11.

$\sin \left(\mathrm{A}\right) + \sin \left(2\mathrm{A}\right) + \sin \left(3\mathrm{A}\right)$ 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

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

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

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

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

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

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

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

What is ${\mathrm{\omega }}^{100} + {\mathrm{\omega }}^{200} + {\mathrm{\omega }}^{300} \mathrm{equal} \mathrm{to}, \mathrm{where} \mathrm{\omega } \mathrm{is} \mathrm{the} \mathrm{cube} \mathrm{root} \mathrm{of} \mathrm{unity} ?$

• 1

• 3$\mathrm{\omega }$

• $3{\mathrm{\omega }}^2$

• 0

Let S be a set of all distinct numbers of the form "q' where p. q $\in$ { 1, 2, 3, 4, 5, 6}. What is the cardinality of the set S ?

• 21

• 23

• 32

• 36

If c > 0 and 4a + c < 2b, then ax² - bx + c = 0 has a root in which one of the following intervals ?

• (0, 2)

• (2, 3)

• (3, 4)

• ( - 2, 0)

If A is a square matrix of order 3 and detA = 5, then what is det [(2A)^-1] equal to ?

• $\frac{1}{10}$

• $\frac{2}{5}$

• $\frac{8}{5}$

• $\frac{1}{40}$