All AQA Level 2 Further Maths topic questions

AQA Level 2 Further Maths · Topic N

Number questions

Practice questions covering number from this specification. Every question below is the real AQA question as it is printed on the paper, with its mark scheme.

Questions to attempt

6 questions on Number, each with its mark scheme and worked method. Reveal the answer when you have had a go.

N1 · Knowledge and use of numbers and the number system including fractions, decimals, percentages, ratio, proportion and order of operations
Q1
ORIGINAL
1
A sum of $\pounds 294$ is divided in the ratio $5:2$ . Work out the larger share.
[2 marks]
N1 · Knowledge and use of numbers and the number system including fractions, decimals, percentages, ratio, proportion and order of operations
Q2
ORIGINAL
2
Work out $\dfrac{3}{5}+0.35$ .
[1 mark]
N2 · The product rule for counting
Q3
ORIGINAL
3
A code consists of one of $5$ letters, followed by one of $4$ digits, followed by one of $3$ symbols. Repetition is allowed. Work out the number of possible codes.
[2 marks]
N2 · The product rule for counting
Q4
ORIGINAL
4
A meal has a choice of $4$ main courses and $3$ desserts. How many different main-course-and-dessert meals are possible?
[1 mark]
N3 · Manipulation of surds, including rationalising the denominator; the use of surds in exact calculations
Q5
ORIGINAL
5
Rationalise the denominator of $\dfrac{5}{\sqrt{7}-2}$ .
[3 marks]
N3 · Manipulation of surds, including rationalising the denominator; the use of surds in exact calculations
Q6
ORIGINAL
6
Simplify $\sqrt{72}$ .
[2 marks]

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Printable question pack — N NumberAll 30 questions across 3 specification points — 24 more than the 6 above — with the full mark scheme. Guest accessible · browser print.

N Number revision notes

Real past-paper questions

16 questions shown as printed, 3 reference-only · newest series first

Question and mark-scheme extracts © AQA. Reproduced for educational use; the original paper is linked on every question. Contact hello@oscarsongtutoring.co.uk for removal.

