Mathematics 4024 · O Level
Algebraic manipulation
100 practice questions on Algebraic manipulation, with worked solutions and instant marking.
Factorise $3-12a^2$ completely.
May/June 2015
The equation can be written as $c = \sqrt{8a - 3b}$.
May/June 2015
Factorise $p^2q - pq$ completely.
May/June 2015
Express as a single matrix $3\begin{pmatrix}1 & 3 \\ -2 & 5\end{pmatrix} - \begin{pmatrix}4 & 0 \\ -1 & 2\end{pmatrix}$.
May/June 2015
Factorise $4x^3-10xy$ completely.
May/June 2015
Write $3xy - 20 + 5x - 12y$ in fully factorised form.
May/June 2016
Rearrange the formula $t=\frac{p+3}{p-4}$ to make $p$ the subject.
May/June 2016
Solve $ \dfrac{p - 1}{7 - p} = 5$.
May/June 2016
$p = \dfrac{8 - 5q}{q}$. In trapezium $A$, the two parallel sides are $x\text{ cm}$ and $(x-2)\text{ cm}$. In trapezium $B$, the two parallel sides are $x\text{ cm}$ and $(x+3)\text{ cm}$. Each trapezium has area $15\text{ cm}^2$.
May/June 2016
Evaluate $2a - b$ when $a = 3$ and $b = -7$.
May/June 2017
The matrices $A=\begin{pmatrix}2 & 0 \\ 4 & -1\end{pmatrix}$ and $B=\begin{pmatrix}2 & -1 \\ 6 & -1\end{pmatrix}$ are provided.
May/June 2017
Factorise $12a^2b-15ab^3$ completely.
May/June 2017
Matrices $A$, $B$ and $C$ are provided.
May/June 2017
$\begin{pmatrix}2 & a\\ -3 & 1\end{pmatrix}\begin{pmatrix}-4 & b\\ 3 & 2\end{pmatrix} = \begin{pmatrix}7 & 10\\ 15 & 2\end{pmatrix}$!
May/June 2018
Factorise completely the expression $2ax - 3by + 6bx - ay$.
May/June 2018
The relation is $s = \sqrt[3]{t + 4}$.
May/June 2018
Across a two-week span, a shopkeeper notes how many packets of two distinct kinds of tea he sells and the profit earned from them. Matrices may be used to represent this information.
May/June 2018
Factorise the expression $25t^2-4$.
May/June 2018
Factorise the expression $25x - 5$.
May/June 2019
Given $T = \begin{pmatrix} 2 & 7 \\ 1 & 5 \end{pmatrix}$, determine $T^{-1}$.
May/June 2019
Starting with $c = \frac{3b^2 - 2a}{5}$, rewrite the equation so that $b$ is the subject.
May/June 2019
Expand then simplify $(x-3)^2$.
May/June 2019
Express $x^2-7x+5$ in the format $(x-a)^2-b$.
May/June 2019
$r=\dfrac{4p+2}{3-p}$.
May/June 2019
$P=\begin{pmatrix}4&0\\-2&3\end{pmatrix}$ and $Q=\begin{pmatrix}1&2\\0&-1\end{pmatrix}$. Evaluate the product $PQ$.
May/June 2019
Write this as one fraction, then give it in simplest form: $\frac{3a}{4b} - \frac{a}{6b}$.
May/June 2019
Factorise the expression.
May/June 2021
Simplify the expression $6x + 15 - 2x + 8$.
May/June 2021
$p = \frac{3q + 5}{r^2}$. Calculate the value of $p$ for $q = 15$ and $r = -4$.
May/June 2021
Simplify the expression $4a - b + 6b - 7a$.
May/June 2021
Expand, then simplify.
May/June 2022
Factorise the expression.
May/June 2022
The equation is $y=\dfrac{3x+2}{2x-1}$.
May/June 2022
Let $M=\begin{pmatrix}1&0\\4&3\end{pmatrix}$ and $N=\begin{pmatrix}k&0\\1&4\end{pmatrix}$. If $MN=NM$, determine the value of $k$.
May/June 2022
Factorise the expression $4x^2+5x-6$.
May/June 2022
Using $A = 3p + q$, find $q$ when $A = 23$ and $p = 5$.
May/June 2022
Expand and simplify the expression $5(3x - 2) - 3(2x - 3)$.
May/June 2023
Starting from $y = \sqrt{\frac{x + 2}{3}}$, rearrange the equation so that $x$ is the subject.
May/June 2023
Factorise $7y + 2xy - 6x - 21$.
May/June 2023
Expand then simplify $(4x - y)(2x + 5y)$.
May/June 2023
Simplify the expression $3u - 6w - 5u + 9w$.
May/June 2023
Each chocolate bar is $p$ cents and each packet of sweets costs $75$ cents. Tanish spends $9.10$ on $5$ chocolate bars and $8$ packets of sweets. Set up an equation and solve it to determine $p$. Show your working.
May/June 2023
The matrix $N$ obeys the equation $3N = N + 5\begin{pmatrix}4 & 0\\6 & -2\end{pmatrix}$.
May/June 2024
$a = \dfrac{5b + 2x}{x - 3}$. Rewrite the formula so that $x$ is the subject.
May/June 2024
Simplify the expression $6r + 7s - r + 3s$.
May/June 2024
$a = 5b + 7$. Find the value of $a$ for $b = -2$.
May/June 2024
The matrix is given by $A = \begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}$.
May/June 2024
Solve $\frac{x}{3}=7$.
May/June 2024
Rewrite the formula so that $x$ is the subject. $ax = \frac{3x + 2}{5}$.
May/June 2025
Expand and simplify the product $(x - 3)(2x + 5)(x + 2)$.
