Lemmara logoLemmaraWorked solution

0580/42 May/Jun 2021, Question 9: Equation of the Tangent Line

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

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

For y = x³ − 4x² + 4x, find a in the factorised form, sketch the curve showing axis intercepts, and find the tangent at x = 4 in y = mx + c form.

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

Part (ai) · 2 marks

Write y = x³ − 4x² + 4x as y = x(x − a)² and find the value of a.

  1. 1x³ − 4x² + 4x = x(x² − 4x + 4) = x(x − 2)²
  2. 2a = 2

Answer2

Part (aii) · 4 marks

Sketch y = x³ − 4x² + 4x on the axes, marking where the graph crosses the x- and y-axes.

  1. 1Positive cubic through O (0, 0)
  2. 2Touches the x-axis at (2, 0) (repeated root) and turns there
  3. 3Label 0 and 2 on the x-axis

AnswerCorrect sketch with curve passing through O and touching (2, 0)

Part (b) · 7 marks

Find the tangent to y = x³ − 4x² + 4x at x = 4, giving it as y = mx + c.

  1. 1dy/dx = 3x² − 8x + 4
  2. 2At x = 4: gradient = 48 − 32 + 4 = 20
  3. 3At x = 4: y = 64 − 64 + 16 = 16, point (4, 16)
  4. 416 = 20 × 4 + c, so c = −64
  5. 5y = 20x − 64

Answery = 20x − 64

Stuck on a step? Read the free lesson: Equation of the Tangent Line

Practise more equation of the tangent line 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 equation of the tangent line

Also searched as: 0580/42/M/J/21 · 0580_s21_qp_42 · 0580 s21 42 · question 9.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.