A unit vector perpendicular to both i + j + k and 2i + j + 3k is

• $\frac{\left(2\mathrm{i} - \mathrm{j} - \mathrm{k}\right)}{\sqrt{6}}$

• $2\mathrm{i} - \mathrm{j} - \mathrm{k}$

• $\frac{\left(3\mathrm{i} + \mathrm{j} - 2\mathrm{k}\right)}{\sqrt{6}}$

The value of ${\int }_0^4\left|\mathrm{x} - 1\right|\mathrm{dx}$ is

• $\frac{5}{2}$

• 5

• 4

• 1

If I_n = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, where where n is apositive integer, then ${\mathrm{I}}_{10} + {\mathrm{I}}_8$ is

• $\frac{1}{9}$

• $\frac{1}{8}$

• $\frac{1}{7}$

• 9

If I_n = $\int {\mathrm{e}}^{\mathrm{x}}\left[\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{1 - {\sin }^2\left(\mathrm{x}\right)}\right]\mathrm{dx}$ is

• $\left({\mathrm{e}}^{\mathrm{x}} . \csc \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

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

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

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

25.

When x > 0, then $\int {\cos }^{-1}\left(\frac{1 - {\mathrm{x}}^2}{1 +\hspace{0.17em}{\mathrm{x}}^2}\right)\mathrm{dx}$ is

• $2\left[{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) - \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)\right] + \mathrm{C}$

• $2\left[{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) + \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)\right] + \mathrm{C}$

• $2{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) + \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right) + \mathrm{C}$

• $2{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) - \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right) + \mathrm{C}$

If the area between y = mx² and x = my² (m > 0) is 1/4 sq units, then the value of m is

• $\pm 3\sqrt{2}$

• $\pm \frac{2}{\sqrt{3}}$

• $\sqrt{2}$

• $\sqrt{3}$

If m and n are degree and order of ${\left(1 + {\mathrm{y}}_1^2\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = {\mathrm{y}}_2$, then the value of $\frac{\mathrm{m} + \mathrm{n}}{\mathrm{m} - \mathrm{n}}$ is

• 3

• 4

• 5

• 2

The general solution of ${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 1 - {\mathrm{x}}^2 - {\mathrm{y}}^2 + {\mathrm{x}}^2{\mathrm{y}}^2$ is

• $2{\sin }^{-1}\left(\mathrm{y}\right) = \mathrm{x}\sqrt{1 - {\mathrm{x}}^2} + {\sin }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• ${\cos }^{-1}\left(\mathrm{y}\right) = {\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• ${\sin }^{-1}\left(\mathrm{y}\right) =\frac{1}{2}{\sin }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• $2{\sin }^{-1}\left(\mathrm{y}\right) = \mathrm{x}\sqrt{1 - {\mathrm{y}}^2} + \mathrm{C}$