AQA AS Level Mathematics Paper 2, June 2025: Question 3

4 marks · Medium difficulty · Multi-step Problem

Find the size of the obtuse angle CAB of a triangular prism's cross-section given side lengths 6 cm and 10 cm, prism length 20 cm, and volume 500 cm³.

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Question

Question 3 presents a triangle ABC representing the cross-section of a triangular prism. Side AB has length 6 cm and side AC has length 10 cm. The question states that the prism has a length of 20 cm and a volume of 500 cm³. Students are instructed to find the size of the obtuse angle CAB to the nearest degree.
Question text

3 ABC is the cross‑section of a triangular prism.

The length of AB is 6 cm and the length of AC is 10 cm as shown in the diagram.

A

6cm 10cm

B C

The prism has length 20 cm and volume 500 cm3

Find the size of the obtuse angle CAB

Give your answer to the nearest degree.

[4 marks]

Mark scheme

Show the mark scheme Mark scheme for Question 3 with a 4-mark breakdown: M1 for equating the area of the triangle multiplied by 20 to 500 (giving area = 25); B1 for using the area formula 1/2 * a * b * sin C with sides 6 and 10 to get 30 sin A; M1 for solving sin A = k where 0 < k < 1 (A = 56.4°); A1 for obtaining the obtuse angle 180 - 56.4° = 124° (AWRT 124°).

Q Marking instructions AO Marks Typical solution

3 Multiplies the area of their 1.1a M1 500

Area of triangle = = 25

triangle by 20 and equates to 20

500 1

5 ×10×6×sinA =25

PI by 25 or sinA = 2

1 sinA =

Uses absin C with 6 and 10 1.1b B1 6

sin A A = 56 4.°

PI by 30

obtuse angle = 180 − 56 4.

Solves equation of the form 1.1a M1 = 124°

sinA = k 0 < k < 1

Obtains AWRT 124° 3.2a A1

Question 3 Total 4

How to answer it

Trigonometry: Cross-Sectional Area & Obtuse Angles

WHAT THIS QUESTION TESTS

Core AS-Level Pure Mathematics Skills:

  • Volume of a Prism: Linking total volume, uniform cross-sectional area, and length ( Volume = Area × Length ).
  • Area of a Triangle: Applying the non-right-angled triangle formula Area = ½ab sin(C) using two sides and an included angle.
  • Trigonometric Equations: Solving sin(θ) = k for values in the range 0° to 180°.
  • Obtuse Angles: Recognising that sin(θ) = sin(180° - θ) to find the obtuse second quadrant solution.

Question 3 Walkthrough (4 Marks)

AQA AS Mathematics — Pure Core

📐 Step-by-Step Calculation

  1. Calculate the cross-sectional area of triangle ABC:
    Since Volume = Area of cross-section × Length:
    500 = Area × 20
    Area = 500 / 20 = 25 cm²
  2. Set up the triangle area formula with the given sides:
    The sides enclosing angle CAB are AB = 6 cm and AC = 10 cm.
    Area = ½ × AB × AC × sin(CAB)
    25 = ½ × 6 × 10 × sin(CAB)
    25 = 30 sin(CAB)
  3. Solve for sin(CAB):
    sin(CAB) = 25 / 30 = 5/6 ≈ 0.8333
  4. Find the acute reference angle:
    CAB = arcsin(5/6) ≈ 56.4427°
  5. Calculate the obtuse angle and round appropriately:
    Because angle CAB is specified as obtuse (90° < CAB < 180°):
    CAB = 180° - 56.4427° = 123.5573°
    Rounding to the nearest degree gives:
    CAB = 124°

✅ Final Answer

Angle CAB = 124°

Must be rounded to the nearest whole degree as instructed in the question.

💡 Key Knowledge

  • Sine Area Rule: Requires the angle to be strictly between the two known sides (included angle).
  • Symmetry of Sine: For any angle θ between 0° and 180°, sin(θ) = sin(180° - θ) . Your calculator always defaults to the acute value when you input arcsin .

🧠 Exam Technique & Mark Breakdown

M1 AO 1.1a: Multiply triangle area by 20 and equate to 500 (implied by 25 or sin A = 5/6).
B1 AO 1.1b: Correctly apply ½ × 6 × 10 × sin(A) , seeing 30 sin(A) .
M1 AO 1.1a: Solve an equation of form sin(A) = k where 0 < k < 1.
A1 AO 3.2a: Correct final obtuse angle: AWRT (answers which round to) 124°.

❌ Common Examiner Traps

  • Stopping at the Acute Angle: Students write 56° or 56.4° and stop, completely missing the explicit word "obtuse" in the question. This loses the final accuracy mark.
  • Premature Rounding: Rounding 56.44° to 56° first, then doing 180 - 56 = 124° happens to yield 124° here, but rounding early often causes intermediate calculation errors in multi-step questions. Keep unrounded figures stored in your calculator!
  • Radian Mode: Ensure your calculator displays a 'D' at the top, not 'R', before starting trigonometry questions requiring degree answers.

Topics

Pure Mathematics · E: Trigonometry

Question and mark scheme from the AQA AS Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.