1.

If A = $\left[\begin{array}{cc}3& 1\\ 7& 5\end{array}\right]$, find the values of x and y such that A² + xI₂ = yA

Find the eccentricity and the coordinates of foci of the hyperbola 25x² - 9y² = 225

Evaluate: $\tan \left(2{\tan }^{-1}\left(\frac{1}{2}\right) - {\cot }^{-1}\left(3\right)\right)$

Using L'Hospital's Rule, evaluate: $\underset{\mathrm{x}\to 0}{\lim } {\left(1 +\hspace{0.17em}\mathrm{x}\right)}^{\cot \left(\mathrm{x}\right)}$

Evaluate: $\int {\mathrm{e}}^{\mathrm{x}}\frac{(2 +\hspace{0.17em}\sin \left(2\mathrm{x}\right)}{{\cos }^2\left(\mathrm{x}\right)}d\mathrm{x}$

Using properties of definite integrals, evaluate: ${\int }_0^{\mathrm{\pi }/2}\frac{\sqrt{\sin \left(\mathrm{x}\right)}}{\sqrt{\sin \left(\mathrm{x}\right)} + \sqrt{\cos \left(\mathrm{x}\right)}}d\mathrm{x}$

For the given lines of regression, 3x - 2y = 5, and x - 4y = 7, Find:

(i) regression coefficient b_yx and b_xy

(ii) coefficient of correlation r(x, y)

Express the complex number $\frac{{\left(1 + \sqrt{3}\mathrm{i}\right)}^2}{\sqrt{3} - \mathrm{i}}$ in the form of a + ib. Hence, find the modulus and argument of the complex number.

A bag contains 20 balls numbered from 1 to 20. One ball is drawn at random from the bag. What is the probability that the ball drawn is marked with a number which is multiple of 3 or 4?

Solve the differential equation:

$\left(\mathrm{x} +\hspace{0.17em}1\right)d\mathrm{y} - 2\mathrm{xy}d\mathrm{x} = 0$