AQA 8365 · June 2025 · Paper 1 · Q6 · 3 marks
3 marks
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Examiner report
This question was very well answered. A number of different approaches were seen but Alternative method 1 was the most efficient. An unusual method used was to write each surd as the product of its prime factors and then to ‘cancel’ by root 3
Text version
Consider the exact surd expression (sqrt(75) - sqrt(27)) / sqrt(48).
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AQA 8365 · June 2025 · Paper 1 · Q9 · 3 marks
3 marks
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Examiner report
This topic continues to be challenging for many students. Awarding of part marks was not common. An SC1 mark was available for students who had an answer of 81. This was for those who did not realise that digits could only be used once in each combination.
Text version
Three-digit integers are made using three different digits chosen from 1, 2, 3, 4, 5, 6, 7, 8 and 9.
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AQA 8365 · June 2024 · Paper 1 · Q1 · 2 marks
2 marks
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Examiner report
Most students scored both marks. Some made errors in their working to score 1 and some were able to get a mark for values embedded in incorrect working (as this is question 1). On a non-calculator paper students should realise that much will cancel out to leave an easy calculation, but many get tangled in standard form or surd answers.
Text version
The value of t is 2.42 x 10^3.
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AQA 8365 · June 2024 · Paper 1 · Q18 · 4 marks
4 marks
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Examiner report
It was possible to rationalise this expression by multiplying first by 3 or 5 which gets rid of a surd on the top and the bottom. They then rationalised this. It is more work but can be done for full marks (most trying to do it this way didn’t manage to get to full marks though). A surprising number of students think that 18 ÷ 2 is 6. Too many students still don’t multiply by the correct surd combination to rationalise and so don’t score anything. Lots of correct answers though on this question.
Text version
This question concerns rationalising a surd denominator.
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AQA 8365 · June 2024 · Paper 2 · Q9 · 3 marks
3 marks
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Examiner report
Another good discriminator with quite a good spread of marks. Most successful attempts involved the product of four integers although some did equivalent work by using the sum of two products of four integers. Some did not include the condition that only multiples of 5 were needed while others did not count zero as an integer. A few misread the question and gave the answer 9795, the largest 4-digit integer that satisfied the criteria.
Text version
A set contains 4-digit integers whose first digit is greater than 6 and whose second digit is less than 8.
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AQA 8365 · June 2023 · Paper 1 · Q4 · 5 marks
5 marks
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Examiner report
Part (a) and part (b) were marked together in case a sketch had been drawn containing information for both parts. Both parts were answered well by most candidates. Errors in part (a) came from numerical errors mainly. A few candidates had change in x at the top and change in y at the bottom. 3 1 was not awarded the A mark as it was not complete. Most of the errors in part (b) came from not being able to simplify root 40. There were very few examples of candidates using the incorrect values from part (a) and gaining the ft mark in part (b).
Text version
S(7, 2) and T(5, -4) lie on a straight line.
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AQA 8365 · June 2023 · Paper 1 · Q10 · 4 marks
4 marks
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Examiner report
Most candidates completed this correctly. A common error was getting 9+5 in the denominator. Some candidates lost the final mark as they simplified incorrectly. There are still some candidates who don’t know what to multiply by to begin the method.  2 5 was condoned.
Text version
Consider (1 + sqrt(5))/(3 - sqrt(5)).
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AQA 8365 · June 2023 · Paper 2 · Q16 · 4 marks
4 marks
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Examiner report
Many students accurately factorised the cubic equation, often using algebraic division. It was very common to see correct use of the quadratic formula to solve their equation, but a large number of students chose rounded decimals as their answer rather than the exact values required. Occasionally students who had worked correctly omitted the solution from the factor given in the question. The question was a good discriminator.
Text version
(2x - 3) is a factor of 6x^3 - 25x^2 + 28x - 6.
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AQA 8365 · June 2023 · Paper 2 · Q22 · 3 marks
3 marks
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Examiner report
There were some fully correct solutions seen following many different approaches. Some students tried to list some or all of the possible integers. Relevant values such as 72, 12 or 54 were often seen but frequently 36 or 48 were given as incorrect answers.
Text version
Five-digit integers use each of 1, 2, 7, 8 and 9 exactly once. The integer must be greater than 40000 and odd.
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AQA 8365 · June 2022 · Paper 1 · Q1 · 3 marks
3 marks
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Examiner report
Not surprisingly this question was answered well. Quite a few students worked out the difference using our SC1 method but went on to get full marks. Most students using an algebraic method followed Alternative method 1 and did the method well but some dropped careless marks when expanding the brackets. A common error came from incorrectly working out 6 – 1.2 . Only a small number of students followed Alt 2.
Text version
The expression x + 1 is increased by 20%. Its new value equals x + 6.
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AQA 8365 · June 2022 · Paper 1 · Q10 · 3 marks
3 marks
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Examiner report
This question was answered well by many candidates who wrote the answer in a neat form. A common error was 3 ÷ 6 = 1 3 . A number of candidates multiply by 3 + 3 . There were also a few candidates who didn’t simplify fully so lost the last mark.
Text version
Consider sqrt(3)/(3+sqrt(3)).
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AQA 8365 · June 2022 · Paper 2 · Q7 · 2 marks
2 marks
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Examiner report
This was a new topic compared to the previous version of this specification and the question was answered quite well. Many students gave a response that showed some understanding with 120, 360, 600 and 720 being seen quite often. A common error was to multiply all the given digits. Others worked out 240 but then did further work and gave a different answer.
Text version
Form six-digit integers between 200000 and 400000 using each of the digits 1,2,3,5,8,9 at most once.
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AQA 8365 · November 2021 · Paper 1 · Q8 · 2 marks
2 marks
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Text version
Consider sqrt(3)(sqrt(75)+sqrt(48)).
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AQA 8365 · November 2021 · Paper 1 · Q20 · 3 marks
3 marks
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Text version
The date 14/09/2006 supplies the digits for a four-digit code. Digits in the code must all be different and the first digit cannot be zero.
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AQA 8365 · November 2021 · Paper 2 · Q2 · 4 marks
4 marks
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Text version
Solve the two equations in the parts below.
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AQA 8365 · sample1 · Paper 2 · Q18 · 3 marks
3 marks
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Text version
Use the digits 2, 4, 6, 7 and 8 without repetition.
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References open the specific AQA question paper when it is available. Missing, login-only or unverified question-paper sources are labelled on the affected row. Question content remains on the exam board's website.

November 20211 question

Paper 1 · Q215 marks · Non-calculator · calculate · hardView question in paper (p.20)Mark scheme (p.24)

sample12 questions

Paper 1: Q7 (3 marks), Q14 (4 marks)

Question extracts are shown for study alongside a link to the original AQA paper. Reference-only rows carry series, paper, question number and marks.

Number questions

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