31.

In a AABC, if $\angle$C = 90°, r and R are the inradius and circumradius of the $∆$ABC respectively, then 2(r + R) is equal to

• b + c

• c + a

• a + b

• a + b + c

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be two distinct roots of $\mathrm{acos}\left(\mathrm{\theta }\right) + \mathrm{bsin}\left(\mathrm{\theta }\right) = \mathrm{c}$  where a, b, c are three real constants and $\mathrm{\theta } \in \left[0, 2\mathrm{\pi }\right]$. Then, $\mathrm{\alpha } + \mathrm{\beta }$ is also a root of the same equation, if

• a + b = c

• b + c = a

• c + a = b

• c = a

If $\cos \left(\mathrm{x}\right) \mathrm{and} \sin \left(\mathrm{x}\right)$ are solutions of the differential equation

${\mathrm{a}}_0\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + {\mathrm{a}}_1\frac{d\mathrm{y}}{d\mathrm{x}} + {\mathrm{a}}_2\mathrm{y} = 0$

where a₀, a₁ and a₂ are real constants, then which of the following is/are always true?

• $\mathrm{Acos}\left(\mathrm{x}\right) + \mathrm{Bsin}\left(\mathrm{x}\right)$ is a solution, where A and B are real constants 

• $\mathrm{Acos}\left(\mathrm{x} + \frac{\mathrm{\pi }}{4}\right)$ is a solution, where A is a real constant

• $\mathrm{Acos}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$ is a solution, where A is a real constant

• $\mathrm{Acos}\left(\mathrm{x} + \frac{\mathrm{\pi }}{4}\right) + \mathrm{Bsin}\left(\mathrm{x} - \frac{\mathrm{\pi }}{4}\right)$ is a souton, where A and B are real constants 

Which of the following statements is /are correct for $0 < \mathrm{\theta } < \frac{\mathrm{\pi }}{2}$

• ${\left(\cos \left(\mathrm{\theta }\right)\right)}^{1/2} \le \cos \left(\frac{\mathrm{\theta }}{2}\right)$

• ${\left(\cos \left(\mathrm{\theta }\right)\right)}^{3/4} \ge \cos \left(\frac{3\mathrm{\theta }}{4}\right)$

• $\cos \left(\frac{5\mathrm{\theta }}{6}\right) \ge {\left(\cos \left(\mathrm{\theta }\right)\right)}^{5/6}$

• $\cos \left(\frac{7\mathrm{\theta }}{8}\right) \le {\left(\cos \left(\mathrm{\theta }\right)\right)}^{7/8}$

The value of $\tan \left(\frac{\mathrm{\pi }}{2}\right) + 2\tan \left(\frac{2\mathrm{\pi }}{5}\right) + 4\cot \left(\frac{4\mathrm{\pi }}{5}\right)$ is

• $\cot \left(\frac{\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{2\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{4\mathrm{\pi }}{5}\right)$

• $\cot \left(\frac{3\mathrm{\pi }}{5}\right)$

The range of the function y = $3\sin \left(\sqrt{\frac{{\mathrm{\pi }}^2}{16} - {\mathrm{x}}^2}\right)$ is

• $\left[0, \sqrt{3/2}\right]$

• [0, 1]

• $\left[0, 3/\sqrt{2}\right]$

• $\left[0, \infty \right]$

In a $∆\mathrm{ABC}$,  a, b, c are the sides of the triangle opposite to the angles A, B, C, respectively. Then, the value of a³sin(B - C) + b³sin(C - A) + c³sin(A - B) is equal to

• 0

• 1

• 3

• 2

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

• 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

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

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

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

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