SECTION-I

Single Correct Answer Type

1. A continuous function f : [0, 1] → R satisfies the

equation ∫f x dx() =+^1 ∫f ( ) ,x dx

3

22

0

1

0

1

then f^1

4

⎛

⎝⎜

⎞

⎠⎟=

(a) 1 (b) 1/2 (c) 1/4 (d) –1/4

2. A continuous function f : [0, 1] → R satisfies the

condition ∫fx x fx dx()(− ()) =^1 ,

0 12

1

then f^1

4

⎛

⎝⎜

⎞

⎠⎟=

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

3. Compute () 2332113 /
   0

1

∫ xxx dx−−+

(a) 0 (b) –1 (c) 1/8 (d) –1/8

4. Consider a one-to-one differentiable solution y to

the equation dy

dx

dx

dy

2

2

2

+= 2 0 such that y(0) = 2 and

y(–1) = –1 then y(1) =

(a) –20 (b) 20 (c) –5 (d) 5

5. Let f : [0, 1] → R be a continuous function with the
   property that x f(y) + y f(x) ≤ 1 for all x, y ∈ [0, 1]
   then

(a) ∫fxdx() =π

0 4

1

(b) ∫fxdx() ≤π

0 4

1

(c) ∫fxdx() ≥π 4

0

1

(d) ∫fxdx() ≥π 2

0

1

6. For every θ ∈ (0, π], we have

(a)^1222

0

∫ +>+cos tdt θθsin

θ

(b)^1222

0

∫ +=+cos tdt θθsin

θ

(c)^1222

0

∫ +<+cos tdt θθsin

θ

(d)^1

1

2

222

0

∫ +=+cos tdt θθsin

θ

7. y(x) satisfies the differential equation
   dy
   dx

=+yyyelog x. If y(0) = 1, then y(1) =

(a) ee (b) e–e (c) e1/e (d) e

8. The function f dx
   x

()

cos

,(,)

/

λ

λ

λ

π

=

−

∫ ∈

1

01

0 2

2

is

(a) increasing (b) decreasing

(c) increasing on (0, 1/2) and decreasing on (1/2, 1)

(d) increasing on (1/2, 1) and decreasing on (0, 1/2)

9. Let f satisfy the equation x = f(x)·ef(x) then
   fxdx

e

∫ () =

0

(a) 1 (b) e – 1 (c) e + 1 (d) 0

10. Let a, b ∈ R, a < b and let f be a differentiable function

on the interval [a, b] then lim ( ) sin

n a

b

f x nxdx

→∞∫

=

(a) 0 (b) 1 (c) b (d) b – a

11. The shortest possible length of an interval [a, b] for
    which^4
    1 +^2
    ∫ =
    x

dx

a

b

π is

(a) 2 (b) 22 (c) 22 2+ (d) 22 2−

SECTION-II

Comprehension Type

Paragraph for Question No. 12-14

Let θk(x) be 0 for x < k and 1 for x ≥ k. The dirac delta

function is defined to be δθkkx d

dx

()= ().x Suppose

 









By : Tapas Kr. Yogi, Mob : 9533632105.