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Padasalai.net Special- Centum Coaching Team Question Paper

(2017-2018)

PART – III

MATHEMATICS

Xll std ( ENGLISH VERSION)

TIME ALLOWED: 3 HRS MAXIMUM MARKS: 200

INSTRUCTIONS : 1) Check the question paper for fairness of printing. If there is

any lack of fairness, inform the hall supervisor immaditely.

2 ) Use Blue or black link to write and pencil to draw diagrams

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 40 x1 =40

.

01. The rank of the matrix

21

42 is a) 1 b) 2 c) 0 d) 8

02. The rank of the diagonal matrix

0

4

0

2

1

a) 0 b) 2 c) 3 d) 5

03. 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

04. 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



06. If p

and q

and qp

are vectors of magnitude then the magnitude of qp

is

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

07.. The projection of

OP on a unit vector

OQ equals thrice the area of parallelogram OPRQ. Then POQ is,

a) 3

1tan 1 b)

10

3cos 1 c)

10

3sin 1 d)

3

1sin 1

08. The vector dcba

is

a) perpendicular to a

, b

, c

and d

b) parallel to the vectors ba

and dc

c) parallel to the line of intersection of the plane containing a

and b

and the plane containing c

and d

d) perpendicular to the line of intersection of the plane containing a

and b

and the plane containing c

and

d

09. If a

, b

, c

are a right handed triad of mutually perpendicular vectors of magnitude a, b, c then the value of

c,b,a

is a) 222 cba b) 0 c) abc

2

1 d) abc

10. Let vu , and w be vector such that 0 wvu

If 4,3 vu

and 5w

then uwwvvu is

a) 25 b) -25 c) 50 d) 5

11. The work done by the force kjiaF in moving the point of application from (1,1,1) to (2,2,2) along

a straight line is given to be 5 units. The value of a is a) -3 b) 3 c)8 d) -8

12. 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

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

a) 23

23

22

22

21

21 bababa b) 2321

2321 bbbaaa

c) 23

23

22

22

21

21 bababa d) 2

32

22

12

32

22

1 bbbaaa

14. 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



15. If 1Z and 2Z are any two complex numbers then which one of the following is false

a) )Re()Re(Z) Z ZRe( 2121 Z b) )Im()Im(Z) Z ZIm( 2121 Z

c) )arg()arg(Z) Z Zarg( 2121 Z d) 2121 ZZZZ

16. The difference between the focal distances of any point on the hyperbola 1b

y

a

x

2

2

2

2

is 24 and the

eccentricity is 2. Then the equation of the hyperbola is.

a) 1432

y

144

x 22

b) 1144

y

432

x 22

c) 1312

y

12

x 22

d) 112

y

312

x 22

17. The area of the triangle formed by the tangent at any point on the rectangular hyperbola xy = 72 and

its asymptotes is. a) 36 b) 18 c) 72 d) 144

18. The normal to the rectangular hyperbola xy = 9 at

2

3,6 meets the curve again at.

a)

24,

8

3 b)

8

3,24 c)

24,

8

3 d)

8

3,24

19. The condition that the line 0 nmylx may be normal to the ellipse 12

2

2

2

b

y

a

xis

a) 02 223 nmalmal b)

2

222

2

2

2

2

n

ba

m

b

l

a c)

2

222

2

2

2

2

n

ba

m

b

l

a d)

2

222

2

2

2

2

n

ba

m

b

l

a

20. The angle between the curves 19

y

25

x 22

and 18

y

8

x 22

is

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

21. The ‘c’ of Lagranges Mean Value Theorem for the function f(x) = x² + 2x -1 ; a = 0, b = 1 is.

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

22. Food pockets were dropped from an helicopter during the flood and distance fallen in “ t “ seconds is given by

22 /8.92

1smggty . Then the speed of the food pocket after it has falled for “ 2 “ seconds is

a) 19.6 m / sec b) 9.8 m / sec c) – 19.6 m / sec d) – 9.8 m / sec

23. An object dropped from the sky follows the law of motion 22 /8.92

1smggtx . The acceleration of the

object when t = 2 is a) 2sec8.9 m

b)

