91.

If for every integer n, ${\int }_{\mathrm{n}}^{\mathrm{n} + 1}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = {\mathrm{n}}^2$, then the value of ${\int }_{- 2}^4\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 16

• 14

• 19

• None of these

The value of integral ${\int }_0^{\mathrm{\pi }}\mathrm{xf}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$ is

• 0

• $\mathrm{\pi }{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$

• $\frac{\mathrm{\pi }}{4}{\int }_0^{\mathrm{\pi }}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$

• None of these

Solution of $$\begin{gathered} \mathrm{Given} \mathrm{equation} \mathrm{is} \frac{d\mathrm{y}}{d\mathrm{x}} = \frac{\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}}{\sin \left(\mathrm{y}\right){\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{ycos}}{\displaystyle \left(\mathrm{y}\right)}} \\ \Rightarrow \left(\sin \left(\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{y}\right)\right)\mathrm{dy} = \left(\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}\right)\mathrm{dx} \\ \mathrm{On} \mathrm{integrating} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ \int \left(\sin \left(\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{y}\right)\right)\mathrm{dy} = \int \left(\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}\right)\mathrm{dx} \\ \Rightarrow - \cos \left(\mathrm{y}\right) + \mathrm{ysin}\left(\mathrm{y}\right) + \cos \left(\mathrm{y}\right) = \frac{{\mathrm{x}}^2}{2}\log \left({\mathrm{x}}^2\right) \end{gathered}$$ is

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2\log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = \mathrm{xog}\left(\mathrm{x}\right) + \mathrm{C}$

$\underset{\mathrm{x}\to \infty }{\lim }\frac{{\int }_0^{2\mathrm{x}}{\mathrm{xe}}^{\mathrm{x}}\mathrm{dx}}{{\mathrm{e}}^{4{\mathrm{x}}^2}}$ equals

• 0

• $\infty$

• 2

• $\frac{1}{2}$

If x^py^q = (x + y)^{p + q}, then $\frac{d\mathrm{y}}{d\mathrm{x}}$ is equal to

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{py}}{\mathrm{qx}}$

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{\mathrm{qy}}{\mathrm{px}}$

Area bounded by the curve y² = 16x and line y = mx is $\frac{2}{3}$, then m is equal to

• 3

• 4

• 1

• 2

The equation $\mathrm{z}\overline{\mathrm{z}} + \mathrm{a}\overline{\mathrm{z}} + \overline{\mathrm{a}}\mathrm{z} + \mathrm{b} = 0$, represents a circle, if

• ${\left|\mathrm{a}\right|}^2 = \mathrm{b}$

• ${\left|\mathrm{a}\right|}^2 > \mathrm{b}$

• ${\left|\mathrm{a}\right|}^2 <\hspace{0.17em}\mathrm{b}$

•  None of the above

Area included between curves y = x² - 3x + 2 and y = - x² + 3x - 2 is

• $\frac{1}{6} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{2}$ sq unit

• 1 sq unit

• $\frac{1}{3}$ sq unit

The solution of $\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right)$ is

• $\mathrm{ysec}\left(\mathrm{x}\right) = \tan \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right) + \mathrm{C}$

• $\tan \left(\mathrm{x}\right) = \mathrm{ytan}\left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{xsec}\left(\mathrm{x}\right) = \mathrm{ytan}\left(\mathrm{y}\right) + \mathrm{C}$

If y = ${\tan }^{-1}\left[\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}\right]$, then $\frac{d\mathrm{y}}{d\mathrm{x}}$ is

• $\frac{1}{2}$

• $\frac{\mathrm{\pi }}{4}$

• 0

• 1