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4PM1/01 Jun 2022, Question 2: Hidden Quadratics and Harder Simultaneous Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

For a flag with a white cross on grey, using given side lengths in x and y, derive area expressions and then solve for x and y with total area 3432 cm² and cross area 1080 cm².

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

Part (a) · 1 mark

Give an expression for the flag's total area H in terms of x and y.

  1. 1Width = 3x + y + 5x = 8x + y
  2. 2Height = 3x + y + 3x = 6x + y
  3. 3H = (8x + y)(6x + y)

AnswerH = (8x + y)(6x + y) (= 48x² + 14xy + y²)

Part (b) · 3 marks

Using the cross area K, prove that K equals 14xy + y².

  1. 1Grey area = 2 × (3x)² + 2 × (5x × 3x) = 2(9x² + 15x²) = 48x²
  2. 2K = H − grey area = 48x² + 14xy + y² − 48x²
  3. 3K = 14xy + y²

AnswerK = 14xy + y² shown

Part (c) · 5 marks

Given H = 3432 cm² and K = 1080 cm², determine the values of x and y.

  1. 148x² + 14xy + y² = 3432 and 14xy + y² = 1080
  2. 2Subtract: 48x² + 1080 = 3432 ⇒ 48x² = 2352 ⇒ x = 7 (x > 0)
  3. 398y + y² = 1080 ⇒ y² + 98y − 1080 = 0
  4. 4(y − 10)(y + 108) = 0 ⇒ y = 10 (y > 0)

Answerx = 7, y = 10

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