21.

Using elementary transformations, find the inverse of the matrx

$\left( \begin{array}{ccc}1 & 3& - 2\\ - 3 & 0 & - 1\\ 2& 1& 0\end{array} \right)$

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Evaluate: $\int \frac{5 \mathrm{x} + 3}{\sqrt{ {\mathrm{x}}^2 + 4 \mathrm{x} + 10}} \mathrm{dx}$

Evaluate: $\int \frac{2\mathrm{x}}{\sqrt{ \left( {\mathrm{x}}^2 + 1 \right) \left( {\mathrm{x}}^2 + 3 \right)}} \mathrm{dx}$

Solve the following differential equation:

e^x tan y dx + ( 1 - e^x ) sec² y dy  = 0

Solve the following differential equation:

${\cos }^2 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = \tan \mathrm{x}$

Find a unit vector perpendicular to each of the vector  $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}$, where

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

Find the angle between the following pair of lines:  

$\frac{- \mathrm{x} + 2}{- 2} = \frac{\mathrm{y} - 1}{7} = \frac{\mathrm{z} + 3}{- 3} \mathrm{and} \frac{\mathrm{x} + 2}{- 1} = \frac{2 \mathrm{y} - 8}{4} = \frac{\mathrm{z} - 5}{4}$

And check whether the lines are parallel or perpendicular.

Probabilities of solving problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$respectively. If both try to solve the problem independently, find the probability that

(i) the problem is solved

(ii) exactly one of them solves the problem.

Using integration find the area of the triangular region whose sides have equations  y=2x+1,  y=3x+1  and  x=4.