$1 + \cos \left(10°\right) \hspace{0.17em}+ \cos \left(20°\right) + \cos \left(30°\right) = ?$

• $4\sin \left(10°\right)\sin \left(20°\right)\sin \left(30°\right)$

• $4\cos \left(5°\right)\cos \left(10°\right)\cos \left(15°\right)$

• $4\cos \left(10°\right)\cos \left(20°\right)\cos \left(30°\right)$

• $4\sin \left(5°\right)\sin \left(10°\right)\sin \left(15°\right)$

$\mathrm{The} \mathrm{value} \mathrm{of} {\left(\frac{1 + \sin \left({\displaystyle \frac{2\mathrm{\pi }}{9}}\right) + \mathrm{icos}\left(\frac{2\mathrm{\pi }}{9}\right)}{1 + \sin \left(\frac{2\mathrm{\pi }}{9}\right) - \mathrm{icos}\left(\frac{2\mathrm{\pi }}{9}\right)}\right)}^3 \mathrm{is} :$

• $- \frac{1}{2}\left(\sqrt{3} - \mathrm{i}\right)$

• $\frac{1}{2}\left(\sqrt{3} - \mathrm{i}\right)$

• $\frac{1}{2}\left(1 - \mathrm{i}\sqrt{3}\right)$

• $- \frac{1}{2}\left(1 - \mathrm{i}\sqrt{3}\right)$

If the equation ${\cos }^4\left(\mathrm{\theta }\right) + {\sin }^4\left(\mathrm{\theta }\right) + \mathrm{\lambda } = 0$ has real solutions for $\mathrm{\theta }, \mathrm{then} \mathrm{\lambda }$ lies in interval :

• $\left( - \frac{5}{4}, - 1\right)$

• $\left[ - 1, - \frac{1}{2}\right]$

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

• $\left[ - \frac{3}{2}, - \frac{5}{4}\right]$

264.

The set of all possible values of $\mathrm{\theta }$ in the interval (0, $\mathrm{\pi }$) for which the points(1, 2) and (sin($\mathrm{\theta )}$, cos($\mathrm{\theta )}$) lie on the same side of the line x + y = 1 is

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(\frac{\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{4}\right)$

• $\left(0, \frac{3\mathrm{\pi }}{4}\right)$

$$\begin{gathered} \mathrm{Solve} \mathrm{L} = {\sin }^2\left(\frac{\mathrm{\pi }}{16}\right) - {\sin }^2\left(\frac{\mathrm{\pi }}{8}\right) \mathrm{and} \\ \mathrm{M} = {\cos }^2\left(\frac{\mathrm{\pi }}{16}\right) - {\cos }^2\left(\frac{\mathrm{\pi }}{8}\right) \end{gathered}$$

• $\mathrm{M} = \frac{1}{4\sqrt{2}} + \frac{1}{4}\cos \left(\frac{\mathrm{\pi }}{8}\right)$

• $\mathrm{M} = \frac{1}{2\sqrt{2}} + \frac{1}{2}\cos \left(\frac{\mathrm{\pi }}{8}\right)$

• L =  - $\mathrm{L} = - \frac{1}{2\sqrt{2}} + \frac{1}{2}\cos \left(\frac{{\displaystyle \mathrm{\pi }}}{8}\right)$

• $\mathrm{L} = \frac{1}{4\sqrt{2}} - \frac{1}{4}\cos \left(\frac{\mathrm{\pi }}{8}\right)$

The angle of elevation of the summit of a mountain from a point on the ground is 45º. After climbing up one km towards the summitat an inclination of 30º from the ground, the angle of elevation of the summit is found to be 60º. Then the height (in km) of the summit from the ground is

• $\frac{\sqrt{3} + 1}{\sqrt{3} - 1}$

• $\frac{1}{\sqrt{3} + 1}$

• $\frac{1}{\sqrt{3} - 1}$

• $\frac{\sqrt{3} - 1}{\sqrt{3} + 1}$