(a) A bucket full of water with a mass of 8kg is pulled out of a well with a light inextensible rope. Find its acceleration when tha tension in the rope is ...

If \(\log_{3} x = \log_{9} 3\), find the value of x.

Given that \(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\), find P and Q.

Solve: \(4(2^{x^2}) = 8^{x}\)

Given that \(f : x \to x^{2}\) and \(g : x \to x + 3\), where \(x \in R\), find \(f o g(2)\).

Find the 3rd term of \((\frac{x}{2} - 1)^{8}\) in descending order of x.

Given that \(P = \begin{pmatrix} -2 & 1 \\ 3 & 4 \end{pmatrix}\) and \(Q = \begin{pmatrix} 5 & -3 \\ 2 & -1 \end{pmatrix}\), find PQ - QP.

Simplify \(\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}}\).

Solve: \(2\cos x - 1 = 0\).