(a) If \(^{k}P_{2} = 72\), find the value of k.

(b) Solve the equation : \(2\cos^{2} \theta - 5\cos \theta = 3; 0° \leq \theta \leq 360°\)

The table shows the distribution of marks scored by some students in a test.

(a)(i) Construct a cumulative frequency table for the distribution ; (ii) Draw a cumulative frequency curve for the distribution.

(b) Use the curve to estimate the :

(i) number of students who scored marks between 32 and 74 ; (ii) pass mark, if 18% of the students failed ; (iii) lowest mark for distinction, if 8% of the students passed with distinction.

(a) Two Mathematics books, 5 different Physics books and 3 different Chemistry are to be arranged on a shelf. How many arrangements are possible if ;

(i) books on the same subject must stand together?  (ii) only the Physics books must stand together?

(b) In a certain community, 13 out of every 20 persons speak English. If 8 persons are selected at random from the community, find, correct to three significant figures, the probability that at least 3 of them speak English.

Four vectors \(r = \alpha i + \beta j\), where \(\alpha \text{ and } \beta\) are constants, \(s = 2i -j, m = 3i + 2j\) and \(n = i + j\) are such that the magnitude of r is three times as s and is parallel to the vactor (m - n).

(a) Find the values of \(\alpha\) and \(\beta\).

(b) Calculate the magnitude and direction of (r - s).

A particle is projected vertically upwards from the ground with speed \(30ms^{-1}\). Calculate the :

(a) maximum height reached by the particle;

(b) time taken by the particle to return to the ground;

(c) time(s) taken for the particle to attain a height of 40m above the ground. [Take \(g = 10ms^{-2}\)]