All AQA Level 2 Further Maths topic questions

AQA Level 2 Further Maths · Topic CG

Coordinate Geometry (2 dimensions only) questions

Practice questions covering coordinate geometry (2 dimensions only) 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 Coordinate Geometry (2 dimensions only), each with its mark scheme and worked method. Reveal the answer when you have had a go.

CG1 · Know and use the definition of a gradient
Q1
ORIGINAL
1
Work out the gradient of the line $3x-2y=12$ .
[2 marks]
CG2 · Know the relationship between the gradients of parallel and perpendicular lines
Q2
ORIGINAL
2
Points $A(-2,1)$ , $B(4,3)$ , $C(1,5)$ and $D(3,-1)$ are given. Show that $AB$ is perpendicular to $CD$ .
[3 marks]
CG3 · Use Pythagoras' theorem to calculate the distance between two points
Q3
ORIGINAL
3
Work out the exact distance between $R(-2,5)$ and $S(6,-1)$ .
[3 marks]
CG4 · Use ratio to find the coordinates of a point on a line given the coordinates of two other points, including the midpoint
Q4
ORIGINAL
4
Point $P$ divides the segment from $A(-4,1)$ to $B(11,16)$ in the ratio $AP:PB=2:3$ . Work out the coordinates of $P$ .
[3 marks]
CG5 · The equation of a straight line: y = mx + c and y - y1 = m(x - x1) and other forms, including interpretation of the gradient and y-intercept
Q5
ORIGINAL
5
Work out the equation of the line with gradient $4$ that passes through $(2,-1)$ . Give your answer as $y=mx+c$ .
[3 marks]
CG6 · Draw a straight line from given information
Q6
ORIGINAL
6
On coordinate axes, draw $2x+3y=6$ for $-3\leq x\leq3$ .
[3 marks]

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Printable question pack — CG Coordinate Geometry (2 dimensions only)All 90 questions across 9 specification points — 84 more than the 6 above — with the full mark scheme. Guest accessible · browser print.

