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

84.

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

• $\infty$

$\mathrm{If} \mathrm{a} > 0, \underset{\mathrm{x}\to \mathrm{a}}{\lim } \frac{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{x}}^{\mathrm{a}}}{{\mathrm{x}}^{\mathrm{x}} - {\mathrm{a}}^{\mathrm{a}}} = - 1, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• 1

• e

• 2e

The value of $\underset{\mathrm{n}\to \infty }{\lim }\frac{1}{{\mathrm{n}}^3}\sum _{\mathrm{k}=1}^{\mathrm{n}}\left({\mathrm{k}}^2\mathrm{x}\right)$ is

• x

• $\frac{\mathrm{x}}{2}$

• $\frac{\mathrm{x}}{3}$

• $\frac{\mathrm{x}}{4}$

If f: $\mathrm{R} \to \mathrm{R}$ is an even function having derivatives of all orders, then an odd function among the following is

• f''

• f'''

• f' + f''

• f'' + f'''