Find \( x \) if \({25^3} = {5^x}\).

Find \(a \) and \(b \) , if $\dfrac{1}{{\sqrt 8 – 2}} + \dfrac{1}{{2\sqrt 8 – 2}} = \dfrac{{a + b\sqrt 2 }}{{14}}$.

The perimeter of a rectangular playing field is 170 m and the length of the diagonals of the field is 65 m. Calculate the dimensions of the field.

Given \(f\left( x \right) = 2x – 3\), \(g\left( x \right) = – x + k\) and \(\left( {g \circ f} \right)\left( x \right) = \left( {f \circ g} \right)\left( x \right)\), find \(k\).

Find the angle of inclination, in radian correcting to two significant figures, of a line having a gradient of 1.2.

Solve  \(\cos 2x – \cos x = 0\)  for  \(0^\circ \le x \le 360^\circ \).   Write answers in ascending order.

Find \(k\),   correcting to two significant figures, such a vector  \(k\left( {\begin{array}{*{20}{c}}
4\\
{ – 3}\\
1
\end{array}} \right)\) is in the same direction as  \(\left( {\begin{array}{*{20}{c}}
4\\
{ – 3}\\
1
\end{array}} \right)\)  and with length 3 units.

Find for the values of \(x\) for which  \(\dfrac{{dy}}{{dx}} = {x^3} – 3x\)  is concave upwards.

Find  \(\int {\left( {\sin \dfrac{x}{2} + \cos \dfrac{x}{3} + 4{x^3} + 1} \right)dx} \).

Find the area of the region bounded by   \(y = {x^3} – 2{x^2}\) and \(y = {x^2} – 2x\),   correcting to one decimal place.