1.

Find the value of k if M = $\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$ and M² - kM - I₂ = 0

Find the equation of an ellipse whose latus rectum is 8 and eccentricity is $\frac{1}{3}$

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

Solve: ${\cos }^{-1}\left(\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right)\right) = \frac{\mathrm{\pi }}{6}$

evaluate: $\int \frac{2{\mathrm{y}}^2}{{\mathrm{y}}^2 + 4}\mathrm{dy}$

Evaluate: $$\begin{gathered} {\int }_0^3\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}, \mathrm{where} \mathrm{f}\left(\mathrm{x}\right) =\left\{ \cos \left(2\mathrm{x}\right), 0 \le \mathrm{x} \le \frac{\mathrm{\pi }}{2} \\ 3, \frac{\mathrm{\pi }}{2} \le \mathrm{x} \le 3 \right\} \end{gathered}$$

The two lines of regression are 4x + 2y - 3 = 0 and 3x + 6y + 5 = 0. Find the correlation co-efficient between x and y.

A card is drawn from a well shuffled pack of playing cards. What is the probability that it is either a space or an ace or both ?

If 1, w and w² are the cube roots of unity, prove that $\frac{\mathrm{a} + \mathrm{bw} + {\mathrm{cw}}^2}{\mathrm{c} + \mathrm{aw} + {\mathrm{bw}}^2}$ = w²

Solve the differential equation: ${\sin }^{-1}\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right) = \mathrm{x} +\hspace{0.17em}\mathrm{y}$