3. $$\mathrm { f } ( x ) = x ^ { 3 } - ( k + 4 ) x + 2 k , \quad \text { where } k \text { is a constant. }$$ （a）Show that，for all values of \(k\) ，the curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,0 )\) ．
（b）Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has exactly two distinct roots． Given that \(k > 0\) ，that the \(x\)－axis is a tangent to the curve with equation \(y = \mathrm { f } ( x )\) ，and that the line \(y = p\) intersects the curve in three distinct points，
（c）find the set of values that \(p\) can take．
\includegraphics[max width=\textwidth, alt={}, center]{a243ceda-8175-4ae0-9bc7-b3048f468d10-3_573_899_343_704} The circle, with centre \(C\) and radius \(r\), touches the \(y\)-axis at \(( 0,4 )\) and also touches the line with equation \(4 y - 3 x = 0\), as shown in Fig. 1.

1. 1. Find the value of \(r\).
   2. Show that \(\arctan \left( \frac { 3 } { 4 } \right) + 2 \arctan \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } \pi\).
      (8) The line with equation \(4 x + 3 y = q , q > 12\), is a tangent to the circle.
2. Find the value of \(q\).
   (4)