A body of mass 5 kg resting on a smooth horizontal plane, is acted upon by force 6i + 2j, 5i + 4j and 4i - j. Calculate the:

(a) velocity of the body

(b) Magnitude of its velocity after 4s.

An object is projected vertically upwards with a velocity of 80 m/s. Find the :

(a) Maximum height reached

(b) Time taken to return to the point of projection. [Take g = \(10 ms^{-2}\)].

(a) Simplify \(\frac{\sqrt{75} - 3}{\sqrt{3} + 1}\), leaving your answer in the form \(a + b\sqrt{c}\); where a, b and c are rational numbers.

(b) The points (7, 3), (2, 8) and (-3, 3) lie on a circle. Find the (i) equation and (ii) radius of the circle.

(a) The gradient of the tangent to the curve \(y = 4x^{3}\) at points P and Q is 108. Find the coordinates of P and Q.

(b) Given that \(A = 45°, B = 30°, \sin (A + B) = \sin A \cos B + \sin B \cos A\) and \(\cos (A + B)  = \cos A \cos B - \sin A \sin B\)

(i) Show that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\) and \(\cos 15° = \frac{\sqrt{6} + \sqrt{2}}{4}\)

(ii) hence find \(\tan 15°\).

(a) Using the substitution \(u = 5 - x^{2}\), evaluate \(\int_{1}^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\).

(b) If \(y = px^{2} + qx; \frac{\mathrm d y}{\mathrm d x} = 6x + 7\) and \(\frac{\mathrm d^{2} y}{\mathrm d x^{2}} = 6\), find the values of p and q.