N2020 P1 Q9

1. By considering the gradients of two lines, explain why \({{\tan }^{-1}}(2)-{{\tan }^{-1}}\left( -\frac{1}{2} \right)=\frac{1}{2}\pi \).[1]

The curves \({{C}_{1}}\) and \({{C}_{2}}\) have equations \(y=\frac{1}{{{x}^{2}}+1}\) and \(y=\frac{k}{3x+4}\) respectively, where \(k\) is a constant and \(k>0\).

2. Find the set of values of \(k\) such that \({{C}_{1}}\) and \({{C}_{2}}\) intersect.[3]

It is now given that \(k=2\).

3. Sketch \({{C}_{1}}\) and \({{C}_{2}}\) on the same graph, giving the coordinates of any points where \({{C}_{1}}\) or \({{C}_{2}}\) cross the axes and the equations of any asymptotes.[3]
4. Find the exact area of the region bounded by \({{C}_{1}}\) and \({{C}_{2}}\), simplifying your answer.[5]