Edexcel International GCSE Mathematics A 4MA1: Geometry and trigonometry – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover Topic 4, Geometry and trigonometry, sections 4.1–4.8, of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series examined on it. Questions 1–7 use content on both tiers. Questions 8–12 are Higher tier only. A calculator may be used on every 4MA1 paper. Diagrams are described in words; sketch each one before you start. Give lengths to 3 significant figures and angles to 1 decimal place unless told otherwise.

Learn the content first in the study guide and the revision notes. The course hub is Edexcel IGCSE Mathematics and the printable checklist lists every statement.

Questions

1. (Both tiers) AB and CD are parallel straight lines, with A and C at their left-hand ends. A straight line crosses AB at P and CD at Q. Angle BPQ = 118°.

(a) Find the size of angle APQ. Give a reason for your answer. [2] (b) Find the size of angle PQD. Give a reason for your answer. [2]

2. (Both tiers) Each exterior angle of a regular polygon is 20°.

(a) Work out the number of sides of the polygon. [1] (b) Write down the size of each interior angle. [1] (c) Work out the sum of the interior angles of the polygon. [2]

3. (Both tiers)

(a) Name a quadrilateral that has exactly one line of symmetry and rotational symmetry of order 1. [1] (b) Write down the number of lines of symmetry and the order of rotational symmetry of a rhombus that is not a square. [2]

4. (Both tiers)

(a) A train leaves at 22:40 and arrives at 03:15 the next day. Work out the journey time. [1] (b) The train travels 385 km. Work out its average speed in km/h. [2] (c) A metal block is a cuboid measuring 8 cm by 5 cm by 2.5 cm. Its mass is 790 g. Work out its density. [2] (d) A crate exerts a force of 750 N on an area of 0.3 m². Using pressure = force ÷ area, work out the pressure. [1]

5. (Both tiers)

(a) The bearing of B from A is 125°. Work out the bearing of A from B. [1] (b) A map has a scale of 1 : 25 000. Two villages are 6.8 cm apart on the map. Work out the real distance in kilometres. [2] (c) Describe how to construct the perpendicular bisector of a line segment XY using only a straight edge and compasses. [2]

6. (Both tiers) A ramp rises 1.4 m over a horizontal distance of 4.8 m. The vertical rise and the horizontal distance meet at a right angle.

(a) Calculate the length of the sloping surface of the ramp. [2] (b) Calculate the angle the ramp makes with the horizontal. [2]

7. (Both tiers)

(a) A circle has centre O and radius 8.5 cm. AB is a chord of length 15 cm and M is the midpoint of AB. Calculate OM. [3] (b) PA and PB are tangents to a circle with centre O, touching it at A and B. Angle APB = 50°. Work out angle AOB. Give reasons. [3]

8. (Higher tier only) A, B, C and D lie on a circle with centre O, in that order. Angle ABC = 112°.

(a) Find angle ADC. Give a reason. [2] (b) Find the obtuse angle AOC. Give a reason. [2] (c) On a different circle, a tangent touches the circle at K. The chord KL makes an angle of 47° with the tangent. M is a point on the circle in the alternate segment. Write down angle KML and name the theorem you used. [2]

9. (Higher tier only)

(a) Chords PQ and RS of a circle meet at X inside the circle. PX = 6 cm, XQ = 4 cm and RX = 3 cm. Calculate the length of RS. [2] (b) From a point T outside a circle, one line meets the circle at A and then B, with TA = 4 cm and AB = 5 cm. Another line from T meets the circle at C and then D, with TC = 3 cm. Calculate CD. [2]

10. (Higher tier only) A triangle has sides 6 cm, 9 cm and 13 cm.

(a) Calculate the size of the largest angle. [3] (b) Calculate the area of the triangle. [2] (c) Explain how your working in (a) shows the angle is obtuse. [1]

11. (Higher tier only) A ship sails from a harbour H for 14 km on a bearing of 060° to a point P. It then sails 9 km on a bearing of 170° to a point Q.

(a) Show that angle HPQ = 70°. [2] (b) Calculate the distance HQ. [3] (c) Calculate the bearing of Q from H, to the nearest degree. [3]

12. (Higher tier only) VABCD is a pyramid with a horizontal square base ABCD of side 10 cm. The vertex V is vertically above M, the centre of the base, and VM = 12 cm. N is the midpoint of AB.

(a) Calculate the length AM. [2] (b) Calculate the length of the edge VA. [2] (c) Calculate the angle between VA and the base ABCD. [2] (d) Calculate the angle of elevation of V from N. [2]

Answers

1. (a) 180° − 118° = 62° [1]; angles on a straight line add up to 180° [1] (b) 180° − 118° = 62° [1]; allied (co-interior) angles add up to 180° (or: angle PQD = angle APQ, alternate angles are equal) [1] Examiner insight: The reason mark needs the property in words; “co-interior” or “allied” alone, or “C angles”, does not earn it.

2. (a) 360 ÷ 20 = 18 [1] (b) 180° − 20° = 160° [1] (c) (2 × 18 − 4) × 90° or 18 × 160° [1] = 2880° [1] Examiner insight: A follow-through mark is usually allowed in (c) from a wrong n in (a), so show the method even if you are unsure of (a).

3. (a) Kite (an isosceles trapezium is also correct) [1] (b) 2 lines of symmetry [1]; rotational symmetry of order 2 [1] Examiner insight: The two values in (b) are marked independently; a rhombus has 2 lines of symmetry (its diagonals), not 4, because the perpendicular bisectors of its sides are only lines of symmetry when it is a square.

