Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 10

The original line meets the horizontal axis at $(-3/2,0)$. A perpendicular line has slope $-1/2$, the negative reciprocal of $2$, soSubstituting $x=-3/2$ verifies the required common point.

For the partial fraction decomposition, write $A/(x-4)+B/(x+9)$. Clearing denominators gives $13(x+1)=A(x+9)+B(x-4)$. Substitution at $x=4$ and $x=-9$ yields $A=5$ and $B=8$, hence