If (A - 2I)(A - 3I) = 0, where A = $\left(\begin{array}{cc}4& 2\\ - 1& \mathrm{x}\end{array}\right) \mathrm{and} \mathrm{I} = \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, find the value of x.

Find the values of k so that the line 2x + y + k = 0 may touch the hyperbola 3x² - y² = 3

Prove that: ${\tan }^{-1}\left(\frac{1}{4}\right) + \hspace{0.17em}{\tan }^{-1}\left(\frac{2}{9}\right) = \frac{1}{2}{\sin }^{-1}\left(\frac{4}{5}\right)$

Using L'Hospital's Rule, evaluate:

$\underset{\mathrm{x}\to 0}{\lim }\left(\frac{{\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{- \mathrm{x}} - 2\mathrm{x}}{\mathrm{x} - \sin \left(\mathrm{x}\right)}\right)$

Evaluate: $\int \frac{1}{\mathrm{x} + \sqrt{\mathrm{x}}}d\mathrm{x}$

Evaluate: ${\int }_0^1\log \left(\frac{1}{\mathrm{x}} - 1\right)d\mathrm{x}$

Two regression lines are represented by 4x + 10y = 9 and 6x + 3y = 4. Find the line of regression of y on x.

If 1, w, and w² are the cube roots of unity, evaluate (1 - w⁴ + w⁸)(1 - w⁸ + w¹⁶)

Solve the differential equation:

$\log \left(\frac{d\mathrm{y}}{d\mathrm{x}}\right) = 2\mathrm{x} - 3\mathrm{y}$

If two balls are drawn from a bag containing three red balls and four blue balls, find the probability that:

(a) They are of the same colour

(b) They are of different colours