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4PM1/01 Jun 2021, Question 11: Definite Integrals and Areas

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 15 marks

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The question, in short

Show a double-angle identity, then find where y = sin x + 2 meets y = cos 2x + 2 at points A, B, C, and compare shaded areas R₁ and R₂.

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

Part (a) · 3 marks

Show that cos 2x equals 1 − 2sin²x using a formula from page 2.

  1. 1cos 2x = cos(x + x) = cos x cos x − sin x sin x
  2. 2= cos²x − sin²x
  3. 3= (1 − sin²x) − sin²x = 1 − 2sin²x

Answercos 2x = 1 − 2sin²x shown

Part (b) · 4 marks

Find the coordinates of the three intersection points A, B and C of the two curves.

  1. 1sin x + 2 = cos 2x + 2 ⇒ sin x = 1 − 2sin²x
  2. 22sin²x + sin x − 1 = 0 ⇒ (2sin x − 1)(sin x + 1) = 0
  3. 3sin x = 1/2: x = π/6, 5π/6; sin x = −1: x = 3π/2
  4. 4y = sin x + 2 gives 5/2, 5/2 and 1

AnswerA(π/6, 5/2), B(5π/6, 5/2), C(3π/2, 1)

Part (c) · 8 marks

Use calculus to find the ratio of the area of R₁ to the area of R₂ in simplest form.

  1. 1R₁ = ∫ from π/6 to 5π/6 of (sin x − cos 2x) dx
  2. 2= [−cos x − (sin 2x)/2] from π/6 to 5π/6 = 3√3/2
  3. 3R₂ = ∫ from 5π/6 to 3π/2 of (cos 2x − sin x) dx
  4. 4= [(sin 2x)/2 + cos x] from 5π/6 to 3π/2 = 3√3/4
  5. 5R₁ : R₂ = 3√3/2 : 3√3/4 = 2 : 1

Answer2 : 1

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