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WMA11/01 Jan 2023, Question 10: Tangent and Normal Problems

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

The question, in short

For the cubic y = (3x + 20)(x + 6)(2x − 3) sketched in Figure 4, find where it is positive, expand it, and locate where the y-axis tangent meets it again.

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

Part (a) · 2 marks

State the x values satisfying f(x) > 0, using the sketch and factorised form.

  1. 1Roots of f(x) = 0: x = −20/3, x = −6, x = 3/2
  2. 2Positive cubic: above the x-axis between the two lower roots and after the largest root
  3. 3f(x) > 0 for −20/3 < x < −6 or x > 3/2

Answer−20/3 < x < −6 or x > 3/2

Part (b) · 3 marks

Expand (3x + 20)(x + 6)(2x − 3) into a polynomial in simplest form.

  1. 1(x + 6)(2x − 3) = 2x² + 9x − 18
  2. 2(3x + 20)(2x² + 9x − 18) = 6x³ + 27x² − 54x + 40x² + 180x − 360
  3. 3= 6x³ + 67x² + 126x − 360

Answer6x³ + 67x² + 126x − 360

Part (c) · 5 marks

Find algebraically the x coordinate of P where the tangent at the y-intercept cuts C again.

  1. 1f′(x) = 18x² + 134x + 126, so the gradient at x = 0 is 126
  2. 2C cuts the y-axis at (0, −360), so l: y = 126x − 360
  3. 36x³ + 67x² + 126x − 360 = 126x − 360
  4. 46x³ + 67x² = 0, so x²(6x + 67) = 0
  5. 5x = 0 is the point of contact, so at P: x = −67/6

Answerx = −67/6

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