\begin{enumerate} \setcounter{enumi}{2} \item Given that \(y = x ^ { 3 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) from first principles. \item The cubic polynomial \(f ( x )\) is given by \(f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b , c\) are constants. The graph of \(f ( x )\) intersects the \(x\)-axis at the points with coordinates \(( - 3,0 ) , ( 2 \cdot 5,0 )\) and \(( 4,0 )\). Find the coordinates of the point where the graph of \(f ( x )\) intersects the \(y\)-axis. \item The points \(A ( 0,2 ) , B ( - 2,8 ) , C ( 20,12 )\) are the vertices of the triangle \(A B C\). The point \(D\) is the mid-point of \(A B\).

1. Show that \(C D\) is perpendicular to \(A B\).
2. Find the exact value of \(\tan C \hat { A } B\).
3. Write down the geometrical name for the triangle \(A B C\). \item In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true while the other is false.
   A Given that \(( 2 c + 1 ) ^ { 2 } = ( 2 d + 1 ) ^ { 2 }\), then \(c = d\).
   B Given that \(( 2 c + 1 ) ^ { 3 } = ( 2 d + 1 ) ^ { 3 }\), then \(c = d\).