex. b) e-x. c) ey. d) e-y.



10. The volume, when the curve y = 2x3 from x = 0 to x = 4 is rotated about x axis is .

a) 100 b) 9

100 c)

3

100 d) 100 / 3

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

a) 3 / 2. b) 5 / 2. c) 1 / 2 d) 7 / 2

12 The value of

2/

03/53/5

3/5

xsinxcos

xcos dx is

a) 2 b) 4 c) 0 d)

13. If rA then which of the following is correct ?

a) all the minors of order r which does not vanish

b) A has atleast one minor of order r which does not vanish

c) A has atleast one (r+1) order minor which vanishes

d) all (r+1) and higher order minors should not vanish

14. The length of the perpendicular from the origin to the plane 261243 kjir is

a)26 b) 26 / 169 c) 2 d) 1 / 2

15. The vector equation of a plane passing through the line of intersection the planes

2211 qnrandqnr is

a) 02211 qnrqnr b) 2121 qqnrnr

c) 2121 qqnrnr d) 2121 qqnrnr

16 The cube roots of unity are

a) in G.P. with common ratio b) in G.P. with common difference 2

c) in A.P. with common difference d) in A.P. with common difference 2

17 The equation of the tangent at (-3,1) to the parabola yx 92 is

a) 0323 yx b) 0332 yx c) 0332 yx d) 0323 yx

18 The distance – time relationship of a moving body is given by y = F ( t ) then the acceleration

of the body is the

a) gradient of the velocity / time graph b) gradient of the distance / time graph

c) gradient of the acceleration / distance graph d) gradient of the velocity / distance graph

19 dxxI nn cos then nI

a) 21 1

sincos1

n

n In

nxx

n b) 2

1 1sincos

n

n In

nxx

c) 21 1

sincos1

n

n In

nxx

n d) 2

1 1sincos

1

n

n In

nxx

n

20. Identify the incorrect statement

a) The order of a differential equation is the order of the highest derivative occurring in it.

b) The degree of the differential equation is the degree of the highest order derivative which

occurs in it (the derivatives are free from radicals and fractions)

c) yxf

yxf

dx

dy

,

,

2

1 is the first order first degree homogeneous differential equation

d) xexydx

dy is a linear differential equation in x.

21 Which of the following is not true?

a) Negation of a negation of a statement is the statement itself

b) If the last column of its truth table contain only T then it is tautology

c) If the last column of its truth table contains only F then it is contradiction d) If p and q are any two statements then qp is a tautology



22. If the length of the diagonal of a square is increasing at the rate of 0.1 cm /sec. What is the rate of

increase of its area when the side is cm2

15 ?

a) 1.5 cm² / sec. b) 3 cm² / sec c) 23 cm³ / sec. d) 0.15 cm³ / sec

23 Which of the following is not true regarding the normal distribution?

a) skewness is zero b) Mean = median = mode

c) the Points of inflection are at X d) maximum height of the curve is 2

1

24. 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

25 The curve 9y² = x² (4-x²) is symmetrical about.

a) y axis. b) x axis. c) y = x d) both the axes

26. In a given semi circle of diameter 4 cm a rectangle is to be inscribed. The maximum area of the

rectangle is.

a) 2 b) 4 c) 8 d) 16 27 The value of c in Rolle’s Theorem for the function f(x) = cos x / 2 on 3, is

a) 0 b) 2 c) 2 d) 23

28 The co-ordinate of the vertices of the rectangular hyperbola xy = 16 are.

a) (4,4) (-4, -4) b) (2,8) , (-2,-8) c) (4,0), (-4,0) d) (8,0 ) (-8,0).

29 If the matrix

541

3k1

231

has an inverse then the values of k

a) k is any real number b) 4k c) 4k d) 4k

30 If 102A , then the rank of TAA is,

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

31. The distance between the foci of the ellipse 180y5x9 22 is.

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

32. The directrix of the parabola 4xy2 is

a) 4

15x b)

4

15x c)

4

17x d)

