31.

Consider the set A of all matrices of order 3 x 3 with entries 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is - 1. Then which one of the following is correct ?

• C is empty

• B as as many elements as C

• B has thrice as many elements as C

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$, then what is A³ = ?

• $\left[\begin{array}{cc}\cos \left(3\mathrm{\theta }\right)& \sin \left(3\mathrm{\theta }\right)\\ - \sin \left(3\mathrm{\theta }\right)& \cos \left(3\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}{\cos }^3\left(\mathrm{\theta }\right)& {\sin }^3\left(\mathrm{\theta }\right)\\ - {\sin }^3\left(\mathrm{\theta }\right)& {\cos }^3\left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(3\mathrm{\theta }\right)& - \sin \left(3\mathrm{\theta }\right)\\ \sin \left(3\mathrm{\theta }\right)& \cos \left(3\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}{\cos }^3\left(\mathrm{\theta }\right)& - {\sin }^3\left(\mathrm{\theta }\right)\\ {\sin }^3\left(\mathrm{\theta }\right)& {\cos }^3\left(\mathrm{\theta }\right)\end{array}\right]$

$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{order} \mathrm{of} \left[\begin{array}{ccc}\mathrm{x}& \mathrm{y}& \mathrm{z}\end{array}\right]\left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] = ?$

• $\mathrm{3 \times 1}$

• $1 \times 1$

• $1 \times 3$

• $3 \times 3$

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right], \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {\mathrm{A}}^4 \mathrm{is}$

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

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

• $\frac{1}{\sqrt{10}}$

• $- \sqrt{\frac{3}{10}}$

• $\frac{\sqrt{3}}{\sqrt{10}}$

• None of the above

$\mathrm{What} \mathrm{is} \frac{1}{\sin \left(10°\right)} - \frac{\sqrt{3}}{\cos \left(10°\right)} = ?$

• 0

• 1

• 2

• 4

From the top of a lighthouse, 100 m high, the angle of depression of a boat is ${\tan }^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse ?

• 120m

• 180m

• 240m

• 360m

The maximum value of $\sin \left(\mathrm{x} + \frac{\mathrm{\pi }}{6}\right) + \cos \left(\mathrm{x} + \frac{\mathrm{\pi }}{6}\right)$$\mathrm{in} \mathrm{the} \mathrm{interval} \left(0, \frac{\mathrm{\pi }}{2}\right) \mathrm{is} \mathrm{attained} \mathrm{at}$

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

$\mathrm{If} \mathrm{k} = \sin \left(\frac{\mathrm{\pi }}{18}\right)\sin \left(\frac{5\mathrm{\pi }}{18}\right)\sin \left(\frac{7\mathrm{\pi }}{18}\right), \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{k} ?$

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{1}{16}$

$\mathrm{The} \mathrm{expression} \frac{\sin \left(\mathrm{\alpha }\right) + \sin \left(\mathrm{\beta }\right)}{\cos \left(\mathrm{\alpha }\right) + \cos \left(\mathrm{\beta }\right)} = ?$

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

• $\cot \left(\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}\right)$

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

• $\cot \left(\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}\right)$