All AQA Level 2 Further Maths topic questions

AQA Level 2 Further Maths · Topic A

Algebra questions

Practice questions covering algebra 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 Algebra, each with its mark scheme and worked method. Reveal the answer when you have had a go.

A1 · The basic processes of algebra, including use of the associative, commutative and distributive laws
Q1
ORIGINAL
1
Use algebraic laws to work out $25\times47+25\times55-25\times2$ without evaluating any product separately.
[2 marks]
A2 · Definition of a function; notation f(x)
Q2
ORIGINAL
2
Given $f(x)=x^2-4x+1$ , work out $f(-2)$ and simplify $f(a+1)$ .
[3 marks]
A3 · Domain and range of a function
Q3
ORIGINAL
3
For $f(x)=x^2+2$ with domain $-2\leq x\leq3$ , work out the range of $f$ .
[3 marks]
A4 · Composite functions (the result of two or more functions acting in succession)
Q4
ORIGINAL
4
Let $f(x)=x-4$ and $g(x)=3x^2$ . Work out and simplify $gf(x)$ .
[3 marks]
A5 · Inverse functions (domains chosen for f to make f one-one)
Q5
ORIGINAL
5
The function $f(x)=4-(x-2)^2$ has domain $x\geq2$ . Work out $f^{-1}(x)$ and state its domain.
[3 marks]
A6 · Expanding brackets and collecting like terms
Q6
ORIGINAL
6
Expand and simplify $(2x-3)(x+5)-(x-1)^2$ .
[3 marks]

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Printable question pack — A AlgebraAll 220 questions across 22 specification points — 214 more than the 6 above — with the full mark scheme. Guest accessible · browser print.

