A binary operation ,*, is defined on the set R, of real numbers by \(a * b = a^{2} + b + ab\). Find the value of x for which \(5 * x = 37\).

The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Determine the general term of the sequence.

If \(2\log_{4} 2 = x + 1\), find the value of x.

Find the range of values of x for which \(2x^{2} + 7x - 15 > 0\).

If \(f(x) = 2x^{2} - 3x - 1\), find the value of x for which f(x) is minimum.

The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Find the common difference of the sequence.

Simplify \(\frac{\sqrt{3} + \sqrt{48}}{\sqrt{6}}\)

The polynomial \(2x^{3} + x^{2} - 3x + p\) has a remainder of 20 when divided by (x - 2). Find the value of constant p.

A function f is defined on R, the set of real numbers, by: \(f : x \to \frac{x + 3}{x - 2}, x \neq 2\), find \(f^{-1}\).