$\mathrm{The} \mathrm{value} \mathrm{of} \cos \left(\frac{2\mathrm{\pi }}{15}\right)\cos \left(\frac{4\mathrm{\pi }}{15}\right)\cos \left(\frac{8\mathrm{\pi }}{15}\right)\cos \left(\frac{14\mathrm{\pi }}{15}\right) \mathrm{is} :$

• $\frac{1}{16}$

• $\frac{1}{8}$

• $\frac{1}{12}$

• $\frac{1}{4}$

If A + B + C = 270°, then
cos(2A) + cos(2B) + cos(2C) is equal to :

• 4sin(A)sin(B)sin(C)

• 4cos(A)cos(B)cos(C)

• 1 - 4sin(A)sin(B)sin(C)

• 1 - 4cos(A)cos(B)cos(C)

If ${\mathrm{x}}_{\mathrm{n}} = \cos \left(\frac{\mathrm{\pi }}{2^{\mathrm{n}}}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{2^{\mathrm{n}}}\right)$, then $\prod _{\mathrm{n} = 1}^{\infty }$xⁿ is equal to

• - 1

• 1

• $\frac{1}{\sqrt{2}}$

• - 3

If n $\in$ N and the period of $\frac{\cos \left(\mathrm{n\pi }\right)}{\sin \left({\displaystyle \frac{\mathrm{x}}{\mathrm{n}}}\right)}$ is 4$\mathrm{\pi }$, then n is equal to

• 4

• 3

• 2

• 1

$\mathrm{The} \mathrm{expression} \mathrm{for} \tan \left(9°\right) - \tan \left(27°\right) - \tan \left(63°\right) + \tan \left(81°\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• 4

• 3

• 2

• 1

If $\mathrm{x} = \log \left[\cot \left(\frac{\mathrm{\pi }}{4} + \mathrm{\theta }\right)\right]$, then the value of $\sin \left(\mathrm{hx}\right)$ is

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

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

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

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

187.

If in a $∆$ABC, r₃ = r₁ + r₂ + r, then $\angle \mathrm{A} + \angle \mathrm{B}$ is equal to

• 120°

• 100°

• 90°

• 80°

In a $∆\mathrm{ABC}$${\left(\mathrm{a} - \mathrm{b}\right)}^2{\cos }^2\left(\frac{\mathrm{C}}{2}\right) + {\left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\right)}^2{\sin }^2\left(\frac{\mathrm{C}}{2}\right)$ is equal to

• a²

• c²

• b²

• a² + b²

In a $∆\mathrm{ABC}$, the correct formulae among the following are

$$\begin{gathered} \mathrm{I}. \mathrm{r} = 4\mathrm{Rsin}\left(\frac{\mathrm{A}}{2}\right)\sin \left(\frac{\mathrm{B}}{2}\right)\sin \left(\frac{\mathrm{C}}{2}\right) \\ \mathrm{II}. {\mathrm{r}}_1 = \left(\mathrm{s} - \mathrm{a}\right)\tan \left(\frac{\mathrm{A}}{2}\right) \\ \mathrm{III}. {\mathrm{r}}_3 = \frac{∆}{\mathrm{s} - \mathrm{c}} \end{gathered}$$

• only I, II

• only II, III

• only I, III

• I, II, III

An aeroplane flying with uniform speed horizontally one km above the ground is observed at an elevation of 60°. After 10 s if the elevation is observed to be 30°, then the speed of the plane (in km/h) is

• $\frac{240}{\sqrt{3}}$

• $200\sqrt{3}$

• $240\sqrt{3}$

• $\frac{120}{\sqrt{3}}$