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WMA11/01 Jun 2023, Question 7: Finding Regions from Multiple Inequalities

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks

The question, in short

Using the shaded region R₁ bounded by 0 ≤ y ≤ 2, y ≤ 10 − 2x and y ≤ kx, with area 27/4 and x = a through y = 2 meeting y = kx, find a and k, then give inequalities for R₂.

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

Part (a) · 4 marks

Find (i) the value of a and (ii) the value of k, given the area of R₁ is 27/4 square units.

  1. 1y = 10 − 2x meets the x-axis at x = 5 and meets y = 2 at x = 4
  2. 2R₁ is a trapezium of height 2 with parallel sides 5 (along the x-axis) and 4 − a (along y = 2)
  3. 3Area = ½ × 2 × (5 + 4 − a) = 9 − a = 27/4
  4. 4a = 9 − 27/4 = 9/4
  5. 5y = kx passes through (a, 2), so k = 2/a = 2 ÷ 9/4 = 8/9

Answer(i) a = 9/4 (ii) k = 8/9

Part (b) · 2 marks

Write down inequalities that define the region R₂ shown shaded in Figure 2.

  1. 1R₂ lies above the line y = (8/9)x, so y ≥ (8/9)x
  2. 2R₂ lies below the line y = 10 − 2x, so y ≤ 10 − 2x
  3. 3R₂ lies to the right of the dashed line x = 9/4, so x > 9/4

Answery ≥ (8/9)x, y ≤ 10 − 2x, x > 9/4

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