If $\tan \left(\mathrm{\theta }\right) + \tan \left(\mathrm{\theta } + \frac{\mathrm{\pi }}{3}\right) + \tan \left(\mathrm{\theta } + \frac{2\mathrm{\pi }}{3}\right) = 3$, then which of the following is equal to 1 ?

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

• $\tan \left(3\mathrm{\theta }\right)$

• ${\tan }^2\left(\mathrm{\theta }\right)$

• ${\tan }^3\left(\mathrm{\theta }\right)$

$\mathrm{If} \mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = 2\mathrm{\theta }, \mathrm{then} \cos \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta } - \mathrm{\alpha }\right) + \cos \left(\mathrm{\theta } - \mathrm{\beta }\right) + \cos \left(\mathrm{\theta } - \mathrm{\gamma }\right) = ?$

• $4\sin \left(\frac{\mathrm{\alpha }}{2}\right) . \cos \left(\frac{\mathrm{\beta }}{2}\right)\cos \left(\frac{\mathrm{\gamma }}{2}\right)$

• $4\sin \left(\frac{\mathrm{\alpha }}{2}\right) . \sin \left(\frac{\mathrm{\beta }}{2}\right)\sin \left(\frac{\mathrm{\gamma }}{2}\right)$

• $4\cos \left(\frac{\mathrm{\alpha }}{2}\right) . \cos \left(\frac{\mathrm{\beta }}{2}\right) . \cos \left(\frac{\mathrm{\gamma }}{2}\right)$

• $4\sin \left(\mathrm{\alpha }\right) . \cos \left(\mathrm{\beta }\right) . \cos \left(\mathrm{\gamma }\right)$

$\left\{\mathrm{x} \in \mathrm{R} : \cos \left(2\mathrm{x}\right) + 2{\cos }^2\left(\mathrm{x}\right) = 2\right\} = ?$

• $\left\{2\mathrm{n\pi } + \frac{\mathrm{\pi }}{3} : \mathrm{n} \in \mathrm{Z} \right\}$

• $\left\{\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{6} : \mathrm{n} \in \mathrm{Z} \right\}$

• $\left\{\mathrm{n\pi } + \frac{\mathrm{\pi }}{3} : \mathrm{n} \in \mathrm{Z} \right\}$

• $\left\{2\mathrm{n\pi } - \frac{\mathrm{\pi }}{3} : \mathrm{n} \in \mathrm{Z} \right\}$

$\frac{1 + \tanh \left({\displaystyle \frac{\mathrm{x}}{2}}\right)}{1 - \tanh \left({\displaystyle \frac{\mathrm{x}}{2}}\right)} = ?$

• ${\mathrm{e}}^{ - \mathrm{x}}$

• ${\mathrm{e}}^{\mathrm{x}}$

• $2{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• $2{\mathrm{e}}^{ - \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

$\mathrm{In} ∆\mathrm{ABC}, \mathrm{if} \frac{1}{\mathrm{b} + \mathrm{c}} + \frac{1}{\mathrm{c} +\hspace{0.17em}\mathrm{a}} = \frac{3}{\mathrm{a} +\hspace{0.17em}\mathrm{b} + \mathrm{c}}, \mathrm{then} \mathrm{C} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $90°$

• $60°$

• $45°$

• $30°$

$$\begin{gathered} \mathrm{Observe} \mathrm{the} \mathrm{following} \mathrm{statements} \\ \left(\mathrm{I}\right) \mathrm{In} ∆\mathrm{ABC}, {\mathrm{bcos}}^2\left(\frac{\mathrm{C}}{2}\right) + {\mathrm{ccos}}^2\left(\frac{\mathrm{B}}{2}\right) = \mathrm{s} \\ \left(\mathrm{II}\right) \mathrm{In} ∆\mathrm{ABC}, \cot \left(\frac{\mathrm{A}}{2}\right) = \frac{\mathrm{b} + \mathrm{c}}{2} \Rightarrow \mathrm{B} = 90° \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

$\mathrm{In} \mathrm{a} \mathrm{triangle}, \mathrm{if} {\mathrm{r}}_1 = 2{\mathrm{r}}_2 = 3{\mathrm{r}}_3, \mathrm{then} \frac{\mathrm{a}}{\mathrm{b}} + \frac{\mathrm{b}}{\mathrm{c}} + \frac{\mathrm{c}}{\mathrm{a} } = ?$

• $\frac{75}{60}$

• $\frac{155}{60}$

• $\frac{176}{60}$

• $\frac{191}{60}$

868.

From the top of a hill h metres high the angles of depressions of the top and the bottom of a pillar are $\mathrm{\alpha }$ and $\mathrm{\beta }$ respectively.The height (in metres) of the pillar is

• $\frac{\mathrm{h}\left(\tan \left(\mathrm{\beta }\right) - \tan \left(\mathrm{\alpha }\right)\right)}{\tan \left(\mathrm{\beta }\right)}$

• $\frac{\mathrm{h}\left(\tan \left(\mathrm{\alpha }\right) - \tan \left(\mathrm{\beta }\right)\right)}{\tan \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{h}\left(\tan \left(\mathrm{\beta }\right) + \tan \left(\mathrm{\alpha }\right)\right)}{\tan \left(\mathrm{\beta }\right)}$

• $\frac{\mathrm{h}\left(\tan \left(\mathrm{\beta }\right) + \tan \left(\mathrm{\alpha }\right)\right)}{\tan \left(\mathrm{\alpha }\right)}$

The period of ${\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)$ is

• $\frac{{\mathrm{\pi }}^4}{2}$

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

$\frac{\cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x} - 2\mathrm{y}\right)} = \mathrm{\lambda } \Rightarrow \tan \left(\mathrm{x} - \mathrm{y}\right)\tan \left(\mathrm{y}\right)$ is equal to

• $\frac{1 + \mathrm{\lambda }}{1 - \mathrm{\lambda }}$

• $\frac{1 - \mathrm{\lambda }}{1 + \mathrm{\lambda }}$

• $\frac{\mathrm{\lambda }}{1 + \mathrm{\lambda }}$

• $\frac{\mathrm{\lambda }}{1 + \mathrm{\lambda }}$