(a) A body of mass 5 kg is placed on a smooth plane inclined at an angle 30° to the horizontal. Find the magnitude of the force: (i) acting parallel to the plane                               (ii) required to keep the body in equilibrium. [Take g = \(10 ms^{-2}\)].

(b) A uniform plank PQ of length 10m and mass m kg rests on two supports A and B, where |PA| = |BQ| = 1m. A load of mass 8 kg is placed on the plank at point C such that |AC| = 3.5 m. If the reaction at B is 100 N, calculate the (i) value of m  (ii) reaction at A.  [Take g = \(10 ms^{-2}\)].

The second term of a geometric progression is 3. If its sum to infinity is \(\frac{25}{2}\), find the value of its common ratio.

Solve \(x^{\frac{2}{3}} - 5x^{\frac{1}{3}} + 6 = 0\).

The normal to the curve \(y = 2x^{2} + x - 3\) at the point (2, 7) meets the x- axis at the point P. Find the coordinates of P.

(a) Write down the first four terms of the binomial expansion of \((2 - \frac{1}{2})^{5}\) in ascending powers of x.

(b) Use your expansion in (a) above to find, correct to two decimal places, the value of \((1.99)^{5}\).