1.

Find the matrix X for which: $\left[\begin{array}{cc}5& 4\\ 1& 1\end{array}\right]\mathrm{X} = \left[\begin{array}{cc}1& - 2\\ 1& 3\end{array}\right]$

Solve for x, if:

$\tan \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \frac{2}{\sqrt{5}}$

Prove that the line 2x - 3y = 9 touches the conics y² = - 8x, also find point of contact.

Using L'Hospital Rule, evaluate: $\underset{\mathrm{x}\to 0}{\lim }\left(\frac{1}{{\mathrm{x}}^2} - \frac{\cot \left(\mathrm{x}\right)}{\mathrm{x}}\right)$

Evaluate: $\int {\tan }^3\left(\mathrm{x}\right)\mathrm{dx}$

Using properties of definite integrals, evaluate:

${\int }_0^{\mathrm{\pi }/2}\frac{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{1 +\hspace{0.17em}\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}d\mathrm{x}$

The two lines of regressions are x + 2y - 5 = 0 and 2x + 3y - 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.

Express $\frac{2 +\hspace{0.17em}\mathrm{i}}{(1 + \mathrm{i})(1 - 2\mathrm{i})}$ in the form of a + ib. Find its modulus and argument.

A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of 8 ?

Solve the Differential equation:

$\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 3{\mathrm{x}}^2 - 2$