A Algebra revision notes

Real past-paper questions

113 questions shown as printed, 28 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 · Q1 · 5 marks
5 marks
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Examiner report
This question was well answered. Some students used incorrect terminology or made an arithmetic error. Question 1 (b) Writing the inequality with x instead of g(x) was a common error. Others used incorrect inequality symbols. Question 1 (c) This question was very well answered. A small number of students left their answer in terms of y. Fewer errors were made by students using Alternative method 1.
Text version
This question is about functions, their domains and ranges, and an inverse function.
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AQA 8365 · June 2025 · Paper 1 · Q3 · 3 marks
3 marks
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Most students were awarded at least two marks. A significant number gave the answer as 167 instead of 168.
Text version
A linear sequence starts 2, 5, 8, 11, ...
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AQA 8365 · June 2025 · Paper 1 · Q5 · 3 marks
3 marks
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Examiner report
Most fully correct responses came from students using Alternative method 1. Those using Alternative method 2 quite often made errors when substituting in a negative value. Very few students used alternative methods 3 or 4.
Text version
A sketch of y = x^2 + bx + c crosses the x-axis at (-1, 0) and (5, 0).
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AQA 8365 · June 2025 · Paper 1 · Q8 · 1 mark
1 mark
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This question was very well answered with most students being able to apply index laws correctly.
Text version
Consider ((x^(9/2)) / (x^(1/2)))^3.
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AQA 8365 · June 2025 · Paper 1 · Q11 · 4 marks
4 marks
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Examiner report
This question was a good discriminator. A significant number of students did not use the common factor of x which often led to only two marks being awarded. The quadratic formula was the most straightforward way to work out the solutions that were in surd form. Some gave answers involving the square root of 68 and therefore had not given their answers simplified as requested. Some students tried to use the quadratic formula or complete the square on the original cubic equation.
Text version
Solve the cubic equation x^3 + 10x^2 + 8x = 0.
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AQA 8365 · June 2025 · Paper 1 · Q13 · 1 mark
1 mark
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This question was very well answered.
Text version
Four sketches are offered for y = 5^x: A is a positive decreasing curve; B is a negative decreasing curve; C is a positive increasing curve through the y-axis above the origin; D is a negative increasing curve.
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AQA 8365 · June 2025 · Paper 1 · Q14 · 4 marks
4 marks
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Examiner report
Those students who factorised correctly were often able to produce a fully correct response. Some students chose to use multiplication and this made the working unnecessarily complicated and this approach usually ran into difficulties.
Text version
Consider the algebraic expression ((2x^2 - 4x - 6)/(x^2 - 3x)) divided by ((4x + 4)/(7x^2)).
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AQA 8365 · June 2025 · Paper 1 · Q15 · 3 marks
3 marks
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This question was very well answered.
Text version
(x - 3)(x + 4)(x + p) is identically equal to x^3 + qx^2 + rx + 24.
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AQA 8365 · June 2025 · Paper 1 · Q17 · 3 marks
3 marks
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Examiner report
Fewer students were attempting full expansions than in previous years. A significant number of students made an error when working out one of the relevant coefficients and were only awarded one mark. The question was a good discriminator.
Text version
The expansions of (4a + x)^4 and (5 - 2x)^3 have the same coefficient of x^3.
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AQA 8365 · June 2025 · Paper 1 · Q18 · 6 marks
6 marks
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This question was very well answered. Students should take extra care with their working when asked to ‘show that’ something is true. Question 18 (b) This question was more challenging and was a good discriminator. Most students awarded full marks used Alternative method 1. Some made errors when differentiating while others gave their answer as x = 15 rather than continuing to find the value of A. There were a relatively high number of non-attempts.
Text version
A 60 cm wire forms three sides of a rectangle and is attached to a fixed rod along the fourth side. Each of the two perpendicular wire sides has length x cm and the remaining wire side has length y cm. The rectangle has area A cm^2.
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AQA 8365 · June 2025 · Paper 1 · Q19 · 5 marks
5 marks
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This question was well answered with algebraic division the most commonly used starting point. Some students ignored the given factor and used Alternative method 4. Some students would have benefitted from checking their answer by expanding their brackets. Question 19 (b) The unfamiliar nature of this question will have contributed to the fact that many found this difficult. There were a high number of non-attempts and some students did a lot of working but were not awarded any marks.
Text version
This question concerns factorisation and a polynomial value with exactly two prime factors.
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AQA 8365 · June 2025 · Paper 1 · Q22 · 6 marks
6 marks
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Examiner report
Some very good responses were seen. Some students attempted a solution from accurate drawing and although this was sometimes successful the solutions were not easy to find using this approach. A significant majority were awarded at least two marks. © 2025 AQA
Text version
A circle has centre (2, 3) and radius 5. A straight line passes through (3, 7) and has gradient 2.
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AQA 8365 · June 2025 · Paper 1 · Q23 · 5 marks
5 marks
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Examiner report
A significant number of students were awarded four marks. A common error was to ignore the lower bound. Students who used the quadratic formula or completing the square on the quadratic inequality rarely managed to get to the answer and on this paper most of the quadratics will factorise into double brackets.
Text version
A trapezium has parallel sides (3x - 8) cm and (x + 5) cm, and perpendicular height 2x cm. Its area is less than 22 cm^2.
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AQA 8365 · June 2025 · Paper 2 · Q1 · 3 marks
3 marks
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This question was very well answered.
Text version
Sequence A has nth term 6n/(7n + 25). Sequence B has nth term 20/(n + 4). The fifth term in sequence A is also a term in sequence B.
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AQA 8365 · June 2025 · Paper 2 · Q4 · 2 marks
2 marks
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Examiner report
Many students gave an answer involving n rather than n2 while others gave a list of integers. Another common error was to only have one side of the inequality correct, usually n2 ⩽ 25 25 ⩽ n2 ⩽ 25 was seen quite often and this answer was not awarded any marks.
Text version
-5 <= n <= 5.
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AQA 8365 · June 2025 · Paper 2 · Q7 · 5 marks
5 marks
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This question was very well answered. Question 7(b) This question was well answered. Some students used c 7 instead of 7 – c but the most common error was when rearranging the terms in a correct linear equation. © 2025 AQA
Text version
This question contains two equations involving indices.
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AQA 8365 · June 2025 · Paper 2 · Q8 · 3 marks
3 marks
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Examiner report
Omission of brackets was the most common error, and students should take extra care with their working when asked for a proof. Having 27k – 9 instead of 27k + 9 was seen quite often. Most students who had correct working leading to 27k + 9 showed sufficient further working to be awarded the final mark as well.
Text version
k is a positive integer.
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AQA 8365 · June 2025 · Paper 2 · Q9 · 2 marks
