A-LevelMathematicsAlgebra9709/32 · May/June 2017 · Q810 Marks

Let f(x) = (5x² - 7x + 4) / ((3x + 2)(x² + 5)).

(i)Express f(x) in partial fractions.[5]
(ii)Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x².[5]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[B1] State or imply the form A/(3x+2) + (Bx+C)/(x²+5)
[M1] Use a relevant method to determine a constant
[A1] Obtain one of the values A = 2, B = 1, C = -3
[A1] Obtain a second value
[A1] Obtain the third value
[M1] Use correct method to find the first two terms of the expansion of (3x + 2)⁻¹, (1 + (3/2)x)⁻¹, (5+x²)⁻¹ or (1+(1/5)x²)⁻¹
[A1FT] Obtain correct unsimplified expansions up to the term in x² of each partial fraction. The FT is on A, B, C. from part (i)
[A1FT] (Second A1FT for the other fraction)
[M1] Multiply out up to the term in x² by Bx + C, where BC ≠ 0
[A1] Obtain final answer 2 - (13/10)x + (237/100)x² , or equivalent

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Topic: Algebra, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, May/June 2017 Paper 3 Variant 2, question 8.

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