1. Let \[f\left(x\right)=\begin{cases}x^{3} \mid \cos\frac{\pi}{2x}\mid& x \neq 0\\+x^{4}\mid\sin\frac{\pi}{x}\mid\\0, & x= 0\end{cases}\]

then f is

a) differentiable at x = 0 and x = 1

b) differentiable at x = 0 not differentiable at x
= 2

c) differentiable at x = 2 and not differentiable
at x = 1

d) differentiable neither at x = 0 nor at x = 2.

Explanation:

2. If \[f\left(x\right)=\cot^{-1}\left(\frac{x^{x}-x^{-x}}{2}\right)\] then f' (1)
equals

a) -1

b) 1

c) log 2

d) – log 2.

Explanation: Putting u = x

^{x}and using logarithmic

3. If \[f\left(x\right)=\left(1+x\right)^{n}\] , then the value of \[f\left(0\right)+f'\left(0\right)+\frac{f"\left(0\right)}{2!}+....+\frac{\left(0\right)}{}\] is

a) n

b) \[2^{n}\]

c) \[2^{n+1}\]

d) none of these

Explanation:

4. The solution set of f '(x) > g'(x) where \[f\left(x\right)=\left(1/2\right)5^{2x+1}\] and \[g\left(x\right)=5^{x}+4x\log 5\] is

a) \[\left(1,\infty\right)\]

b) \[\left(0,1\right)\]

c) \[\left[0,\infty\right)\]

d) \[\left(0,\infty\right)\]

Explanation:

5.Let \[f\left(x\right)=\sin x;g\left(x\right)=x^{2}\] and h(x) = log x.
If u (x) = h( f (g(x))), then \[\frac{d^{2}u}{dx^{2}}\mid x=\sqrt{\pi}/2\] is

a) \[2-\pi\]

b) \[2-2\pi\]

c) \[\pi-2\]

d) \[2\pi-2\]

Explanation:

6. If \[y=\tan^{-1}\frac{1}{1+x+x^{2}}+\tan^{-1}\frac{1}{x^{2}+3x+3}+\tan^{-1}\frac{1}{x^{2}+5x+7}+....+\]

up to n terms, then y'(0) is equal to

a) \[-\frac{1}{1+n^{2}}\]

b) \[-\frac{n^{2}}{1+n^{2}}\]

c) \[\frac{n}{1+n^{2}}\]

d) none of these

Explanation:

7. Let f and g be functions satisfying

\[f\left(x\right)=e^{x}g\left(x\right),f\left(x+y\right)=f\left(x\right)+f\left(y\right),g\left(0\right)=0 , g'\left(0\right)=4\]

g and g' are continuous at 0.
Then

a) f (x) = 0 for all x

b) f (x) = x for all x

c) f (x) = x + 4 for all x

d) f (x) = 4 x for all x

Explanation:

8. Let \[f:R\rightarrow R\] is a function which is defined
by \[f\left(x\right)=\max\left\{x,x^{3}\right\}\] . The set of all points on which f (x) is
not differentiable is

a) \[\left\{-1,1\right\}\]

b) \[\left\{-1,0\right\}\]

c) \[\left\{0,1\right\}\]

d) \[\left\{-1,0\right\}\]

Explanation: The graphs of y = x and y = x

^{3}are given in the

9. Which of the following functions is differentiable
at x = 0?

a) cos (|x|) + |x|

b) cos (|x|) – |x|

c) sin |x| + |x|

d) sin (|x|) – |x|

Explanation: cos |x| = cos x or cos (– x). Thus, n any case

10. If f is a differentiable function at a point 'a’
and \[f'\left(a\right)\neq0\] then which of the following is true.

a) \[-f'\left(a\right)=\lim_{h \rightarrow 0}\frac{f\left(a\right)-f\left(a-h\right)}{h}\]

b) \[\frac{1}{2}f'\left(a\right)=\lim_{h \rightarrow 0}\frac{f\left(a+2h\right)-f\left(a+h\right)}{2h}\]

c) \[f'\left(a\right)=\lim_{h \rightarrow 0}\frac{f\left(a+2h\right)-f\left(a\right)}{h}\]

d) \[3f'\left(a\right)=\lim_{h \rightarrow \infty}\frac{f\left(a+3h\right)-f\left(a+h\right)}{h}\]

Explanation: