Find \(a\) and \(b\),   if  \(\dfrac{{{{\left( {{x^2} + 1} \right)}^3}}}{{{x^2} – 2}} = \dfrac{{ax{{\left( {{x^2} + 1} \right)}^2}\left( {2{x^2} + b} \right)}}{{{{\left( {{x^2} – 2} \right)}^2}}}\).

Express \({\log _b}108\) in terms of \(x,y\) or \(z\) , given \(x = {\log _b}2,y = {\log _b}3,z = {\log _b}5\).

The mass of bacteria \(t\) hours after establishment is \({A_t} = 500 \times {1.6^{1.1t}}\) grams. Find the time for the mass to reach 600 grams. Answer correct to the nearest minute.

Find \(a \) and \(b \) , if $\dfrac{{\sqrt 7 + \sqrt 8 }}{{3\sqrt 7 – 3\sqrt 8 }} + \dfrac{{2\sqrt 8 – 2\sqrt 7 }}{{3\sqrt 8 + 3\sqrt 7 }} = a + b\sqrt {14} $.

Given \(f\left( x \right) = ax + b\left( {a \ne 0} \right)\), \(\left( {f \circ f} \right)\left( x \right) = x\) and \(f\left( 5 \right) = – 4\), find \(f\left( 1 \right)\).

Two numbers differ by 3 such that three times the larger number plus twice the smaller number is equal to 54.   Find the numbers.

Given  \(P\left( A \right) = \dfrac{1}{2}\) and \(P\left( B \right) = \dfrac{1}{3}\),   find  \(P\left( {B’} \right)\),   correcting to one decimal place.

Find the smallest angle in the triangle, correcting to the nearest degree.

Find the positive angle of inclination, in radian correcting to three significant figures, of a line having a gradient of -2.

Solve  \(\tan x = \dfrac{1}{{\sqrt 3 }}\)  for  \(- 360^\circ \le x \le 360^\circ \).   Write four answers in ascending order.