If y = $\sqrt{\mathrm{x} + \sqrt{\mathrm{y} + \sqrt{\mathrm{x} + \sqrt{\mathrm{y} + ... \infty }}}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{y} + \mathrm{x}}{{\mathrm{y}}^2 - 2\mathrm{x}}$

• $\frac{{\mathrm{y}}^3 - \mathrm{x}}{2{\mathrm{y}}^2 - 2\mathrm{xy} - 1}$

• $\frac{{\mathrm{y}}^3 + \mathrm{x}}{2{\mathrm{y}}^2 - \mathrm{x}}$

• None of these

If f : R $\to$ R is defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{2\sin \left(\mathrm{x}\right) - \sin \left(2\mathrm{x}\right)}{2\mathrm{xcos}\left(\mathrm{x}\right)}, \mathrm{if} \mathrm{x}\ne 0 \\ \mathrm{a}, \mathrm{if} \mathrm{x} = 0, \mathrm{if} \mathrm{x} = 0\right\} \end{gathered}$$

then the value of a so that f is continuous at 0 is

• 2

• 1

• - 1

• 0

$\mathrm{y} = {\mathrm{e}}^{{\mathrm{asin}}^{-1}\left(\mathrm{x}\right)} \Rightarrow \left(1 - {\mathrm{x}}^2\right){\mathrm{y}}_{\mathrm{n} + 2} - \left(2\mathrm{n} + 1\right){\mathrm{xy}}_{\mathrm{n} + 1}$ is equal to

• $- \left({\mathrm{n}}^2 + {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 - {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 + {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $- \left({\mathrm{n}}^2 - {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

The value of f(0) so that $\frac{\left(- {\mathrm{e}}^{\mathrm{x}} + 2^{\mathrm{x}}\right)}{\mathrm{x}}$  may be continuous at x = 0 is

• $\log \left(\frac{1}{2}\right)$

• 0

• 4

• - 1 + $\log \left(2\right)$

95.

Let [ ] denotes the greatest integer function and f(x) = [tan²(x)] Then,

• $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ does not exist

• f(x) is continuous at x = 0

• f(x) is not differentiable at x = 0

• f(x) = 1

If (x + y)$\sin \left(\mathrm{u}\right) = {\mathrm{x}}^2{\mathrm{y}}^2, \mathrm{then} \mathrm{x}\frac{\partial \mathrm{u}}{\partial \mathrm{x}} + \mathrm{y}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}$ is equal to

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{4}{{\mathrm{e}}^4}$

If $\mathrm{x} = \frac{2\mathrm{at}}{1 + {\mathrm{t}}^3} \mathrm{and} \mathrm{y} = \frac{2{\mathrm{at}}^2}{{\left(1 + {\mathrm{t}}^3\right)}^2}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

• ax

• a²x²

• $\frac{\mathrm{x}}{\mathrm{a}}$

• $\frac{\mathrm{x}}{2\mathrm{a}}$

If f(x) = ${\log }_{{\mathrm{x}}^3}\left({\log }_{\mathrm{e}}\left({\mathrm{x}}^2\right)\right)$, then f'(x) at x = e is

• $\frac{1}{3\mathrm{e}}\left(1 - {\log }_{\mathrm{e}}\left(2\right)\right)$

• $\frac{1}{3\mathrm{e}}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• $\frac{1}{3\mathrm{e}}\left(- 1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• $- \frac{1}{3\mathrm{e}}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

If f(x) = (x - 2)(x - 4)(x - 6) ... (x - 2n), then f'(2) is

• (- 1)ⁿ2^{n - 1} (n - 1)!

• (- 2)^{n - 1} (n - 1)!

• (- 2)ⁿ n!

• (- 1)^{n - 1}2ⁿ (n - 1)!

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 - \cos \left(\mathrm{x}\right)}{\mathrm{x}}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$  is continuous at x = 0, then the value of k is

• 0

• 1/2

• $\frac{1}{4}$

• - $\frac{1}{2}$