A-Level Maths Preparation

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:

  1. \(3x-2y+4y\)
  2. \(5-3y-6y-2\)
  3. \(5x+2y-4y-x^2\)
  4. \(x^2+3x^2-4x^2+5x\)
  5. \(2y^2-y(x-y)\)
  6. \(8pq-9p^2-3pq\)
  7. \(x^3-2x^2+x^2-4x+5x+7\)
  8. \(3x(x-2)+4(3x-5)\)
  9. \(7+3(x-1)\)
  10. \(7b(a+2)-a(3b+3)\)
Solutions
  1. \(3x+2y\)
  2. \(3-9y\)
  3. \(5x-2y-x^2\)
  4. \(5x\)
  5. \(3y^2-xy\)
  6. \(5pq-9p^2\)
  7. \(x^3-x^2+x+7\)
  8. \(3x^2+6x-20\)
  9. \(3x+4\)
  10. \(4ab-3a+14b\)

Solving Linear Equations

Find the value of \(x\) in the following equations:

  1. \(2x-5=11\)
  2. \(8=7+3x\)
  3. \(-7=2x-10\)
  4. \(\frac{x}{5}=7\)
  5. \(\frac{x}{10}=5\)
  6. \(\frac{x}{2}=\frac{1}{3}\)
  7. \(\frac{3x}{2}=-5\)
  8. \(x-3=3x+7\)
  9. \(5x-4=3-x\)
  10. \(5x-16=16-2x\)
Solutions
  1. \(x=8\)
  2. \(x=\frac{1}{3}\)
  3. \(x=\frac{3}{2}\)
  4. \(x=35\)
  5. \(x=50\)
  6. \(x=\frac{2}{3}\)
  7. \(x=-\frac{10}{3}\)
  8. \(x=-5\)
  9. \(x=\frac{7}{6}\)
  10. \(x=\frac{32}{7}\)

Expanding Brackets

Expand and collect like terms in each of the following:

  1. \((x+3)(x-2)\)
  2. \((x-5)(x-1)\)
  3. \((2x+y)(x-y)\)
  4. \((2x-3)(3x-1)\)
  5. \((2x+1)(x+2)\)
  6. \((x+2)(x-2)\)
  7. \(2(x+3)(x-1)\)
  8. \((2x-3)(2x+3)\)
  9. \((5-x)(5+x)\)
  10. \((x+7)^2\)
Solutions
  1. \(x^2+x-6\)
  2. \(x^2-6x+5\)
  3. \(2x^2-xy-y^2\)
  4. \(6x^2-11x+3\)
  5. \(2x^2+5x+2\)
  6. \(x^2-4\)
  7. \(2x^2+4x-6\)
  8. \(4x^2-9\)
  9. \(25-x^2\)
  10. \(x^2+14x+49\)

Factorising

Factorise each of the following. Note – you can check your answers by expansion:

  1. \(x^2+3x+2\)
  2. \(x^2-3x+2\)
  3. \(x^2+5x+6\)
  4. \(x^2+7x+6\)
  5. \(x^2+4x\)
  6. \(x^2-x-12\)
  7. \(x^2-2x-3\)
  8. \(2x^2+3x-5\)
  9. \(x^2-9\)
  10. \(4x^2-1\)
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:

  1. \(b\times 5b^5\)
  2. \(3c^2\times 2c^5\)
  3. \(b^2c\times bc^3\)
  4. \(2n^6\times\left(-6n^2\right)\)
  5. \(8n^8\div\left(2n^3\right)\)
  6. \(d^{11}\div d^9\)
  7. \(\left(a^3\right)^2\)
  8. \(\left(-d^4\right)^3\)
  9. \(\left(25g^{12}\right)^\frac{1}{2}\)
  10. \({\left(64h^{-3}\right)}^\frac{1}{3}\)
Solutions
  1. \(5b^6\)
  2. \(6c^7\)
  3. \(b^3 c^4\)
  4. \(-12n^8\)
  5. \(4n^5\)
  6. \(d^2\)
  7. \(a^6\)
  8. \(-d^{12}\)
  9. \(5g^6\)
  10. \(4h^{-1}\)

Solving Quadratic Equations

Solve the following equations:

  1. \(x^2+10-7x=0\)
  2. \(15-x^2-2x=0\)
  3. \(x^2-3x=4\)
  4. \(12-7x+x^2=0\)
  5. \(2x-1+3x^2=0\)
  6. \(x\left(x+7\right)+6=0\)
  7. \(2x^2-4x=0\)
  8. \(x\left(4x+5\right)=-1\)
  9. \(2-x=3x^2\)
  10. \(6x^2+3x=0\)
Solutions
  1. \(x=2\ \mathrm{or}\ x=5\)
  2. \(x=-5\ \mathrm{or}\ x=3\)
  3. \(x=-1\ \mathrm{or}\ x=4\)
  4. \(x=3\ \mathrm{or}\ x=4\)
  5. \(x=-1\ \mathrm{or}\ x=\frac{1}{3}\)
  6. \(x=-6\ \mathrm{or}\ x=-1\)
  7. \(x=0\ \mathrm{or}\ x=2\)
  8. \(x=-1\ \mathrm{or}\ x=-\frac{1}{4}\)
  9. \(x=-1\ \mathrm{or}\ x=\frac{2}{3}\)
  10. \(x=-\frac{1}{2}\ \mathrm{or}\ x=0\)

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:

  1. \(\left\{\begin{align*}x+y&=12\\x-y&=6\end{align*}\right.\)
  2. \(\left\{\begin{align*}2x+y&=10\\x-y&=2\end{align*}\right.\)
  3. \(\left\{\begin{align*}4x+y&=10\\3x+y&=9\end{align*}\right.\)
  4. \(\left\{\begin{align*}2x+y&=7\\3x+y&=10\end{align*}\right.\)
  5. \(\left\{\begin{align*}2x+3y&=19\\2x+y&=9\end{align*}\right.\)
Solutions
  1. \(x=9,\ y=3\)
  2. \(x=4,\ y=2\)
  3. \(x=1,\ y=6\)
  4. \(x=3,\ y=1\)
  5. \(x=2,\ y=5\)

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.

  1. \(y=7x-1\)
  2. \(y=\frac{x+5}{4}\)
  3. \(4y=\frac{x}{3}-2\)
  4. \(y=\frac{4\left(3x-5\right)}{9}\)
  5. \(ax+3=bx+c\)
  6. \(3\left(x+a\right)=k\left(x-2\right)\)
  7. \(y=\frac{2x+3}{5x-2}\)
  8. \(\frac{x}{a}=1+\frac{x}{b}\)
Solutions
  1. \(x=\frac{y+1}{7}\)
  2. \(x=4y-5\)
  3. \(x=12y+6\)
  4. \(x=\frac{3}{4}y+\frac{5}{3}\)
  5. \(x=\frac{c-3}{a-b}=\frac{3-c}{b-a}\)
  6. \(x=-\frac{3a+2k}{3-k}=\frac{3a+2k}{k-3}\)
  7. \(x=\frac{3+2y}{5y-2}\)
  8. \(x=\frac{ab}{b-a}\)

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