By considering the gradients of two lines, explain why ${{ an }^{-1}}(2)-{{ an }^{-1}}\left( -\fra

Question

By considering the gradients of two lines, explain why ${{	an }^{-1}}(2)-{{	an }^{-1}}\left( -\frac{1}{2} ight)=\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 $k>0$. Find the set of values of $k$ such that ${{C}_{1}}$ and ${{C}_{2}}$ intersect. [3] It is now given that $k=2$. 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] Find the exact area of the region bounded by ${{C}_{1}}$ and ${{C}_{2}}$, simplifying your answer. [5]

Accepted Answer

$\frac{\pi}{2} - \frac{4}{3}\ln 2$ units²

N2020 P1 Q9

N2020 P1 Q9

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