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WMA12/01 Oct 2022, Question 9: Stationary Points and Turning Points

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 12 marks

The question, in short

Use calculus on the cubic y = x³ − 5x² + 3x + 14 and its minimum point T to find T, the circle centred at T through A (x = 2), the tangent at A, and the area of region R.

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

Part (a) · 3 marks

Find the coordinates of T, the minimum turning point of the cubic curve C₁, using calculus.

  1. 1dy/dx = 3x² − 10x + 3
  2. 2Set to 0: (3x − 1)(x − 3) = 0, so x = 1/3 or x = 3
  3. 3The minimum is the larger root, x = 3
  4. 4y = 27 − 45 + 9 + 14 = 5, so T = (3, 5)

AnswerT(3, 5)

Part (b) · 3 marks

Given that C₁ meets circle C₂ at A with x-coordinate 2, find an equation of C₂.

  1. 1At A: y = 8 − 20 + 6 + 14 = 8, so A = (2, 8)
  2. 2r² = (2 − 3)² + (8 − 5)² = 1 + 9 = 10
  3. 3(x − 3)² + (y − 5)² = 10

Answer(x − 3)² + (y − 5)² = 10

Part (c) · 3 marks

Show that the tangent l to circle C₂ at A has equation y = (1/3)x + 22/3.

  1. 1Gradient of TA = (8 − 5)/(2 − 3) = −3
  2. 2Tangent is perpendicular: gradient = 1/3
  3. 3y − 8 = (1/3)(x − 2)
  4. 4y = (1/3)x − 2/3 + 8 = (1/3)x + 22/3

AnswerShown: gradient of TA = −3, so tangent gradient = 1/3; y − 8 = (1/3)(x − 2) gives y = (1/3)x + 22/3

Part (d) · 3 marks

Find the exact area of the shaded region R bounded by C₁, line l and the y-axis.

  1. 1Area under C₁ from 0 to 2: [x⁴/4 − 5x³/3 + 3x²/2 + 14x] from 0 to 2
  2. 2= 4 − 40/3 + 6 + 28 = 74/3
  3. 3Area under l from 0 to 2 (trapezium): (1/2)(22/3 + 8) × 2 = 46/3
  4. 4Area of R = 74/3 − 46/3 = 28/3

Answer28/3

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