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WMA11/01 Jun 2019, Question 3: Quadratic Inequalities

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Use the L-shaped garden plan with side lengths in x to work with its perimeter and area, showing a lower bound on x and solving a quadratic inequality for its area.

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

Part (a) · 3 marks

Using the six edge lengths from the diagram, write the perimeter and show that it being greater than 29 m gives x > 1.5 m.

  1. 1Perimeter = 2 × 5x + 2 × (6x − 2) = 22x − 4
  2. 222x − 4 > 29
  3. 322x > 33
  4. 4x > 33/22 = 1.5

AnswerShown: perimeter = 22x − 4 > 29 gives 22x > 33, so x > 1.5

Part (b) · 5 marks

Split the L-shape into rectangles, express the area in terms of x, and form and solve the inequality area < 72 m².

  1. 1Area = 2x(2x − 1) + 3x(6x − 2) = 22x² − 8x
  2. 222x² − 8x < 72, so 22x² − 8x − 72 < 0
  3. 3Critical values: 11x² − 4x − 36 = 0, (11x + 18)(x − 2) = 0, x = −18/11 or x = 2
  4. 4Inside region: −18/11 < x < 2

Answer22x² − 8x − 72 < 0, giving −18/11 < x < 2

Part (c) · 1 mark

Combine the results of parts a and b to state the full range of possible x values.

  1. 1Combine x > 1.5 with −18/11 < x < 2
  2. 21.5 < x < 2

Answer1.5 < x < 2

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