(a) Without using mathematical tables or calculator, evaluate \(\frac{\frac{3}{2}\log 27 - 3\log 5\sqrt{5}}{\log 0.6}\)

(b) Two linear transformations A and B in the \(O_{xy}\) plane, are defined by :

\(A : (x, y)   (x + 2y, -x + y)\)

\(B : (x, y)   (2x + 3y, x + 2y)\).

(i) Write down the matrices A and B; (ii) Find the image of the point P(-2, 2) under the linear transformation A followed by B.

(a)(i) Write down the expansion of \((1 + x)^{7}\) in ascending powers of x.

(ii) If the coefficients of the fifth, sixth and seventh terms in the expansion in (a)(i) above form a linear sequence(A.P), find the common difference of the A.P.

(b) Using the trapezium rule with ordinates at 1, 2, 3, 4 and 5, calculate, correct to two decimal places, 

\(\int_{1}^{5} \sqrt{(2x + 8x^{2})} \mathrm {d} x\).

(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.