Lemmara logoLemmaraWorked solution

4PM1/01 Jun 2019, Question 11: Equation of the Tangent Line

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 17 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Curve C: 3y = x² + 2, point P on C with x = 4. Find tangent and normal equations at P, a triangle area, and a volume of revolution.

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

Part (a) · 6 marks

Find the tangent line k to C at P, expressed as ay = bx + c with integer coefficients.

  1. 1x = 4: y = (16 + 2)/3 = 6
  2. 2dy/dx = 2x/3, at x = 4 gradient = 8/3
  3. 3y − 6 = (8/3)(x − 4)
  4. 43y = 8x − 14

Answer3y = 8x − 14

Part (b) · 2 marks

Find the normal line l to C at P, expressed as dy = ex + f with integer coefficients.

  1. 1Gradient of normal = −3/8
  2. 2y − 6 = −(3/8)(x − 4)
  3. 38y = −3x + 60

Answer8y = −3x + 60

Part (c) · 3 marks

Calculate the area of the triangle formed by lines k, l and the x-axis.

  1. 1k meets the x-axis at x = 7/4; l meets it at x = 20
  2. 2Base = 20 − 7/4 = 73/4, height = 6
  3. 3Area = ½ × 73/4 × 6 = 219/4 = 54.75

Answer219/4 (54.75)

Part (d) · 6 marks

Rotate the region bounded by C, l, both axes fully about the x-axis; find the volume by integration to the nearest whole number.

  1. 1V = π ∫ from 0 to 4 of ((x² + 2)/3)² dx + volume of cone (radius 6, height 16)
  2. 2π ∫ (1/9)(x⁴ + 4x² + 4) dx = (π/9)[x⁵/5 + 4x³/3 + 4x] from 0 to 4
  3. 3= (π/9)(1024/5 + 256/3 + 16)
  4. 4Cone = (1/3)π × 6² × 16 = 192π
  5. 5Total = 710 (nearest whole number)

Answer710

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

Practise more equation of the tangent line questions

Real 4PM1 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: 4PM1 1 June 2019 · 4PM1/01 · 4pm1 jun19 1 · question 11.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.