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4PM1/02R Jan 2022, Question 6: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 11 marks

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The question, in short

For curve y = 4√x with point A(9, 12), find where the tangent and normal at A meet the x-axis, then find the area of triangle ATN.

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

Part (a) · 5 marks

Find the coordinates of T, where the tangent to the curve at A(9, 12) crosses the x-axis.

  1. 1dy/dx = 2x^(−1/2) = 2/√x
  2. 2At x = 9: dy/dx = 2/3
  3. 3Tangent: y − 12 = (2/3)(x − 9)
  4. 4y = 0: −12 = (2/3)(x − 9), x − 9 = −18
  5. 5x = −9, so T(−9, 0)

AnswerT(−9, 0)

Part (b) · 4 marks

Find the coordinates of N, where the normal to the curve at A(9, 12) crosses the x-axis.

  1. 1Gradient of normal = −3/2
  2. 2y − 12 = −(3/2)(x − 9)
  3. 3y = 0: −12 = −(3/2)(x − 9), x − 9 = 8
  4. 4x = 17, so N(17, 0)

AnswerN(17, 0)

Part (c) · 2 marks

Calculate the area of triangle ATN using the points found in parts a and b.

  1. 1Base TN = 17 − (−9) = 26, height = 12
  2. 2Area = ½ × 26 × 12 = 156

Answer156

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