If  $\left( \begin{array}{cc}2 & 3\\ 5 & 7\end{array} \right) \left(\begin{array}{cc}1& - 3\\ - 2 & 4\end{array} \right) = \left( \begin{array}{cc}- 4& 6\\ - 9& \mathrm{x}\end{array} \right)$,  write the value of x.

Let * be a ‘binary’ operation on N given by a * b = LCM  ( a, b ) for all  a, b $\in$ N. Find 5 * 7.

Write the principal value of  ${\cos }^{-1} \left( \frac{1}{2} \right) - 2 {\sin }^{-1} \left( - \frac{1}{2} \right).$

Simplify: $\cos \mathrm{\theta } \left[ \begin{array}{cc}\cos \mathrm{\theta } & \sin \mathrm{\theta }\\ - \sin \mathrm{\theta } & \cos \mathrm{\theta }\end{array} \right] + \sin \mathrm{\theta } \left[ \begin{array}{cc}\sin \mathrm{\theta } & - \cos \mathrm{\theta }\\ \cos \mathrm{\theta } & \sin \mathrm{\theta }\end{array} \right]$

If $∆ =\left| \begin{array}{ccc}5 & 3& 8 \\ 2 & 0 & 1 \\ 1 & 2& 3 \end{array}\right|$,  white the cofactor of the element a₃₂.

Evaluate $\int ( 1 - \mathrm{x} ) \sqrt{ \mathrm{x}} \mathrm{dx}$

Evaluate: ${\int }_2^3 \frac{1}{\mathrm{x}} \mathrm{dx}$

Find ' $\mathrm{\lambda }$ ' when the projection of  $\overrightarrow{\mathrm{a}} = \mathrm{\lambda } \hat{\mathrm{i}} + \hat{\mathrm{j}} + 4 \hat{\mathrm{k}} \mathrm{on} \overrightarrow{\mathrm{b}} = 2 \hat{\mathrm{i}} + 6 \hat{\mathrm{j}} + 3 \hat{\mathrm{k}}$ is  4  units.

9.

If a line has direction ratios 2,-1,-2 then what are its direction cosines?

Find the sum of the following vectors: 

$\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - 2 \hat{\mathrm{j}} , \overrightarrow{\mathrm{b}} = 2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}} , \overrightarrow{\mathrm{c}} =2 \hat{\mathrm{i}} + 3 \hat{\mathrm{k}}.$