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4PM1/01 Jun 2023, Question 11: Definite Integrals and Areas

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 14 marks

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The question, in short

Rewrite f(x) = 10 + 6x − x² in completed-square form A(x + B)² + C, then use it to find the maximum, find where y = f(x) meets y = x² − x + 13, and integrate to get the enclosed area.

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

Part (a) · 4 marks

Find the constants A, B and C in the completed-square form A(x + B)² + C of f(x) = 10 + 6x − x².

  1. 1f(x) = −(x² − 6x) + 10
  2. 2= −[(x − 3)² − 9] + 10
  3. 3= −(x − 3)² + 19
  4. 4A = −1, B = −3, C = 19

AnswerA = −1, B = −3, C = 19

Part (b) · 2 marks

Using your completed square, state the x value giving f(x) its greatest value and state that greatest value.

  1. 1−(x − 3)² ≤ 0, greatest when x = 3
  2. 2Greatest value = 19

Answer(i) x = 3 (ii) 19

Part (c) · 3 marks

Find the x coordinates of the two points where y = f(x) and y = x² − x + 13 intersect.

  1. 1x² − x + 13 = 10 + 6x − x²
  2. 22x² − 7x + 3 = 0
  3. 3(2x − 1)(x − 3) = 0
  4. 4x = 1/2, 3

Answerx = 1/2 and x = 3

Part (d) · 5 marks

Set up and evaluate an exact definite integral for the area of the finite region between the two curves.

  1. 1Area = ∫ from 1/2 to 3 of [(10 + 6x − x²) − (x² − x + 13)] dx
  2. 2= ∫ from 1/2 to 3 of (−2x² + 7x − 3) dx
  3. 3= [−2x³/3 + 7x²/2 − 3x] from 1/2 to 3
  4. 4= (−18 + 31.5 − 9) − (−1/12 + 7/8 − 3/2) = 9/2 + 17/24
  5. 5= 125/24

Answer125/24

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