If a and b are unit vectors and $\mathrm{\theta }$ is the angle between them, then the value of $\left|\cos \left(\frac{\mathrm{\theta }}{2}\right)\right|$ is

• $\frac{1}{2}\left|\mathrm{a} + \mathrm{b}\right|$

• $\frac{1}{2}\left|\mathrm{a} - \mathrm{b}\right|$

• $\frac{\left|\mathrm{a} - \mathrm{b}\right|}{\left|\mathrm{a} + \mathrm{b}\right|}$

• $\frac{\left|\mathrm{a} + \mathrm{b}\right|}{\left|\mathrm{a} - \mathrm{b}\right|}$

$\int {\mathrm{xsec}}^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

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

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

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

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

$\int \frac{\mathrm{t}}{{\mathrm{e}}^{3{\mathrm{t}}^2}}\mathrm{dt}$ is equal to

• $\frac{1}{6}{\mathrm{e}}^{3{\mathrm{t}}^2} + \mathrm{c}$

• $- \frac{1}{6}{\mathrm{e}}^{3{\mathrm{t}}^2} + \mathrm{c}$

• $\frac{1}{6}{\mathrm{e}}^{- 3{\mathrm{t}}^2} + \mathrm{c}$

• $- \frac{1}{6}{\mathrm{e}}^{- 3{\mathrm{t}}^2} + \mathrm{c}$

${\int }_0^{\mathrm{\pi }}\log \left({\sin }^2\left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• $2{\mathrm{\pi log}}_{\mathrm{e}}\left(\frac{1}{2}\right)$

• ${\mathrm{\pi log}}_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{2}{\log }_{\mathrm{e}}\left(\frac{1}{2}\right)$

• None of these

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

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

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

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

• None of these

The area bounded by the curves y² - x = 0 and y - x² = 0, is

• 7/3 sq unit

• 1/3 sq unit

• 5/3 sq unit

• 1 sq unit

${\int }_0^2\left[{\mathrm{x}}^2\right]\mathrm{dx}$ is equal to

• $2 - \sqrt{2}$

• $2 + \sqrt{2}$

• $\sqrt{2} - 1$

• $- \sqrt{2} - \sqrt{3} + 5$

Area between the curve y = cos(x) and X - axis, when $0 \le \mathrm{x} \le 2\mathrm{\pi }$, is

• 0 sq units

• 2 sq units

• 3 sq units

• 4 sq units

39.

In = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, then $\underset{\mathrm{n}\to \infty }{\lim }\mathrm{n}\left[{\mathrm{I}}_{\mathrm{n}} + {\mathrm{I}}_{\mathrm{n} + 2}\right]$ equals

• 1/ 2

• 2 sq units

• 3 sq units

• 4 sq uits

The order of the differential equation

$\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{\mathrm{x}}{\left({\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^3$ is

• 1

• 2

• 3

• 4