Circle theorems I Notes and examples: past paper questions

Real questions on circle theorems i notes and examples

The newest 20 of 44 written questions. Open one to see it on the real paper.

1. 4024/11 · October/November 2025 · Q10[2]
   10 A circle has diameter $12$ cm. Find the area of the circle. Give your answer in terms of $\pi$.See it on the paper →
2. 4024/12 · October/November 2025 · Q14[5]
   14 [Diagram: A circle with centre $O$. Points $A,B,C,D$ are on the circumference. $EF$ is a tangent line to the circle at point $A$. The angle between the tangent segment $AF$ and the chord $AB$ is marked as $65^{\circ }$. The angle at the centre, $\angle AOD$, is marked as $80^{\circ }$. Chords $AB,BC,CD,DA$ and radii...See it on the paper →
3. 4024/21 · October/November 2025 · Q3b[1]
   3 In this question, all lengths are in centimetres. The diagram shows the net of a cuboid. [Diagram: A net of a cuboid consisting of a horizontal row of four rectangles and one rectangle attached above the third rectangle from the left. The top edge of the upper rectangle is labelled 3. The right edge of the upper rect...See it on the paper →
4. 4024/21 · October/November 2025 · Q7[2]
   7 Work out the size of the interior angle of a regular decagon.See it on the paper →
5. 4024/22 · October/November 2025 · Q22b[4]
   22 $ABCD$ is a field. A straight path crosses the field from $B$ to $D$. [Diagram: Quadrilateral $ABCD$ with a diagonal $BD$. Angle $DAB$ is a right angle. Angle $ABD=42^{\circ }$. Length $BD=495\text{ m}$. Length $DC=620\text{ m}$. Length $BC=580\text{ m}$. Label "NOT TO SCALE".] (b) Show that angle $BDC = 61.5^\...See it on the paper →
6. 4024/12 · May/June 2025 · Q23[5]

   5-mark “Give” question — its maths and diagrams are best seen on the paper.

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7. 4024/13 · May/June 2025 · Q20a[3]

   3-mark question — its maths and diagrams are best seen on the paper.

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8. 4024/13 · May/June 2025 · Q20b[3]

   3-mark “Calculate” question — its maths and diagrams are best seen on the paper.

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9. 4024/11 · May/June 2024 · Q16b(i)[1]

   1-mark “Find” question — its maths and diagrams are best seen on the paper.

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10. 4024/21 · May/June 2024 · Q3b[3]
    [Diagram: A trapezium $ABCD$ with parallel sides $AB$ and $DC$ indicated by arrows. Diagonals $AC$ and $BD$ intersect at point $E$. Labelled "NOT TO SCALE".] The diagonals of trapezium $ABCD$ meet at $E$. Show that triangle $ABE$ is similar to triangle $CDE$. Give a reason for each statement you make.See it on the paper →
11. 4024/12 · October/November 2023 · Q21a[3]
    Diagram $A$ shows a sector of a circle, centre $D$ and radius $\frac{3}{4}y$ cm. The obtuse angle $EDF=6x^{\circ }$. Diagram $B$ shows a sector of a circle, centre $P$ and radius $y$ cm. The sector angle is $x^{\circ }$. (a) The length of the major arc $EF$ is 9 times the length of the arc $QR$. Show that $x=20$.See it on the paper →
12. 4024/12 · October/November 2023 · Q21b[2]
    Diagram $A$ shows a sector of a circle, centre $D$ and radius $\frac{3}{4}y$ cm. The obtuse angle $EDF=6x^{\circ }$. Diagram $B$ shows a sector of a circle, centre $P$ and radius $y$ cm. The sector angle is $x^{\circ }$. (b) Find the value of $y$ when the area of sector $QPR$ is equal to $2\pi {\text{ cm}}^2$.See it on the paper →
13. 4024/21 · October/November 2023 · Q10b[5]

    5-mark “Calculate” question — its maths and diagrams are best seen on the paper.

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14. 4024/22 · October/November 2023 · Q9b[5]
    (b) [Diagram: Triangle $PQR$ with point $S$ on $PR$. Angle $QPS=42^{\circ }$, Angle $QRS=35^{\circ }$. Length $PS=10.3$, Length $QR=12.6$. Label "NOT TO SCALE".] $PQR$ is a triangle and $S$ is a point on $PR$. $PS=10.3\text{cm}$, $QR=12.6\text{cm}$, $\widehat{QPS}=42^{\circ }$ and $\widehat{QRS}=35^{\circ }$. Calculate $QS...See it on the paper →
15. 4024/11 · May/June 2023 · Q19[3]
    [Diagram: A circle with centre $O$. Points $A$ and $B$ are on the circumference. Lines $TA$ and $TB$ are tangents to the circle at $A$ and $B$ respectively. Lines $OA$, $OB$, and $OT$ are drawn. The diagram is labelled NOT TO SCALE.] $A$ and $B$ are points on the circumference of a circle, centre $O$. $TA$ and $TB$ are...See it on the paper →
16. 4024/21 · May/June 2023 · Q3a[3]
    3 (a) [Diagram: Two parallel horizontal lines $PQ$ (top) and $RS$ (bottom). A straight line $ABCD$ intersects $PQ$ at $C$ and $RS$ at $B$. A line segment $CE$ connects $C$ to a point $E$ on $RS$. Tick marks indicate $BE=CE$. The angle $\widehat{ABE}$ is marked as $110^{\circ }$.] $PQ$ is parallel to $RS$. $ABCD$ is a straig...See it on the paper →
17. 4024/21 · May/June 2023 · Q3b[4]
    [Diagram: A circle with centre $O$. Points $U,V,W,X,Y$ are on the circumference. $UY$ is a horizontal diameter. $Z$ is a point outside the circle to the right of $Y$ such that $U,Y,Z$ are collinear. $ZX$ is a tangent to the circle at $X$. Lines connect $U$ to $V$, $V$ to $W$, $W$ to $X$, $X$ to $Y$, $U$ to $X$, a...See it on the paper →
18. 4024/22 · May/June 2023 · Q8b(i)[2]

    2-mark “Find” question — its maths and diagrams are best seen on the paper.

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19. 4024/12 · October/November 2022 · Q4b[1]

    1-mark question — its maths and diagrams are best seen on the paper.

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20. 4024/12 · October/November 2022 · Q5[2]

    2-mark “Find” question — its maths and diagrams are best seen on the paper.

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What examiners said

This question was found challenging by most candidates and fully correct solutions were rare. Those who started with a correct equation often rearranged it correctly to reach the required solution of x = 20. Those who did not start by cancelling 360, π and y to simplify the equation often used very long methods that contained errors. Candidates who knew the correct formula for the length of an arc...

— Examiner report, October/November 2023, 4024/12 Q21(a) (Cambridge)

As expected, this question was very challenging for most candidates. The main issue was that candidates seemed to not realise the need to find angle BOC or an equivalent angle. The use of trigonometry in finding a required angle was essential in this question to gain any marks. Candidates could use right-angled trigonometry or the cosine rule to find an angle, but assuming the size of an angle did...

— Examiner report, October/November 2023, 4024/21 Q10(b) (Cambridge)

Just under a third of the candidates were able to calculate an accurate value for the length QS. Candidates did not always use the cosine rule in the most useful format for finding a side and this sometimes resulted in incorrect rearrangements or inaccuracies. Several candidates made some progress by correctly calculating PQ or PR, however, not all were then able to proceed correctly as assumption...

— Examiner report, October/November 2023, 4024/22 Q9(b) (Cambridge)