CG Coordinate Geometry (2 dimensions only) revision notes

Real past-paper questions

26 questions shown as printed, 9 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 · Q4 · 4 marks
4 marks
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Examiner report
This question was very well answered. Question 4 (b) This question was quite well answered. Not many students recognised that it could be answered using knowledge of the Pythagorean triple 8, 15, 17. There were quite a few arithmetic errors when applying Pythagoras’ theorem.
Text version
AC is a straight line. A is (-3, 2). B is the midpoint of AC and B is (12, 10).
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AQA 8365 · June 2025 · Paper 1 · Q10 · 4 marks
4 marks
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Examiner report
The skills needed for this question are ones that are on the GCSE Mathematics specification and this question was accessible to most students. Some errors were made when working out gradients. Those students using Alternative method 2 generally found it difficult to obtain the correct answer due to the complexity of the algebra involved.
Text version
P(0, a), Q(3, 12) and R(9, 0) form a right-angled triangle with PQ perpendicular to QR.
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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 2 · Q2 · 4 marks
4 marks
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Examiner report
This question was very well answered. The most common error was when attempting to work out the area of triangle COD with students either using an incorrect formula or having an incorrect value for OD.
Text version
C and E lie on the circle x^2 + y^2 = 144, centred at O. C is the point on the positive y-axis and E is the point on the positive x-axis. D lies on OE, CD is a straight line, and OD : DE = 5 : 1. The shaded region is the quarter-circle sector COE with triangle COD removed.
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AQA 8365 · June 2025 · Paper 2 · Q3 · 3 marks
3 marks
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Examiner report
This question was well answered. Some students stopped after working out a = 4 and others gave the gradient as 8 instead of – 8
Text version
The equation of a straight line is 2ax + y = 25, where a is a constant. The line passes through (3, 1).
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AQA 8365 · June 2025 · Paper 2 · Q6 · 3 marks
3 marks
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Examiner report
This question was a good discriminator for this stage in the paper. A majority of students were awarded full marks. Some of those who correctly worked out that C was (11, 6) gave y = 6 11 x as their answer. Others tried to work out the equation of AB. A common error was to have C as (13.5, 4.5) but many of these went on to give the correct follow through equation and were awarded one mark.
Text version
C lies on line AB. A = (-2.5, 1.5), B = (20, 9), and AC : CB = 3 : 2. O is the origin.
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AQA 8365 · June 2024 · Paper 2 · Q1 · 3 marks
3 marks
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Examiner report
Very well answered with the majority of lines drawn having the correct gradient. The most common error was to draw a line passing through (–1 , 1). Only a few lines were not the required length.
Text version
A straight line passes through (-1, -1) and has gradient 3/2. A coordinate grid is supplied.
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AQA 8365 · June 2024 · Paper 2 · Q12 · 6 marks
6 marks
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Examiner report
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 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 · Q8 · 6 marks
6 marks
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Examiner report
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 2 · Q4 · 4 marks
4 marks
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Examiner report
A wide variety of methods were used but many correct solutions were seen. Some students worked out r correctly but then used an incorrect expression for the gradient with the difference in the x-coordinates divided by the difference in the y-coordinates. The equation 1 2 r − = 5 was sometimes seen which gave r = 11 but often the rest of the method was correct. Occasionally students only had an equation using the gradient in terms of r and made no progress.
Text version
A line passes through P(1,k) and Q(r,6). The midpoint of PQ has x-coordinate 5 and the line gradient is 2.
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AQA 8365 · June 2023 · Paper 2 · Q6 · 2 marks
2 marks
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Examiner report
The vast majority of students gave a fully correct response. Sometimes the signs were incorrect for the coordinates of the centre and occasionally the radius was given as 36 or was not fully evaluated.
Text version
The circle has equation (x + 7)^2 + (y - 4)^2 = 36.
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AQA 8365 · June 2022 · Paper 1 · Q2 · 2 marks
2 marks
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Examiner report
Most students followed Alt 1 and most of these gained both marks. Most of the lost marks on this question came from arithmetic errors. Quite a few students found c then continued with further incorrect working so didn’t score both marks. A small number of students gained both marks by looking at steps but about half of these confused x and y and found the point at (0 , 0) scoring no marks.
Text version
The point (-6,-4) lies on a straight line with gradient 3/2.
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AQA 8365 · June 2022 · Paper 1 · Q7 · 3 marks
3 marks
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Examiner report
The question is written in fractions. The candidates that did it using fractions usually went on to get all 3 marks. The ones who changed everything to decimals ended up doing lots of long multiplication and rarely managed to get to the correct answer. A small number of candidates spotted that it was a 3, 4, 5 triangle and quickly gained all 3 marks. Some candidates worked out the gradient or the midpoint (so didn’t read the question). A small number of candidates wrote ± 5 4 which was condoned.
Text version
ABC is right-angled with A(-1,5), B(-2,5) and C(-1,23/4).
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AQA 8365 · June 2022 · Paper 1 · Q17 · 6 marks
6 marks
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Examiner report
[17a] Some students gained 1 mark in more than one way but didn’t get 2 marks on any method. It was common to use trigonometry to get one coordinate and then use Pythagoras to get the other one.
Text version
P lies in the first quadrant on x^2+y^2=16. OP makes an angle of 60 degrees with the positive x-axis.
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AQA 8365 · June 2022 · Paper 2 · Q4 · 4 marks
4 marks
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Examiner report
The common error was to use a gradient of 4 or 4 1 or – 4 1 for line B. Substituting (2d, d) into their equation for B was also seen quite often. There were many fully correct solutions.
Text version
Line A has equation y+4x=6. Line B is parallel to A and passes through (2,1). The point (d,2d) lies on B.
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AQA 8365 · June 2022 · Paper 2 · Q13 · 3 marks
3 marks
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Examiner report
Many students worked out the centre correctly but some used (3, 4). A significant number were unsure whether 5 was the radius or the diameter. Some used the diameter of 10 in their equation of the circle. Fully correct equations were seen in a small majority of responses.
Text version
A(-2,5) and B(4,13) are endpoints of a diameter of a circle.
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AQA 8365 · November 2021 · Paper 1 · Q1 · 2 marks
2 marks
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Text version
A = (-3, 7) and B = (5, 1).
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AQA 8365 · November 2021 · Paper 1 · Q12 · 5 marks
5 marks
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Text version
A circle with centre C=(4,-2) passes through the origin and A=(8,0). A tangent is drawn at A.
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AQA 8365 · November 2021 · Paper 2 · Q4 · 3 marks
3 marks
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Text version
Two circles centred at the origin have equations x^2+y^2=100 and x^2+y^2=36. The shaded region is the first-quadrant quarter of the annulus between them.
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AQA 8365 · November 2021 · Paper 2 · Q5 · 3 marks
3 marks
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Text version
SQR is right-angled at S. P=(1,6) lies on SQ, Q=(1,1), R=(k,15), angle SPR=45 degrees, and M is the midpoint of QR.
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AQA 8365 · November 2021 · Paper 2 · Q12 · 5 marks
5 marks
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Text version
A and B lie on y=3x+2. B, C and D=(5,0) lie on line L. C is on the y-axis and OA:AC=1:4, with A between O and C.
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AQA 8365 · sample1 · Paper 2 · Q1 · 3 marks
3 marks
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Text version
A sketch shows the lines y=2x and y=6. They meet at Q, and P is (0,6). O is the origin.
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AQA 8365 · sample1 · Paper 2 · Q2 · 2 marks
2 marks
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Text version
A circle with centre (0,0) has circumference 20pi.
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AQA 8365 · sample1 · Paper 2 · Q3 · 2 marks
2 marks
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Text version
M(p,-1) is the midpoint of A(-2,3) and B(7,r).
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AQA 8365 · sample1 · Paper 2 · Q7 · 3 marks
3 marks
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Text version
A straight line is parallel to 2y=x and intersects the x-axis at (4,0).
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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 20243 questions

Paper 1 · Q42 marks · Non-calculator · prove · easyView question in paper (p.3)Mark scheme (p.6)
Paper 1 · Q63 marks · Non-calculator · calculate · easyView question in paper (p.4)Mark scheme (p.8)
Paper 2 · Q205 marks · Calculator · calculate · hardView question in paper (p.20)

June 20231 question

Paper 2: Q10 (p.10, 3 marks)

June 20221 question

Paper 2: Q2 (p.2, 2 marks)

November 20211 question

Paper 2: Q13 (p.11, 4 marks)

sample13 questions

Paper 1: Q4 (4 marks), Q11 (3 marks), Q15 (9 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.

Coordinate Geometry (2 dimensions only) questions

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