Three forces \(-63j , 32.14i + 38.3j\) and \(14i - 24.25j\) act on a body of mass 5kg. Find, correct to one decimal place, the :

(a) magnitude of the resultant force ;

(b) acceleration of the body.

Simplify \(^{n + 1}C_{4} - ^{n - 1}C_{4}\)

= \(\frac{(n + 1)!}{4! (n - 3)!} - \frac{(n - 1)!}{4! (n - 5)!}\)

= \(\frac{(n + 1)(n)(n - 1)(n - 2)(n - 3)!}{4! (n - 3)!} - \frac{(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)!}{4! (n - 5)!}\)

= \(\frac{(n + 1)(n)(n - 1)(n - 2)}{4!} - \frac{(n - 1)(n - 2)(n - 3)(n - 4)}{4!}\)

= \(\frac{(n - 1)(n - 2) [n(n + 1) - (n - 3)(n - 4)]}{4!}\)

= \(\frac{(n - 1)(n - 2) [n^{2} + n - n^{2} + 7n - 12]}{24}\)

= \(\frac{(n - 1)(n - 2)[8n -  12]}{24}\)

= \(\frac{(n - 1)(n - 2)(2n - 3)}{6}\)

The marks scored by 35 students in a test are given in the table below.

Draw a histogram for the distribution.

(a) The polynomial \(f(x) = x^{3} + px^{2} - 10x + q\) is exactly divisible by \(x^{2} + x - 6\). Find the :

(i) values of p and q ; (ii) third factor.

(b) The volume of a cube is increasing at the rate of \(2\frac{1}{2} cm^{3} s^{-1}\). Find the rate of change of the side of the base when its length is 2cm.

(a) Write down the matrix A of the linear transformation \(A(x, y) \to (2x -y, -5x + 3y)\).

(b) If \(B = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\), find :

(i) \(A^{2} - B^{2}\) ; (ii) matrix \(C = B^{2} A\) ; (iii) the point \(M(x, y)\) whose image under the linear transformation \(C\) is \(M' (10, 18)\).

(c) What is the relationship between matrix A and matrix C?