2sec8.9 m c) 2sec6.19 m

d) 2sec6.19 m



24. Identify the true statements in the following

i. if a curve is symmetrical about the origin, then it is symmetrical about both axes.

ii. if a curve is symmetrical about the both the axes, then it is symmetrical about the origin.

iii. A curve f(x,y) =0 is symmetrical about the line y = x if f(x,y) = f(y, x)

iv. for the curve f(x,y) =0, if f(x,y) = f(-y,-x), then it is symmetrical about the origin

a) (ii), (iii) b) (i), (iv) c) (i), (iii) d) (ii), (iv)

25. If u = f ( x , y ) is a differentiable function of x and y ; x and y are differentiable functions of t then

a)t

y

y

f

t

x

x

f

dt

du

b)

t

y

y

t

dt

dx

x

f

dt

du

c)

dt

dy

y

f

dt

dx

x

f

dt

du

d)

t

y

y

f

t

x

x

f

t

u

26. Volume of solid obtained by revolving the area of the ellipse 1b

y

a

x

2

2

2

2

about major and minor

axes are in the ratio.

a) b² : a² b) a² : b² c) a : b d) b : a

27. The area bounded by the parabola y² = x and its latus rectum is.

a) 4 / 3 b) 1 / 6 c) 2 / 3 d) 8 / 3

28. The value of

2/

2/ xcos2

xsin dx is. a) 0 b) 2 c) log 2 d) log 4

29. dxxI nn sin then nI

a) 21 1

cossin1

n

n In

nxx

n b) 2

1 1cossin

1

n

n In

nxx

n

c) 21 1

cossin1

n

n In

nxx

n d) n

n In

nxx

n

1cossin

1 1

30. The differential equation xy5dy

dx 3/12

is

a) of order 2 and degree 1. b) of order 1 and degree 2 c) of order 1 and degree 6

d) of order 1 and degree 3.

31. The differential equation of all non-vertical lines in a plane is.

a) 0dx

dy b) 0

dx

yd

2

2

c) mdx

dy d) m

dx

yd

2

2



32. The particular integral of the differential equation f(D) y = eax where f(D) = (D-a) g(D), g(a)

a) meax b) eax / g(a) c) g(a) eax d) xeax / g(a)

33. 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)xexy

dx

dy is a linear differential equation in x.

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

a) k1 b) 1 c) kn d) kn

35. In the set of integers under the operation * defined by a * b = a + b -1, the identity element is.

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

36. The truth values of the following statements are

i. Ooty is in Tamilnadu and 3 + 4 = 8 ii. Ooty is in Tamilnadu and 3 + 4 = 7

iii. Ooty is in Kerala and 3 + 4 = 7 iv. Ooty is in Kerala and 3 + 4 = 8

a) F,T,F,F b) F,F,F,T c) T,T,F,F d) T,F,T,F

37. In 55 ,0 Z the order of 10 is a) 1 b) 2 c) 3 d) 4

38. A random variable X has the following probability mass function as follows

x -2 3 1

P(X=x) 6 4 12

Then the value of is… a) 1 b) 2 c) 3 d) 4

39. The marks secured by 400 students in a Mathematics 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



40. If X is a discrete random variable then which of the following is correct? a) 10 xF

b) 0F and 1F

c) 1 nnn xFxFxXP d) xF is a constant function

PART-B

NOTE (i) ANSWER ANY TEN QUESTIONS

(ii) Q.NO 55 IS COMPULSORY AND CHOOSE ANY NINE QUESTION FROM

THE REMAINING 10 x6 =60

(41) For

544

434

221

A show that 1 AA .

(42) Solve the following non-homogeneous system of linear equations by determinant method:

18108;954 yxyx

(43) a) If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is .3

b)If the points ( , 0,3 ) , (1,3,-1) and (-5,-3,7) are collinear then find .

44) Find the vector and Cartesian equation of the sphere whose centre is (1,2,3) and which passes through he point

(5,5,3)

45) Prove that the complex numbers iii 3333,33,33 are the vertices of an equilateral triangle in the

complex plane.

