A-level · Algebra & functions

Algebraic fractionsExam-style questions

Fractions with letters in them, handled by the rules of number fractions. What may cancel and why; factorising before cancelling; adding and subtracting over a common denominator, with the minus sign kept on the whole top; multiplying, and dividing by the reciprocal; solving equations with fractions and rejecting answers that make a denominator zero; and writing a top-heavy fraction as a quotient plus a remainder.

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20 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$\text{f}(x) = \dfrac{9x^3 - 18x^2 + 14x + 2}{3x - 2} \qquad x \in \mathbb{R},\ x > \dfrac{2}{3}$
(a) Express $\text{f}(x)$ in the form

$Ax^2 + Bx + C + \dfrac{D}{3x - 2}$

where $A$ , $B$ , $C$ and $D$ are integers to be found.

(4 marks)
(b) Using your answer to part (a) and algebraic integration, find the exact value of

$\int_1^2 \text{f}(x)\,dx$

Write your answer in the form $p + q\ln 2$ , where $p$ and $q$ are integers.

(5 marks)
Level 4 · 9 marksTry it
2
$\text{g}(x) = \dfrac{2x^3 - x^2 + 12x - 3}{x^2 + 3} \qquad x \in \mathbb{R}$
(a) Write $\text{g}(x)$ in the form

$Ax + B + \dfrac{Cx}{x^2 + 3}$

where $A$ , $B$ and $C$ are constants to be found.

(3 marks)
(b) Hence, using algebraic integration, show that

$\int_1^3 \text{g}(x)\,dx = 6 + \ln 27$

(4 marks)
Level 4 · 7 marksTry it