171.

$\tan \left(\mathrm{\theta }\right) + \cot \left(\mathrm{\theta }\right) = 2$ = 2, then $\sin \left(\mathrm{\theta }\right)$ is equal to

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{2}$

• 1

$$\begin{gathered} \mathrm{If} \mathrm{\theta } = \frac{\mathrm{\pi }}{6}, \mathrm{then} \mathrm{the} 10\mathrm{th} \mathrm{term} \mathrm{of} 1 + \left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right) + {\left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right)}^2 + \\ {\left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right)}^3 + ...\mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\mathrm{i}$

• - 1

• 1

• $- \mathrm{i}$

$\frac{\sin \left(5\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $16co{s}^4\left(\theta \right) - 12co{s}^2\left(\theta \right)+ 1$

• $16{\cos }^4\left(\mathrm{\theta }\right)+ 12{\cos }^2\left(\mathrm{\theta }\right) + 1$

• $16{\cos }^4\left(\mathrm{\theta }\right) - 12{\cos }^2\left(\mathrm{\theta }\right) - 1$

• $16{\cos }^4\left(\mathrm{\theta }\right) + 12{\cos }^2\left(\mathrm{\theta }\right) - 1$

${\cos }^2\left(\frac{\mathrm{\pi }}{6} + \mathrm{\theta }\right) - {\sin }^2\left(\frac{\mathrm{\pi }}{6} + \mathrm{\theta }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}\cos \left(2\mathrm{\theta }\right)$

• 0

• $- \frac{1}{2}\cos \left(2\mathrm{\theta }\right)$

• $\frac{1}{2}$

$\mathrm{In} ∆\mathrm{ABC}, \frac{\cos \left(\mathrm{C}\right) + \cos \left(\mathrm{A}\right)}{\mathrm{c} + \mathrm{a}} + \frac{\cos \left(\mathrm{B}\right)}{\mathrm{b}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\mathrm{a}}$

• $\frac{1}{\mathrm{b}}$

• $\frac{\mathrm{c} + \mathrm{a}}{\mathrm{b}}$

• 1

$\mathrm{In} ∆\mathrm{ABC}, \frac{\mathrm{a}}{{\mathrm{b}}^2 - {\mathrm{c}}^2} + \frac{\mathrm{c}}{{\mathrm{b}}^2 - {\mathrm{a}}^2} = 0, \mathrm{then} \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3}$

$\mathrm{In} ∆\mathrm{ABC},{\mathrm{a}}^2\sin \left(2\mathrm{C}\right) + {\mathrm{c}}^2\sin \left(2\mathrm{A}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $∆$

• $2∆$

• $3∆$

• $4∆$

$$\begin{gathered} \mathrm{The} \mathrm{shadow} \mathrm{of} \mathrm{a} \mathrm{tower} \mathrm{standing} \mathrm{on} \mathrm{a} \mathrm{level} \mathrm{ground} \mathrm{is} \mathrm{found} \mathrm{to} \mathrm{be} 60\mathrm{m} \mathrm{longer} \mathrm{when} \mathrm{the} \mathrm{sun}\text{'}\mathrm{s} \\ \mathrm{altitude} \mathrm{is} 30° \mathrm{than} \mathrm{when} \mathrm{it} \mathrm{is} 45°.\mathrm{The} \mathrm{height} \mathrm{of} \mathrm{the} \mathrm{tower} \mathrm{is} \end{gathered}$$

• $30\mathrm{m}$

• $90\mathrm{m}$

• $60\sqrt{3}\mathrm{m}$

• $30\left(\sqrt{3} + 1\right)\mathrm{m}$

$\mathrm{If} \mathrm{cosec\theta } = \frac{\mathrm{p} + \mathrm{q}}{\mathrm{p} - \mathrm{q}},\mathrm{then} \cot \left(\frac{\mathrm{\pi }}{4} + \frac{\mathrm{\theta }}{4}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\sqrt{\mathrm{p}}}{\mathrm{q}}$

• $\sqrt{\frac{\mathrm{q}}{\mathrm{p}}}$

• $\sqrt{\mathrm{pq}}$

• $\mathrm{pq}$

From a point on the level ground, the angle of elevation of top of a pole is $30°$ on moving 20metres nearer,the angle of elevation is $45°$. Then,the height of the pole $\left(\mathrm{in} \mathrm{metres}\right)$, is

• $10\left(\sqrt{3} - 1\right)$

• $10\left(\sqrt{3} - 1\right)$

• 15

• 20