1. If f(x) = I x2-5x+6 I, then f(x) equalsa) 2x-5 for 2<x<3 b) 5-2x for 2<x<3 c) 2x-5 for x>2 d) 5-2x for x<3

2. If f(0) = 0, f’(0) = 2, then the derivative of y = f(f(f(f(x))) at x = 0 isa) 2 b) 8 c) 16 d) 4

3. If f(x) = √1+cos2(x¿¿2) ,¿ then f’(√π2

) is

a) √π / 6 b) -√π /6 c) 1/√6 d) π/√64. Let g(x) be the inverse of an invertible function f(x) which is differentiable

at x = c. then g’(f(c)) equals

a) f’(c) b) 1

f '(c) c) f(c) d) none of these

5. If f(x) = x4tan(x3)-x in (1+x2), then the value of d4( f ( x ))d x4

at x = 0 is

a) 0 b) 6 c) 12 d) 246. If f(x) satisfies the relation

F (5x−3 y2 ) =

5 f ( x )−3 f ( y)2

∀ x , y∈R , and f(0) = 3 and f’(0) = 2, then the period of sin(f(x))

isa) 2π b) π c) 3π d) 4π

7. A function f:R→R satisfies c cos y (f(2x+2y)-f(2x-2y)) = cos x sin y (f(2x+2y)+f(2x-2y)).

if f’(0) = 12 ,then

a) f’’(x) = f(x) = 0 b) 4f’’(x) + f(x) = 0 c) f’’(x) + f(x) = 0 d) 4f’’(x)-f(x) = 0

8. If limt→ x

e t f ( x )−ex f (t)(t−x)( f ( x ))2

= 2 and f(0) = 12 , then find the value of f’(0).

a) 4 b) 2 c) 0 d) 1

9. If f(x) = √ x−sinxx+cos2 x , then lim

n→∞f ( x ) is

a) 0 b) ∞ c) 1 d) none of these

10. If G(x) = -√25−x2, then limx→1

G ( x )−G(1)x−1

is

a) 1/24 b) 1/5 c) -√24 d) none of these

11. limn→∞

{ 11−n2

+ 21−n2

+…+ n1−n2

} is equal to

a) 0 b) -12 c)

12 d) none of these

12. If f(x) = sin [ x ][ x ]

, for [ x ]≠0

0 , for [x ]=0, where [x] denotes the greatest integer less than or equal

to x, then limx→0

f ( x ) is

a) 1 b) 0 c) -1 d) none of these

13. limx→0

x tan 2x−2x tan x¿¿¿ ¿ is equal to

a) 2 b) -2 c) ½ d) -1/2

14. limx→0

sin (πcos2x )x2

is equal to

a) –π b) π c) π/2 d) 1

15. If limx→0

{ (a−n ) nx−tan x }sin nxx2

= 0, where n is nonzero real number, than a is

a) 0 b) n+1n c) n d) n+

1n

16. The value of limx→ 0¿¿¿ + (1+x)sin x) = 0, where x>0, is

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

17. If limx→0

[1+ xin (1+b2 )]1/x = 2b sin2θ,b>0, s m f θ ∈(-π,π), then the value of θ is

a) ±π4 b) ±π3 c) ±π6 d) ±π218. If lim

x→∞( x

2+x+1x+1

¿−ax−b)=4 ,¿ then

a) a = 1, b = 4 b) a = 1, b = -4 c) a = 2, b = -3 d) a =2, b = 319. The function f:N→N(N is the set of natural numbers) defined by f(n) = 2n+3 is

a) surjective only b) injective only c) bijective d) none of these20. Let f:[-π/3,2π/3]→[0,4] be a function defined as f(x) = √3sin x –cos x +2. Then f-1(x) is given

by

a) sin-1(x−22

¿−π6 b) sin-1(

x−22

¿+ π6 c) cos-1(

x−22

¿+ 2π3 d) none of these

21. The period of the function I sin3 x2I+ I cos5 x

5I is

a) 2π b) 10 π c) 8 π d) 5 π22. If f is periodic, g is polynomial function, f(g(x)) is periodic, g(2) = 3, and g(4) = 7, then g(6) is

a) 13 b) 15 c) 11 d) none of these23. If [x] and {x} represent the integral and fractional parts of x, respectively,

then the value of ∑r=1

2000 {x+r }2000

is

a) x b) [x] c) {x} d) x+200124. F:N→N, where f(x) = x-(-1)x. then f is

a) one – one and into b) many-one and into c) one-one and onto d) many-one and onto 25. If f:R→R is an invertible function such that

f(x) and f-1(x) are symmetric about the line y=-x thena) f(x) is odd b) f(x) and f-1(x) may not be symmetric about the line y = xc) f(x) may not be odd d) none of these

26. If g(x) = x2+x-2 and 12gof(x) = 2x2-5x+2, then which is not a possible f(x)?

a) 2x-3 b) -2x+2 c) x-3 d) none of these27. Which of the following functions have finite number of points of discontinuity in R ([.]

represents the greatest integer function)?

a) tan x b) x[x] c) I x Ix d) sin [πx]

28. If f(x) = 8x−4 x−2x+1

x2x>0

ex sin x+πx+¿4 , x≤0 is continuous at x = 0, then the value of is

a) 4log e2 b) 2log e2 c) log e2 d) none of these29. Which of the following function is non-differentiable?

a) f(x) = (ex-1) I e2x-1 I in R b) f(x) = x−1x2+1

in R d) f(x) = 3(x-2)1/3 + 3 in R

30. The number of points non-differentiability for f(x) = max{IIxI-1I, ½} isa) 4 b) 3 c) 2 d) 5