$\int \sqrt{\frac{\mathrm{x} - 1}{\mathrm{x} + 1}}\mathrm{dx}$ is equal to

• $2\sqrt{{\mathrm{x}}^2 + 1} + {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\sqrt{{\mathrm{x}}^2 - 1} - {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $2\sqrt{{\mathrm{x}}^2 - 1} + {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{\sqrt{{\mathrm{x}}^2 - 1}}{2} + {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

The solution of the differential equation (1 + y) tan^-1(x)dx + y(1 + x²)dy = 0 is

• $\log \left(\frac{{\tan }^{-1}\left(\mathrm{x}\right)}{\mathrm{x}}\right) + \mathrm{y}\left(1 + {\mathrm{x}}^2\right) = \mathrm{c}$

• $\log \left(1 + {\mathrm{y}}^2\right) + {\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 = \mathrm{c}$

• $\log \left(1 + {\mathrm{x}}^2\right) + \log \left({\tan }^{-1}\left(\mathrm{y}\right)\right) = \mathrm{c}$

• $\left({\tan }^{-1}\left(\mathrm{x}\right)\right)\left(1 + {\mathrm{y}}^2\right) + \mathrm{c} = 0$

The value of ${\int }_{- 1}^1\log \left(\frac{\mathrm{x} - 1}{\mathrm{x} +\hspace{0.17em}1}\right)\mathrm{dx}$ is

• 1

• 2

• 0

• 4

Considering four sub-intervals, the value of ${\int }_0^4{2}^{\mathrm{x}}\mathrm{dx}$  by Simpson's rule is

• $\frac{64}{8}$

• $\frac{65}{3}$

• $\frac{62}{12}$

• $\frac{61}{8}$

If $\left|\overrightarrow{\mathrm{a}}\right| = \left|\overrightarrow{\mathrm{b}}\right| = 1 \mathrm{and} \left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = \sqrt{3}$, then the value of $\left(3\overrightarrow{\mathrm{a}} - 4\overrightarrow{\mathrm{b}}\right) . \left(2\overrightarrow{\mathrm{a}} +\hspace{0.17em}5\overrightarrow{\mathrm{b}}\right)$ is

• - 21

• $- \frac{21}{2}$

• 21

• $\frac{21}{2}$

A unit vector in the plane of $\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}$ and $\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$ and perpendicular to $2\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}$, is

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

• $\frac{\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{j}} -\hspace{0.17em}\hat{\mathrm{k}}}{\sqrt{2}}$

If $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$, $\left|\overrightarrow{\mathrm{a}}\right|$ = 2, $\left|\overrightarrow{\mathrm{b}}\right|$ = 3, $\left|\overrightarrow{\mathrm{c}}\right|$ = 4 and the angle between $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ is $\frac{2\mathrm{\pi }}{3}$, then [$\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$] is equal to

• $4\sqrt{3}$

• $6\sqrt{3}$

• $12\sqrt{3}$

• $18\sqrt{3}$

If $\overrightarrow{\mathrm{a}}$, $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are perpendicular to $\overrightarrow{\mathrm{b}} +\hspace{0.17em}\overrightarrow{\mathrm{c}}$, $\overrightarrow{\mathrm{c}} +\hspace{0.17em}\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{a}} +\hspace{0.17em}\overrightarrow{\mathrm{b}}$ respectively and if $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|$ = 6, $\left|\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ = 8 and $\left|\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right|$ = 10 then $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is equal to

• $5\sqrt{2}$

• 50

• $10\sqrt{2}$

• 10

19.

If I₁ = $\int {\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ and I₂ = $\int {\sin }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)\mathrm{dx}$, then

• I₁ = I₂

• ${\mathrm{I}}_2 = \frac{\mathrm{\pi }}{2}{\mathrm{I}}_1$

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}\mathrm{x}$

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}$

$\int \frac{\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)}{\sqrt{\sin \left(2\mathrm{\theta }\right)}}\mathrm{d\theta }$ is equal to

• $\log \left|\cos \left(\mathrm{\theta }\right) - \sin \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right| + \mathrm{c}$

• $\log \left|\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right| + \mathrm{c}$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)\right) + \mathrm{c}$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right) + \mathrm{c}$