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WMA12/01 Oct 2021, Question 8: Definite Integrals and Areas

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 10 marks

The question, in short

Cubic curve with constant k has maximum point M at x = 2; show k = 28, find where y increases, and find exact shaded area R bounded by curve, horizontal line through M and y-axis.

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

Part (a) · 3 marks

Differentiate the cubic, use that the derivative is zero at x = 2, and solve for k to show k = 28.

  1. 1dy/dx = 4x² − 22x + k
  2. 2At M, x = 2 and dy/dx = 0
  3. 316 − 44 + k = 0
  4. 4k = 28

AnswerShown: dy/dx = 4x² − 22x + k; at x = 2, 16 − 44 + k = 0 ⇒ k = 28

Part (b) · 2 marks

Solve the inequality where the derivative is positive to give the x-values for which y is increasing.

  1. 1dy/dx = 4x² − 22x + 28 = 2(2x² − 11x + 14)
  2. 22(2x − 7)(x − 2) = 0 ⇒ x = 2, x = 7/2
  3. 3y is increasing when dy/dx > 0 (outside the roots)
  4. 4x < 2 or x > 7/2

Answerx < 2 or x > 7/2

Part (c) · 5 marks

Integrate algebraically the difference between the horizontal line through M and the curve from x = 0 to x = 2 to find the exact area of R.

  1. 1y at M: (4/3)(8) − 11(4) + 28(2) = 32/3 + 12 = 68/3
  2. 2∫((4/3)x³ − 11x² + 28x) dx = x⁴/3 − 11x³/3 + 14x²
  3. 3∫₀² = 16/3 − 88/3 + 56 = 32
  4. 4Rectangle under l from x = 0 to 2: 2 × 68/3 = 136/3
  5. 5R = 136/3 − 32 = 40/3

Answer40/3

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