2 marks
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A challenging question for many students, partly due to the unfamiliar nature of the question. Some students worked with equations in x and p and many of these did not go on to substitute 1 for x. Some used inequalities, presumably because there were inequalities in the domain of the function. A majority of students were awarded full marks with only a few not able to solve a correct linear equation in p.
Text version
f(x) = x^2 + px for x <= 1, and f(x) = 2p - x/2 for x > 1. The supplied sketch shows the two pieces meeting at x = 1.
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AQA 8365 · June 2025 · Paper 2 · Q10 · 2 marks
2 marks
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This question was very well answered.
Text version
The first four terms of a quadratic sequence are -1, 11, 31 and 59.
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AQA 8365 · June 2025 · Paper 2 · Q11 · 6 marks
6 marks
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This question was well answered. Students should ensure their answer is written unambiguously especially when the fraction has an algebraic denominator. Question 11(b) Many students did not use (x – 1) 8 as a factor. Many of those who did could not work out the other factor or did not simplify this factor. Question 11(c) This question was quite well answered and was a good discriminator. Common errors included dividing by 2 at the outset and leaving the answer in surd form.
Text version
Simplify or factorise each expression fully.
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AQA 8365 · June 2025 · Paper 2 · Q13 · 4 marks
4 marks
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This question was well answered and most students who showed correct work after the first step went on to produce a fully correct response. © 2025 AQA
Text version
x = 3k^2/(5k^2 - 2).
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AQA 8365 · June 2025 · Paper 2 · Q14 · 5 marks
5 marks
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Examiner report
Most students who started by writing Pythagoras’ theorem as (3w + 2)2 = (2w + 5)2 + w2 subsequently showed enough correct working to be awarded both marks. Mistakes were more likely when students attempted to expand straight away or wrote Pythagoras’ theorem in a less standard way. Question 14(b) Some errors were made when factorising or using the quadratic formula. The other common error was to link an expression to an incorrect trigonometric ratio. Overall, the question was well answered.
Text version
A right-angled triangle has perpendicular sides w cm and (2w + 5) cm, hypotenuse (3w + 2) cm, and angle x between the side (2w + 5) and the hypotenuse.
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AQA 8365 · June 2025 · Paper 2 · Q17 · 5 marks
5 marks
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Examiner report
Although many students needed more than one attempt, overall, the question was well answered. Most students showed their working as instructed. Many different approaches were seen including some matrix methods. The majority of these had the method shown and were correct. The question was quite a good discriminator.
Text version
Solve the simultaneous equations 2a + 4b + c = 12, 3a - b + 2c = -2, and a - 5b - 3c = 6.
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AQA 8365 · June 2025 · Paper 2 · Q19 · 4 marks
4 marks
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Examiner report
gf(x) was correct more often than f(2x). Common errors for f(2x) included omitting brackets or working out 2f(x). Some students did not include the + 8 when attempting gf(x). Most students who equated two correct expressions solved correctly but some made a sign error.
Text version
f(x) = 4x^3 - 7 and g(x) = 6x + 8.
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AQA 8365 · June 2024 · Paper 1 · Q3 · 1 mark
1 mark
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About 80% of students gained the mark. There was a mix of incorrect answers including a small number of students choosing more than one option.
Text version
The nth term of a sequence is (3n + 4)/n.
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AQA 8365 · June 2024 · Paper 1 · Q8 · 3 marks
3 marks
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Examiner report
Too many students left the equation factorised in brackets without solving so lost the final mark. Similarly, quite a large group of students cancelled through by x without ever stating that it could be a solution and could only gain 1 mark.
Text version
This question requires a cubic equation to be solved by factorising.
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AQA 8365 · June 2024 · Paper 1 · Q10 · 5 marks
5 marks
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Lots of students managed to get to the two simultaneous equations but then made careless errors costing them one or both A marks. It is a concern how many students couldn’t get the coordinates of point P.
Text version
A function f is defined by f(x) = -x/2 + 21 for 0 <= x <= 2 and f(x) = ax^2 + bx for 2 < x <= 6. The two pieces meet at P, and Q = (6, 12) lies on the quadratic piece.
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AQA 8365 · June 2024 · Paper 1 · Q11 · 6 marks
6 marks
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[11b] Students that followed alt method 1 virtually always continued on to gain full marks. Students trying to use other methods (possibly because they hadn’t realised that it was a factor theorem question) rarely gained full marks but quite often gained the first M mark. It was interesting to see a number of students using a grid reduction method to solve this question with a small number gaining full marks (on the whole most using this method rarely scored more than 1 mark though).
Text version
This question concerns polynomial factors.
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AQA 8365 · June 2024 · Paper 1 · Q13 · 4 marks
4 marks
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This question gave a good spread of marks. Lots miss 3 < x which costs them the B mark. Some lose marks (between 1 and 3) for using T&I – evidence of working needs to be seen. There were some careless errors in calculations. Some use = instead of < but most of these recover it in the answer line (some don’t though and then only score 1). 15(x – 2) < 30 was seen a few times which is equivalent to M2.
Text version
A patio is formed from two non-overlapping rectangles. One rectangle has dimensions 7 m by x m. The other has dimensions 8 m by (x - 3) m. The total area is less than 51 m^2.
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AQA 8365 · June 2024 · Paper 1 · Q14 · 4 marks
4 marks
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[14b] Surprisingly poorly done. Many students expand and then try to factorise or use the formula. Others solve incorrectly. Too many work it out correctly and then forget there are two answers so lose the mark. Some students found the turning point and should remember to read the question. Being only worth 1 mark would suggest it’s a lot easier than many of the students gave it credit for.
Text version
This question concerns completing the square and solving an equation.
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AQA 8365 · June 2024 · Paper 1 · Q17 · 3 marks
3 marks
$AQA 8365 · June 2024 · Paper 1 · Q17 · 3 marks — This question concerns simplifying an algebraic fraction.$
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Too many students think that 7(3x + 4) = 21x + 4. Alt method 1 is the most efficient method and most students who followed this method scored at least 2 marks. Fortunately, the vast majority followed our instruction to show that it simplified so there were very few cases of SC1 being awarded.
Text version
This question concerns simplifying an algebraic fraction.
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AQA 8365 · June 2024 · Paper 1 · Q23 · 5 marks
5 marks
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The slightly unusual B scheme worked well on this question. Due to the AO2 nature of this question there were lots of different ways to solve it and often combinations of different methods were used. Careless arithmetic cost many students.
Text version
The nth term of a quadratic sequence is an^2 - 5n + c, where a and c are integers. The first four terms are 2, x, 16, y.
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AQA 8365 · June 2024 · Paper 2 · Q2 · 2 marks
2 marks
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This question was not well answered. Many simplified the inequality to, for example, n2 = 7.2, but did not continue to work out integer values. Another common error was to only give the positive integer values of 1 and 2. Some students did not include zero while others gave an inequality for their answer.
