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

• 4

• 3

• 2

• 1

$\mathrm{In} ∆\mathrm{ABC}, \cos \left(\frac{\mathrm{B} + 2\mathrm{C} + 3\mathrm{A}}{2}\right) + \cos \left(\frac{\mathrm{A} - \mathrm{B}}{2}\right) \mathrm{is}$

• - 1

• 0

• 1

• 2

$\mathrm{The} \mathrm{value} \mathrm{of} \mathrm{series} \cos \left(12°\right) + \cos \left(84°\right) + \cos \left(132°\right) + \cos \left(156°\right) \mathrm{is}$

• $\frac{1}{2}$

• $\frac{1}{4}$

• $- \frac{1}{4}$

• $- \frac{1}{2}$

$\mathrm{For} \mathrm{x} \in \mathrm{IR}, 3\cos \left(4\mathrm{x} - 5\right) + 4 \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

• [1, 7]

• [4, 7]

• [0, 7]

• [2, 7]

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

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

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

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

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

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

• $\frac{240}{\sqrt{3}}$

• $200\sqrt{3}$

• $240\sqrt{3}$

• $\frac{120}{\sqrt{3}}$

830.

If the distance between the points (acos$\mathrm{\theta }$, asin$\mathrm{\theta }$) and  (acos$\mathrm{\varphi }$, asin$\mathrm{\varphi }$) is 2a, then $\mathrm{\theta }$ is equal to

• $2\mathrm{n\pi } \pm \mathrm{\pi } + \mathrm{\varphi }, \mathrm{n} \in \mathrm{Z}$

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

• $n\mathrm{\pi } - \mathrm{\varphi }, \mathrm{n} \in \mathrm{Z}$

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