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WMA11/01 Oct 2021, Question 8: Lines Meeting Curves: the Discriminant

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

The question, in short

For the curve y = 4 + 12x − 3x², rewrite the quadratic in completed-square form, use it to locate the maximum point M, then find the equation of the line l₂ that touches the curve and runs parallel to line l₁ through O and M.

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

Part (ai) · 3 marks

Express 4 + 12x − 3x² as a + b(x + c)², giving the constants a, b and c.

  1. 14 + 12x − 3x² = 4 − 3(x² − 4x)
  2. 2= 4 − 3[(x − 2)² − 4]
  3. 3= 16 − 3(x − 2)²

Answer16 − 3(x − 2)² (a = 16, b = −3, c = −2)

Part (aii) · 2 marks

Using your completed-square result, state the coordinates of the curve's maximum point M.

  1. 1Maximum when (x − 2)² = 0, so x = 2
  2. 2y = 16, so M = (2, 16)

AnswerM(2, 16)

Part (b) · 5 marks

Line l₂ is tangent to C and parallel to l₁, which joins O to M. Determine an equation for l₂.

  1. 1Gradient of l₁ = 16/2 = 8, so l₂: y = 8x + k
  2. 24 + 12x − 3x² = 8x + k
  3. 33x² − 4x + k − 4 = 0
  4. 4Touches, so b² − 4ac = 0: 16 − 12(k − 4) = 0
  5. 5k = 16/3, so l₂: y = 8x + 16/3

Answery = 8x + 16/3

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Also searched as: WMA11 October 2021 · WMA11/01 · wma11 oct21 · IAS Maths Pure Mathematics 1 October 2021 · question 8.
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