May/June 2025
Factorise the expression $7hx + 6fy - 21fx - 2hy$.
May/June 2025
Factorise the expression $5x^2 + 15xy$.
May/June 2025
Simplify the expression $5a + 3b + 2a - 7b$.
May/June 2025
The matrices $A = \begin{pmatrix} 1 & 3 \\ -2 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} -1 & 2 \\ -3 & 2 \end{pmatrix}$ are provided.
Oct/Nov 2015
Factorise completely the expression $5 - 20t^2$.
Oct/Nov 2016
The matrix is $A = \begin{pmatrix}2 & -1 \\ 1 & 3\end{pmatrix}$.
Oct/Nov 2016
Evaluate the expression $\sqrt[3]{\dfrac{543}{28.6 - 1.35}}$.
Oct/Nov 2016
Simplify the expression $\dfrac{3a^2}{10bc} \div \dfrac{9a}{5b^2c}$.
Oct/Nov 2016
$A=\begin{pmatrix}2&0\\3&1\end{pmatrix}$ together with $B=\begin{pmatrix}1&2\\-1&3\end{pmatrix}$.
Oct/Nov 2016
Anya manufactures T-shirts. Matrix $M$ displays how many T-shirts of each type she makes in one week.
Oct/Nov 2017
Express $\frac{4}{x-2}-\frac{5}{x+1}$ as a single fraction in simplest form.
Oct/Nov 2017
Factorise $15a + 3ab$ completely.
Oct/Nov 2018
Simplify the expression $4c - 3(2c - 5)$.
Oct/Nov 2018
Here $A = \begin{pmatrix}3 & 1 \\ -2 & 0\end{pmatrix}$ while $B = \begin{pmatrix}-2 & 1 \\ 3 & 0\end{pmatrix}$.
Oct/Nov 2019
Factorise $1 - 36p^2$
Oct/Nov 2019
Express $3\begin{pmatrix}1 & -1 \\ 0 & 2\end{pmatrix} - 2\begin{pmatrix}-1 & 1 \\ 2 & 1\end{pmatrix}$ in a single matrix.
Oct/Nov 2019
Factorise this expression.
Oct/Nov 2019
$x^2 + 7x - 13 = (x+a)^2 + b$ Work out the value of $a$ and the value of $b$.
Oct/Nov 2019
Rearrange $v = \frac{3}{p+5}$ so that $p$ is the subject.
Oct/Nov 2019
Factorise $4p^2 - 1$ into factors.
Oct/Nov 2020
Matrix $A$ is given by $A = \begin{pmatrix}2 & 1 \\ -3 & -2\end{pmatrix}$.
Oct/Nov 2020
Factorise $12t^2 - 4t$ by removing the common factor $4t$.
Oct/Nov 2020
Simplify the expression $3(3a-4) + 2(2-a)$.
Oct/Nov 2020
Algebra, then simplification.
Oct/Nov 2020
Factorise $4b^2 - c^2$.
Oct/Nov 2021
Using $A = \begin{pmatrix}-6 & 2 \\ 1 & 4\end{pmatrix}$, find $A^2$.
Oct/Nov 2021
Simplify the expression $3a - a + 2a$.
Oct/Nov 2021
Factorise the expression $3xy - qy + 6px - 2pq$.
Oct/Nov 2021
Find the inverse of $\begin{pmatrix} 3 & -2 \\ 1 & 2 \end{pmatrix}$.
Oct/Nov 2021
Expand and simplify the expression $3(2x + 1) - 2(4x + 3)$.
Oct/Nov 2022
$A = \begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix}$ and $A + 2B = \begin{pmatrix} 1 & 5 \\ 10 & 12 \end{pmatrix}$.
Oct/Nov 2022
Factorise $3a^2 + 12a$.
Oct/Nov 2022
Factorise the expression $9p^2 - q^2$.
Oct/Nov 2022
Adam and Ben purchase cinema and theatre tickets.
Oct/Nov 2022
Take $A = \begin{pmatrix}-2 & 1 \\ 4 & 3\end{pmatrix}$ and $B = \begin{pmatrix}3 & 2 \\ -1 & 1\end{pmatrix}$.
Oct/Nov 2023
The relation is $T = \sqrt{P - 4}$.
Oct/Nov 2023
Factorise this expression.
Oct/Nov 2023
Simplify this expression. $7a - 4b - 2a + b$.
Oct/Nov 2023
Express as a single matrix: $2\begin{pmatrix}3 & -1 \\ 2 & 4\end{pmatrix} - \begin{pmatrix}1 & 3 \\ -2 & 5\end{pmatrix}$.
Oct/Nov 2024
Simplify the expression $2a - 3b + 4b - 5a$.
Oct/Nov 2024
Factorise the expression $4m^{2}-14m$.
Oct/Nov 2024
Simplify this expression.
Oct/Nov 2024
The inverse of matrix $A$ is $\frac{1}{20}\begin{pmatrix}m & 7\\-1 & k\end{pmatrix}$. Here, $m$ and $k$ are positive integers with $m<k$. The determinant of matrix $A$ equals $20$.
Oct/Nov 2024
Solve $\frac{y}{4}=8$ for $y$.
Oct/Nov 2024
Solve for $x$ in $4x + 7 = 16$.
Oct/Nov 2024
Factorise the expression $6x^3+5x^2-4x$.
Oct/Nov 2025
Expand and simplify $(3+\sqrt{2})(1+4\sqrt{2})$.
Oct/Nov 2025
Expand, then simplify $(x-1)(x+3)(3x-4)$.
Oct/Nov 2025
Factorise $20x^2-5xy$ into a factorised form.
Oct/Nov 2025
Expand and simplify the expression $4(3x+2)+5(x+1)$.
Oct/Nov 2025