Three linear transformations, P, Q, and R in the oxy plane are defined by

P: (x, y) → (-4x - y, 2x)

Q: (x, y) → (y, 6x - 9y)

R: (x, y) → (x - 2y, 3x + 5y)

(a) write down the matrices of P, Q, and R 

(b) Find:

(i) 2P - 3R + Q;

(ii) QR;

(iii) the inverse of the matrix R.

(a) Express \(\frac{9x}{(2x + 1)(x^2 + 1)}\) in partial fraction

(b) If \(^{2m}P_2\) - 10 = \(^m P_2\), find the positive value of m.

The data shows the ordered marks scored by students in a test: 11, 12, (2x + y), (x + 2y), 14, and ((y\(^2\) - 2x). Given that the median is 13\(\frac{1}{2}\) and y is greater than x by 1, find:

(a) the values of x and y

(b) correct to three significant figures, the standard deviation of the distribution.

In an examination, 70% of the candidates passed. If 12 candidates are selected at random, find the probability that:

(a) at least two of them failed;

(b) exactly half of them passed;

(c) not more than one - six of them failed.

                                                                                 PART II

A particle of weight 12 N lying on a horizontal ground is acted by forces F\(_1\) = (10 N, 090º), F\(_2\) = (16 N, 180º), F\(_3\) = (7 N, 300º) and F\(_4\) = (12N, 030º)

(a) Express all the forces acting on the particle as column vectors

(b) Find, correct to two decimal places, the magnitude of the:

(i) resultant forces;

(ii) acceleration with which the particle starts to move.[Take g = 10 ms\(^{-2}\)]