A binary operation \(*\) is defined on the set, R, of real numbers by \(m * n = m + n + 2\). Find the :

(a) identity element under the operation ;

(b) inverse of n under the operation .

Given that (5, 2), (-4, k) and (2, 1) lie on a straight line, find the value of k.

(a) If \(f(x + 2) = 6x^{2} + 5x - 8\), find \(f(5)\).

(b) Express \(\frac{7\sqrt{2} + 3\sqrt{3}}{4\sqrt{2} - 2\sqrt{3}}\) in the form \(p + q\sqrt{r}\), where p, q and r are rational numbers. 

When \(f(x) = 2x^{3} + mx^{2} + nx + 11\) is divided by \(x^{2} + 5x + 1\), the quotient is \(2x - 5\) and the remainder is \(30x + 16\). Find the values of m and n.

The probabilities that Ago, Sulley and Musa will gain admission to a certain university are \(\frac{4}{5}, \frac{3}{4}\) and \(\frac{2}{3}\) respectively. Find the probability that :

(a) none of them will gain admission ; 

(b) only Ago and Sulley will gain admission.