The following exercises cover vital skills from GCSE Maths and can help your A-Level Maths preparation. For successful study at A-Level, you must be highly proficient at applying these skills. You can become skilled by practising and getting help where necessary.
Collecting Like Terms
Simplify each of these expressions as far as possible:
- \(3x-2y+4y\)
- \(5-3y-6y-2\)
- \(5x+2y-4y-x^2\)
- \(x^2+3x^2-4x^2+5x\)
- \(2y^2-y(x-y)\)
- \(8pq-9p^2-3pq\)
- \(x^3-2x^2+x^2-4x+5x+7\)
- \(3x(x-2)+4(3x-5)\)
- \(7+3(x-1)\)
- \(7b(a+2)-a(3b+3)\)
Solutions
- \(3x+2y\)
- \(3-9y\)
- \(5x-2y-x^2\)
- \(5x\)
- \(3y^2-xy\)
- \(5pq-9p^2\)
- \(x^3-x^2+x+7\)
- \(3x^2+6x-20\)
- \(3x+4\)
- \(4ab-3a+14b\)
Solving Linear Equations
Find the value of \(x\) in the following equations:
- \(2x-5=11\)
- \(8=7+3x\)
- \(-7=2x-10\)
- \(\frac{x}{5}=7\)
- \(\frac{x}{10}=5\)
- \(\frac{x}{2}=\frac{1}{3}\)
- \(\frac{3x}{2}=-5\)
- \(x-3=3x+7\)
- \(5x-4=3-x\)
- \(5x-16=16-2x\)
Solutions
- \(x=8\)
- \(x=\frac{1}{3}\)
- \(x=\frac{3}{2}\)
- \(x=35\)
- \(x=50\)
- \(x=\frac{2}{3}\)
- \(x=-\frac{10}{3}\)
- \(x=-5\)
- \(x=\frac{7}{6}\)
- \(x=\frac{32}{7}\)
Expanding Brackets
Expand and collect like terms in each of the following:
- \((x+3)(x-2)\)
- \((x-5)(x-1)\)
- \((2x+y)(x-y)\)
- \((2x-3)(3x-1)\)
- \((2x+1)(x+2)\)
- \((x+2)(x-2)\)
- \(2(x+3)(x-1)\)
- \((2x-3)(2x+3)\)
- \((5-x)(5+x)\)
- \((x+7)^2\)
Solutions
- \(x^2+x-6\)
- \(x^2-6x+5\)
- \(2x^2-xy-y^2\)
- \(6x^2-11x+3\)
- \(2x^2+5x+2\)
- \(x^2-4\)
- \(2x^2+4x-6\)
- \(4x^2-9\)
- \(25-x^2\)
- \(x^2+14x+49\)
Factorising
Factorise each of the following. Note – you can check your answers by expansion:
- \(x^2+3x+2\)
- \(x^2-3x+2\)
- \(x^2+5x+6\)
- \(x^2+7x+6\)
- \(x^2+4x\)
- \(x^2-x-12\)
- \(x^2-2x-3\)
- \(2x^2+3x-5\)
- \(x^2-9\)
- \(4x^2-1\)
Solutions
- \((x+1)(x+2)\)
- \((x-1)(x-2)\)
- \((x+2)(x+3)\)
- \((x+6)(x+1)\)
- \(x(x+4)\)
- \((x-4)(x+3)\)
- \((x-3)(x+1)\)
- \((2x+5)(x-1)\)
- \((x+3)(x-3)\)
- \((2x-1)(2x+1)\)
Laws of Indices
Simplify the following:
- \(b\times 5b^5\)
- \(3c^2\times 2c^5\)
- \(b^2c\times bc^3\)
- \(2n^6\times\left(-6n^2\right)\)
- \(8n^8\div\left(2n^3\right)\)
- \(d^{11}\div d^9\)
- \(\left(a^3\right)^2\)
- \(\left(-d^4\right)^3\)
- \(\left(25g^{12}\right)^\frac{1}{2}\)
- \({\left(64h^{-3}\right)}^\frac{1}{3}\)
Solutions
- \(5b^6\)
- \(6c^7\)
- \(b^3 c^4\)
- \(-12n^8\)
- \(4n^5\)
- \(d^2\)
- \(a^6\)
- \(-d^{12}\)
- \(5g^6\)
- \(4h^{-1}\)
Solving Quadratic Equations
Solve the following equations:
