A binary operation \(\ast\) is defined on the set of rational numbers by \(m \ast n = \frac{m^{2} - n^{2}}{2mn}, m \neq 0 ; n \neq 0\).
(a) Find \(-3 \ast 2\).
(b) Show whether or not \(\ast\) is associative.
If \(\alpha\) and \(\beta\) are the roots of \(3x^{2} + 5x + 1 = 0\), evaluate \(27(\alpha^{3} + \beta^{3})\).
Find the gradient of \(xy^{2} + x^{2} y = 4xy\) at the point (1, 3).
The position vectors of P, Q and R are \(11i + j, 5i + \frac{13}{3}j\) and \(2i + 6j\) respectively.
(a) Show that P, Q and R lie on a straight line.
(b) Find the ratio of \(|\overrightarrow{PQ}| : |\overrightarrow{QR}|\)
