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

• ${\mathrm{e}}^{ - \mathrm{x}}$

• ${\mathrm{e}}^{\mathrm{x}}$

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

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

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

• $90°$

• $60°$

• $45°$

• $30°$

253.

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

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

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

• $\frac{75}{60}$

• $\frac{155}{60}$

• $\frac{176}{60}$

• $\frac{191}{60}$

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

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

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

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

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

$\cos \left(\mathrm{A}\right)\cos \left(2\mathrm{A}\right)\cos \left(4\mathrm{A}\right) ... \cos \left(2^{\mathrm{n} - 1}\mathrm{A}\right)$ equals

• $\frac{\sin \left(2^{\mathrm{n}}\mathrm{A}\right)}{2^{\mathrm{n}}\sin \left(\mathrm{A}\right)}$

• $\frac{2^{\mathrm{n}}\sin \left(2^{\mathrm{n}}\mathrm{A}\right)}{\sin \left(\mathrm{A}\right)}$

• $\frac{2^{\mathrm{n}}\sin \left(2^{\mathrm{n}}\mathrm{A}\right)}{\sin \left(2^{\mathrm{n}}\mathrm{A}\right)}$

• $\frac{\sin \left(\mathrm{A}\right)}{2^{\mathrm{n}}\sin \left(2^{\mathrm{n}}\mathrm{A}\right)}$

If 3cos(x) $\ne$ sin x, then the general solution of sin²(x) - cos(2x) = 2 - sin(2x) is

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\left(4\mathrm{n} \pm 1\right)\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\left(2\mathrm{n} - 1\right)\mathrm{\pi }, \mathrm{n} \in \mathrm{Z}$

$$\begin{gathered} \mathrm{In} \mathrm{a} ∆\mathrm{ABC} \\ \frac{\left( \mathrm{a} + \mathrm{b} + \mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} + \mathrm{a} - \mathrm{b}\right)\left( \mathrm{a} + \mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2} = ? \end{gathered}$$

• cos²A

• cos²B

• sin²A

• sin²B