The graph shows how the magnification $m$ produced by a thin lens varies with image distance $v$. What is the focal length of the lens used?

As the graph between magnification $\left(m\right)$ and image distance $\left(v\right)$ varies linearly, then
$$\begin{gathered} m=k_{1}v+k_2 \\ \Rightarrow \frac{v}{u}=k_{1}v+k_2 \\ \Rightarrow \frac{1}{u}=k_1+\frac{k_2}{v} \\ \Rightarrow \frac{k_2}{v}-\frac{1}{u}=k_1 \end{gathered}$$
Clearly, from above, we have
     $k_1=\frac{1}{f}$ and $k_2=1$
$\Rightarrow f=\frac{1}{slope of m-v graph}=\frac{b}{c}$