Find the domain of \(f(x) = \frac{x}{3 - x}, x \in R\), the set of real numbers.

Find the value of \(\cos(60° + 45°)\) leaving your answer in surd form.

If \(\frac{5}{\sqrt{2}} - \frac{\sqrt{8}}{8} = m\sqrt{2}\), where m is a constant. Find m.

If \(16^{3x} = \frac{1}{4}(32^{x - 1})\), find the value of x.

Simplify \(\frac{\log_{5} 8}{\log_{5} \sqrt{8}}\).

The coefficient of the 7th term in the binomial expansion of \((2 - \frac{x}{3})^{10}\) in ascending powers of x is

The roots of a quadratic equation are \((3 - \sqrt{3})\) and \((3 + \sqrt{3})\). Find its equation.

If (x - 3) is a factor of \(2x^{2} - 2x + p\), find the value of constant p.

If \(\sin x = -\sin 70°, 0° < x < 360°\), determine the two possible values of x.

For what values of x is \(\frac{x^{2} - 9x + 18}{x^{2} + 2x - 35}\) undefined?

Calculate, correct to one decimal place, the length of the line joining points X(3, 5) and Y(5, 1).

If \(y = 2(2x + \sqrt{x})^{2}\), find \(\frac{\mathrm d y}{\mathrm d x}\).

Calculate, correct to one decimal place, the acute angle between the lines 3x - 4y + 5 = 0 and 2x + 3y - 1 = 0.

Evaluate \(\int_{1}^{2} \frac{4}{x^{3}} \mathrm {d} x\)

If \(\begin{vmatrix} 3 & x \\ 2 & x - 2 \end{vmatrix} = -2\), find the value of x.

Given that \(P = {x : \text{x is a factor of 6}}\) is the domain of \(g(x) = x^{2} + 3x - 5\), find the range of x.

The third of geometric progression (G.P) is 10 and the sixth term is 80. Find the common ratio.

Find the axis of symmetry of the curve \(y = x^{2} - 4x - 12\).

Find the equation of the tangent to the curve \(y = 4x^{2} - 12x + 7\) at point (2, -1).

The mean age of 15 pupils in a class is 14.2 years. One new pupil joined the class and the mean changed to 14.1 years. Calculate the age of the new pupil.

The distance s metres of a particle from a fixed point at time t seconds is given by \(s = 7 + pt^{3} + t^{2}\), where p is a constant. If the acceleration at t = 3 secs is \(8 ms^{-2}\), find the value of p.

The probabilities that a husband and wife will be alive in 15 years time are m and n respectively. Find the probability that only one of them will be alive at that time.

In a class of 50 pupils, 35 like Science and 30 like History. What is the probability of selecting a pupil who likes both Science and History?

P, Q, R, S are points in a plane such that PQ = 8i - 5j, QR = 5i + 7j, RS = 7i + 3j  and PS = xi + yj. Find (x, y).

Find the least value of n for which \(^{3n}C_{2} > 0, n \in R\).

If \(\overrightarrow{OA} = 3i + 4j\) and \(\overrightarrow{OB} = 5i - 6j \) where O is the origin and M is the midpoint of AB, find OM.

Find the direction cosines of the vector \(4i - 3j\).

Yomi was asked to label four seats S, R, P, Q. What is the probability he labelled them in alphabetical order?

Two forces (2i - 5j)N and (-3i + 4j)N act on a body of mass 5kg. Find in \(ms^{-2}\), the magnitude of the acceleration of the body.

Two particles are fired together along a smooth horizontal surface with velocities 4 m/s and 5 m/s. If they move at 60° to each other, find the distance between them in 2 seconds.

Two forces \(F_{1} = (7i + 8j)N\) and \(F_{2} = (3i + 4j)N\) act on a particle. Find the magnitude and direction of \(F_{1} - F_{2}\).

A stone is thrown vertically upwards and its height at any time t seconds is \(h = 45t - 9t^{2}\). Find the maximum height reached.

Given that \(\frac{\mathrm d y}{\mathrm d x} = 3x^{2} - 4\) and y = 6 when x = 3, find the equation for y.

If \(h(x) = x^{3} - \frac{1}{x^{3}}\), evaluate \(h(a) - h(\frac{1}{a})\).

A company took delivery of 12 vehicles made up of 7 buses and 5 saloon cars for two of its departments; Personnel and General Administration. If the Personnel department is to have at least 3 saloon cars, in how many ways can these vehicles be distributed equally between the departments?