(a) Simplify; \(\frac{log_2 ^8 + log_2 ^{16} - 4 log_2 ^2}{log_4^{16}}\)

(b) The first, third, and seventh terms of an Arithmetic Progression (A.P) from three consecutive terms of a Geometric Progression (G.P). If the sum of the first two terms of the A.P is 6, find its:

(I) first term; (ii) common difference.

(a) An association is made up of 6 farmers and 8 traders. If an executive body of 4 members is to be formed, find the probability that it will consist of at least two farmers. (b) The probability of an accident occurring in a given month in factories X, Y, and Z are \(\frac{1}{5}, \frac{1}{12} \) and \(\frac{1}{6}\) respectively.

Find the probability that the accident will occur in:

i) none of the factories;

(ii) all the factories;

(iii) at least one factory.

The essays of 10 candidates were ranked by three examiners as shown in the table.

a) Calculate the Spearman's rank correlation coefficient of the ranks assigned by:

(i) Examiners I and lI;

(ii) Examiners I and III

(iii) Examiners II and II.

(b) Using the results in (a), state which two examiners agree most.

(a) A car is moving with a velocity of 10ms\(^{-1}\) It then accelerates at 0.2ms\(^{-2}\) for 100m. Find, correct to two decimal places the time taken by the car to cover the distance.

(b) A particle moves along a straight line such that its distance S metres from a fixed point O is given by S = t\(^2\) - 5t + 6, where t is the time in seconds. Find its:

(i) initial velocity;

(ii) distance when it is momentarily at rest

The diagram is that of a light inextensible string of length 4.2m, whose ends are attached to two fixed points X and Y, 3m apart, and on the same horizontal level. A body of mass 800g is hung on the string at a point O, 2.4m from Y. If the system is kept in equilibrium by a horizontal force P acting on the body and the tensions are equal, calculate:

(a) < XOY;

(b) the magnitude of the force P;

(c) the tension T in the string.