Find \( x \) and \( y \) if \({12^3} = {2^x} \times {3^y}\).

Factorise \({4^x} – {2^x} – 20\).

Solve $\dfrac{{x + \sqrt x }}{{x – \sqrt x }} = 2$.

Find all factors of \( 2x^4 + 10x^3 – 4x^2 – 48x \).

Given \(f\left( x \right) = 2x – 1\), \(g\left( x \right) = x + 1\), \(\left( {f \circ f} \right)\left( 2 \right) = a\), \(\left( {g \circ f} \right)\left( 1 \right) = b\), \(\left( {f \circ g} \right)\left( c \right) = 3\) and \(\left( {g \circ g} \right)\left( d \right) = 1\) find the domain and range of \(a + b + c + d\).

If the domain of \(y = 2x – 1\) is \(1 \le x < 2\), state the range of \({y^{ - 1}}\).

Solve \(1 + 12{e^{ – x}} = {e^x}\).

The equation of a straight line perpendicular to \(y = – \dfrac{x}{2} + 4\) and passing through \(\left( {3,1} \right)\) is \(y = Ax + B\).

Find all possible angles   \(\theta \)  with  \(0 \le \theta \le 2\pi \),  such that  \(\sin \theta = 1\).

For normally distributed data \(X \sim N\left( {19,{3^2}} \right)\), find the value of \(k\) for \(\Pr \left( {X < k} \right) = 0.878\), correcting to two decimal places.