$$\begin{gathered} \mathrm{If} \mathrm{A} = \left(\cos \left(12°\right) - \cos \left(36°\right)\right)\left(\sin \left(96° + \sin \left(24°\right)\right)\right) \mathrm{and} \\ \mathrm{B} = \left(\sin \left(60° - \sin \left(12°\right)\right)\right)\left(\cos \left(48°\right) - \cos \left(72°\right)\right), \mathrm{then} \mathrm{what} \mathrm{is} \frac{\mathrm{A}}{\mathrm{B}} = ? \end{gathered}$$

•  - 1

• 0

• 1

• 2

$\sin \left(\mathrm{A}\right) + \sin \left(2\mathrm{A}\right) + \sin \left(3\mathrm{A}\right)$ is equal to which of the following ?

$1. 4\sin \left(2\mathrm{A}\right){\cos }^2\left(\frac{\mathrm{A}}{2}\right)$

$2. 2\sin \left(2\mathrm{A}\right){\left(\sin \left(\frac{\mathrm{A}}{2}\right) + \cos \left(\frac{\mathrm{A}}{2}\right)\right)}^2$

3. $8\sin \left(\mathrm{A}\right)\cos \left(\mathrm{A}\right){\cos }^2\left(\frac{\mathrm{A}}{2}\right)$

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

If $\mathrm{x} = \sin \left(70°\right) . \sin \left(50°\right) \mathrm{and} \mathrm{y} = \cos \left(60°\right) . \cos \left(80°\right),$ then what is xy = ?

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$16$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

$\mathrm{If} \sin \left({\mathrm{\theta }}_1\right) + \sin \left({\mathrm{\theta }}_2\right) + \sin \left({\mathrm{\theta }}_3\right) + \sin \left({\mathrm{\theta }}_4\right) = 4, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} :$

$\cos \left({\mathrm{\theta }}_1\right) + \cos \left({\mathrm{\theta }}_2\right) + \cos \left({\mathrm{\theta }}_3\right) + \cos \left({\mathrm{\theta }}_4\right) ?$

• 0

• 1

• 2

• 4

What is the value of :

$\left(1 + \cos \left(\frac{\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{3\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{5\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{7\mathrm{\pi }}{8}\right)\right) ?$

• $\frac{1}{2}$

• $\frac{1}{2} + \frac{1}{2\sqrt{2}}$

• $\frac{1}{2} - \frac{1}{2\sqrt{2}}$

• $\frac{1}{8}$

If $\mathrm{xcos}\left(\mathrm{\theta }\right) + \mathrm{ysin}\left(\mathrm{\theta }\right) = \mathrm{z},$ then what is the value of

${\left(\mathrm{xsin}\left(\mathrm{\theta }\right) - \mathrm{ycos}\left(\mathrm{\theta }\right)\right)}^2 ?$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - {\mathrm{z}}^2$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 - {\mathrm{z}}^2$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 + {\mathrm{z}}^2$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2$

67.

$\mathrm{If} \mathrm{A} = \left\{\mathrm{x} \in \mathrm{ℝ} : {\mathrm{x}}^2 + 6\mathrm{x} - 7 < 0\right\} \mathrm{and} \mathrm{B} = \left\{\mathrm{x} \in \mathrm{ℝ} : {\mathrm{x}}^2 + 9\mathrm{x} + 14 > 0\right\},$

then which of the following is/are correct ?

$$\begin{gathered} 1. \mathrm{A} \cap \mathrm{B} = \left\{\mathrm{x} \in \mathrm{R} : - 2 < \mathrm{x} < 1\right\} \\ 2. \mathrm{A}\overline{)\mathrm{B}} = \left\{\mathrm{x} \in \mathrm{R} : - 7 < \mathrm{x} < - 2\right\} \end{gathered}$$

Select the correct answer using the code given below :

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

If $\sin \left(18°\right) = \frac{\sqrt{5} - 1}{4},$ then what is the value of sin 81° ?

• $\frac{\sqrt{3 + \sqrt{5}} + \sqrt{5 - \sqrt{5}}}{4}$

• $\frac{\sqrt{3 +\hspace{0.17em}\sqrt{5}} + \sqrt{5 + \sqrt{5}}}{4}$

• $\frac{\sqrt{3 -\hspace{0.17em}\sqrt{5}} + \sqrt{5 - \sqrt{5}}}{4}$

• $\frac{\sqrt{3 +\hspace{0.17em}\sqrt{5}} - \sqrt{5 - \sqrt{5}}}{4}$

What is $\frac{1 - \tan \left(2°\right) . \cot \left(62°\right)}{\tan \left(152°\right) - \cot \left(88°\right)} = ?$

• $\sqrt{3}$

• $- \sqrt{3}$

• $\sqrt{2} - 1$

• $1 - \sqrt{2}$

What is $\tan \left(25°\right)\tan \left(15°\right) + \tan \left(15°\right)\tan \left(50°\right)$ $+ \tan \left(25°\right)\tan \left(50°\right) = ?$

• 0

• 1

• 2

• 4