1. Let f be a differentiable function on R taking non negative values such that \[\lim_{x \rightarrow 0}\frac{f'\left(x\right)}{x}\] exists and satisfy

\[f\left(x\right)f\left(y\right)=f\left(x_{1}\right)f\left(y_{1}\right)\]

For all \[x,y,x_{1},y_{1}\] such that \[x^{2}+y^{2}=x_1^2+y_1^2\] such that
f (0) = 1, then

a) f is infinitely many times differentiable

b) f is an exponetial function

c) \[\lim_{x \rightarrow 0}f\left(x\right)=1\]

d) All of the Above

Explanation: Taking logarithm of both sides,

2. Let [x] denote the greatest integer less than or equal to x.
If \[f\left(x\right)=\left[x\sin\pi x\right]\] , then f (x) is

a) continuous at x = 0

b) continuous in (–1, 0)

c) differentiable in (– 1, 1)

d) All of the Above

Explanation: By definition of [x], we have f(x) = [x sin \[\pi\] x]

3. The function \[f\left(x\right)=\begin{cases}\mid x-3 \mid& x \geq 1\\\frac{x^{2}}{4}-\frac{3x}{2}+\frac{13}{4} & x < 1\end{cases}\]

is

a) continuous at x = 1

b) continuous at x = 3

c) differentiable at x = 1

d) All of the Above

Explanation: The given function can be written as

4. If \[f\left(x\right)=\begin{bmatrix}x^{n} &\sin x & -\cos x \\n! &\sin\left(n\pi/2\right)& \cos\left(n\pi/2\right)\\a &a^{2} & a^{3} \end{bmatrix}\]

the value of \[\frac{d^{n}}{dx^{n}}\left(f\left(x\right)\right)\] at x = 0 for n = 2m + 1 is

a) independent of a

b) 0

c) \[a^{6}\]

d) Both a and b

Explanation:

5. If \[f\left(x\right)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\]

then

a) f is a differentiable at all points of its domain except x= 4

b) f is differentiable on \[\left(2,\infty\right)\sim\left\{4\right\}\]

c) f' (x) = 0 for all \[x\epsilon\] [2,4)

d) All of the Above

Explanation:

6. If f'(5) = 4 then \[\lim_{x \rightarrow \infty}\frac{f\left(5+h^{2}\right)-f\left(5-h^{2}\right)}{2h^{2}}\]

is

a) 2

b) 4

c) 3

d) 1/4

Explanation:

7. If \[y=\tan^{-1}\left(\frac{\log ex^{-3}}{\log ex^{3}}\right)+\tan^{-1}\left(\frac{3+3\log x}{1-9\log x}\right)\]

then \[\frac{d^{2} y}{dx^{2}}\] is

a) 2

b) 1

c) 0

d) -1

Explanation:

8. If \[f\left(x\right)=\frac{\tan x+\sec x -1}{\tan x-\sec x+1}\] then \[f'\left(\pi/4\right)\] is equal to

a) \[\sqrt{2}\left(\sqrt{2}-1\right)\]

b) \[\sqrt{2}+1\]

c) \[\sqrt{2}\left(\sqrt{2}+1\right)\]

d) \[\sqrt{2}-1\]

Explanation:

9. If \[f\left(x\right)=x^{1/x}\] then f'' (e) is equal to

a) \[e^{1/\left({e-3}\right)}\]

b) \[e^{1/e}\]

c) \[e^{1/\left({e-2}\right)}\]

d) \[e^{1/e-3}\]

Explanation:

10. If \[y=f\left(\frac{3x+4}{5x+6}\right)\] and \[f"\left(x\right)=\tan x^{2}\] then \[\frac{dy}{dx}\] is
equal to

a) \[-2\tan\left(\frac{3x+4}{5x+6}\right)^{2}\times\frac{1}{\left(5x+6\right)^{2}}\]

b) \[f\left(\frac{3\tan x^{2}+3}{5\tan x^{2}+6}\right) \tan x^{2}\]

c) \[ \tan x^{2}\]

d) \[ \tan \left(\frac{3x+4}{5x+6}\right)^{2}\]

Explanation: