$$\begin{gathered} \sin \left(\mathrm{A}\right) + \sin \left(\mathrm{B}\right) = \sqrt{3}\left(\cos \left(\mathrm{B}\right) - \cos \left(\mathrm{A}\right)\right) \\ \Rightarrow \sin \left(3\mathrm{A}\right) + \sin \left(3\mathrm{B}\right) = ? \end{gathered}$$

• 0

• 2

• 1

• - 1

$\sec \left({\mathrm{h}}^{ - 1}\right)\sin \left(\mathrm{\theta }\right) = ?$

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

• $\log \left(\sin \left(\frac{\mathrm{\theta }}{2}\right)\right)$

• $\log \left(\cos \left(\frac{\mathrm{\theta }}{2}\right)\right)$

• $\log \left(\cot \left(\frac{\mathrm{\theta }}{2}\right)\right)$

243.

If two angles of  $∆$ABC are 45° and 60°, then the ratio of the smallest and the greatest sides are

• $\left(\sqrt{3} - 1\right) : 1$

• $\sqrt{3 } : \sqrt{2}$

• $1 : \sqrt{3}$

• $\sqrt{3} : 1$

$\mathrm{In} ∆ \mathrm{ABC}, \left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\hspace{0.17em}+ \mathrm{c}\right)\left(\tan \left(\frac{\mathrm{A}}{2}\right) + \tan \left(\frac{\mathrm{B}}{2}\right)\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $2\mathrm{ccot}\left(\frac{\mathrm{A}}{2}\right)$

• $2\mathrm{acot}\left(\frac{\mathrm{A}}{2}\right)$

• $2\mathrm{bcot}\left(\frac{\mathrm{B}}{2}\right)$

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

In $∆$ABC, with usual notation, observe the two statements given below :

$$\begin{gathered} \left(\mathrm{I}\right) {\mathrm{rr}}_1{\mathrm{r}}_2{\mathrm{r}}_3 = {∆}^2 \\ \left(\mathrm{II}\right) {\mathrm{r}}_1{\mathrm{r}}_{2 } + {\mathrm{r}}_2{\mathrm{r}}_3 + {\mathrm{r}}_3{\mathrm{r}}_1 = {\mathrm{s}}^2 \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

$\sqrt{3}\csc \left(20°\right) - \sec \left(20°\right)$ is equal to

• 2

• $2\sin \left(20°\right) - \csc \left(40°\right)$

• 4

• $4\sin \left(20°\right) . \csc \left(40°\right)$

If A = 35°, B = 15° and C = 40°, then tan(A) · tan(B) + tan(B) . tan(C) + tan(C) . tan(A) is equal to

• 0

• 1

• 2

• 3

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 ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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