21.

The value of $\int \frac{\sin \left(2\mathrm{x}\right)}{{\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)}\mathrm{dx}$ is

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

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

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

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

The value of $\int \sqrt{1 + \sec \left(\mathrm{x}\right)}\mathrm{dx}$ is

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

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

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

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

The value of $\int \frac{\left({\mathrm{x}}^2 + 1\right)}{{\mathrm{x}}^4 +\hspace{0.17em}{\mathrm{x}}^2 + 1}\mathrm{dx}$ is

• $\frac{1}{\sqrt{3}}{\tan }^{-1}\left\{\frac{\mathrm{x} - 1/\mathrm{x}}{\sqrt{3}}\right\} + \mathrm{C}$

• $\frac{1}{2\sqrt{3}}\log \left\{\frac{\left(\mathrm{x} - 1/\mathrm{x}\right) - \sqrt{3}}{\left(\mathrm{x} - 1/\mathrm{x}\right) + \sqrt{3}}\right\} + \mathrm{C}$

• ${\tan }^{-1}\left\{\frac{\mathrm{x} + 1/\mathrm{x}}{\sqrt{3}}\right\} + \mathrm{C}$

• ${\tan }^{-1}\left\{\frac{\mathrm{x} - 1/\mathrm{x}}{\sqrt{3}}\right\} + \mathrm{C}$

The value of ${\int }_0^1{\mathrm{x}}^2{\left(1 - {\mathrm{x}}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ is

• $\frac{1}{32}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{32}$

The value of ${\int }_0^{\infty }\frac{\mathrm{x}}{\left(1 + \mathrm{x}\right)\left({\mathrm{x}}^2 + 1\right)}\mathrm{dx}$ is

• $2\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{32}$

The area of the region bounded by the curves, y² = 8x and y = x is

• $\frac{64}{3}$

• $\frac{32}{3}$

• $\frac{16}{3}$

• $\frac{8}{3}$

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

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

$\mathrm{i} . \left(\mathrm{j} \times \mathrm{k}\right) + \mathrm{j} . \left(\mathrm{k} \times \mathrm{i}\right) + \mathrm{k} . \left(\mathrm{j} \times \mathrm{i}\right)$ is equal to

• 3

• 2

• 1

• 0

One card is drawn at random from a pack of playing cards the probability that it is an ace or black king or the queen of the heart will be

• $\frac{3}{52}$

• $\frac{7}{52}$

• $\frac{6}{52}$

• $\frac{1}{52}$

15 coins are tossed, then the probability of getting 10 heads will be

• $\frac{511}{32768}$

• $\frac{1001}{32768}$

• $\frac{3003}{32768}$

• $\frac{3005}{32768}$