If \(P = \begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix}\), find \((P^{2} + P)\).

Which of the following binary operations is not commutative?

Find the coefficient of \(x^{4}\) in the binomial expansion of \((2 + x)^{6}\).

Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.

Solve \(x^{2} - 2x - 8 > 0\).

The coordinates of the centre of a circle is (-2, 3). If its area is \(25\pi cm^{2}\), find its equation.

If (x + 3) is a factor of the polynomial \(x^{3} + 3x^{2} + nx - 12\), where n is a constant, find the value of n.

Given \(\sin \theta = \frac{\sqrt{3}}{2}, 0° \leq \theta \leq 90°\), find \(\tan 2\theta\) in surd form.

The line \(y = mx - 3\) is a tangent to the curve \(y = 1 - 3x + 2x^{3}\) at (1, 0). Find the value of the constant m.