Text version
The integer n satisfies 5n^2 + 2 < 38.
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AQA 8365 · June 2024 · Paper 2 · Q6 · 2 marks
2 marks
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Many students do not realise that the range for a quadratic function cannot always be worked out by using the smallest and largest x values in the domain. Most scored only one mark for 50 The question was a good discriminator with about a third of the students scoring both marks.
Text version
The function f is given by f(x) = 3x^2 + 2 with domain -1 <= x <= 4.
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AQA 8365 · June 2024 · Paper 2 · Q12 · 6 marks
6 marks
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Both parts were challenging for a significant number of students and this question was a good discriminator. Students could score marks for relevant working even if this was seen in the wrong part. In part (a) many presented their work well and with sufficient detail to show what was required. Many could expand one bracket correctly with most errors being made if their method involved expanding (7 – x)2. A significant number thought they needed to use the centre of the circle and others attempted a solution to the given quadratic equation without making any attempt to answer (a). In part (b) most attempted a solution using the quadratic formula with those using completing the square rarely being successful. Using –182 instead of (–18)2 was the most common error. Some students gave both x values as their answer without working out a y-coordinate. Most gave answers to two decimal places.
Text version
The circle (x - 2)^2 + (y + 3)^2 = 16 and the line y = 4 - x intersect at points A and B.
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AQA 8365 · June 2024 · Paper 2 · Q13 · 3 marks
3 marks
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Very well answered with many fully correct responses. Most students scored at least two marks. Some thought that the answer had to have a positive coefficient of x3.
Text version
This question concerns expansion of three linear factors.
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AQA 8365 · June 2024 · Paper 2 · Q16 · 3 marks
3 marks
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This question produced a good spread of marks and was a good discriminator. Those who tried to eliminate the denominator sometimes divided both parts of the numerator. Other errors included writing an incorrect power of x for 3 x and multiplying powers when they should have been adding them. Those who multiplied numerator and denominator by x were rarely able to score marks as processing errors were common.
Text version
This question concerns powers and roots.
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AQA 8365 · June 2024 · Paper 2 · Q18 · 4 marks
4 marks
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Examiner report
Most students eliminated the fraction correctly. Some sign errors were subsequently made when rearranging the terms. Writing 3m3k instead of 3mk3 was quite common. Transcription errors from one line of working to the next were also quite often made. Overall, the question was well answered but a number of responses were poorly presented. For example, when taking the cube root sometimes only the numerator was inside the cube root symbol.
Text version
The equation is m = (2k^3 - 7)/(5m - 3k^3).
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AQA 8365 · June 2024 · Paper 2 · Q19 · 4 marks
4 marks
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Many attempted a full expansion. The most common error was to have a in the two relevant terms rather than a2 and a4. After correctly writing 10 × 33 a significant number of students evaluated this to 90. Many set up incorrect equations in a and x rather than using the coefficients.
Text version
In the expansion of (3 + ax)^5, where a is a non-zero constant, eight times the coefficient of x^2 equals the coefficient of x^4.
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AQA 8365 · June 2024 · Paper 2 · Q21 · 6 marks
6 marks
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This question was answered quite well and there was a good spread of marks. Most worked out the inverse function correctly and knew how to start working out the composite function. Errors made included not including the 18 and expansion errors in 2 3 2       −x . Some forgot to add the inverse function to the composite function at the end.
Text version
The functions are f(x) = (2 - x)/3 and g(x) = 18x^2 + 15x. The expression f^(-1)(x) + gf(x) simplifies to ax^2 + bx + c.
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AQA 8365 · June 2023 · Paper 1 · Q1 · 4 marks
4 marks
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[1a] Far too many candidates substituted x = –5 into the function rather than realising the function itself equalled –5. This resulted in 0 marks. A number of candidates think that –5 – 1 is –4 unfortunately. There were a small number of SC1s from a likely miscopy.
Text version
The function f is given by f(x) = 2x + 1. The function g is given by g(x) = x^2.
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AQA 8365 · June 2023 · Paper 1 · Q2 · 2 marks
2 marks
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This question was done well by most candidates. Some did not factorise fully or made an error but could still gain B1. Quite a few candidates missed the final bracket off which was condoned. (3xy)(2x+7) was condoned for B2. There were instances of all three SC1s seen.
Text version
Consider the expression 6x^2y + 21xy.
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AQA 8365 · June 2023 · Paper 1 · Q5 · 3 marks
3 marks
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Alternative method one was the most elegant proof but not many candidates tried to use this method. This question was done well by most candidates however. It was also pleasing to see how many of them completed their proof with a closing statement (although we didn’t require one with this being so early in the paper). Too many candidates are dropping brackets though. Some of these managed to recover them by reaching 3n+3 but some didn’t. Being a proof it was necessary to show each stage of the process and some candidates lost marks by going straight from expanding the brackets to 3n+3 without showing any evidence of subtraction taking place.
Text version
X_n = (n - 1)(n + 1) and Y_n = (n + 1)(n + 2).
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AQA 8365 · June 2023 · Paper 1 · Q7 · 4 marks
4 marks
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Many correct solutions. Some candidates don’t know how to find the area of a triangle sadly which limited them to a maximum of 1 mark. Too many candidates were expanding 4( 1 2 y2x) as 4 x 1 2 y2 + or x 4x. A small number of candidates found the correct answer but then rationalised the root pi in the denominator. As long as this was done correctly we didn’t penalise this.
Text version
A cone has radius r and perpendicular height x. A triangular prism has cross-section base y and perpendicular height y, and length x. The cone volume is four times the prism volume. Use volume of cone = (1/3) x base area x perpendicular height.
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AQA 8365 · June 2023 · Paper 1 · Q8 · 6 marks
6 marks
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Too many candidates think that x − 2 25 is x – 5. A number of candidates ignored the guidance to not use trial and improvement and tried to draw a graph. This was not accepted as a valid method due to the lack of accuracy involved. Some candidates went for guesswork and managed to get the correct answers but without working this was worth no marks (although some of them still gained the first mark). Sadly a small number of candidates worked it all out correctly and then put the coordinates in the wrong order (so lost the last A mark).
Text version
A circle has centre (0,0) and radius 5. A line has equation 2y = x + 5.
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AQA 8365 · June 2023 · Paper 1 · Q9 · 4 marks
4 marks
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A large majority of candidates can do these very neatly and quickly and once again it regularly scored highly. The A mark was sometimes lost due to not writing y = on the answer line or for not rooting the term. Most candidates gained at least one mark.
Text version
w = (y^2 + 5)/(y^2 - 2).
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AQA 8365 · June 2023 · Paper 1 · Q13 · 3 marks
3 marks
