Continuity and Differentiability

$\mathrm{If} \mathrm{u} = {\sin }^{-1}\left(\frac{{\mathrm{x}}^4 + {\mathrm{y}}^4}{\mathrm{x} + \mathrm{y}}\right), \mathrm{then} \mathrm{x}\frac{\partial \mathrm{u}}{\partial \mathrm{x}} + \mathrm{y}\frac{\partial \mathrm{u}}{\partial \mathrm{y}} = ?$

• 3u

• 4u

• 3sin(u)

• 3tan(u)

$\mathrm{If} \mathrm{y} = \frac{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}{\mathrm{x}} \mathrm{and} \mathrm{z} = {\log }_{\mathrm{e}}\left(\mathrm{x}\right), \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dz}} = ?$

• ${\mathrm{e}}^{ - \mathrm{z}}$

• $2{\mathrm{e}}^{ - \mathrm{z}}$

• ${\mathrm{ze}}^{ - \mathrm{z}}$

• $- {\mathrm{e}}^{ - \mathrm{z}}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \log \left({\mathrm{e}}^{\mathrm{x}}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} + 2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\right), \mathrm{then} \mathrm{f}\text{'}\left(0\right) = ?$

• $\frac{1}{4}$

• 4

• $\frac{- 3}{4}$

• 1

$$\begin{gathered} \mathrm{If} \mathrm{xy} \ne 0, \mathrm{x} + \mathrm{y} \ne 0 \mathrm{and} {\mathrm{x}}^{\mathrm{m}}{\mathrm{y}}^{\mathrm{n}} = {\left(\mathrm{x} + \mathrm{y}\right)}^{\mathrm{m} + \mathrm{n}}, \\ \mathrm{where} \mathrm{m}, \mathrm{n} \mathrm{not} \mathrm{belongs} \mathrm{to} \mathrm{N}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{x} + \mathrm{y}}{\mathrm{xy}}$

• xy

• $\frac{\mathrm{x}}{\mathrm{y}}$

$$\begin{gathered} \mathrm{If} {\mathrm{x}}^2 + {\mathrm{y}}^2 = \mathrm{t} + \frac{1}{\mathrm{t}}, {\mathrm{x}}^4 + {\mathrm{y}}^4 = {\mathrm{t}}^2 + \frac{1}{{\mathrm{t}}^2}, \mathrm{then} \\ {\mathrm{x}}^3\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• - 1

• 1

• 0

• t

If the volume of a sphere increases at the rate of 2$\mathrm{\pi }$ cm³/ s, then the rate of increase of its radius(in cm/s), when the volume is 288$\mathrm{\pi }$ cm³, is

• $\frac{1}{36}$

• $\frac{1}{72}$

• $\frac{1}{18}$

• $\frac{1}{9}$

Let f be a non-zero real valued continuous function satisfying f(x + y) = f(x) · f(y) for all x, y ∈ R. If f(2) = 9, then f(6) is equal

• 3²

• 3⁶

• 3⁴

• 3³

$\underset{\mathrm{x}\to 0}{\lim } \frac{{\tan }^3\left(\mathrm{x}\right) - {\sin }^3\left(\mathrm{x}\right)}{{\mathrm{x}}^5} = ?$

• $\frac{5}{2}$

• $\frac{3}{2}$

• $\frac{3}{5}$

• $\frac{2}{5}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{1 + {\displaystyle \frac{1}{\mathrm{x}}}} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = \frac{1}{1 + {\displaystyle \frac{1}{\mathrm{f}\left(\mathrm{x}\right)}}}, \mathrm{then} \mathrm{g}\text{'}\left(2\right) = ?$

• $\frac{1}{5}$

• $\frac{1}{25}$

• 5

• $\frac{1}{16}$

$\mathrm{If} \sqrt{\frac{\mathrm{y}}{\mathrm{x}}} + \sqrt{\frac{\mathrm{x}}{\mathrm{y}}} = 2, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

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

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

• 1

• 2