Lemmara logoLemmaraWorked solution

0580/43 Oct/Nov 2024, Question 10: Equation of the Tangent Line

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 11 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

For the curve y = 3 + 2x − x² with points A(−1, 0) and B(2, 3), differentiate, work with tangents and perpendicular lines, and locate the maximum.

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

Part (a) · 2 marks

Differentiate the given quadratic expression with respect to x.

  1. 1Multiply by the power, then take 1 off the power: axⁿ → naxⁿ⁻¹
  2. 23 → 0
  3. 32x → 2
  4. 4−x² → −2x
  5. 5dy/dx = 2 − 2x

Answer2 − 2x

Part (bi) · 3 marks

Verify that the tangent line at A(−1, 0) has equation y = 4x + 4.

  1. 1dy/dx = 2 − 2x
  2. 2At A, x = −1: gradient = 2 − 2(−1) = 4
  3. 3Put A(−1, 0) into y = 4x + c: 0 = −4 + c, so c = 4
  4. 4Tangent at A: y = 4x + 4

AnswerGradient = 2 − 2(−1) = 4, 0 = 4 × (−1) + c, c = 4 leading to y = 4x + 4

Part (bii) · 3 marks

Line L is perpendicular to y = 4x + 4 and goes through B(2, 3); find L in y = mx + c form.

  1. 1Perpendicular gradient = −1/4
  2. 23 = −1/4 × 2 + c
  3. 3c = 3 + 1/2 = 7/2
  4. 4y = −1/4 x + 7/2

Answery = −1/4 x + 7/2

Part (c) · 3 marks

Determine the coordinates of the highest point on the curve y = 3 + 2x − x².

  1. 1At the maximum, 2 − 2x = 0
  2. 2x = 1
  3. 3y = 3 + 2(1) − 1² = 4
  4. 4Maximum point (1, 4)

Answer(1, 4)

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: 0580/43/O/N/24 · 0580_w24_qp_43 · 0580 w24 43 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.