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A number of candidates tried to split the question into two parts. This method can work well but too many candidates forgot about the 1 and either didn’t deal with it all or just added it to the numerator without considering the common denominator (resulting in 0 marks). Some candidates gained the answer but then did further working (many getting x −10 3 3 ) so lost the A mark.
Text version
Consider (2x - 3)/x - 1/(3x) + 1.
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AQA 8365 · June 2023 · Paper 1 · Q15 · 4 marks
4 marks
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This question was very poorly done and proved to be the hardest question on the paper. Most candidates tried to expand the brackets with just 2 terms and forgot about the middle terms of the expansion. Those that did expand properly usually worked it through for full marks. Most of the solutions that were awarded 1 mark were for either factorising an incorrect cubic equation or the SC1 for writing x = 0 which may have come from observation, incorrect working or a lucky guess. Some candidates chose to change the indices into surd form but this was rarely successful.
Text version
Consider the equation (x^(1/2) - x^(3/2))^2 = x^2 + x.
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AQA 8365 · June 2023 · Paper 1 · Q16 · 4 marks
4 marks
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There are still some candidates adding within terms for the expansion rather than multiplying ie. 6(12) + 12x2 which didn’t gain any marks. There were a large number of non-attempts on this question which would suggest that some centres are not teaching this topic. Some candidates attempted to expand the brackets rather than using the Binomial expansion which usually used up a lot of time and rarely delivered results.
Text version
The expansions of (1 + 12x)^4 and (a + 4x)^3 have the same coefficient of x^2.
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AQA 8365 · June 2023 · Paper 1 · Q19 · 3 marks
3 marks
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Examiner report
The B scheme worked very well for this question. Candidates not gaining two of the values could still gain a mark by completing the square or by showing an understanding of equating identities. Although the numbers used in this question were not easy ones to manipulate the question in general was done quite well.
Text version
8x^2 + 20x + n is identically equal to c(x + d)^2 + 3, where c, d and n are constants.
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AQA 8365 · June 2023 · Paper 2 · Q1 · 3 marks
3 marks
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Many fully correct solutions were seen with some making an arithmetic or sign error. A fairly common incorrect approach was to equate numerators and denominators to give two equations in d and two solutions.
Text version
Consider the equation (8d - 3)/(3d - 7) = 5/2.
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AQA 8365 · June 2023 · Paper 2 · Q2 · 4 marks
4 marks
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Part (a) was very well answered. The most common incorrect answers were n + 3.5 and 3.5n – 11.5. In part (b) most students managed to work out that the 36th term was the first negative term. The most frequent error was to give 36 as the answer, sometimes after –6 had been correctly evaluated. A misconception was to substitute n = –1 into 318 – 9n giving an answer of 327.
Text version
This question concerns two linear sequences.
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AQA 8365 · June 2023 · Paper 2 · Q7 · 3 marks
3 marks
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Many correct answers were seen in part (a) with the common incorrect responses being 4 and 4.5. In part (b) some students gave the correct inequality for the negative section of the quadratic curve and some stated the roots of the quadratic. Others tried to combine the two correct inequalities into a single invalid inequality such as –4 > x > 5.
Text version
An upward-opening quadratic y = ax^2 + bx + c crosses the x-axis at (-4,0) and (p,0). Its turning point has x-coordinate 0.5.
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AQA 8365 · June 2023 · Paper 2 · Q11 · 4 marks
4 marks
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A great deal of correct algebraic expansion work was seen. Occasionally students made errors and some students misinterpreted the second part of the expression and squared all the terms. A common incorrect approach was to expand both parts and then multiply the results rather than add them. Students sometimes miscopied their own work.
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Consider (2x^3 - 9)(3x^2 + 4) + x(x - 4)^2.
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AQA 8365 · June 2023 · Paper 2 · Q14 · 4 marks
4 marks
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Those who started by factorising and cancelling were usually more successful than those who expanded the numerator and the denominator, although this latter approach was seen more frequently. Those who expanded often simply cancelled by 3 or 3a or a and thought they had finished, not realising there were common factors to eliminate. A frequently seen incomplete answer was ( ) ( )( ) 20 2 8 2 a a a a − + − and some cancelled the factor of (2 – a) with (a – 2), losing the –1. The question was a good discriminator.
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Consider [15a^2/(a^2 + 6a - 16)] x [(8 - 4a)/(3a)].
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AQA 8365 · June 2023 · Paper 2 · Q15 · 4 marks
4 marks
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Most students were able to substitute into the equation and work out the values of a and b. Some then interpreted the resulting equation as y = 1 3 2 x        and worked out incorrect values for the range. Although many students did work out the values of 1 6 and 9 2 , it was quite common to see x used in place of g(x) in the final answer. Some students confused the idea of range with a statistical range and subtracted the values. There were quite a few non-attempts.
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g(x)=a x b^x has domain -1 <= x <= 2. Points P(0,1/2) and Q(1,3/2) lie on y=g(x).
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AQA 8365 · June 2023 · Paper 2 · Q16 · 4 marks
4 marks
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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.
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(2x - 3) is a factor of 6x^3 - 25x^2 + 28x - 6.
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AQA 8365 · June 2023 · Paper 2 · Q17 · 4 marks
4 marks
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It was rare to see a fully correct solution for this question. Most students did expand the brackets correctly even if they did not know how to proceed after that. However, a good number differentiated correctly and many of those did state that the result had to be > 0. Some thought that it was the second derivative that would be positive. Many who formed the correct inequality then either rearranged and gave an answer for a in terms of x or substituted x = 1 and gave the final answer a < 5 7 − .
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h(x)=ax(3x^2-2)+5x, where a is a positive constant. The function is increasing for all real x.
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AQA 8365 · June 2023 · Paper 2 · Q19 · 3 marks
3 marks
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Many students were able to correctly eliminate one of the variables, often y. However, the algebra involved sometimes proved too much and mistakes were made in rearranging and simplifying. Mistakes in simplifying the expression for y were frequently seen with students multiplying the numerator and denominator by 2k or incorrectly cancelling. Occasionally students found an equation for k rather than for x or y.
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k is positive. The simultaneous equations are 3x + 4y = k and y = 2kx.
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AQA 8365 · June 2022 · Paper 1 · Q1 · 3 marks
3 marks
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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.
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The expression x + 1 is increased by 20%. Its new value equals x + 6.
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AQA 8365 · June 2022 · Paper 1 · Q5 · 3 marks
3 marks
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This question was answered well and most students expanded the brackets and then equated coefficients. A small number of students tried to use substitutions which works but they ended up with simultaneous equations to solve if they didn’t choose to use x = 0. A common error was 2 × –4 = 8.
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(3x+a)(5x-4) is identically equal to 15x^2-2x+b.
