If $$x = {e^t}\sin t$$, $$y = {e^t}\cos t$$ then $${{{d^2}y} \over {d{x^2}}}$$ at x = $$\pi$$ is

The value of $${{dy} \over {dx}}$$ at $$x = {\pi \over 2}$$, where y is given by $$y = {x^{\sin x}} + \sqrt x $$ is

Select the correct statement from (a), (b), (c), (d). The function $$f(x) = x{e^{1 - x}}$$

The function $$f(x) = {e^{ax}} + {e^{ - ax}},a > 0$$ is monotonically increasing for

The second order derivative of a sin³t with respect to a cos³t at $$t = {\pi \over 4}$$ is

If $$y = {\tan ^{ - 1}}\sqrt {{{1 - \sin x} \over {1 + \sin x}}} $$, then the value of $${{dy} \over {dx}}$$ at $$x = {\pi \over 6}$$ is

If x² + y² = 4, then $$y{{dy} \over {dx}} + x = $$

If $$y = {A \over x} + B{x^2}$$, then $${x^2}{{{d^2}y} \over {d{x^2}}}$$ =

Let $$f(x) = ta{n^{ - 1}}x$$. Then $$f'(x) + f''(x)=0$$, when x is equal to

If $$y = {\tan ^{ - 1}}{{\sqrt {1 + {x^2}} - 1} \over x}$$, then y'(1) =

If $$y = 2{x^3} - 2{x^2} + 3x - 5$$, then for x = 2 and $$\Delta$$x = 0.1 the value of $$\Delta$$y is

The approximate value of $$\root 5 \of {33} $$ correct to 4 decimal places is

Let $$f(x) = {x^3}{e^{ - 3x}},\,x > 0$$. Then the maximum value of f(x) is

If ' $f$ ' is the inverse function of ' $g$ ' and $g^{\prime}(x)=\frac{1}{1+x^n}$, then the value of $f^{\prime}(x)$ is

Let $f(x)$ be a second degree polynomial. If $f(1)=f(-1)$ and $p, q, r$ are in A.P., then $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are

If $$\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)$$ denotes the $$\mathrm{n}^{\text {th }}$$ derivative $$(\mathrm{n}=1,2)$$ of $$\mathrm{U}(x)=\frac{\mathrm{L} x+\mathrm{M}}{x^2-2 \mathrm{~B} x+\mathrm{C}}$$ (L, M, B, C are constants), then $$\mathrm{PU}_2+\mathrm{QU}_1+\mathrm{RU}=0$$, holds for

$$ \text { If } y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \cdot \log _e x}\right] \text {, then } \frac{d^2 y}{d x^2}= $$

Suppose $$f:R \to R$$ be given by $$f(x) = \left\{ \matrix{ 1,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,x = 1 \hfill \cr {e^{({x^{10}} - 1)}} + {(x - 1)^2}\sin {1 \over {x - 1}},\,\mathrm{if}\,x \ne 1 \hfill \cr} \right.$$

then

Let $${\cos ^{ - 1}}\left( {{y \over b}} \right) = {\log _e}{\left( {{x \over n}} \right)^n}$$, then $$A{y_2} + B{y_1} + Cy = 0$$ is possible for, where $${y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}$$

The function $$y = {e^{kx}}$$ satisfies $$\left( {{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}}} \right)\left( {{{dy} \over {dx}} - y} \right) = y{{dy} \over {dx}}$$. It is valid for

If $$y = {\log ^n}x$$, where $${\log ^n}$$ means $${\log _e}{\log _e}{\log _e}\,...$$ (repeated n times), then $$x\log x{\log ^2}x{\log ^3}x\,.....\,{\log ^{n - 1}}x{\log ^n}x{{dy} \over {dx}}$$ is equal to

If $$x = \sin \theta $$ and $$y = \sin k\theta $$, then $$(1 - {x^2}){y_2} - x{y_1} - \alpha y = 0$$, for $$\alpha=$$

If $$y = {e^{{{\tan }^{ - 1}}x}}$$, then

Let $$f(x) = {x^m}$$, m being a non-negative integer. The value of m so that the equality $$f'(a + b) = f'(a) + f'(b)$$ is valid for all a, b > 0 is