4. (a) 1 h 20 min to midnight + 3 h 15 min = 4 h 35 min [1] (b) 4 h 35 min = 4 + 35/60 = 4.583… h [1]; 385 ÷ 4.583… = 84 km/h [1] (c) Volume = 8 × 5 × 2.5 = 100 cm³ [1]; 790 ÷ 100 = 7.9 g/cm³ [1] (d) 750 ÷ 0.3 = 2500 N/m² [1] Examiner insight: Using 4.35 hours in (b) gives 88.5… km/h and loses both marks; the method mark needs a correct conversion of minutes.

5. (a) 125° + 180° = 305° [1] (b) 6.8 × 25 000 = 170 000 cm [1]; = 1.7 km [1] (c) With compasses set to more than half of XY, draw arcs from X and from Y, above and below the line [1]; join the two points where the arcs cross [1] Examiner insight: In a drawn construction the arcs are the evidence; a correct line with no visible arcs scores no marks.

6. (a) √(4.8² + 1.4²) = √25 [1] = 5 m [1] (b) tan θ = 1.4 ÷ 4.8 [1]; θ = 16.3° [1] Examiner insight: The method mark needs the correct ratio for the chosen function; sin θ = 1.4 ÷ 5 is equally valid, and uses your own answer from (a) on follow-through.

7. (a) OM is perpendicular to AB, so AM = 7.5 cm [1]; OM² = 8.5² − 7.5² = 16 [1]; OM = 4 cm [1] (b) Angle OAP = angle OBP = 90° because a tangent is perpendicular to the radius [1]; angles in a quadrilateral add up to 360° [1]; AOB = 360° − 90° − 90° − 50° = 130° [1] Examiner insight: In (b) the reason marks need both properties named; a correct 130° with no reasons gets only the accuracy mark.

8. (a) 180° − 112° = 68° [1]; opposite angles of a cyclic quadrilateral add up to 180° [1] (b) 2 × 68° = 136° [1]; the angle at the centre is twice the angle at the circumference [1] (c) 47° [1]; the alternate segment theorem [1] Examiner insight: Doubling 112° in (b) gives the reflex angle AOC (224°); read which arc the angle stands on before applying the theorem.

9. (a) 6 × 4 = 3 × XS, XS = 8 cm [1]; RS = 3 + 8 = 11 cm [1] (b) TB = 4 + 5 = 9; 4 × 9 = 3 × TD, TD = 12 cm [1]; CD = 12 − 3 = 9 cm [1] Examiner insight: In (b) the product uses the whole distance TB, not AB; writing 4 × 5 = 3 × TD is a method error and scores 0.

10. (a) cos C = (6² + 9² − 13²) ÷ (2 × 6 × 9) [1] = −52/108 [1]; C = 118.8° [1] (b) ½ × 6 × 9 × sin 118.8° [1] = 23.7 cm² [1] (c) The cosine is negative, which only happens for angles between 90° and 180° [1] Examiner insight: Keep the full calculator value of the angle for (b); an answer in the accepted range from correct working gets full marks, but early rounding can push it outside that range.

11. (a) The bearing of H from P is 060° + 180° = 240° [1]; angle HPQ = 240° − 170° = 70° [1] (b) HQ² = 14² + 9² − 2 × 14 × 9 × cos 70° [1] = 190.8… [1]; HQ = 13.8 km [1] (c) sin(PHQ) ÷ 9 = sin 70° ÷ 13.81… [1]; angle PHQ = 37.8° [1]; bearing = 60° + 37.8° = 97.8°, so 098° [1] Examiner insight: On a “show that” the given 70° cannot be used to justify itself; the mark needs the 240° back bearing (or equivalent parallel-line reasoning) written down.

12. (a) AC = √(10² + 10²) = 14.14… cm [1]; AM = 7.07 cm [1] (b) VA = √(12² + 7.071²) [1] = 13.9 cm [1] (c) tan⁻¹(12 ÷ 7.071) [1] = 59.5° [1] (d) MN = 5 cm, angle VNM = tan⁻¹(12 ÷ 5) [1] = 67.4° [1] Examiner insight: In (c) the angle is between VA and its projection AM on the base; using the slant height VN or the edge AB gives a different triangle and no marks.

Where marks are usually lost

• Naming angle facts by letter shapes (“Z angles”) instead of the standard wording.
• Omitting reasons when the question says “give reasons”, or giving only one reason when two properties were used.
• Treating 35 minutes as 0.35 hours in speed calculations.
• Writing bearings with fewer than three figures, or adding instead of subtracting 180°.
• Forgetting that the perpendicular from the centre halves the chord before using Pythagoras.
• Doubling the wrong angle when using “angle at the centre” (Higher).
• Using AB instead of TB in the external intersecting-chords result (Higher).
• Rounding intermediate lengths too early in cosine rule and sine rule questions (Higher).
• Choosing the wrong right-angled triangle for the angle between a line and a plane (Higher).

Next steps

• Refresh the facts: revision notes
• Re-learn anything you missed: study guide
• Course hub: Edexcel IGCSE Mathematics
• Track coverage: printable checklist
• Check other topics: all free 10-minute diagnostics
• Want help from a teacher? Book a free trial class

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 4, Geometry and trigonometry, sections 4.1–4.8 (Foundation and Higher tier content).