If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x + n = 0\), such that \(\alpha\beta = 2\), find the value of n.

Solve \(\log_{2}(12x - 10) = 1 + \log_{2}(4x + 3)\).

Find the coefficient of \(x^{3}\) in the binomial expansion of \((x - \frac{3}{x^{2}})^{9}\).

The general term of an infinite sequence 9, 4, -1, -6,... is \(u_{r} = ar + b\). Find the values of a and b.

If \(\begin{vmatrix}  k & k \\ 4 & k \end{vmatrix} + \begin{vmatrix}  2 & 3 \\ -1 & k \end{vmatrix} = 6\), find the value of the constant k, where k > 0.