Exact calculation with fractions, surds and π - Edexcel GCSE Maths (Calculator) Exam Questions

Practise 19 exam-style questions on Exact calculation with fractions, surds and \\\\u03c0 for Edexcel GCSE Maths (Calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

19 questions
1–5 marks each
Instant marking
No sign-up needed

19 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

Simplify fully \(\sqrt{48} + \sqrt{27} - \sqrt{12}\)

Model answer

\(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)
\(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\)
\(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)
\(4\sqrt{3} + 3\sqrt{3} - 2\sqrt{3} = 5\sqrt{3}\)

Explanation

Write each surd as a multiple of \(\sqrt{3}\) by taking out the largest square factor. Surds with the same root can then be collected like algebra: \(4\sqrt{3} + 3\sqrt{3} - 2\sqrt{3} = 5\sqrt{3}\). You cannot add the numbers under the roots: \(\sqrt{48} + \sqrt{27}\) is not \(\sqrt{75}\).

Mark scheme

1 mark
M1 for two of \(4\sqrt{3}\), \(3\sqrt{3}\), \(2\sqrt{3}\)
correct answer with no working scores full marks
1 mark
A1 for \(5\sqrt{3}\)
cao

Question 2

3 marks

A water tank is \(\frac{3}{8}\) full.
After 55 litres of water are added, the tank is \(\frac{5}{6}\) full.

Work out the capacity of the tank.

Model answer

\(\frac{5}{6} - \frac{3}{8} = \frac{20}{24} - \frac{9}{24} = \frac{11}{24}\)
So \(\frac{11}{24}\) of the tank is 55 litres.
\(\frac{1}{24}\) of the tank is \(55 \div 11 = 5\) litres.
Capacity \(= 5 \times 24 = 120\) litres

Explanation

The 55 litres filled the gap between \(\frac{3}{8}\) and \(\frac{5}{6}\) of the tank, so first subtract the fractions using a common denominator of 24. Then find one twenty-fourth and multiply by 24. A common mistake is to work out \(\frac{11}{24}\) of 55 instead of dividing 55 by \(\frac{11}{24}\).

Mark scheme

1 mark
M1 for \(\frac{5}{6} - \frac{3}{8}\) or \(\frac{11}{24}\)
oe, e.g. \(\frac{20}{24} - \frac{9}{24}\); correct answer with no working scores full marks
1 mark
M1 for \(55 \div \frac{11}{24}\)
oe, e.g. \(55 \div 11 \times 24\)
1 mark
A1 for 120 litres
cao

Question 3

3 marks

A solid hemisphere has radius \(3\sqrt{2}\) cm.

Work out the exact volume of the hemisphere.
Give your answer in the form \(a\sqrt{2}\pi\), where a is an integer.

Volume of a sphere = \(\frac{4}{3}\pi r^3\)

Model answer

\((3\sqrt{2})^3 = 3^3 \times (\sqrt{2})^3 = 27 \times 2\sqrt{2} = 54\sqrt{2}\)
Volume \(= \frac{1}{2} \times \frac{4}{3} \times \pi \times 54\sqrt{2} = \frac{2}{3} \times 54\sqrt{2}\pi = 36\sqrt{2}\pi\) cm³

Explanation

A hemisphere is half a sphere, so its volume is \(\frac{2}{3}\pi r^3\). Cube the 3 and the \(\sqrt{2}\) separately: \((\sqrt{2})^3 = \sqrt{2} \times \sqrt{2} \times \sqrt{2} = 2\sqrt{2}\). A common mistake is to write \((\sqrt{2})^3 = 2\) or \(3\sqrt{2}\).

Mark scheme

1 mark
M1 for \((3\sqrt{2})^3 = 54\sqrt{2}\)
oe, e.g. \(27 \times 2\sqrt{2}\); correct answer with no working scores full marks
1 mark
M1 for \(\frac{1}{2} \times \frac{4}{3} \times \pi \times \text{their } (3\sqrt{2})^3\)
oe, e.g. \(\frac{2}{3}\pi(3\sqrt{2})^3\)
1 mark
A1 for \(36\sqrt{2}\pi\) cm³
cao; do not accept 159.9... as the final answer

Question 4

4 marks

Show that
\[\frac{\sqrt{3} + 1}{\sqrt{3} - 1} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} = 4\]

Model answer

Common denominator: \((\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2\)
\(\dfrac{(\sqrt{3} + 1)^2 + (\sqrt{3} - 1)^2}{2}\)
\((\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}\)
\((\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}\)
\(\dfrac{(4 + 2\sqrt{3}) + (4 - 2\sqrt{3})}{2} = \dfrac{8}{2} = 4\)

Explanation

Add the fractions over the common denominator \((\sqrt{3} - 1)(\sqrt{3} + 1)\), which is a difference of two squares equal to 2. The \(2\sqrt{3}\) terms cancel when the two squared brackets are added. In a "show that" question every step must be written down; a calculator check alone earns no marks.

Mark scheme

1 mark
M1 for a common denominator \((\sqrt{3} - 1)(\sqrt{3} + 1) = 2\), or for rationalising one fraction correctly
1 mark
M1 for \((\sqrt{3} + 1)^2 = 4 + 2\sqrt{3}\) or \((\sqrt{3} - 1)^2 = 4 - 2\sqrt{3}\)
oe, e.g. \(2 + \sqrt{3}\) or \(2 - \sqrt{3}\) as a rationalised fraction
1 mark
M1 for the sum of the numerators \(= 8\), or for \((2 + \sqrt{3}) + (2 - \sqrt{3})\)
1 mark
C1 for a complete, correct argument leading to 4
answer only, or calculator values, score 0

Ready to try the rest? Every answer you write is marked against the mark scheme, so you can see exactly where the marks are.

AI Tutor Hints first, answers after you try

Hi! Stuck? Ask me for a hint and I’ll help you work it out. Once you’ve checked your answer I can explain exactly where the marks are.