651.

$\cos \left(\frac{2\mathrm{\pi }}{7}\right) + \cos \left(\frac{4\mathrm{\pi }}{7}\right) + \cos \left(\frac{6\mathrm{\pi }}{7}\right)$

• is equal to zero

• lies between 0 and 3

• is a negative number

• lies between 3 and 6

The minimum value of $2^{\sin \left(\mathrm{x}\right)} + 2^{\cos \left(\mathrm{x}\right)}$ is

• $2^{1 - 1/\sqrt{2}}$

• $2^{1 + 1/\sqrt{2}}$

• $2^{\sqrt{2}}$

• 2

If $\mathrm{p} = \left(\begin{array}{cc}\cos \left(\frac{\mathrm{\pi }}{4}\right)& - \sin \left(\frac{\mathrm{\pi }}{4}\right)\\ \sin \left(\frac{\mathrm{\pi }}{4}\right)& \cos \left(\frac{\mathrm{\pi }}{4}\right)\end{array}\right) \mathrm{and} \mathrm{X} = \left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$. Then, p³X is equal to

• $\left(\begin{array}{c}0\\ 1\end{array}\right)$

• $\left(\begin{array}{c}- \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$

• $\left(\begin{array}{c}- 1\\ 0\end{array}\right)$

• $\left(\begin{array}{c}- \frac{1}{\sqrt{2}}\\ - \frac{1}{\sqrt{2}}\end{array}\right)$

For $0 \le \mathrm{P}, \mathrm{Q} \le \frac{\mathrm{\pi }}{2}$, if $\sin \left(\mathrm{P}\right) + \cos \left(\mathrm{Q}\right) = 2$, then the value of $\tan \left(\frac{\mathrm{P} + \mathrm{Q}}{2}\right)$ is equal to

• 1

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

The value of

${\cos }^2\left(75°\right) + {\cos }^2\left(45°\right) + {\cos }^2\left(15°\right) - {\cos }^2\left(30°\right) - {\cos }^2\left(60°\right)$ is

• 0

• 1

• $\frac{1}{2}$

• $\frac{1}{4}$

The maximum and minimum values of ${\cos }^6\left(\mathrm{\theta }\right) + {\sin }^6\left(\mathrm{\theta }\right)$  are respectively

• $1 \mathrm{and} \frac{1}{4}$

• 1 and 0

• 2 and 0

• 1 and $\frac{1}{2}$

Let $\mathrm{f}\left(\mathrm{\theta }\right) = \left(1 + {\sin }^2\left(\mathrm{\theta }\right)\right)\left(2 - {\sin }^2\left(\mathrm{\theta }\right)\right)$. Then, for all values of $\mathrm{\theta }$

• $\mathrm{f}\left(\mathrm{\theta }\right) > \frac{9}{4}$

• $\mathrm{f}\left(\mathrm{\theta }\right) < 2$

• $\mathrm{f}\left(\mathrm{\theta }\right) > \frac{11}{4}$

• $2 \le \mathrm{f}\left(\mathrm{\theta }\right) \le \frac{9}{4}$

If P, Q and R are angles of an isosceles triangle and $\angle \mathrm{P} = \frac{\mathrm{\pi }}{2}$,  then the value of

$$\begin{gathered} {\left(\cos \left(\frac{\mathrm{P}}{3}\right) - \mathrm{isin}\left(\frac{\mathrm{P}}{3}\right)\right)}^3 + \left(\cos \left(\mathrm{Q}\right) + \mathrm{isin}\left(\mathrm{Q}\right)\right)\left(\cos \left(\mathrm{R}\right) - \mathrm{isin}\left(\mathrm{R}\right)\right) \\ + \left(\cos \left(\mathrm{P}\right) - \mathrm{isin}\left(\mathrm{P}\right)\right)\left(\cos \left(\mathrm{Q}\right) - \mathrm{isin}\left(\mathrm{Q}\right)\right)\left(\cos \left(\mathrm{R}\right) - \mathrm{isin}\left(\mathrm{R}\right)\right) \end{gathered}$$

• i

• - i

• 1

• - 1

If $\mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right) + 2{\cos }^2\left(\mathrm{x}\right), \frac{\mathrm{\pi }}{4} \le \mathrm{x} \le \frac{3\mathrm{\pi }}{4}$. Then, f attains its

• $\mathrm{minimum} \mathrm{at} \mathrm{x} = \frac{\mathrm{\pi }}{4}$

• maximum at x = $\frac{\mathrm{\pi }}{2}$

• minimum x = $\frac{\mathrm{\pi }}{2}$

• mamum at x = ${\sin }^{-1}\left(\frac{1}{4}\right)$

If ${\sin }^2\left(\mathrm{\theta }\right) + 3\cos \left(\mathrm{\theta }\right) = 2 \mathrm{then} {\cos }^3\left(\mathrm{\theta }\right) + {\sec }^3\left(\mathrm{\theta }\right)$ is equal to

• 1

• 4

• 9

• 18