A-LevelMathematicsSeries9709/33 · May/June 2025 · Q78 Marks

Let f(x) = (3a-5x) / ((3a+2x)(2a-x)), where a is a positive constant.

(a)Express f(x) in partial fractions.[3]
(b)Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x².[4]
(c)State the set of values of x for which the expansion in part (b) is valid.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] State or imply the form A/(3a+2x) + B/(2a-x) and use a correct method to find a constant.
[A1] Obtain one of A = 3 and B = -1.
[A1] Obtain the second value. ISW
[M1] Use a correct method to obtain the first two terms in the expansion of (3a+2x)^(-1), (2a-x)^(-1), or (1-x/2a)^(-1).
[A1] Obtain the correct unsimplified expansion for 3/(3a+2x) up to x² term. 3/3a(1-2x/3a + (2x/3a)^2 - ...)
[A1] Obtain the correct unsimplified expansion for -1/(2a-x) up to x² term. -1/2a(1+x/2a + (x/2a)^2 + ...)
[A1] Obtain final answer 1/(2a) - 11/(12a²) x + 23/(72a³) x². Ignore terms in higher powers of x. Do not ISW. Allow reverse order.
[B1] State |x| < a. OE

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