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4PM1/02 Jun 2022, Question 11: Rational Functions and Asymptotes

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 17 marks

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The question, in short

For curve y = ((2a − 1)x + 1)/(ax − 6), find the derivative, use a normal at the y-axis intercept to determine a and the curve equation, sketch C, and find where line l meets C again.

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

Part (a) · 3 marks

Differentiate y = ((2a − 1)x + 1)/(ax − 6) with respect to x.

  1. 1Quotient rule: u = (2a − 1)x + 1, u′ = 2a − 1; v = ax − 6, v′ = a
  2. 2dy/dx = ((2a − 1)(ax − 6) − a((2a − 1)x + 1))/(ax − 6)²

Answerdy/dx = ((2a − 1)(ax − 6) − a((2a − 1)x + 1))/(ax − 6)²

Part (bi) · 4 marks

Using the normal 66y − 72x + 11 = 0 at point A on the y-axis, show that a = 3.

  1. 1Normal gradient = 72/66 = 12/11, so tangent gradient at A = −11/12
  2. 2At x = 0: dy/dx = (−6(2a − 1) − a)/36 = (−13a + 6)/36
  3. 3(−13a + 6)/36 = −11/12 ⇒ −13a + 6 = −33
  4. 4a = 3

Answera = 3 shown

Part (bii) · 1 mark

Show that with a = 3 the curve becomes y = (5x + 1)/(3x − 6), x ≠ 2.

  1. 1Substitute a = 3: y = ((2 × 3 − 1)x + 1)/(3x − 6) = (5x + 1)/(3x − 6), x ≠ 6/3 = 2

Answery = (5x + 1)/(3x − 6) shown

Part (c) · 5 marks

Sketch C, labelling asymptotes with equations and the coordinates where C crosses the axes.

  1. 1Vertical asymptote: 3x − 6 = 0 ⇒ x = 2
  2. 2Horizontal asymptote: y → 5/3 as x → ±∞
  3. 3x-intercept: 5x + 1 = 0 ⇒ (−1/5, 0); y-intercept: (0, −1/6)
  4. 4Sketch the two branches approaching the asymptotes

AnswerTwo-branch curve with asymptotes x = 2 and y = 5/3; one branch in the first quadrant (right of x = 2, above y = 5/3), the other through quadrants 2, 3 and 4 crossing the axes at (−1/5, 0) and (0, −1/6)

Part (d) · 4 marks

Find the x coordinate of D, where line l meets C again, as an improper fraction.

  1. 1l: y = (12/11)x − 1/6
  2. 2(12/11)x − 1/6 = (5x + 1)/(3x − 6)
  3. 3(3x − 6)(72x − 11) = 66(5x + 1) ⇒ 72x² − 265x = 0
  4. 4x(72x − 265) = 0; x = 0 is A, so x = 265/72

Answerx = 265/72

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