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0580/42 May/Jun 2023, Question 3: Constructing Equations

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 13 marks

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The question, in short

Right-angled triangle ABC has legs (x + 3) cm and (2x + 5) cm; use its 60 cm² area to form and solve a quadratic, then find an angle and work with similar triangle DEF of area 93.75 cm².

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

Part (ai) · 3 marks

Using the two leg lengths and area 60 cm², show the quadratic equation 2x² + 11x − 105 = 0.

  1. 1Area = 1/2 × (x + 3)(2x + 5) = 60
  2. 2(x + 3)(2x + 5) = 120
  3. 32x² + 5x + 6x + 15 = 120
  4. 42x² + 11x − 105 = 0

Answer(x + 3)(2x + 5)/2 = 60, 2x² + 6x + 5x + 15 = 120, leading to 2x² + 11x − 105 = 0

Part (aii) · 3 marks

Solve 2x² + 11x − 105 = 0 by factorising.

  1. 12x² + 11x − 105 = 0
  2. 2(2x + 21)(x − 5) = 0
  3. 3x = −10.5 or x = 5

Answer−10.5 and 5

Part (aiii) · 3 marks

Given x = 5, calculate angle ACB of the right-angled triangle.

  1. 1x = 5 (a length must be positive): AC = 5 + 3 = 8 cm, AB = 2 × 5 + 5 = 15 cm
  2. 2tan ACB = 15/8
  3. 3Angle ACB = tan⁻¹(15/8) = 61.92…° ≈ 61.9°

Answer61.9 or 61.92 to 61.93

Part (bi) · 1 mark

Triangle DEF is similar to ABC and has area 93.75 cm²; find its smallest angle.

  1. 1Similar triangles have the same angles
  2. 2Smallest angle = 90 − 61.92… = 28.07…° ≈ 28.1°

Answer28.1 or 28.07 to 28.08

Part (bii) · 3 marks

Find the shortest side length of triangle DEF.

  1. 1Area scale factor = 93.75 ÷ 60 = 1.5625
  2. 2Length scale factor = √1.5625 = 1.25
  3. 3Shortest side = 8 × 1.25 = 10 cm

Answer10

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Also searched as: 0580/42/M/J/23 · 0580_s23_qp_42 · 0580 s23 42 · question 3.
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