Primitive of cos^-1(x) w.r.t. x is

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) - \frac{1}{2}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) - \sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) + \sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) + \frac{1}{2}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

If y = log(cot(x)), then ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{ydx}$ is equal to

• 1

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

A force $\overrightarrow{\mathrm{F}} = 3\hat{\mathrm{i}} - \hat{\mathrm{j}}$ acts on a point R (0, 1, 1), then the moment of a force about the point P(0, 1, 0) is

• $3\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}}$

• $- \hat{\mathrm{i}} -\hspace{0.17em}3\hat{\mathrm{j}}$

• $\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

$\int \frac{1}{{\sin }^2\left(\mathrm{x}\right) + {\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \mathrm{c}$

• $\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\tan \left(\mathrm{x}\right) - \cot \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

   

15.

Let $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are non-zero and non-collinear vectors. If there exists scalars $\mathrm{\alpha }, \mathrm{\beta }$ such that $\mathrm{\alpha }\overrightarrow{\mathrm{a}} + \mathrm{\beta }\overrightarrow{\mathrm{b}} = \overrightarrow{0}$, then

• $\mathrm{\alpha } = \mathrm{\beta } \ne 0$

• $\mathrm{\alpha } + \mathrm{\beta } = 0$

• $\mathrm{\alpha } = \mathrm{\beta } = 0$

• $\mathrm{\alpha } \ne \mathrm{\beta }$

Primitive of $\frac{1}{4\sqrt{\mathrm{x}} + \mathrm{x}}$ is equal to

• $2\log \left|1 + 4\sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}\log \left|4 - \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $2\log \left| 4 + \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}\log \left| 4 + \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

If G is centroid of $∆$ABC, then

• $\overrightarrow{\mathrm{G}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $\overrightarrow{\mathrm{G}} = \frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{2}$

• $3\overrightarrow{\mathrm{G}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $3\overrightarrow{\mathrm{G}} = \frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{2}$

$\int {\mathrm{e}}^{\mathrm{x}}\left(\log \left(\mathrm{x}\right) + \frac{1}{\mathrm{x}}\right)\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\mathrm{x}}\log \left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\log \left(\mathrm{x}\right)}+ \mathrm{c}$

• $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$

${\int }_0^1\frac{{\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4} - 1$

• $1 - \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$

• $1 - \frac{\mathrm{\pi }}{4}$

Minimize : z = 3x + y, subject to 2x + 3y $\le$ 6, x + y $\ge$ 1, $\mathrm{x} \ge 0, \mathrm{y} \ge 0$

• x = 1, y = 1

• x = 0, y = 1

• x = 1, y = 0

• x = - 1, y = - 1