If \(P = {x : -2 < x < 5}\) and \(Q = {x : -5 < x < 2}\) are subsets of \(\mu = {x : -5 \leq x \leq 5}\), where x is a real number, find \((P \cup Q)\).

Express \(\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}\) in the form \(p\sqrt{3} + q\sqrt{2}\).

An operation * is defined on the set, R, of real numbers by \(p * q = p + q + 2pq\). If the identity element is 0, find the value of p for which the operation has no inverse.

Consider the statements:

p : Musa is short

q : Musa is brilliant

Which of the following represents the statement "Musa is short but not brilliant"?

If \(f(x) = \frac{4}{x} - 1, x \neq 0\), find \(f^{-1}(7)\).

If \(y = 4x - 1\), list the range of the domain \({-2 \leq x \leq 2}\), where x is an integer.

Factorize completely: \(x^{2} + x^{2}y + 3x - 10y + 3xy - 10\).

If the solution set of \(x^{2} + kx - 5 = 0\) is (-1, 5), find the value of k.

The remainder when \(x^{3}  - 2x + m\) is divided by \(x - 1\) is equal to the remainder when \(2x^{3} + x - m\) is divided by \(2x + 1\). Find the value of m.

If (2t - 3s)(t - s) = 0, find \(\frac{t}{s}\).

Solve for x in the equation \(5^{x} \times 5^{x + 1} = 25\).

If \(\log_{10}y + 3\log_{10}x \geq \log_{10}x\), express y in terms of x.

Simplify \(\frac{^{n}P_{5}}{^{n}C_{5}}\).

Given n = 3, evaluate \(\frac{1}{(n-1)!} - \frac{1}{(n+1)!}\)

Find the coefficient of \(x^{3}\) in the expansion of \([\frac{1}{3}(2 + x)]^{6}\).

Find the fourth term in the expansion of \((3x - y)^{6}\).

The 3rd and 6th terms of a geometric progression (G.P.) are \(\frac{8}{3}\) and \(\frac{64}{81}\) respectively, find the common ratio.

Given that \(-6, -2\frac{1}{2}, ..., 71\) is a linear sequence , calculate the number of terms in the sequence. 

If \(\begin{vmatrix}  m-2 & m+1 \\ m+4 & m-2 \end{vmatrix} = -27\), find the value of m.

If \(P = \begin{pmatrix} 1 & 2 \\ 5 & 1 \end{pmatrix}\) and \(Q = \begin{pmatrix} 0 & 1 \\ 1 & 3 \end{pmatrix}\), find PQ.

Evaluate \(\cos 75°\), leaving the answer in surd form.

Given that \(\tan x = \frac{5}{12}\), and \(\tan y = \frac{3}{4}\), Find \(\tan (x + y)\).

Find the equation of the line which passes through (-4, 3) and parallel to line y =  2x + 5.

Points E(-2, -1) and F(3, 2) are the ends of the diameter of a circle. Find the equation of the circle.

The lines \(2y + 3x - 16 = 0\) and \(7y - 2x - 6 = 0\) intersect at point P. Find the coordinates of P.

Find \(\lim\limits_{x \to 3} \frac{2x^{2} + x - 21}{x - 3}\).

Find the gradient to the normal of the curve \(y = x^{3} - x^{2}\) at the point where x = 2.

Find the minimum value of \(y = 3x^{2} - x - 6\).

The radius of a circle increases at a rate of 0.5\(cms^{-1}\). Find the rate of change in the area of the circle with radius 7cm. \([\pi = \frac{22}{7}]\)

Find an expression for y given that \(\frac{\mathrm d y}{\mathrm d x} = x^{2}\sqrt{x}\)

Given that \(n = 10\) and \(\sum d^{2} = 20\), calculate the Spearman's rank correlation coefficient.

Find the variance of 11, 12, 13, 14 and 15.

A fair coin is tossed 3 times. Find the probability of obtaining exactly 2 heads.

A box contains 14 white balls and 6 black balls. Find the probability of first drawing a black ball and then a white ball without replacement.

Given that \(r = 3i + 4j\) and \(t = -5i + 12j\), find the acute angle between them.