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

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

42.

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

Which of the following real valued functions is/are not even functions?

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3\sin \left(\mathrm{x}\right)$

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 \cos \left(\mathrm{x}\right)$

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}}{\mathrm{x}}^3\sin \left(\mathrm{x}\right)$

• f(x) = x - [x], where [x] denotes the greatest integer less than or equal to x.

Number of solutions of the equation tan(x) + sec(x) = 2cos(x), x $\in$ [0, $\mathrm{\pi }$] is

• 0

• 1

• 2

• 3

If ${\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}{\sin }^{-1}\left(\mathrm{y}\right) + {\sin }^{-1}\left(\mathrm{z}\right) = \frac{3\mathrm{\pi }}{2}$, then the value of x⁹ + y⁹ + z⁹ - $\frac{1}{{\mathrm{x}}^9{\mathrm{y}}^9{\mathrm{z}}^9}$ is equal to

• 0

• 1

• 2

• 3