Mathematics : Continuity and Differentiability

$\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

$$\begin{gathered} \mathrm{If} \frac{\mathrm{d}}{\mathrm{dx}}\left[\left(\mathrm{x} + 1\right)\left({\mathrm{x}}^2 + 1\right)\left({\mathrm{x}}^4 + 1\right)\left({\mathrm{x}}^8 + 1\right)\right] \\ = \left(15{\mathrm{x}}^{\mathrm{p}} - 16{\mathrm{x}}^{\mathrm{q}} + 1\right){\left(\mathrm{x} - 1\right)}^{ - 2}, \mathrm{then} \left(\mathrm{p}, \mathrm{q}\right) = ? \end{gathered}$$

• (12, 11)

• (15, 14)

• (16, 14)

• (16, 15)

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1 + {\mathrm{x}}^2} - \sqrt{1 - \mathrm{x} + {\mathrm{x}}^2}}{3^{\mathrm{x}} - 1} = ?$

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

• ${\log }_{\mathrm{e}}\left(9\right)$

• $\frac{1}{{\log }_{\mathrm{e}}\left(9\right)}$

• ${\log }_{\mathrm{e}}\left(3\right)$