$\mathrm{What} \mathrm{is} \frac{\sin \left(5\mathrm{x}\right) - \sin \left(3\mathrm{x}\right)}{\cos \left(5\mathrm{x}\right) + \cos \left(3\mathrm{x}\right)} = ?$

• $\sin \left(\mathrm{x}\right)$

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\cot \left(\mathrm{x}\right)$

$\mathrm{What} \mathrm{is} \sin \left(105°\right) + \cos \left(105°\right) = ?$

• $\sin \left(50°\right)$

• $\cos \left(50°\right)$

• $\frac{1}{\sqrt{2}}$

• 0

In a triangle ABC if a = 2, b = 3 and sin(A) = z, then what is angle B = ?

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

ABC is a triangle inscribed in a circle with centre O. Let a = $\angle$BAC, where 45° < a < 90°. Let p = $\angle$BOC. Which one of the following is correct ?

• $\cos \left(\mathrm{\beta }\right) = \frac{1 - {\tan }^2\left(\mathrm{\alpha }\right)}{1 + {\tan }^2\left(\mathrm{\alpha }\right)}$

• $\cos \left(\mathrm{\beta }\right) = \frac{1 + {\tan }^2\left(\mathrm{\alpha }\right)}{1 - {\tan }^2\left(\mathrm{\alpha }\right)}$

• $\cos \left(\mathrm{\beta }\right) = \frac{2\tan \left(\mathrm{\alpha }\right)}{1 + {\tan }^2\left(\mathrm{\alpha }\right)}$

• $\sin \left(\mathrm{\beta }\right) = 2{\sin }^2\left(\mathrm{\alpha }\right)$

If a flag-staff of 6m height placed on the top of a tower throws a shadow of 2$\sqrt{3}$ m along the ground,then what is the angle that the sun makes with the ground?

• 60⁰

• 45⁰

• $30°$

• 15

A spherical balloon of radius r subtends an angle a at the eye of an observer, while the angle of elevation of its centre is $\mathrm{\beta }$. What is the height of the centre of the balloon (neglecting the height of the observer) ?

• $\frac{\mathrm{rsin}\left(\mathrm{\beta }\right)}{\sin \left({\displaystyle \frac{\mathrm{\alpha }}{2}}\right)}$

• $\frac{\mathrm{rsin}\left(\mathrm{\beta }\right)}{\sin \left({\displaystyle \frac{\mathrm{\alpha }}{4}}\right)}$

• $\frac{\mathrm{rsin}\left({\displaystyle \frac{\mathrm{\beta }}{2}}\right)}{\sin \left({\displaystyle \mathrm{\alpha }}\right)}$

• $\frac{\mathrm{rsin}\left(\mathrm{\alpha }\right)}{\sin \left({\displaystyle \frac{\mathrm{\beta }}{2}}\right)}$

27.

$\mathrm{If} \frac{\sin \left(\mathrm{x} + \mathrm{y}\right)}{\sin \left(\mathrm{x} - \mathrm{y}\right)} = \frac{\mathrm{a} + \mathrm{b}}{\mathrm{a} - \mathrm{b}}, \mathrm{then} \mathrm{what} \mathrm{is} \frac{\tan \left(\mathrm{x}\right)}{\tan \left(\mathrm{y}\right)} = ?$

• $\frac{\mathrm{a}}{\mathrm{b}}$

• $\frac{\mathrm{b}}{\mathrm{a}}$

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

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

$\mathrm{If} \sin \left(\mathrm{\alpha }\right) + \sin \left(\mathrm{\beta }\right) = 0 = \cos \left(\mathrm{\alpha }\right) + \cos \left(\mathrm{\beta }\right), \mathrm{where} 0 < \mathrm{\beta } < \mathrm{\alpha } < 2\mathrm{\pi }, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{correct} ?$

• $\mathrm{\alpha } = \mathrm{\pi } - \mathrm{\beta }$

• $\mathrm{\alpha } = \mathrm{\pi } + \mathrm{\beta }$

• $\mathrm{\alpha } = 2\mathrm{\pi } - \mathrm{\beta }$

• $2\mathrm{\alpha } = \mathrm{\pi } + 2\mathrm{\beta }$

Suppose cos(A) is given. If only one value of $\cos \left(\frac{\mathrm{A}}{2}\right)$ is possible, then A must be :

• An odd multiple of 90⁰

• A multiple of 90°

• An odd multiple of 180⁰

• A multiple of 180°

$$\begin{gathered} \mathrm{If} \cos \left(\mathrm{\alpha }\right) + \cos \left(\mathrm{\beta }\right) + \cos \left(\mathrm{\gamma }\right) = 0,\mathrm{where} 0 < \mathrm{\alpha } \le \frac{\mathrm{\pi }}{2}, 0 < \mathrm{\beta } \le \frac{\mathrm{\pi }}{2}, 0 < \mathrm{\gamma } \le \frac{\mathrm{\pi }}{2}, \\ \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \sin \left(\mathrm{\alpha }\right) + \sin \left(\mathrm{\beta }\right) + \sin \left(\mathrm{\gamma }\right) = ? \end{gathered}$$

• 0

• 3

• $\frac{5\sqrt{2}}{2}$

• $\frac{3\sqrt{2}}{2}$