31.

The maximum value of $\sin \left(\mathrm{x} + \frac{\mathrm{\pi }}{5}\right) + \cos \left(\mathrm{x} + \frac{\mathrm{\pi }}{5}\right), \mathrm{where} \mathrm{x} \in \left(0, \frac{\mathrm{\pi }}{2}\right), \mathrm{is} \mathrm{attained} \mathrm{at} :$

• $\frac{\mathrm{\pi }}{20}$

• $\frac{\mathrm{\pi }}{15}$

• $\frac{\mathrm{\pi }}{10}$

• $\frac{\mathrm{\pi }}{2}$

What is the period of the function f(x) = sin(x) ?

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• 2$\mathrm{\pi }$

What is $\frac{2\tan \left(\mathrm{\theta }\right)}{1 + {\tan }^2\left(\mathrm{\theta }\right)} = ?$

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

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

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

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

If A + B + C = 180°, then what is sin(2A) - sin(2B) - sin(2C) equal to ?

• $- 4\sin \left(\mathrm{A}\right)\sin \left(\mathrm{B}\right)\sin \left(\mathrm{C}\right)$

• $- 4\cos \left(\mathrm{A}\right)\sin \left(\mathrm{B}\right)\cos \left(\mathrm{C}\right)$

• $- \cos \left(\mathrm{A}\right)\cos \left(\mathrm{B}\right)\sin \left(\mathrm{C}\right)$

• $- \sin \left(\mathrm{A}\right)\cos \left(\mathrm{B}\right)\cos \left(\mathrm{C}\right)$

A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is 48°. If the height of the balloon above the bridge is 122 m, then what is the length of the bridge ?

• $122\sin \left(48°\right) \mathrm{m}$

• $122\tan \left(42°\right) \mathrm{m}$

• $122\cos \left(48°\right) \mathrm{m}$

• $122\tan \left(48°\right) \mathrm{m}$

A is an angle in the fourth quadrant. If satisfies the trigonometric equation 3(3 - tan²(A) - cot(A))² = 1.

Which one of the following is a value of A?

• $300°$

• $315°$

• $330°$

• $345°$

$$\begin{gathered} \mathrm{What} \mathrm{is}/\mathrm{are} \mathrm{the} \mathrm{solution}\left(\mathrm{s}\right) \mathrm{of} \mathrm{the} \mathrm{trigonometric} \mathrm{equation} \\ \mathrm{cosec}\left(\mathrm{x}\right) + \cot \left(\mathrm{x}\right) = \sqrt{3}, \mathrm{where} 0 < \mathrm{x} < 2\mathrm{\pi } ? \end{gathered}$$

• $\frac{5\mathrm{\pi }}{3} \mathrm{only}$

• $\frac{\mathrm{\pi }}{3} \mathrm{only}$

• $\mathrm{\pi } \mathrm{only}$

• $\mathrm{\pi }, \frac{\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3}$

$$\begin{gathered} \mathrm{If} \mathrm{\theta } = \frac{\mathrm{\pi }}{8}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} {\left(2\cos \left(\mathrm{\theta }\right) + 1\right)}^{10} \\ {\left(2\cos \left(2\mathrm{\theta }\right) - 1\right)}^{10}{\left(2\cos \left(\mathrm{\theta }\right) - 1\right)}^{10}{\left(2\cos \left(4\mathrm{\theta }\right) - 1\right)}^{10} ? \end{gathered}$$

• 0

• 1

• 2

• 4

$\mathrm{If} \mathrm{A} = {\sin }^2\left(\mathrm{\theta }\right) + {\cos }^4\left(\mathrm{\theta }\right), \mathrm{then} \mathrm{for} \mathrm{all} \mathrm{real} \mathrm{\theta }, \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $1 \le \mathrm{A} \le 2$

• $\frac{3}{4} \le \mathrm{A} \le 1$

• $\frac{13}{16} \le \mathrm{A} \le 1$

• $\frac{3}{4} \le \mathrm{A} \le \frac{13}{16}$

$\mathrm{If} \sin \left(\mathrm{A}\right) = \frac{3}{5}, \mathrm{where} 450° < \mathrm{A} < 540°, \mathrm{then} \cos \left(\frac{\mathrm{A}}{2}\right) = ?$

• $\frac{1}{\sqrt{10}}$

• $- \sqrt{\frac{3}{10}}$

• $\frac{\sqrt{3}}{\sqrt{10}}$

• None of the above