Limits, Continuity and Differentiability Questions and Answers Part-16

1. suppose that f is a differentiable function with the property that f (x + y) = f (x) + f (y) + xy and f '(0) = 5 then
a) f is a linear function
b) $f\left(x\right)=3x+x^{2}$
c) $f\left(x\right)=5x+x^{2}/2$
d) $f\left(x\right)=x+\frac{5x^{2}}{2}$

Explanation:

2. If $f\left(a\right)=a^{2},\phi\left(a\right)=b^{2}$    and $f'\left(a\right)=5\phi' (a)$    then $\lim_{x \rightarrow 0}\frac{\sqrt{f\left(x\right)}-a}{\sqrt{\phi\left(x\right)}-b}$
is
a) $b^{2}/a^{2}$
b) b/a
c) 2b/a
d) 5b/a

Explanation:

3. If $y=\cot^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$
then $\frac{d^{2} y}{dx^{2}}$   is equal to
a) 0
b) 1/2
c) $\frac{1}{1+\sin x}$
d) $\frac{1}{\sqrt{1+\sin x}}+\frac{1}{\sqrt{1-\sin x}}$

Explanation:

4. If $x=\cos\theta,y=\sin^{3}\theta$    , then $\left(\frac{d y}{dx}\right)^{2}+y\frac{d^{2} y}{dx^{2}}$    at $\theta=\pi/4$  is
a) 1
b) 2
c) -2
d) 9/4

Explanation:

5. If $y=\tan^{-1}\frac{1}{1+x+x^{2}}+\tan^{-1}\frac{1}{x^{2}+3x+3}+\tan^{-1}\frac{1}{x^{2}+5x+7}+....+$
upto n terms, then y'' (0) is equal to
a) $-1/\left(n^{2}+1\right)^{2}$
b)$-n^{2}/\left(n^{2}+1\right)^{2}$
c) $n^{2}/\left(n^{2}+1\right)^{2}$
d) 0

Explanation:

6. If $\left(\sin x\right)\left(\cos y\right)=\frac{1}{2}$    then $d^{2}y/dx^{2}$   at $\left(\pi/4,\pi/4\right)$   equal to
a) -4
b) -2
c) -6
d) 0

Explanation:

7. If x = sin t, y = sin kt satisfies $\left(1-x^{2}\right)y_{2}-xy_{1}+Ay=0$
then A is equal to
a) k
b) 1
c) $k^{2}$
d) 1+k

Explanation:

8. Let $f\left(x\right)=\begin{cases}x^{2}/2 & if 0\leq x\leq1\\2x^{2}-3x+3/2 &if 1\leq x\leq 2\end{cases}$
then
a) f' is not a continuous function
b) f'' is not continuous at x = 1
c) f is not differentiable at x = 1
d) f is not continuous at x = 1

Explanation:

9. Let f and g be differentiable function satisf ying g'(a) = 2, g(a) = b and f o g = I (identity function). Then f '(b) is equal to
a) 1/2
b) 2
c) 2/3
d) 3/4

10. If $y=x^{n}\left(a\cos\left(\log x\right)+b\sin\left(\log x\right)\right)$       and y satisfies $y_{2}+\left(1-2n\right)xy_{1}+Ay=0$
b) $1+n^{2}$
d) $1-n^{2}$