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WMA12/01 Oct 2022, Question 7: Fractional Indices

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 9 marks

The question, in short

Rewrite the curve equation in the specified power-of-x form, then use calculus to derive a cubic for turning point x-coordinates and find the other two exactly given x = 1.

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

Part (a) · 3 marks

Expand and simplify the given quotient to express y as a sum of terms with powers x^(7/2), x^(5/2), x^(3/2), x^(1/2), giving constants a, b, c, d.

  1. 1Numerator = 12x⁴ − 84x³ + 182x² − 210x
  2. 2Divide each term by 21x^(1/2)
  3. 3y = (12/21)x^(7/2) − (84/21)x^(5/2) + (182/21)x^(3/2) − (210/21)x^(1/2)
  4. 4y = (4/7)x^(7/2) − 4x^(5/2) + (26/3)x^(3/2) − 10x^(1/2)

Answery = (4/7)x^(7/2) − 4x^(5/2) + (26/3)x^(3/2) − 10x^(1/2)

Part (b) · 3 marks

Differentiate the curve, set the derivative to zero, and rearrange to show the stated cubic equation 2x³ − 10x² + 13x − 5 = 0.

  1. 1dy/dx = 2x^(5/2) − 10x^(3/2) + 13x^(1/2) − 5x^(−1/2)
  2. 2Turning points: dy/dx = 0
  3. 3Multiply through by x^(1/2) (x > 0): 2x³ − 10x² + 13x − 5 = 0

AnswerShown: dy/dx = 2x^(5/2) − 10x^(3/2) + 13x^(1/2) − 5x^(−1/2) = 0; multiply by x^(1/2) to get 2x³ − 10x² + 13x − 5 = 0

Part (c) · 3 marks

Using x = 1 as one root, factorise the cubic algebraically and solve to find the exact x-coordinates of the remaining two turning points.

  1. 1(x − 1) is a factor: 2x³ − 10x² + 13x − 5 = (x − 1)(2x² − 8x + 5)
  2. 2Solve 2x² − 8x + 5 = 0: x = (8 ± √(64 − 40))/4
  3. 3x = (8 ± √24)/4 = (8 ± 2√6)/4 = (4 ± √6)/2

Answerx = (4 + √6)/2 and x = (4 − √6)/2

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