21.

The area of the portion of the circle x² + y² = 64 which is exterior to the parabola y² = 12x, is

• $\left(8\mathrm{\pi } - \sqrt{3}\right) \mathrm{sq} \mathrm{units}$

• $\frac{16}{3}\left(8 - \sqrt{3}\right) \mathrm{sq} \mathrm{units}$

• $\frac{16}{3}\left(8\mathrm{\pi } - \sqrt{3}\right) \mathrm{sq} \mathrm{units}$

• None of the above

$\underset{\mathrm{n}\to \infty }{\lim }\frac{1}{\mathrm{n}}\left(\frac{1}{\mathrm{n} + 1} + \frac{2}{\mathrm{n} + 2} + ... + \frac{3\mathrm{n}}{4\mathrm{n}}\right)$ is equal to

• log(4)

• - log(4)

• 1 - log(4)

• None of these

The value of the integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\sin }^2\left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) +\hspace{0.17em}\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\sqrt{2}\left(\log \left(\sqrt{2}\right)\right)$

• $\sqrt{2}\left(\sqrt{2} + 1\right)$

• $\log \left(\sqrt{2} + 1\right)$

• None of the above