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Two real functions are defined as $$\begin{aligned} & f ( x ) = \frac { 8 } { x - 4 } \quad \text { for } \quad ( - \infty < x < 4 ) \cup ( 4 < x < \infty ) ,
& g ( x ) = ( x - 2 ) ^ { 2 } \quad \text { for } \quad - \infty < x < \infty . \end{aligned}$$ a) i) Find an expression for \(f g ( x )\).
ii) Determine the values of \(x\) for which \(f g ( x )\) does not exist.
b) Find an expression for \(f ^ { - 1 } ( x )\). A curve \(C\) has equation \(f ( x ) = 5 x ^ { 3 } + 2 x ^ { 2 } - 3 x\). a) Find the \(x\)-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary.
b) Determine the range of values of \(x\) for which \(C\) is concave. The rate of change of a variable \(y\) with respect to \(x\) is directly proportional to \(y\). a) Write down a differential equation satisfied by \(y\).
b) When \(x = 1\) and \(y = 0 \cdot 5\), the rate of change of \(y\) with respect to \(x\) is 2 . Find \(y\) when \(x = 3\). The curve \(C _ { 1 }\) has parametric equations \(x = 3 p + 1 , y = 9 p ^ { 2 }\). The curve \(C _ { 2 }\) has parametric equations \(x = 4 q , y = 2 q\).
Find the Cartesian coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). a) Use integration by parts to evaluate \(\int _ { 0 } ^ { 1 } ( 3 x - 1 ) \mathrm { e } ^ { 2 x } \mathrm {~d} x\). b) Use the substitution \(u = 1 - 2 \cos x\) to find \(\int \frac { \sin x } { 1 - 2 \cos x } \mathrm {~d} x\). END OF PAPER