21.

The value of $\int \frac{{\mathrm{e}}^{\mathrm{x}}\left(1 +\hspace{0.17em}\mathrm{x}\right)}{{\cos }^2\left({\mathrm{e}}^{\mathrm{x}} . \mathrm{x}\right)}\mathrm{dx}$ is equal to

• $- \cot \left({\mathrm{ex}}^{\mathrm{x}}\right)$ + C

• $\tan \left({\mathrm{e}}^{\mathrm{x}} . \mathrm{x}\right) + \mathrm{C}$

• $\tan \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

• $\cot \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

The vate of $\int \frac{{\mathrm{e}}^{\mathrm{x}}\left({\mathrm{x}}^2{\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) + 1\right)}{{\mathrm{x}}^2 + 1}\mathrm{dx}$ is equal to

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

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

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

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

The value of ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\sin }^{103}\left(\mathrm{x}\right) . {\cos }^{101}\left(\mathrm{x}\right)\mathrm{dx}$ is

• ${\left(\frac{\mathrm{\pi }}{4}\right)}^{103}$

• ${\left(\frac{\mathrm{\pi }}{4}\right)}^{101}$

• 2

• 0

The value of $\int \frac{{\mathrm{e}}^{6\log \left(\mathrm{x}\right)} - {\mathrm{e}}^{5\log \left(\mathrm{x}\right)}}{{\mathrm{e}}^{4\log \left(\mathrm{x}\right)} - {\mathrm{e}}^{3\log \left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to

• 0 + C

• $\frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

• $\frac{3}{{\mathrm{x}}^3} + \mathrm{C}$

• $\frac{1}{\mathrm{x}} + \mathrm{C}$

The differential coefficient of log₁₀(x) with respect to log_x(10) is

• 1

• $- {\left({\log }_{10}\left(\mathrm{x}\right)\right)}^2$

• ${\left({\log }_{\mathrm{x}}\left(10\right)\right)}^2$

• $\frac{{\mathrm{x}}^2}{100}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\sin }^{1000}\left(\mathrm{x}\right)}{{\sin }^{1000}\left(\mathrm{x}\right) + {\cos }^{1000}\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• 1000

• 1

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

The solution for the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{dx}}{\mathrm{x}} = 0$ is

• $\frac{1}{\mathrm{y}} + \frac{1}{\mathrm{x}} = \mathrm{C}$

• $\log \left(\mathrm{x}\right)\log \left(\mathrm{y}\right) = \mathrm{C}$

• xy = C

• x + y = C

The order and degree of the differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + \sin \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} = \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$

• order = 2, degree = 3

• order = 2, degree = 4

• oreder = 2, degree = $\frac{3}{4}$

• order = 2, degree = not defined

If a and b are unit vectors, then what is the angle between a and b for  $\sqrt{3}\mathrm{a} - \mathrm{b}$ to be unit vector?

• $30°$

• $45°$

• $60°$

• 90$°$

Suppose $\mathrm{a} + \mathrm{b} + \mathrm{c} = 0, \left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 5, \left|\mathrm{c}\right| = 7$, then the angle between a and b is

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$