(a) A manufacturer produces light bulbs which are tested in the following way. A batch is accepted in either of the following cases:

(i) a first sample of 5 bulbs contains no faulty bulbs ; (ii) a first sample of 5 bulbs contains at least one faulty bulb but a second sample of size 5 has no faulty bulb. If 10% of the bulbs are faulty, what is the probability that the batch is accepted?

(b) A bag contains 15 identical marbles of which 3 are black, Keshi picks a marble at random from the bag and replaces it. If this is repeated 10 times; what is the probability that he :

(i) did not pick a black ball? (ii) picked a black ball at most three times?

(a) Two ships M and N, moving with constant velocities, have position vectors (3i + 7j) and (4i + 5j) respectively. If the velocities of M and N are (5i + 6j) and (2i + 3j) and the distance covered by the ships after t seconds are in metres, find (i) MN ; (ii) |MN|, when t = 3 seconds.

(b) A particle is acted upon by forces \(F_{1} = 5i + pj ; F_{2} = qi + j ; F_{3} = -2pi + 3j\) and \(F_{4} = -4i + qj\), where p and q are constants. If the particle remains in equilibrium under the action of these forces, find the values of p and q.

A particle moves from point O along a straight line such that its acceleration at any time, t seconds is \(a = (4 - 2t) ms^{-2}\). At t = 0, its distance from O is 18 metres while its velocity is \(5 ms^{-1}\).

(a) At what time will the velocity be greatest?

(b) Calculate the : (i) time ; (ii) distance of the particle from O when the particle is momentarily at rest.

(a) The position vectors of the points X and Y are \(x = (-2i + 5j)\) and \(y = (i - 7j)\) respectively. Find :

(i) (3x + 2y) ; (ii) \(|(y - 2x)|\) ; (iii) the angle between x and y ; (iv) the unit vector in the direction of \((x + y)\).

(b) A bullet of mass 0.084kg is fired horizontally into a 20 kg block of wood at rest on a smooth floor. If they both move at a velocity of \(0.24 ms^{-1}\) after impact; Calculate, correct to two decimal places, the initial velocity of the bullet.

(a) Solve : \(2x^{2} + x - 6 < 0\)

(b) Express \(\frac{5 - 2\sqrt{10}}{3\sqrt{5} + \sqrt{2}}\) in the form \(m\sqrt{2} + n\sqrt{5}\) where m and n are rational numbers.