Further Mathematics Past Questions and Answers

(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.

candidates	A	B	C	D	E	F	G	H	I	J
Examiner I	1st	3rd	6th	2nd	10th	9th	7th	4th	8th	5th
Examiner II	2nd	1st	3rd	9th	7th	4th	8th	10th	5th	6th
Examiner III	3rd	2nd	1st	6th	9th	8th	7th	5th	4th	10th

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.