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4PM1/02R Jun 2019, Question 8: Trigonometry in 3D: Cuboids

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 13 marks

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The question, in short

Right prism ABCDEF with rectangular base ABCD and triangular ends; given AB = 12 cm, BC = 8 cm, AE = BF = 10 cm, angles 60°; N on BC with FN perpendicular to BC. Find lengths and angles.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Show that BN = 5 cm using triangle BFN with BF = 10 cm and angle FBC = 60°.

  1. 1Triangle BNF is right-angled at N
  2. 2BN = BF cos 60° = 10 × 0.5
  3. 3= 5 cm

AnswerBN = 5 cm shown

Part (b) · 3 marks

Find the length EN in cm to 3 significant figures.

  1. 1FN² = 10² − 5² = 75 (FN = 5√3)
  2. 2EF = 12 and EN² = EF² + FN²
  3. 3NE = √(12² + 75) = √219 = 14.798… ≈ 14.8 cm

Answer14.8 cm

Part (c) · 2 marks

X and Y are midpoints of BF and FC; find the angle between planes ABCD and AXYD in degrees to one decimal place.

  1. 1Let R be the midpoint of FN, so R lies on XY at height √75/2 above N
  2. 2P is the point on AD opposite N, PN = 12
  3. 3tan ∠RPN = (√75/2)/12
  4. 4∠RPN = 19.84… ≈ 19.8°

Answer19.8°

Part (d) · 6 marks

Find the size of angle AYE in degrees to one decimal place.

  1. 1Let Z be the foot of the perpendicular from Y to BC: BZ = 6.5, ZY = √75/2
  2. 2AY = √(12² + 6.5² + (√75/2)²) = √205
  3. 3FC = √(75 + 3²) = √84 ⇒ FY = √21
  4. 4EY = √(12² + 21) = √165
  5. 5cos AYE = (205 + 165 − 100)/(2 × √205 × √165)
  6. 6∠AYE = 42.77… ≈ 42.8°

Answer42.8°

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