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4PM1/02R Jun 2025, Question 10: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 18 marks

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The question, in short

For curve y = (5x − 2)/(ax + b), use the vertical asymptote x = −4/3 and y-intercept (0, −1/4) to find a and b, then analyse asymptotes, intercepts, gradient, sketch, and distance EF.

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

Part (a) · 3 marks

Show b = 8 using the y-intercept, then find a using the vertical asymptote x = −4/3.

  1. 1(i) At x = 0: y = −2/b
  2. 2−2/b = −1/4, so b = 8
  3. 3(ii) Vertical asymptote where ax + 8 = 0
  4. 4−(4/3)a + 8 = 0 (or −b/a = −4/3)
  5. 5a = 6

Answer(i) b = 8 shown; (ii) a = 6

Part (b) · 1 mark

State the equation of the horizontal asymptote of C.

  1. 1As x → ∞, y → 5/6

Answery = 5/6

Part (c) · 1 mark

Find the coordinates where C meets the x-axis.

  1. 15x − 2 = 0, so x = 2/5

Answer(2/5, 0)

Part (d) · 4 marks

Use calculus to show the gradient of C is positive everywhere.

  1. 1Quotient rule: dy/dx = [5(6x + 8) − 6(5x − 2)]/(6x + 8)²
  2. 2= 52/(6x + 8)²
  3. 3Numerator 52 > 0 and (6x + 8)² > 0 for x ≠ −4/3
  4. 4So the gradient is always positive

Answerdy/dx = 52/(6x + 8)², which is always positive

Part (e) · 3 marks

Sketch C, labelling both asymptotes and the axis intercept coordinates.

  1. 1Vertical asymptote x = −4/3, horizontal asymptote y = 5/6
  2. 2Gradient always positive, so each branch rises
  3. 3Right branch passes through (0, −1/4) and (2/5, 0)
  4. 4Left branch lies above y = 5/6

AnswerTwo-branch reciprocal-type curve with asymptotes x = −4/3 and y = 5/6, increasing on each branch; crosses the axes at (0, −1/4) and (2/5, 0) on the right-hand branch

Part (f) · 6 marks

Given gradient 13/100 at points E and F, find the length of EF to 2 decimal places.

  1. 152/(6x + 8)² = 13/100
  2. 2(6x + 8)² = 400, so 6x + 8 = ±20 (3x² + 8x − 28 = 0)
  3. 3x = 2, y = 2/5 and x = −14/3, y = 19/15
  4. 4EF = √((2 + 14/3)² + (2/5 − 19/15)²)
  5. 5EF = 6.72

AnswerEF = 6.72

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