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AQA 8365 · June 2022 · Paper 1 · Q6 · 3 marks
3 marks
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A very common error on this question was not expanding the brackets before differentiating. It was common to see the part on the outside of the brackets differentiated and then the part on the inside and at that point students attempted to multiply. For those that did expand the brackets first there were quite a few who were multiplying the powers rather than adding them. If they didn’t gain the M mark they couldn’t gain any A marks so whilst there were a few students who managed to gain the middle term in the answer this merited no marks. Students attempting the product rule were few and far between but those that did use this method usually gained full marks.
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y=2x^4(x^3+2-3/x).
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AQA 8365 · June 2022 · Paper 1 · Q9 · 6 marks
6 marks
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[9a] This question was answered very well for the most part. A growing number of students are using Alternative method 3. They simply find the first and second differences then memorise expressions with a, b and c that match up to the terms. Marks were often lost through arithmetic errors.
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This question concerns quadratic sequences.
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AQA 8365 · June 2022 · Paper 1 · Q12 · 3 marks
3 marks
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[12b] This question was answered well. We condoned answers such as 3 tending to infinity or 3 tending to n or n tending to 3 etc.
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A sequence has nth term 3n^2/(n^2+2).
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AQA 8365 · June 2022 · Paper 1 · Q13 · 3 marks
3 marks
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This question was answered poorly. Most students gained the 2 and 3. Many also managed the x terms. The y terms however were not done well. Some students were penalised for further incorrect working (quite often adding the terms together to get 5xnym).
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Consider (6x^3y^(-2)+9x^5y) divided by 3x^2y^(-3).
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AQA 8365 · June 2022 · Paper 1 · Q14 · 3 marks
3 marks
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This question was done really well. We condoned missing e = on the answer line as long as they had written the complete answer correctly on the last line of their working. Very few students dropped a mark by losing the e too early.
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ef=(5e+4)/3.
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AQA 8365 · June 2022 · Paper 1 · Q16 · 6 marks
6 marks
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Some students tried to factorise or use the formula without getting the quadratic into an integer form. If they managed this successfully the M marks had been recovered and it was possible to still be awarded all 6 marks. Most didn’t manage to recover though and, in most cases, only gained the first M mark.
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Solve x-y=19/4 and xy=-3.
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AQA 8365 · June 2022 · Paper 1 · Q19 · 3 marks
3 marks
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Quite a few students were using b = 2 and following through from that. This was sometimes continued with other incorrect attempts but usually they just left it at that and didn’t get any marks. Very few students were using Alternative method 2 but those that did usually gained full marks.
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Consider 6x^2-24x+17.
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AQA 8365 · June 2022 · Paper 2 · Q1 · 2 marks
2 marks
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This question was answered very well. A few students only partially factorised.
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Consider 12w+18w^2.
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AQA 8365 · June 2022 · Paper 2 · Q5 · 3 marks
3 marks
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Most students could write a simplified correct inequality. Many of these could then progress to the three correct integers but some gave x < 4 as their answer. Others included –4 in their list. Some gave an inequality alongside the three correct integers.
Text version
Consider 3x^2<48.
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AQA 8365 · June 2022 · Paper 2 · Q6 · 3 marks
3 marks
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Common errors were omitting the bracket after the subtraction or expanding their brackets incorrectly and not dealing with the minus sign. Some students misused the equals sign and showed rows of working that were not equal and this meant that full marks could not be awarded. Showing each step is important when asked to prove or show that something is true.
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n is an integer. Consider ((2n+1)^2-(2n-1)^2)/4.
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AQA 8365 · June 2022 · Paper 2 · Q9 · 4 marks
4 marks
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This question was very well answered. Most who didn’t score divided the correct expression and gave 5 + 2k as their answer. Question 9(b) Well answered although there were very few who gained part marks. The most common error was to use the 101st term rather than the 100th term.
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A linear sequence begins 30, 30+4k, 30+8k, where k is constant.
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AQA 8365 · June 2022 · Paper 2 · Q10 · 1 mark
1 mark
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This multiple-choice question was very well answered. The most common incorrect choice was graph C.
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Four sketches are offered for y=3 * 2^x: A decreases through (0,6), B decreases through (0,3), C increases through (0,6), and D increases through (0,3).
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AQA 8365 · June 2022 · Paper 2 · Q11 · 2 marks
2 marks
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This novel question was not well answered. Many worked out 60° but never showed an inequality. Others substituted a = 6 and worked out 41.4°. The most common wrong answer was x > 60 with some of these scoring one mark by having cos x > 2 1 in the working lines.
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A right-angled triangle has hypotenuse 4 cm, side adjacent to acute angle x equal to a-3 cm, and a>5.
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AQA 8365 · June 2022 · Paper 2 · Q14 · 3 marks
3 marks
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Different approaches were seen when forming an equation but most were correct. Some errors were made when eliminating brackets. A few thought the two angles were equal.
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PQRS is cyclic. Angle PSR=4(x+15) degrees, and angle PQR is 40 degrees smaller than angle PSR.
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AQA 8365 · June 2022 · Paper 2 · Q15 · 5 marks
5 marks
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This question was answered quite well. Most students simplified the brackets first and this was usually completed correctly. Most knew how to deal with division but there were quite a lot of errors made when simplifying the square root. Some squared everything but rarely took a square root at the end to recover this. Obtaining a single fraction but not simplifying fully was quite common.
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Consider (x/2+3x/5) divided by sqrt(x^6/4).
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AQA 8365 · June 2022 · Paper 2 · Q17 · 5 marks
5 marks
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This was a new topic compared to the previous version of this specification and the question was answered quite well. Many students were able to work accurately although sometimes their working was hard to follow with equations all over the page and no logical order. Generally those who used elimination rather than substitution were more accurate and progressed further. Some students eliminated terms too early and were left with incorrect equations. There was evidence of students using trial and improvement if their algebraic method had gone wrong.
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Solve a+3b-2c=4, 4a-3b+5c=-5, and 2a+b+3c=9.
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AQA 8365 · June 2022 · Paper 2 · Q19 · 3 marks
