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4PM1/01R Jun 2024, Question 11: Solving Trig Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Given f(θ) = (2cos θ − sin θ)(2sin θ + cos θ), prove a double-angle form, then use the curve y = f(θ) + 2 to find an intercept and an exact shaded area.

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

Part (a) · 3 marks

Show that f(θ) simplifies to (3/2)sin 2θ + 2cos 2θ.

  1. 1f(θ) = 4sin θ cos θ + 2cos²θ − 2sin²θ − sin θ cos θ
  2. 2= 3sin θ cos θ + 2(cos²θ − sin²θ)
  3. 3= (3/2)sin 2θ + 2cos 2θ

Answerf(θ) = (3/2)sin 2θ + 2cos 2θ shown

Part (b) · 5 marks

Given S crosses the θ-axis at (a, 0), show that a = π/2 using sin²θ + cos²θ = 1.

  1. 1(3/2)sin 2θ + 2cos 2θ + 2 = 0, so 3sin 2θ + 4cos 2θ + 4 = 0
  2. 23sin 2θ + 4(cos²θ − sin²θ) + 4(sin²θ + cos²θ) = 0
  3. 36sin θ cos θ + 8cos²θ = 0
  4. 4cos θ(6sin θ + 8cos θ) = 0
  5. 5cos θ = 0 gives θ = π/2 (tan θ = −4/3 gives a larger root), so a = π/2

Answera = π/2 shown

Part (c) · 3 marks

By algebraic integration, find the exact shaded area between S, the positive θ-axis and positive y-axis.

  1. 1Area = ∫ from 0 to π/2 of ((3/2)sin 2θ + 2cos 2θ + 2) dθ
  2. 2= [−(3/4)cos 2θ + sin 2θ + 2θ] from 0 to π/2
  3. 3= (3/4 + π) − (−3/4) = 3/2 + π

Answer3/2 + π

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Also searched as: 4PM1 1R June 2024 · 4PM1/01R · 4pm1 jun24 1r · question 11.
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