Lemmara logoLemmaraWorked solution

0580/41 Oct/Nov 2024, Question 10: Equations That Become Quadratics

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 14 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Rectangle with sides 2x+3 and x+1 has area 190 cm²; sector OAB radius r, angle 50°, shaded segment area 30 cm². Form equations and find x, perimeter, r and segment perimeter.

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

Part (ai) · 4 marks

Form an equation from the rectangle area 190 cm² and solve it to show x = 8.5.

  1. 1(2x + 3)(x + 1) = 190
  2. 22x² + 5x + 3 = 190
  3. 32x² + 5x − 187 = 0
  4. 4(2x − 17)(x + 11) = 0
  5. 5x = 8.5 or x = −11 (length cannot be negative)
  6. 6x = 8.5

AnswerLeading to x = 8.5 with no errors

Part (aii) · 2 marks

Using x = 8.5, calculate the perimeter of the rectangle.

  1. 1Area: (2x + 3)(x + 1) = 190, so 2x² + 5x − 187 = 0
  2. 2Solve: x = (−5 + √1521)/4 = (−5 + 39)/4 = 8.5
  3. 3Perimeter = 2(2x + 3) + 2(x + 1) = 6x + 8
  4. 46 × 8.5 + 8 = 59 cm

Answer59

Part (bi) · 4 marks

Use the shaded segment area 30 cm² and angle 50° to show r = 23.7 cm to 1 decimal place.

  1. 1Sector area = 50/360 × π × r²
  2. 2Triangle area = 1/2 × r² × sin 50°
  3. 350/360 × π × r² − 1/2 × r² × sin 50° = 30
  4. 4r² × 0.05331 = 30, so r² = 562.7…
  5. 5r = 23.72… = 23.7 cm (1 d.p.)

Answer23.709 to 23.72…

Part (bii) · 4 marks

Calculate the perimeter of the shaded segment, including chord AB and arc AB.

  1. 1Chord AB = 2 × 23.7 × sin 25° = 20.03… cm
  2. 2Arc AB = 50/360 × 2 × π × 23.7 = 20.68… cm
  3. 3Perimeter = 20.03… + 20.68… = 40.71… ≈ 40.7 cm

Answer40.7 or 40.8 or 40.71 to 40.75…

Stuck on a step? Read the free lesson: Equations That Become Quadratics

Practise more equations that become quadratics questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on equations that become quadratics

Also searched as: 0580/41/O/N/24 · 0580_w24_qp_41 · 0580 w24 41 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.