3 marks
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This question was well answered. Some students made sign errors or miscopied their working from one line to the next.
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Consider (3x+4)(2x-3)(5x-2).
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AQA 8365 · June 2022 · Paper 2 · Q22 · 4 marks
4 marks
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There were some very good responses with most students showing their working. Those who used powers of 2 often got as far as a quadratic equation but sometimes still had = 1 from their rearrangement, not realising that 20 = 1. Some made a slip rearranging and factorising at the end. Those who worked with powers of 16 rarely made any significant progress.
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Consider (16^x)^x=1/(2^(3x)).
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AQA 8365 · November 2021 · Paper 1 · Q3 · 1 mark
1 mark
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Four sketch options are shown for a straight line: A has negative gradient and both intercepts negative; B has negative gradient and both intercepts positive; C has positive gradient, negative x-intercept and positive y-intercept; D has positive gradient, positive x-intercept and negative y-intercept.
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AQA 8365 · November 2021 · Paper 1 · Q4 · 5 marks
5 marks
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Functions f, g and h are defined as stated in each part.
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AQA 8365 · November 2021 · Paper 1 · Q5 · 3 marks
3 marks
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The nth term of a sequence is (2n+47)/(n+1).
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AQA 8365 · November 2021 · Paper 1 · Q7 · 3 marks
3 marks
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Consider the inequality 2x^2 + 4 > (2x-3)(x+1).
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AQA 8365 · November 2021 · Paper 1 · Q9 · 3 marks
3 marks
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Consider (2x-5)(3x-4)(x+2).
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AQA 8365 · November 2021 · Paper 1 · Q10 · 3 marks
3 marks
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The first four terms of a quadratic sequence are 0, 1, 0, -3.
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AQA 8365 · November 2021 · Paper 1 · Q13 · 3 marks
3 marks
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The curve is y=k^x where k>0. Point A=(2, 3 1/16) lies on the curve.
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AQA 8365 · November 2021 · Paper 1 · Q14 · 5 marks
5 marks
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The simultaneous equations are 4a-b+3c=27, 3a+2b-c=5 and 2a-5c=-7.
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AQA 8365 · November 2021 · Paper 1 · Q16 · 5 marks
5 marks
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f(x)=200x^3+100x^2-18x-9.
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AQA 8365 · November 2021 · Paper 1 · Q19 · 4 marks
4 marks
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y=f(x) is cubic. It is negative for x<5 and non-negative for x>=5. It is increasing for x<-1, decreasing for -1<x<2, and increasing for x>2.
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AQA 8365 · November 2021 · Paper 1 · Q22 · 4 marks
4 marks
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8^x = (2^56 - 4^26)/30.
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AQA 8365 · November 2021 · Paper 2 · Q1 · 3 marks
3 marks
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Consider 5(2x-1)+4(11-x).
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AQA 8365 · November 2021 · Paper 2 · Q2 · 4 marks
4 marks
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Solve the two equations in the parts below.
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AQA 8365 · November 2021 · Paper 2 · Q3 · 3 marks
3 marks
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A rectangle with side lengths 2d and 12de has the same area as a right triangle with perpendicular sides 9d and 8e^2.
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AQA 8365 · November 2021 · Paper 2 · Q6 · 3 marks
3 marks
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y=sqrt((x+2w)/3).
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AQA 8365 · November 2021 · Paper 2 · Q7 · 4 marks
4 marks
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Use the laws of indices in each part.
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AQA 8365 · November 2021 · Paper 2 · Q8 · 4 marks
4 marks
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f(x)=4x for x<0; f(x)=x^2-8x for 0<=x<=8; and f(x)=16-2x for x>8.
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AQA 8365 · November 2021 · Paper 2 · Q9 · 4 marks
4 marks
$AQA 8365 · November 2021 · Paper 2 · Q9 · 4 marks — Simplify the algebraic fractions in the parts below.$
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Simplify the algebraic fractions in the parts below.
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AQA 8365 · November 2021 · Paper 2 · Q10 · 3 marks
3 marks
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A sphere has radius 3k/2 cm and volume 972pi cm^3. Use V=(4/3)pi r^3.
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AQA 8365 · November 2021 · Paper 2 · Q11 · 3 marks
3 marks
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Consider (5x+3y^2)(4x-y^2).
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AQA 8365 · November 2021 · Paper 2 · Q16 · 3 marks
3 marks
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The coefficient of x^4 in (a+2x)^6 is 1500.
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AQA 8365 · November 2021 · Paper 2 · Q19 · 3 marks
3 marks
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Use composition notation in the order shown.
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AQA 8365 · November 2021 · Paper 2 · Q21 · 6 marks
6 marks
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3x^2+2bx+8a can be written as 3(x+a)^2+b+2.
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AQA 8365 · sample1 · Paper 2 · Q4 · 2 marks
2 marks
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Choose the correct solution set in each part.
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AQA 8365 · sample1 · Paper 2 · Q5 · 2 marks
2 marks
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A quadratic graph intersects the x-axis at A(-1,0) and B and has maximum point (0.5,6).
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AQA 8365 · sample1 · Paper 2 · Q8 · 4 marks
4 marks
$AQA 8365 · sample1 · Paper 2 · Q8 · 4 marks — Simplify each algebraic fraction.$
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Simplify each algebraic fraction.
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AQA 8365 · sample1 · Paper 2 · Q11 · 4 marks
4 marks
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Sequence A has nth term n/(14n+30). Sequence B has nth term 2/n. Their kth terms are equal.
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AQA 8365 · sample1 · Paper 2 · Q12 · 7 marks
7 marks
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An L-shaped region is made from two rectangles. Its labelled boundary lengths are 30x, 15x, 10x, 20x and y centimetres, and its perimeter is 252 cm. Its area is A cm^2.
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AQA 8365 · sample1 · Paper 2 · Q13 · 4 marks
4 marks
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f(x)=3x^2+6 for all x, and g(x)=sqrt(x-5) for x>=5.
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AQA 8365 · sample1 · Paper 2 · Q15 · 4 marks
4 marks
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The curve is y=ab^(-x), where a and b are positive constants. The points (0,3) and (2,0.48) lie on the curve.
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AQA 8365 · sample1 · Paper 2 · Q16 · 5 marks
5 marks
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Consider (8x^3-50x)/[2x(6x^2-x-35)].
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AQA 8365 · sample1 · Paper 2 · Q17 · 4 marks
4 marks
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For x>0, solve 2x^(3/2)-3x^(1/2)=7x^(-1/2) by multiplying both sides by x^(1/2).
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AQA 8365 · sample1 · Paper 2 · Q19 · 5 marks
5 marks
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f(x)=3x^3-2x^2-7x-2.
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AQA 8365 · sample1 · Paper 2 · Q21 · 4 marks
4 marks
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Let n be an integer.
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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.

