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4PM1/02R Jun 2020, Question 6: Equation of the Tangent Line

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 15 marks

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The question, in short

For the curve y = x² − 5x + 4 meeting the x-axis at A and B, work out the intercepts, where the tangents and normals at A and B intersect, and the area of quadrilateral ATBN.

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

Part (a) · 3 marks

Work out the coordinates of the two points where the curve crosses the x-axis.

  1. 1x² − 5x + 4 = (x − 4)(x − 1) = 0
  2. 2x = 1 or 4
  3. 3A(1, 0), B(4, 0)

AnswerA(1, 0), B(4, 0)

Part (b) · 6 marks

Work out where the tangent lines at A and B intersect, giving point T.

  1. 1dy/dx = 2x − 5; at A (x = 1) gradient −3
  2. 2Tangent at A: y = −3(x − 1)
  3. 3By symmetry the tangents meet on x = (1 + 4)/2 = 5/2
  4. 4y = −3(5/2 − 1) = −9/2
  5. 5T = (5/2, −9/2)

AnswerT(5/2, −9/2)

Part (c) · 3 marks

Work out where the normal lines at A and B intersect, giving point N.

  1. 1Normal at A has gradient 1/3: y = (1/3)(x − 1)
  2. 2Normals meet on x = 5/2
  3. 3y = (1/3)(5/2 − 1) = 1/2
  4. 4N = (5/2, 1/2)

AnswerN(5/2, 1/2)

Part (d) · 3 marks

Work out the area of the four-sided shape ATBN.

  1. 1AB = 4 − 1 = 3 and NT = 1/2 + 9/2 = 5
  2. 2AB and NT are perpendicular
  3. 3Area = ½ × 3 × 5 = 15/2

Answer15/2

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