Edexcel IAL P4 Partial Fractions Questions
Partial fractions with distinct linear and repeated linear factors in the denominator (improper fractions by division first). Irreducible-quadratic denominators are not on the IAL specification.
- P4 · Pure Mathematics 4
- Calculator allowed
- 43 practice questions
What the questions test
The 43 P4 partial fractions practice questions cover:
Worked example 1
Consider the rational function $f(x) = \frac{3x}{(x-1)(x+2)}$.
- Express $f(x)$ in partial fractions.
- Hence, evaluate exactly the definite integral $\int_2^3 f(x) \,dx$, giving your answer in the form $\ln(k)$ where $k$ is a rational number.
Mark scheme
- M1 for setting up the partial fraction decomposition: $\frac{3x}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$.
M1 for clearing the denominators: $3x = A(x+2) + B(x-1)$.
M1 for substituting strategic values of $x$ (e.g., $x=1$ and $x=-2$) to solve for constants.
A1 for finding $A = 1$ and $B = 2$, yielding $\frac{1}{x-1} + \frac{2}{x+2}$. - M1 for setting up the integral and determining the anti-derivatives: $\left[ \ln|x-1| + 2\ln|x+2| \right]_2^3$.
M1 for substituting the limits: $(\ln 2 + 2\ln 5) - (\ln 1 + 2\ln 4)$.
A1 for using logarithm laws to combine into a single logarithm: $\ln 2 + \ln 25 - 0 - \ln 16 = \ln\left(\frac{50}{16}\right) = \ln\left(\frac{25}{8}\right)$.
Worked example 2
Express $\dfrac{x^3 + 1}{x^2 - 1}$ in the form $Ax + B + \dfrac{C}{x-1} + \dfrac{D}{x+1}$ and find $A, B, C, D$.
Mark scheme
Polynomial division: $\tfrac{x^3 + 1}{x^2 - 1} = x + \tfrac{x + 1}{x^2 - 1} = x + \tfrac{1}{x - 1}$ (since $\tfrac{x+1}{(x+1)(x-1)} = \tfrac{1}{x-1}$) M1 M1 A1. So $A = 1$, $B = 0$, $C = 1$, $D = 0$ A1 A1 A1 A1.
FAQ
What partial fractions is in Edexcel IAL P4?
Partial fractions with distinct linear and repeated linear factors in the denominator (improper fractions by division first). Irreducible-quadratic denominators are not on the IAL specification.
Is P4 a calculator paper?
Yes, a calculator is allowed in P4, but method marks are only given for working you write down.
How are P4 partial fractions questions marked?
With M marks for a correct method, A marks for accurate answers that depend on the M mark before them, and B marks for independent results, as in the worked examples on this page.