Lemmara logoLemmaraWorked solution

0580/41 May/Jun 2024, Question 8: Simultaneous Equations with a Curve

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 15 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Sketch y = x² + 7x − 18 on axes, marking axis intercepts, then differentiate and locate a turning point, and find where a line meets a parabola to 2 decimal places.

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

Part (a) · 4 marks

Draw the curve y = x² + 7x − 18 on the given axes, labelling where it crosses both axes.

  1. 1x² + 7x − 18 = (x + 9)(x − 2), so it meets the x-axis at −9 and 2
  2. 2When x = 0, y = −18
  3. 3Draw a U-shaped curve through −9 and 2 on the x-axis and −18 on the y-axis, minimum below the x-axis to the left of the y-axis

AnswerCorrect sketch: U-shaped parabola with roots at x = −9 and x = 2, y-intercept −18, minimum in 3rd quadrant

Part (bi) · 2 marks

Differentiate y = x² − 3x − 28 with respect to x.

  1. 1Multiply by the power, then take 1 off the power: axⁿ → naxⁿ⁻¹
  2. 2x² → 2x
  3. 3−3x → −3
  4. 4−28 → 0
  5. 5dy/dx = 2x − 3

Answer2x − 3

Part (bii) · 3 marks

Work out the coordinates of the stationary point on y = x² − 3x − 28.

  1. 12x − 3 = 0
  2. 2x = 1.5
  3. 3y = 1.5² − 3 × 1.5 − 28 = −30.25
  4. 4Turning point (1.5, −30.25)

Answer(1.5, −30.25)

Part (c) · 6 marks

Solve where y = 5 − 2x meets y = x² − 3x − 28, giving points P and Q to 2 decimal places with full working.

  1. 1x² − 3x − 28 = 5 − 2x
  2. 2x² − x − 33 = 0
  3. 3x = (1 ± √(1 + 132))/2 = (1 ± √133)/2
  4. 4x = 6.266… or x = −5.266…
  5. 5y = 5 − 2(6.266) = −7.53, y = 5 − 2(−5.266) = 15.53
  6. 6(−5.27, 15.53) and (6.27, −7.53)

Answer(−5.27, 15.53 or 15.54) and (6.27, −7.53 or −7.54)

Stuck on a step? Read the free lesson: Simultaneous Equations with a Curve

Practise more simultaneous equations with a curve 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 simultaneous equations with a curve

Also searched as: 0580/41/M/J/24 · 0580_s24_qp_41 · 0580 s24 41 · question 8.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.