Express \(4 x ^ { 2 } + 12 x + 10\) in the form \(( a x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
Functions f and g are both defined for \(x > 0\). It is given that \(\mathrm { f } ( x ) = x ^ { 2 } + 1\) and \(\mathrm { fg } ( x ) = 4 x ^ { 2 } + 12 x + 10\). Find \(\mathrm { g } ( x )\).
Find \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and give the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
(a) Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are 6 and \(6 r\) respectively. The first and second terms of \(Q\) are 12 and \(- 12 r\) respectively. Find the value of the common sum to infinity.
(b) The first term of an arithmetic progression is \(\cos \theta\) and the second term is \(\cos \theta + \sin ^ { 2 } \theta\), where \(0 \leqslant \theta \leqslant \pi\). The sum of the first 13 terms is 52 . Find the possible values of \(\theta\).