Lemmara logoLemmaraWorked solution

WMA11/01 Jun 2023, Question 10: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Sketch shows quadratic C and lines l₁, l₂; use given equations to find line equations, point P in terms of b, and the value of b.

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

Part (ai) · 1 mark

Write the equation of line l₁ with gradient 1/2 passing through C's negative x-intercept.

  1. 1C meets the negative x-axis at x = −2, so l₁ passes through (−2, 0)
  2. 2Gradient 1/2: y = (1/2)(x + 2)

Answery = (1/2)(x + 2)

Part (aii) · 3 marks

Find the coordinates of point P where l₁ meets C, giving them in terms of b.

  1. 1Set −(1/4)(x + 2)(x − b) = (1/2)(x + 2)
  2. 2At P, x ≠ −2, so divide by (x + 2): −(1/4)(x − b) = 1/2
  3. 3x − b = −2, so x = b − 2
  4. 4y = (1/2)(b − 2 + 2) = b/2, so P is (b − 2, b/2)

AnswerP(b − 2, b/2)

Part (b) · 2 marks

Find an equation for l₂, perpendicular to l₁, through C's positive x-intercept, in terms of b.

  1. 1l₂ is perpendicular to l₁, so its gradient is −2
  2. 2C meets the positive x-axis at (b, 0)
  3. 3y = −2(x − b)

Answery = −2(x − b)

Part (c) · 2 marks

Given l₂ also passes through P, show its equation can be written as y = −2x + 5b/2 − 4.

  1. 1l₂ has gradient −2 and passes through P(b − 2, b/2)
  2. 2y − b/2 = −2(x − (b − 2)) = −2x + 2b − 4
  3. 3y = −2x + 5b/2 − 4

AnswerShown: line through P(b − 2, b/2) with gradient −2 is y − b/2 = −2(x − b + 2), giving y = −2x + 5b/2 − 4

Part (d) · 2 marks

Use the results to determine the numerical value of b.

  1. 1Both equations describe l₂: −2x + 2b = −2x + 5b/2 − 4
  2. 22b = 5b/2 − 4, so b/2 = 4
  3. 3b = 8

Answerb = 8

Stuck on a step? Read the free lesson: When a Line Meets, Touches or Misses a Curve

Practise more when a line meets, touches or misses a curve questions

Real XMA01 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 when a line meets, touches or misses a curve

Also searched as: WMA11 June 2023 · WMA11/01 · wma11 jun23 · IAS Maths Pure Mathematics 1 June 2023 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.