- \(x^2+10-7x=0\)
- \(15-x^2-2x=0\)
- \(x^2-3x=4\)
- \(12-7x+x^2=0\)
- \(2x-1+3x^2=0\)
- \(x\left(x+7\right)+6=0\)
- \(2x^2-4x=0\)
- \(x\left(4x+5\right)=-1\)
- \(2-x=3x^2\)
- \(6x^2+3x=0\)
Solutions
- \(x=2\ \mathrm{or}\ x=5\)
- \(x=-5\ \mathrm{or}\ x=3\)
- \(x=-1\ \mathrm{or}\ x=4\)
- \(x=3\ \mathrm{or}\ x=4\)
- \(x=-1\ \mathrm{or}\ x=\frac{1}{3}\)
- \(x=-6\ \mathrm{or}\ x=-1\)
- \(x=0\ \mathrm{or}\ x=2\)
- \(x=-1\ \mathrm{or}\ x=-\frac{1}{4}\)
- \(x=-1\ \mathrm{or}\ x=\frac{2}{3}\)
- \(x=-\frac{1}{2}\ \mathrm{or}\ x=0\)
Simultaneous Equations
Solve the following pairs of equations. Remember – you can check your answer by substituting your \(x\) and \(y\) into the equations to see if they balance:
- \(\left\{\begin{align*}x+y&=12\\x-y&=6\end{align*}\right.\)
- \(\left\{\begin{align*}2x+y&=10\\x-y&=2\end{align*}\right.\)
- \(\left\{\begin{align*}4x+y&=10\\3x+y&=9\end{align*}\right.\)
- \(\left\{\begin{align*}2x+y&=7\\3x+y&=10\end{align*}\right.\)
- \(\left\{\begin{align*}2x+3y&=19\\2x+y&=9\end{align*}\right.\)
Solutions
- \(x=9,\ y=3\)
- \(x=4,\ y=2\)
- \(x=1,\ y=6\)
- \(x=3,\ y=1\)
- \(x=2,\ y=5\)
Changing the Subject
Make \(x\) the subject of each of these formulae. Hint: when \(x\) appears in more than one place in the formula, collect the terms involving \(x\) on one side of the equation and move the other terms to the other side. Then factorise out the common factor of \(x\) before making \(x\) the subject.
- \(y=7x-1\)
- \(y=\frac{x+5}{4}\)
- \(4y=\frac{x}{3}-2\)
- \(y=\frac{4\left(3x-5\right)}{9}\)
- \(ax+3=bx+c\)
- \(3\left(x+a\right)=k\left(x-2\right)\)
- \(y=\frac{2x+3}{5x-2}\)
- \(\frac{x}{a}=1+\frac{x}{b}\)
Solutions
- \(x=\frac{y+1}{7}\)
- \(x=4y-5\)
- \(x=12y+6\)
- \(x=\frac{3}{4}y+\frac{5}{3}\)
- \(x=\frac{c-3}{a-b}=\frac{3-c}{b-a}\)
- \(x=-\frac{3a+2k}{3-k}=\frac{3a+2k}{k-3}\)
- \(x=\frac{3+2y}{5y-2}\)
- \(x=\frac{ab}{b-a}\)
More Suggestions
- Make sure you are highly fluent in GCSE Grade 7/8/9 skills. Use the following questions from Maths Genie: https://www.mathsgenie.co.uk/gcse.html.
- Play the Bridge It! Gamefrom MEI: https://mei.org.uk/bridgeit. Bridge It! is built in Flash and so will not run on devices that do not allow Flash applications. It is best played on a laptop or desktop using a mouse or finger pad.
- Work through some Senior Maths Challenge questions from UKMT: Senior Mathematical Challenge | UK Mathematics Trust (ukmt.org.uk). These are excellent for developing your problem-solving skills.