46) Show that for any polynomial equation 0xP with real coefficients imaginary roots occur in conjugate pairs

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

0

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

48) Verify

xy

u

yx

u

22

for the following function:

(i) 22 3 yxyxu

49) Evaluate the following Problems using properties of integration.

dxxx cossin2

0

3



50) The temperature T of a cooling object drops at a rate proportional to the difference T - S, where S is

constant temperature of surrounding medium. If initially CT 150 , find the temperature of the cooling object at any

time t.

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 qp and pq are not equivalent.

53) Two bad oranges are accidentally mixed with ten good ones. Three oranges are drawn at random without

replacement from this lot. Obtain the probability distribution for the number of bad oranges.

54) If a publisher of non-technical books takes a great pain to ensure that his books are free of typological errors,

so that the probability of any given page containing atleast one such error is 0.005 and errors are independent from

page to page (i) what is the probability that one of Rs. 400 page novels will contain exactly one page with error. (ii)

atmost three pages with errors. .819.0;1363.0 2.02 ee

55) a) Prove that the inequality nxx

n 11 is true whenever 0x and .1n

(or)

b) Find the equation of the locus of all points such that the differences of their distances from (4,0) and (-4,0) is

always equal to 2.

PART- C

NOTE : (i) ANSWER ANY TEN QUESTIONS

(ii) Q.NO 70 IS COMPULSORY AND CHOOSE ANY NINE QUESTIONS

FROM THE REMAINING

10X10=100

56) For what value of the equations 022,034,03 zyxzyxzyx have a (i) trivial

solution, (ii) non-trivial solution.

57) Verify dcbacdbadcba where kjia ; kib 2 ; kjic 2 ;

kjid 2

58 ) Find the vector and Cartesian equations of the plane passing through the points (-1,1,1) and (1,-1,1) and

perpendicular to the plane 522 zyx

59) Solve the equation 01459 xxx



60) Show that the line 04 yx is a tangent to the ellipse .123 22 yx Find the co-ordinates of the point of

contact.

61) Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes a parabolic path.

The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the line of the pipe, the flow of

water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line

will the water strike the ground?

62) Find the eccentricity, centre, foci and vertices of the hyperbola 01996418169 22 yxyx and also trace

the curve

63) A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If water is

being pumped into the tank at a rate of 2m3 / min , find the rate at which the water level is rising when the water is

3m deep.

64) Using Euler’s theorem prove the following : If

yx

yxu

331tan Prove that .2sin u

y

uy

x

ux

65) Find the length of the curve 1

3232

a

y

a

x

66) Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a distance a

and b from the centre of the sphere is abr 2 and hence deduct the surface area of the sphere. ab .

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 of 213t 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) Solve the following : 12 dx

dyyx

69) The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean

value 31 psi and standard deviation 0.2 psi.

(i) What is the probability that the pressure for a randomly selected tyre (a) between 30.5 and 31.5 psi

(b) between 30 and 32 psi

(ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi?



70) a) A closed (cuboid) box with a square base is to have a volume of 2000 c.c. The material for the top and

bottom of the box is to cost Rs. 3 per square cm. and the material for the sides is to cost Rs. 1.50 per square cm. If

the cost of the materials is to be the least, find the dimensions of the box.

(0r)

b) Show that the set G of all rational numbers except -1 forms an abelian group with respect to the operation *

given by abbaba * for all ., Gba

******** ALL THE BEST ********

PREPARED BY

S. BALAKRISHNAN , M.SC, B.ED , (PG ASST IN MATHS)

SHRI MARUTHI HIGHER SECONDARY SCHOOL

V. MADEPALLI ,veppanapalli KRISHNAGIRI –(DT) -635121

ANSWER SCRIPTS SEND :

S. BALAKRISHNAN , M.SC, B.ED , (PG ASST IN MATHS)

S/0 B. SAMPANGIRAMAN (PWD)

V. MADEPALLI –X-ROAD , VEPPANAPALLI

KRISHNAGIRI –(DT) -635121

CONTACT : MOBILE :: 9159652220

E.MAIL ID:: [email protected]


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