11.

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be three vectors. Then, scalar triple product $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$ is equal to :

• $\left[\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{c}}\right]$

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{b}}\right]$

• $\left[\overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}}\right]$

• $\left[\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{a}}\right]$

$\int \frac{{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^3}{\left(1 + {\mathrm{x}}^2\right)}\mathrm{dx}$ is equa to :

• $3{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

• $\frac{{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^4}{4} + \mathrm{c}$

• ${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^4 + \mathrm{c}$

• None of these

The area between the parabola y = x² and the line y = x is : 

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

• $\frac{1}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{2} \mathrm{sq} \mathrm{unit}$

• None of these

${\int }_0^{\mathrm{\pi }}\frac{\mathrm{xdx}}{1 + \sin \left(\mathrm{x}\right)}$ is equal to :

• $- \mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• None of these

The solution of the differential equation sec²(x)tan(y))dx + sec²(y)tan(x))dy = 0 is :

• $\tan \left(\mathrm{y}\right)\tan \left(\mathrm{x}\right) = \mathrm{c}$

• $\frac{\tan \left(\mathrm{y}\right)}{\tan \left(\mathrm{x}\right)} = \mathrm{c}$

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

• None of these

Cosine of the angle between two diagonals of a cube is equal to :

• $\frac{2}{\sqrt{6}}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• None of these

$\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are three vectors with magnitude $\left|\overrightarrow{\mathrm{a}}\right| = 4, \left|\overrightarrow{\mathrm{b}}\right| = 4, \left|\overrightarrow{\mathrm{c}}\right| = 2$ and such that $\overrightarrow{\mathrm{a}}$ is perpendicular to $\left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right), \overrightarrow{\mathrm{b}}$, is perpendicular to $\left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right) \mathrm{and} \overrightarrow{\mathrm{c}}$ is perpendicular to $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$. It follows that $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is equal to :

• 9

• 6

• 5

• 4

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three non-coplanar vectors, then

$\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) . \left[\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{c}}\right)\right]$ is :

• 0

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $- \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^5 + 1\right)}$ is equal to :

• $\frac{1}{5}\log \left({\mathrm{x}}^5\right)\left({\mathrm{x}}^5 + 1\right) + \mathrm{c}$

• $\frac{1}{5}\log \left(\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5}\right) + \mathrm{c}$

• $\frac{1}{5}\log \left(\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 1}\right) + \mathrm{c}$

• None of these

$\int \frac{\mathrm{x} + \sin \left(\mathrm{x}\right)}{1 +\hspace{0.17em}\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• ${\mathrm{xsec}}^2\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

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

• None of these