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4PM1/01 Jan 2022, Question 4: Inequalities and Linear Programming

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks

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The question, in short

On a grid with x from −6 to 6 and y from −4 to 8, draw three straight lines, shade a region R from three inequalities, and find the least value of P = x − 2y over R.

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

Part (a) · 3 marks

Draw the lines y = 2x − 4, 2x + 3y = 12 and y + 2x + 2 = 0 on the grid.

  1. 1y = 2x − 4 meets the axes at (0, −4) and (2, 0)
  2. 22x + 3y = 12 meets the axes at (0, 4) and (6, 0)
  3. 3y + 2x + 2 = 0 meets the axes at (0, −2) and (−1, 0)
  4. 4Draw each line with a ruler across the grid

Answery = 2x − 4 through (0, −4) and (2, 0); 2x + 3y = 12 through (0, 4) and (6, 0); y + 2x + 2 = 0 through (0, −2) and (−1, 0)

Part (b) · 1 mark

Shade the region R satisfying y ≥ 2x − 4, 2x + 3y ≤ 12 and y + 2x + 2 ≥ 0.

  1. 1y ≥ 2x − 4: above (left of) y = 2x − 4
  2. 22x + 3y ≤ 12: below 2x + 3y = 12
  3. 3y + 2x + 2 ≥ 0: above (right of) y + 2x + 2 = 0
  4. 4The region is the closed triangle between the three lines

AnswerR is the triangle with vertices (0.5, −3), (3, 2) and (−4.5, 7)

Part (c) · 4 marks

For points (x, y) in R, find the least value of P = x − 2y.

  1. 1Vertices of R: (0.5, −3), (3, 2), (−4.5, 7)
  2. 2P = x − 2y at each: 0.5 + 6 = 6.5; 3 − 4 = −1; −4.5 − 14 = −18.5
  3. 3Least value of P = −18.5

Answer−18.5

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