$\mathrm{If} \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = 7 + \cos \left(5\mathrm{x} +\hspace{0.17em}3\right) \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \mathrm{the} \mathrm{period} \mathrm{of} \mathrm{f} \mathrm{i}$

• $2\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{5}$

• $\frac{2\mathrm{\pi }}{5}$

$\cos \left(\mathrm{A}\right) = \frac{3}{4} \Rightarrow 32\sin \left(\frac{\mathrm{A}}{2}\right)\sin \left(\frac{5\mathrm{A}}{2}\right) = ?$

• 7

• 8

• 13

• 11

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\sin }^6\left(\mathrm{x}\right) +\hspace{0.17em}{\cos }^6\left(\mathrm{x}\right) \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{Then} \mathrm{f}\left(\mathrm{x}\right) \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

• $\left[\frac{7}{5}, \frac{5}{4}\right]$

• $\left[\frac{1}{2}, \frac{5}{8}\right]$

• $\left[\frac{1}{4}, 1\right]$

• $\left[\frac{1}{4}, \frac{1}{2}\right]$

The most general value of 0 which satisfies both the equations tan 8 = - 1 and $\cos \left(\mathrm{\theta }\right) = \frac{1}{\sqrt{2}}$ is

• $\mathrm{n\pi } + \frac{7\mathrm{\pi }}{4}$

• $2\mathrm{n\pi } + \frac{7\mathrm{\pi }}{4}$

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{7\mathrm{\pi }}{4}$

• $\frac{7\mathrm{n\pi }}{4}$

$\mathrm{In} \mathrm{a} ∆\mathrm{ABC}, \mathrm{if} \mathrm{a} {\cos }^2\frac{\mathrm{C}}{2} + {\mathrm{ccos}}^2\frac{\mathrm{A}}{2} = \frac{3\mathrm{B}}{2}, \mathrm{then} \mathrm{the} \mathrm{sides} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{are} \mathrm{in}$

• an arithmetic progression

• a geometric progression

• a harmonic progression

• an arithmetico-geometric progression

$\mathrm{In} \mathrm{a} ∆\mathrm{ABC}, \mathrm{if} \frac{\cos \left(\mathrm{A}\right)}{\mathrm{a}} = \frac{\cos \left(\mathrm{B}\right)}{\mathrm{b}} = \frac{\cos \left(\mathrm{C}\right)}{\mathrm{c}}, \mathrm{then} ∆\mathrm{ABC} \mathrm{is}$

• right angled

• isosceles right angled

• equilateral

• Scalene

277.

The angle of elevation of a stationary cloud from a point 2500m above a lake is 15° and from the same point the angle of depression of its reflection in the lake is 45°. The height(in metres) of the cloud above the lake, given that cot(15°) = 2 + $\sqrt{3}$, is

• 2500

• $2500\sqrt{2}$

• $2500\sqrt{3}$

• 5000

$\mathrm{The} \mathrm{minimum} \mathrm{value} \mathrm{of} 27{\tan }^2\left(\mathrm{\theta }\right) + 3{\cot }^2\left(\mathrm{\theta }\right) \mathrm{is}$

• 15

• 18

• 24

• 30

$\cos \left(36°\right) - \cos \left(72°\right) = ?$

• 1

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

$$\begin{gathered} \mathrm{If} \tan \left(\mathrm{x}\right) + \tan \left(\mathrm{x} + \frac{\mathrm{\pi }}{3}\right) + \tan \left(\mathrm{x} + \frac{2\mathrm{\pi }}{3}\right) = 3, \\ \mathrm{then} \tan \left(3\mathrm{x}\right) = ? \end{gathered}$$

• 3

• 2

• 1

• 0