If \(f(x) = 6x^{3} + 13x^{2} + 2x - 5\) and \(f(-1) = 0\), find the factors of f(x).

(a) Find, from first principles, the derivative of \(f(x) = (2x + 3)^{2}\). (b) Evaluate : \(\int_{1} ^{2} \frac{(x + 1)(x^{2} - 2x + 2)}{x^{2}} \mathrm {d} x\)

A tyre manufacturing company researched into the life span of one type of their motorcycle tyres. The results were as follows : Distance (100km) 10-19 20-29 ...

A particle of mass 400g is moving under the action of two forces \(F_{1} = (35N, 210°), F_{2} = (35\sqrt{3} N, 300°)\) and a resistance of 40N. Find the magn...

(a) Evaluate \(\frac{^{9}P_{3}}{^{15}C_{3}} + \frac{^{5}C_{3}}{^{3}P_{2}}\) correct to two decimal places. (b) A committee of 2 tutors and 5 pupils is to be ...

(a) Write the following as column vectors: \(r = (10N, 090°) ; q = (8N, 135°)\). (b) Use your answer in (a) to find \((r + q)\).

If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 7x + 4 = 0\), find the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\be...

Solve \(2^{(2y + 2)} - 9(2^{y}) = -2\).

(a) Write down the binomial expansion of \((2 - x)^{5}\) in ascending powers of x. (b) Use your expansion in (a) to evaluate \((1.98)^{5}\) correct to four d...