Edexcel IAL Mathematics P4.2.1 partial fractions

Partial-fraction questions test the setup as much as the constants: choose the right form, clear denominators and compare coefficients or substitute roots.

Exam points

Clear denominators before comparing coefficients to find A, B, C or D.
Substitute convenient roots to solve constants in linear-factor decompositions.
Divide improper rational functions before writing the remaining partial fractions.

[Maximum number: 3]

Figure 2

Figure 2 shows a sketch of the curve defined by the parametric equations

x=t^{2}+2 t \quad y=\frac{2}{t(3-t)} \quad a \leqslant t \leqslant b

where a and b are constants.
The ends of the curve lie on the line with equation y=1

Write $\frac{t+1}{t(3-t)}$ in partial fractions.

(ii)

$\frac{t+1}{t(3-t)}=\frac{A}{t}+\frac{B}{3-t}$

M1

$\Rightarrow t+1=A(3-t)+B t$

E.g. $t=0 \Rightarrow 1=3 A \Rightarrow A=\frac{1}{3} ; t=3 \Rightarrow 4=3 B \Rightarrow B=\frac{4}{3}$

M1

$\frac{t+1}{t(3-t)}=\frac{1}{3 t}+\frac{4}{3(3-t)}$

A1

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P4.2.1 - Decompose rational functions into Partial fractions question 1