81.

The value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left({\mathrm{x}}^4\right) - {\mathrm{x}}^4\cos \left({\mathrm{x}}^4\right) + {\mathrm{x}}^{20}}{{\mathrm{x}}^4\left({\mathrm{e}}^{2{\mathrm{x}}^4} - 1 - 2{\mathrm{x}}^4\right)}$ is equal to

• 0

• $- \frac{1}{6}$

• $\frac{1}{6}$

• Does not exist

The equation $\sin \left(\mathrm{x}\right) + 2\sin \left(2\mathrm{x}\right) + 3\sin \left(3\mathrm{x}\right) = \frac{8}{\mathrm{\pi }}$ has atleast one root in

• $\left[\mathrm{\pi }, \frac{3\mathrm{\pi }}{2}\right]$

• $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \mathrm{\pi }\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{4}\right]$

If xy = e^{y - x}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ =

• $\frac{\mathrm{y}\left(\mathrm{x} - 1\right)}{\mathrm{x}\left(\mathrm{y} + 1\right)}$

• $\frac{\mathrm{y}\left(\mathrm{x} + 1\right)}{\mathrm{x}\left(\mathrm{y} - 1\right)}$

• $\frac{\mathrm{x} + 1}{\mathrm{y} - 1}$

• $\frac{\mathrm{x} - 1}{\mathrm{y} + 1}$

If y = ${\cos }^{-1}\left(\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\sqrt{2}}\right), - \frac{\mathrm{\pi }}{4} < \mathrm{x} < \frac{\mathrm{\pi }}{4}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ =

• - 1

• 1

• 0

• None of these

$\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{x} + \mathrm{a}}{\mathrm{x} + \mathrm{b}}\right)}^{\mathrm{a} + \mathrm{b}}$ is equal to

• 1

• 1^{b - a}

• e^{a - b}

• e^b

$\underset{\mathrm{x}\to 0}{\lim }\left(\frac{\mathrm{x} . {10}^{\mathrm{x}} - \mathrm{x}}{1 - \cos \left(\mathrm{x}\right)}\right)$ is equal to

• $\log \left(10\right)$

• $2\log \left(10\right)$

• $3\log \left(10\right)$

• $4\log \left(10\right)$

If y_k is the kth derivative of y with respect to 'x', and y = cos(sin(x)), then ${\mathrm{y}}_1\sin \left(\mathrm{x}\right) + {\mathrm{y}}_2\cos \left(\mathrm{x}\right)$ is equal to

• ${\mathrm{ysin}}^3\left(\mathrm{x}\right)$

• $- {\mathrm{ysin}}^3\left(\mathrm{x}\right)$

• ${\mathrm{ycos}}^3\left(\mathrm{x}\right)$

• $- {\mathrm{ycos}}^3\left(\mathrm{x}\right)$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left(\mathrm{x}\right){\sin }^{-1}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}$ is equal to

• 0

• 1

• - 1

• $\infty$

$\underset{\mathrm{x} \to 0}{\lim } \frac{4^{\mathrm{x}} - 9^{\mathrm{x}}}{\mathrm{x}\left(4^{\mathrm{x}} + 9^{\mathrm{x}}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\log \left(\frac{2}{3}\right)$

• $\log \left(\frac{3}{2}\right)$

• $\frac{1}{2}\log \left(\frac{2}{3}\right)$

• $\frac{1}{2}\log \left(\frac{3}{2}\right)$

 $\mathrm{If} \mathrm{z} = \sec \left(\mathrm{y} - \mathrm{ax}\right) + \tan \left(\mathrm{y} + \mathrm{ax}\right), \frac{{\partial }^2\mathrm{z}}{\partial {\mathrm{x}}^2} - {\mathrm{a}}^2\frac{{\partial }^2\mathrm{z}}{\partial {\mathrm{y}}^2} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• - z

• z

• 2x