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WMA12/01 Jan 2023, Question 9: Definite Integrals and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

For curve y = x² − 4x + 5 and line y = 2, find point D on the y-axis, intersections E and F, then use integration to find the exact ratio k of areas R₁ and R₂.

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

Part (a) · 1 mark

State the coordinates of D, where the curve crosses the y-axis.

  1. 1At x = 0: y = 0 − 0 + 5 = 5
  2. 2D = (0, 5)

Answer(0, 5)

Part (b) · 2 marks

Find the x coordinates of the two intersection points E and F of the curve with y = 2.

  1. 1x² − 4x + 5 = 2 ⇒ x² − 4x + 3 = 0
  2. 2(x − 1)(x − 3) = 0
  3. 3x = 1 at E and x = 3 at F

AnswerE: x = 1, F: x = 3

Part (c) · 5 marks

Using algebraic integration, find the exact value of k = area R₁ ÷ area R₂ as a simplified fraction.

  1. 1R₁ = ∫₀¹ (x² − 4x + 3) dx = [x³/3 − 2x² + 3x]₀¹
  2. 2R₁ = 1/3 − 2 + 3 = 4/3
  3. 3Triangle with vertices D(0, 5), (0, 2), F(3, 2): area = ½ × 3 × 3 = 9/2
  4. 4R₂ = 9/2 − R₁ = 9/2 − 4/3 = 19/6
  5. 5k = (4/3) ÷ (19/6) = 8/19

Answerk = 8/19

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