(i) 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 $y=\frac{k}{3 x+4}$ respectively, where $k$ is a constant and $k>0$.

(ii) Find the set of values of $k$ such that $C_{1}$ and $C_{2}$ intersect.

[3]

It is now given that $k=2$.

(iii) 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]

(iv) Find the exact area of the region bounded by $C_{1}$ and $C_{2}$, simplifying your answer.

[5]