RT3-Paper2-22-07-12

8/12/2019 RT3-Paper2-22-07-12

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Section I (Straight Objective Type)

.This section contains 5 questions. Each question has 4 choices (a), (b), (c) and (d ) for itsanswer, out of which ONLY ONE is correct.

1. The tangent at a point P on the curve y = ln

2 +

p 4 x2

2 p

4 x2

!

p 4 x2 meets the

y-axis at T ; then P T 2 equals to

(a) 2 (b) 4 (c) 8 (d) 16

2. If f (x) is a thrice differential function such that

limx!0

f (4x) 3(f (3x) + 3f (2x) f (x)

x3 = 12;

then the value of f

000

(0) equals to

(a) 0 (b) 1 (c) 12 (d) none of these

3. Let

y = 1

1 + (tan )sincos + (cot )coscot +

1

1 + (tan )cossin + (cot )sincot

+ 1

1 + (tan )coscot + (cot )cotsin

then dy

dx at =

3 is

(a) 0 (b) 1 (c)p

3 (d) None of these

4. Let

f (x) =

Z x3x2

dt

ln t; for x > 1

g(x) =

Z x1

(2t2 ln t)f (t) dt; for x > 1

(a) g is increasing on (1;

1)

(b) g is decreasing on (1; 1)

(c) g is increasing on (1; 2) and decreasing on (2; 1)

(d) g is decreasing on (1; 2) and increasing on (2; 1)

1

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AS-MA/T1 – Page 2 of 4 – Name:

5. If ˛;ˇ;;ı 2 R satisfy (˛ + 1)2 + (ˇ + 1)2 + ( + 1)2 + (ı + 1)2

˛ + ˇ + + ı = 4. If biquadratic

equation a0x4 + a1x3 + a2x2 + a3x + a4 = 0 has the roots

˛ +

1

ˇ 1

,

ˇ +

1

1

,

+ 1

ı 1, ı +

1

˛ 1. Then the value of

a2

a0

is

(a) 4 (b) 4 (c) 6 (d) none of these

Section II (Multiple Correct Objective Type)

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

6. Given that f (x) is a non-constant linear function. Then the curves

(a) y = f (x) and y = f 1(x) are orthogonal

(b) y = f (x) and y = f 1(x) are orthogonal

(c) y = f (x) and y = f 1(x) are orthogonal

(d) y = f (x) and y = f 1(x) are orthogonal

7. If a = maxf(x + 2)2 + (y 3)2g and b = minf(x + 2)2 + (y 3)2g where x; y satisfyx2 + y2 + 8x 10y 40 = 0, then

(a) a + b = 18 (b) a + b = 178 (c) a b = 4p 2 (d) a b = 72p 28. Let f (x) =

Z x0

et2

(t2 1)t2(t + 1)2011(t 2)2012 dt for (x > 0), then

(a) The number of point of inflection is 1 (b) The number of points of inflection is 3

(c) The number of points of local maximais 1

(d) The number of point local minima is 1

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9. Let f (x) be a polynomial function of degree 3 where a < b < c and f (a) = f (b) =f (c). If the graph of f (x) is as shown, then which of the following statements is/areINCORRECT? (where c > jaj)

(a)

Z ca

f (x) dx =

Z cb

f (x) dx +

Z ba

f (x) dx

(b)

Z c

a

f (x) dx < 0

(c)

Z ba

f (x) dx <

Z bc

f (x) dx

(d) 1

b a

Z ba

f (x) d x > 1

c b

Z cb

f (x) dx

10. Let T n =

3n1

Xr=2n

r

r2 + n2

, S n =

3n

Xr=2n+1

r

r2 + n2

, then 8

n

2 f1; 2; 3; : : :

g

(a) T n > 1

2 ln 2 (b) S n <

1

2 ln 2 (c) T n <

1

2 ln 2 (d) S n >

1

2 ln 2

Section III (Matrix Matches)

.This section contains 2 questions. Each question contains statements given in two columns,which have to be matched. The statements in Column I are labelled A, B, C and D, whilestatements in Column I are labelled p , q , r , s and t . Any given statement in Column I canhave correct matching with ONE OR MORE statement(s) in Column II.

11.

Column-I Column-II

(A) If

Z 0

log sin x

cos2 x dx = K , then the value of

3k

is greater

than

(P) 0

(B) If ex+y + eyx = 1 and y00 (y

0

)2 + K = 0, then K is

equal to

(Q) 1

(C) If f (x) = x ln x, then 2 (f 1)0

(ln 4) is more than (R) 2

(D) limx!1

(x ln x)1/(x2+1) is less than (S) 4

(T) 5

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12.

Column-I Column-II

(A) If ˛; ˇ are roots of x3 3x2 + 2x + 4 = 0 and y =

1 + ˛

x ˛

+ ˇx

(x ˛)(x ˇ)

+ x2

(x ˛)(x ˇ)(x )

,

then value of y at x = 2 is

(P) 2

(B) If x3 + ax + 1 and x4 + ax + 1 have a common root,then the value of jaj can be equal to

(Q) 3

(C) The number of local maxima of the function x2 +4 cosx + 5 is more than

(R) 4

(D) If f (x) = 2jxj3 + 3x2 12jxj + 1, where x 2 [1; 2],then the greatest value of f (x) is less than

(S) 5

(T) 0

Section IV (Integer Type)

.This section contains 4 questions. The answer to each question is a single-digit integer,ranging from 0 to 9.

13. The solution set of the equation

r hx +

hx

2ii

+hp fxg +

hx

3ii

= 3

is [a; b). Find the product a b. (where [] denotes greatest integer function and fgdenotes fractional part function.)

14. Let a function f (x) satisfy f (2 x) = f (2 + x) and f (4 x) = f (4 + x). Also the

function f (x) satisfies

Z 20

f (x) dx = 5. If

Z 500

f (x) dx = I . Find [p

I ]. (where []denotes greatest integer function).

15. Let limn!1

n1

2(1+ 1

n)

11 22 33 nn

1

n2 = ep

q , where p and q are relatively prime pos-

itive integers. Find the value of (p + q)2

.

16. Let I n =

Z 11

jxj

1 + x + x

2 +

x2

2 +

x3

3 + +

x2n

2n

dx . If lim

n!1I n can be expressed as

rational p

q in its lowest form, then find the value of p q(p + q).

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RT3-Paper2-22-07-12