If \(\alpha\) and \(\beta\) are the roots of \(3x^{2} + 5x + 1 = 0\), evaluate \(27(\alpha^{3} + \beta^{3})\).

The table shows the distribution of the lengths of 20 iron rods measured in metres : Length (m) 1.0 - 1.1 1.2 - 1.3 1.4 - 1.5 1.6 - 1.7 1.8 - 1.9 Frequency 2...

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) Fin...

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\...

(a) Differentiate \((x - 3)(x^{2} + 5)\) with respect to x. (b) If \((x + 1)^{2}\) is a factor of \(f(x) = x^{3} + ax^{2} + bx + 3\), where a and b are const...

Solve \((\log_{2} m)^{2} - \log_{2} m^{3} = 10\).

Find the direction of the resultant of the forces in the diagram.

A committee of 3 is formed from a panel of 5 men and 3 women. Find the : (a) number of ways of forming the committee ; (b) probability that at least one woma...