If \(Px^{2} + (P+1)x + P = 0\) has equal roots, find the values of P.

Integrate \((x - \frac{1}{x})^{2}\) with respect to x.

Given that \(AB = \begin{pmatrix} 4 \\ 3 \end{pmatrix}\) and \(AC = \begin{pmatrix} 2 \\ -3 \end{pmatrix}\), find |BC|.

Find the angle between \((5i + 3j)\) and \((3i - 5j)\).

Find the coefficient of \(x^3\) in the binomial expansion of \((3x + 4)^4\) in ascending powers of x.