Partial fractions – A-level Maths exam questions with mark scheme

1CoreCalculator3 marks

Express $\dfrac{5x+1}{x^{2}-1}$ in partial fractions. [3]

Factorises $x^{2}-1=(x+1)(x-1)$ and writes $5x+1\equiv A(x-1)+B(x+1)$, then substitutes a value of $x$ or compares coefficientsM1
One of $A=2$ (for $\frac{A}{x+1}$) or $B=3$ (for $\frac{B}{x-1}$)A1
$\dfrac{2}{x+1}+\dfrac{3}{x-1}$A1
Answer: $\dfrac{2}{x+1}+\dfrac{3}{x-1}$
Examiner tip: Always write the final answer as partial fractions – stopping at '$A=2$, $B=3$' can lose the last mark.

2StandardCalculator9 marks

$$\mathrm{f}(x)=\frac{3x^{2}+3x+3}{(x+2)(x-1)^{2}}$$

(a) Express $\mathrm{f}(x)$ in partial fractions. [4]

(b) Hence, or otherwise, find the series expansion of $\mathrm{f}(x)$, in ascending powers of $x$, up to and including the term in $x^{2}$. Give each coefficient as a simplified fraction. [5]

(a) Correct form $\dfrac{A}{x+2}+\dfrac{B}{x-1}+\dfrac{C}{(x-1)^{2}}$B1
Uses $3x^{2}+3x+3\equiv A(x-1)^{2}+B(x+2)(x-1)+C(x+2)$ and substitutes values or compares coefficientsM1
Two of $A=1,\ B=2,\ C=3$A1
All three: $\dfrac{1}{x+2}+\dfrac{2}{x-1}+\dfrac{3}{(x-1)^{2}}$A1
(b) Attempts a binomial expansion of one term, e.g. $\tfrac12\left(1+\tfrac{x}{2}\right)^{-1}$ or $3(1-x)^{-2}$ or $-2(1-x)^{-1}$M1
Correct expansion of one term, e.g. $\tfrac12-\tfrac14x+\tfrac18x^{2}$A1
Correct expansion of a second term, e.g. $-2-2x-2x^{2}$ or $3+6x+9x^{2}$A1
Adds three expansions (of the right form) and collects like termsM1
$\tfrac32+\tfrac{15}{4}x+\tfrac{57}{8}x^{2}$A1
Answer: (a) $\dfrac{1}{x+2}+\dfrac{2}{x-1}+\dfrac{3}{(x-1)^{2}}$ (b) $\tfrac32+\tfrac{15}{4}x+\tfrac{57}{8}x^{2}$
Examiner tip: Rewrite $\dfrac{2}{x-1}$ as $-2(1-x)^{-1}$ before expanding – the sign change is where most marks are lost.

3HardCalculator7 marks

$$\mathrm{f}(x)=\frac{4x^{2}+7x}{2x^{2}+x-1},\quad x>1$$

(a) Given that $\mathrm{f}(x)$ can be written in the form $$A+\frac{B}{x+1}+\frac{C}{2x-1}$$ find the values of the constants $A$, $B$ and $C$. [4]

(b) Hence show that $\mathrm{f}$ is a decreasing function. [3]

(a) Divides or uses $4x^{2}+7x\equiv A(x+1)(2x-1)+B(2x-1)+C(x+1)$M1
$A=2$B1
Substitutes $x=-1$ or $x=\tfrac12$ (or compares coefficients) to find $B$ or $C$M1
$B=1$ and $C=3$A1
(b) Differentiates their partial fractions: $\mathrm{f}'(x)=-\dfrac{B}{(x+1)^{2}}-\dfrac{2C}{(2x-1)^{2}}$M1
$\mathrm{f}'(x)=-\dfrac{1}{(x+1)^{2}}-\dfrac{6}{(2x-1)^{2}}$A1
States both terms negative for $x>1$ (squares positive), so $\mathrm{f}'(x)<0$ and f is decreasingA1
Answer: (a) $A=2,\ B=1,\ C=3$ (b) $\mathrm{f}'(x)=-\dfrac{1}{(x+1)^{2}}-\dfrac{6}{(2x-1)^{2}}<0$
Examiner tip: The fraction is improper (degree 2 over degree 2) – you must include the constant $A$.

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