4

17x

33. If is a cube root of unity then the value of 4242 )1()1( is

a) 0 b) 32 c) -16 d) -32

34. The polar form of the complex number 325i is

a) 2

sin2

cos

i b) sincos i c) sincos i d) 2

sin2

cos

i

35. If 122 yx then the value of iyx

iyx

1

1 is

a) iyx b) x2 c) iy2 d) iyx

36 If k3j2ia

and k2ji3b

then a unit vector perpendicular to a and bis,

a) 3

kji

b) 3

kji

c) 3

k2ji

d) 3

kji

37. If a

, b, c

are non-coplanar and ac,cb,ba

= ac,cb,ba

then the value of c,b,a

is,

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

38. If the projection of aon b

and the projection of b

on a

are equal then the angle between ba

and ba

is,

a) 2

b)

3

c)

4

d)

3

2



39. If a and b

are unit vectors and is the angle between them, then )ba(

is a unit vector if

a) 3

b)

4

c)

2

d)

3

2

40. If A and B are any two matrices such that AB = 0 and A is non-singular, then

a) B = 0 b) B is singular c) B is non-singular d) B = A

Part –B

Note (i) Answer any ten questions

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

from the remaining 10x6 =60

41 If

11

21A and

21

10B verify that 111

ABAB .

42 Find the rank of the matrix

11715

4312

3111

43 Find the vector and Cartesian equations of the sphere whose centre is kji 22 and

radius is 3.

44 If n is a positive integer, prove that 6

cos233 1 nii nnn

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

46 Find the angle between the asymptotes to the hyperbola 01417253 22 yxyxyx

47 (a) Evaluate the limit for the following if exists.x

x

x 2

sinlim

2

(b) Prove that log x is strictly increasing function on ,0

48 Determine where the curve 133 xxy is cancave upward and where it is concave

downward. Also find the inflection points.

49 Use differentials to find an approximate value of cube roots of 127

50 Solve xydx

dy

51 Show that qpqp is a tautology.

52

10

01,

10

01,

10

01,

10

01 form an abelian group, under multiplication of

matrices.

53 Find the Mean and Variance for the following probability density functions:

(i)

otherwise

xxf

,0

1212,24

1

54 Marks in an aptitude test given to 800 students of a school was found to be normally

distributed. 10% of the students scored below 40 marks and 10% of the students scored above

90 marks. Find the number of students scored between 40 and 90.

55 (a) (i) For any vector a prove that akakjajiai 2

(ii) If 0.,0.,0. cxbxax and 0x then show that cba ,, are

coplanar (or)



(b) Find the volume of the solid that result when the region enclosed by 12

2

2

2

b

y

a

xis

revolved about major axis .0 ba

Part –C

Note (i) Answer any ten questions

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

from the remaining

10x10=100

56 If

221

212

122

3

1A , prove that

TAA 1 .

57 Show that the lines 31

1

1

1 zyx

and

1

1

2

1

1

2

zyxintersect and find their point of intersection.

58 Derive the equation of the plane in the intercept form.( both in vector and cartesion form )

59 If P represents the variable complex number z. Find the locus of P, if 23

1arg

z

z

60 Find the eccentricity, centre, foci, vertices of the ellipse and draw the diagram:

0681684 22 yxyx

61 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.

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

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

63 A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight river.

He needs no fence

along the river. What ar the dimensions of the field that has the largest area?

64 If veuw 2 where

y

xu and ,log xyv find

x

w

and

y

w

65 Find the area bounded by the curve yx 42 and the line x = 4y - 2

66 Find the surface area of the solid generated by revolving the cycloid tayttax cos1,sin

about its base ( x- axis ).

67 A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A

patient is administered 10 mg of drug at time t = 0 , which is excreted at a Rate of213t mg/h.

(i) What is the general equation for the amount of drug in the patient at time t > 0 ?

(ii) When will the patient be drug free?

68 Show that the set QbabaG ,/2 is an infinite abelian group with respect to addition.

69 Obtain k , and 2 of the normal distribution whose probability distribution function is given by

.42 2

Xkexf xx



70 (a) Solve : 02 dxydyxxy (or)

(b) 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?

All the best

M. Anand M.Sc.,M.Phil.,B.Ed.,P.G.D.C.A.,

P.G,Teacher

Government Higher Secondary

School

Ayyampettai

Thanjavur District 614201

Send your Answer scripts to the following address

M. Anand

P.G.Teacher ( Maths )

No. 8 Vijaya Nagar

Srinivasapuram

Thanjavur

98947129851


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xii std mathematics part a · pdf filexii std mathematics ... f0, 48 determine where the ......

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