If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are unit vectors such that $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right] = \frac{1}{4}$, then angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

32.

The order and degree of the differential equation $\sqrt{\sin \left(\mathrm{x}\right)}\left(\mathrm{dx} + \mathrm{dy}\right) = \sqrt{\cos \left(\mathrm{x}\right)}\left(\mathrm{dx} - \mathrm{dy}\right)$ is :

• (1, 2)

• (2, 2)

• (1, 1)

• (2, 1)

${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to :

• $\frac{1}{2}$

• - 2

• 1

• - 1

$\int \left(\frac{\sin \left(2\mathrm{x}\right)}{\sin \left(3\mathrm{x}\right)\sin \left(5\mathrm{x}\right)}\right)\mathrm{dx}$ is equal to :

• $\frac{1}{5}{\log }_{\mathrm{e}}\left|\sin \left(5\mathrm{x}\right)\right| - \frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right| + \mathrm{c}$

• $\frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right| - \frac{1}{5}{\log }_{\mathrm{e}}\left|\sin \left(5\mathrm{x}\right)\right|$

• $\frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right| + \frac{1}{5}{\log }_{\mathrm{e}}\left|\sin \left(5\mathrm{x}\right)\right|$

• $- \frac{1}{2}\cos \left(2\mathrm{x}\right) + \frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right|$

$\int {\mathrm{e}}^{\mathrm{x}}\left\{\log \left(\sin \left(\mathrm{x}\right)\right) + \cot \left(\mathrm{x}\right)\right\}\mathrm{dx}$ is equal to

• e^xcot(x) + c

• e^xlog(sin(x)) + c

• e^xlog(sin(x)) + tan(x) + c

• e^x + sin(x) + c

${\int }_{- 10}^{10}\log \left(\frac{\mathrm{a} + \mathrm{x}}{\mathrm{a} - \mathrm{x}}\right)\mathrm{dx}$ is equal to :

• 0

• - 2log(a + 10)

• $2\log \left(\frac{\mathrm{a} + 10}{\mathrm{a} - 10}\right)$

• 2log(a + 10)

The point of intersection of the line $\overrightarrow{\mathrm{r}} = 7\hat{\mathrm{i}} + 10\hat{\mathrm{j}} + 13\hat{\mathrm{k}} + \mathrm{s}\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{r}} = 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 7\hat{\mathrm{k}} + \mathrm{s}\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ is :

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$

Define f(x) = ${\int }_0^{\mathrm{x}}\sin \left(\mathrm{t}\right)\mathrm{dt}, \mathrm{x} \ge 0$, Then :

• f is increasing only in the interval $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• f is decreasing in the interval $\left[0, \mathrm{\pi }\right]$

• f attains maximum at x = $\frac{\mathrm{\pi }}{2}$

• f attains minimum at x = $\mathrm{\pi }$

Let f(x) = $\frac{{\sin }^2\left(\mathrm{\pi x}\right)}{1 + {\mathrm{\pi }}^2}$. Then, $\int \left(\mathrm{f}\left(\mathrm{x}\right) + \mathrm{f}\left(- \mathrm{x}\right)\right)\mathrm{dx}$ is equal to :

• 0

• x + c

• $\frac{\mathrm{x}}{2} - \frac{\cos \left(\mathrm{\pi x}\right)}{2\mathrm{\pi }} + \mathrm{c}$

• $\frac{\mathrm{x}}{2} - \frac{\sin \left(2\mathrm{\pi x}\right)}{4\mathrm{\pi }} + \mathrm{c}$

The interior angles of a polygon are AP. The smallest angle is 120° and the common difference is 5°. The number of sides of the polygon is :

• 9

• 10

• 16

• 5