If z = ${\sec }^{-1}\left(\frac{{\mathrm{x}}^4 + {\mathrm{y}}^4 - 8{\mathrm{x}}^2{\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right)$, then $\mathrm{x}\frac{\partial \mathrm{z}}{\partial \mathrm{y}} + \mathrm{y}\frac{\partial \mathrm{z}}{\partial \mathrm{y}}$ is equal to

• $\cot \left(\mathrm{z}\right)$

• $2\cot \left(\mathrm{z}\right)$

• $2\tan \left(\mathrm{z}\right)$

• $2\sec \left(\mathrm{z}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{2\sin \left(\mathrm{x}\right) - \sin \left(2\mathrm{x}\right)}{2\mathrm{xcos}\left(\mathrm{x}\right)}, \mathrm{if} \mathrm{x} \ne 0, \\ \mathrm{a} , \mathrm{if} \mathrm{x} = 0\right\}, \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{a} \mathrm{so} \mathrm{that} \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{at} 0 \mathrm{is} \end{gathered}$$

• 2

• 0

• 1

• 2

${\text{x = cos}}^{-1}\left(\frac{1}{\sqrt{1 +\hspace{0.17em}{\mathrm{t}}^2}}\right), \mathrm{y} = {\sin }^{-1}\left(\frac{\mathrm{t}}{\sqrt{1 +\hspace{0.17em}{\mathrm{t}}^2}}\right) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• 0

• tan(t)

• 1

• sin(t)cos(t)

$\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{a} {\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{blog}\left(\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right)\right] = \frac{1}{{\mathrm{x}}^4 - 1} \Rightarrow \mathrm{a} - 2\mathrm{b} = ?$

• 1

• 0

• - 1

• 2

$\mathrm{y} = {\mathrm{e}}^{{\mathrm{asin}}^{-1}\left(\mathrm{x}\right)} \Rightarrow \left(1 - {\mathrm{x}}^2\right){\mathrm{y}}_{\mathrm{n} +\hspace{0.17em}2} - \left(2\mathrm{n} +\hspace{0.17em}1\right){\mathrm{xy}}_{\mathrm{n} +\hspace{0.17em}1} \mathrm{is}$equal to

• $-\left({\mathrm{n}}^2 +\hspace{0.17em}{\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 -\hspace{0.17em}{\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 +\hspace{0.17em}{\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $-\left({\mathrm{n}}^2 -\hspace{0.17em} {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 + 3{\mathrm{x}}^2 - \cos 2\mathrm{x}}{{\mathrm{x}}^2}, \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{for} \mathrm{x}= 0\right\} \\ \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{then} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• 1

• 5

• 6

• 0

427.

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left(\cos \left(\mathrm{x}\right)\right)\left(\cos \left(2\mathrm{x}\right)\right). . . \left(\cos \left(\mathrm{nx}\right)\right), \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{x}\right) + \sum _{\mathrm{r} = 1}^{\mathrm{n}} \left(\mathrm{rtan}\left(\mathrm{rx}\right)\right)\mathrm{f}\left(\mathrm{x}\right) = ?$

• f(x)

• 0

• - f(x)

• 2f(x)

$$\begin{gathered} \mathrm{If} \mathrm{y} = {\cos }^{-1}\left(\frac{{\mathrm{a}}^2 - {\mathrm{x}}^2}{{\mathrm{a}}^2 + {\mathrm{x}}^2}\right) + {\sin }^{-1}\left(\frac{2\mathrm{ax}}{{\mathrm{a}}^2 + {\mathrm{x}}^2}\right), \\ \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

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

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

• $\frac{4\mathrm{a}}{{\mathrm{x}}^2 + {\mathrm{a}}^2}$

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

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right), \\ \mathrm{then} \mathrm{f}\left(\frac{\mathrm{\pi }}{4}\right){\mathrm{f}}^{\left(\mathrm{iv}\right)}\left(\frac{\mathrm{\pi }}{4}\right) = ? \end{gathered}$$

• 1

• 2

• 3

• 4

$$\begin{gathered} \mathrm{If} \mathrm{y} = \sin \left({\mathrm{msin}}^{-1}\left(\mathrm{x}\right)\right), \mathrm{then} \left(1 - {\mathrm{x}}^2\right){\mathrm{y}}_2 - {\mathrm{xy}}_1 = ? \\ \left(\mathrm{Here}, {\mathrm{y}}_{\mathrm{n}} \mathrm{denotes} \frac{{\mathrm{d}}^{\mathrm{n}}\mathrm{y}}{{\mathrm{dx}}^{\mathrm{n}}}\right) \end{gathered}$$

• m²y

• - m²y

• 2m²y

• - 2m²y