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AS-MA/T1 Page 2 of 5 Name:

7.Let p andq be real numbers such thatp 0,p3 q andp3 q. If and are nonzerocomplex numbers satisfying + =p and 3 +3 = q , then a quadratic equation having

and

as its roots is

(a) (p3 + q)x2 (p3 + 2q)x+ (p3 +q) = 0

(b) (p3

+ q)x2

(p3

2q)x+ (p3

+ q) = 0(c) (p3 q)x2 (5p3 2q)x+ (p3 q) = 0(d) (p3 q)x2 (5p3 + 2q)x+ (p3 q) = 0

8.Let f, g andh be real valued functions defined on the interval [0; 1] by f(x) = ex2

+ex2

,

g(x) = xex2

+ex2

. Ifa; b andc denote respectively, the absolute maximum off, g andhon[0; 1], then

(a) a = b and c b (b)a = c anda b(c) a

band c

b (d)a = b = c

Section II (Multiple Correct Objective Type)

.This section contains 5 multiple choice questions.. Each question has choices (a),(b),(c)and(d)for its answer, out of which ONE or MOREis/are correct.

9.LetAandB be two distinct points on the parabola y2 = 4x. If the axis of the parabola touchesa circle of radius r havingAB as its diameter, then the slope of the line joining A and B canbe

(a) 1r

(b) 1r

(c) 2r

(d) 2r

10.Let ABC be a triangle such that ACB = 6

and leta,b and c denote the lengths of the sides

opposite toA,B andCrespectively. The value(s) ofx for whicha = x2 + x+ 1,b = x2 1andc = 2x+ 1 is(are)

(a)

2 +p

3

(b)1 +p

3 (c) 2 +p

3 (d)4p

3

11.Let z1and z2be two distinct complex numbers and let z= (1t )z1+tz2for some real numbert with0 < t < 1. Ifarg(w) denotes the principal argument of nonzero complex number w ,

then(a) jz z1j + jz z2j = jz1 z2j(b) arg(z z1) =arg(z z2)

(c)

z z1 z z1z2 z1 z2 z1

= 0

(d) arg(z z1) =arg(z2 z1)

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12.Let f be a real-valued function defined on the interval (0; 1) by f(x) =ln x+Z x0

p1 +sin t dt .

Then which of the following statement(s) is (are) true?

(a) f 00

(x)exists for allx2 (0; 1)(b) f

0

(x)exists for allx2 (0; 1)but not differentiable on(0; 1)(c) there exists >1 such that jf 0(x)j < jf(x)j for allx2 (; 1)(d) there exists >0 such that jf(x)j + jf 0(x)j for allx2 (0; 1)

13.The value(s) of

Z 1

0

x4(1 x)41 +x2

dx is/are

(a)22

7 (b) 2

105 (c) 0 (d)

71

153

2Section III (Linked Comprehension Type)

.This section contains 2 passages. First passage has 3 multiple choice questions based on it,

whereas the second passage has 2 multiple choice question based on it. Each question has 4

choices(a),(b),(c)and (d)for its answer, out of whichONE or MOREis/are correct.

Passage for Questions 14 to 16

Letp be an odd prime number andTp be the following set of2 2matrices:

Tp =

A=

a bc a

: a; b; c2 f0; 1; 2; : : : ; p 1g

14.The number ofAinTp such thatAis either symmetric or skew-symmetric or both and det(A)divisible byp is

(a) (p 1)2 (b)2(p 1) (c) (p 1)2 + 1 (d)2p 115.The number ofA in Tp such that the trace ofAis not divisible by p but det(A)is divisible by

pis[Note:The trace of a matrix is the sum of its diagonal entries

(a) (p 1)(p2 p + 1)(b)p3 (p 1)2 (c) (p 1)2 (d)(p 1)(p2 2)16.The number ofAinTp such thatdet(A)is not divisible byp is

(a) 2p3

(b)p3

5p (c) p3

3p (d)p3

p2

Passage for Questions 17 to 18

The circlex2 + y2 8x= 0and hyperbola x2

9 y

2

4 = 1intersect at the pointsAandB

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17.Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

(a) 2x p

5y 20 = 0 (b)2x p

5y+ 4 = 0

(c) 3x 4y+ 8 = 0 (d)4x 3y+ 4 = 018.Equation of the circle withAB as its diameter is

(a) x2 + y2

12x+ 24 = 0 (b)x2 + y2 + 12x+ 24 = 0

(c) x2 + y2 + 24x 12 = 0 (d)x2 + y2 24x 12 = 0Section IV (Integer Type)

.This section contains 10 questions. The answer to each question is a single-digit integer, ranging

from0to9.

19.The number of values ofin the interval

2;

2such that

n

5 forn = 0;

1;

2and

tan =cot 5as well assin 2=cos 4 is

20.The maximum value of the expression1

sin2 + 3 sin cos + 5 cos2 is

21.If #a and#

b are vectors in space given by #a =i 2jp

5and

#

b = 2i+ j + 3kp

14, then the value

of

2 #a + #

b

h

#a #b

#a 2 #b

iis

22.The line 2x+y = 1 is tangent to the hyperbolax2

a2

+ y2

b2

= 1. If this line passes through

the point of intersection of the nearest directrix and the x-axis, then the eccentricity of thehyperbola is

23.If the distance between the planeAx 2y + z= dand the plane containing the lines x 12

=

y 23

= z 3

4 and

x 23

= y 3

4 =

z 45

isp

6, then jdj is

24. For any real number x, let [x] denote the largest integer less than or equal to x. Let f be

real valued function defined on the interval [10; 10]by f(x) =(

x [x] if[x]is odd,1 + [x] x if[x]is even .

Then the value of 2

10

Z 1010

f(x) cos x d x

25.Let! be the complex numbercos2

3 + isin

2

3. The the number of distinct complex numbers

zsatisfying

z+ 1 ! !2! z+ !2 1

!2 1 z+ !

= 0is equal to

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26.Let Sk, k = 1; 2; : : : 100 denote the sum of the infinite geometric series whose first term isk 1

k! and the common ratio is

1

k. Then the value of

1002

100!+

P100

k=1

(k2 3k+ 1) Sk

is

27.The number of all possible values of, where0 < < , for which the system of equations

(y+ z) cos 3= (xyz ) sin 3

x sin 3=2 cos3y

+2 sin 3z

(xyz ) sin 3= (y+ 2z) cos3+ ysin 3

have a solution(x0; y0; z0)withy0z0 0is28.Let f be a real valued differentiable function on R (the set of all real numbers) such that

f(1) = 1. If the y -intercept of the tangent at any point P(x; y) on the curve y = f(x) isequal to the cube of the abscissa ofP, then the value off(3)is equal to

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2010-Paper1