June 20247 questions

Paper 1 · Q21 mark · Non-calculator · calculate · easyView question in paper (p.2)Mark scheme (p.5)
Paper 1 · Q204 marks · Non-calculator · calculate · hardView question in paper (p.18)Mark scheme (p.21)
Paper 1 · Q215 marks · Non-calculator · calculate · hardView question in paper (p.20)Mark scheme (p.22)
Paper 2 · Q44 marks · Calculator · calculate · easyView question in paper (p.4)Mark scheme (p.8)
Paper 2 · Q72 marks · Calculator · multi · easyView question in paper (p.6)
Paper 2 · Q143 marks · Calculator · calculate · standardView question in paper (p.13)
Paper 2 · Q153 marks · Calculator · calculate · standardView question in paper (p.14)

June 20234 questions

Paper 1: Q14 (p.14, 4 marks), Q18 (p.17, 5 marks)

Paper 2: Q9 (p.8, 7 marks), Q13 (p.13, 6 marks)

June 20225 questions

Paper 1: Q3 (p.3, 6 marks), Q11 (p.9, 4 marks)

Paper 2: Q8 (p.6, 4 marks), Q12 (p.10, 5 marks), Q20 (p.18, 6 marks)

November 20211 question

Paper 1: Q17 (p.16, 3 marks)

sample111 questions

Paper 1: Q1 (4 marks), Q2 (2 marks), Q3 (4 marks), Q6 (3 marks), Q8 (3 marks), Q10 (5 marks), Q13 (5 marks), Q15 (9 marks), Q16 (4 marks), Q17 (5 marks), Q18 (3 marks)

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