A-Level Mathematics October/November 2024 Q8(b): Hence obtain the expansion of f(x) in ascending powers of x, up to and including the te…

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  Assuming from part (a) that $f(x) = \frac{2a}{a-2x} + \frac{a}{3a+x}$.

  First, expand each term using the binomial theorem $(1+y)^n = 1 + ny + \frac{n(n-1)}{2!}y^2 + \dots$

  For the first term: $\frac{2a}{a-2x} = 2a(a-2x)^{-1} = 2a[a(1-\frac{2x}{a})]^{-1} = 2a \cdot a^{-1}(1-\frac{2x}{a})^{-1} = 2(1-\frac{2x}{a})^{-1}$

  $2(1-\frac{2x}{a})^{-1} = 2 \left[ 1 + (-1)(-\frac{2x}{a}) + \frac{(-1)(-2)}{2}(-\frac{2x}{a})^2 + \dots \right]$ $= 2 \left[ 1 + \frac{2x}{a} + \frac{4x^2}{a^2} + \dots \right]$ $= 2 + \frac{4x}{a} + \frac{8x^2}{a^2} + \dots$

  For the second term: $\frac{a}{3a+x} = a(3a+x)^{-1} = a[3a(1+\frac{x}{3a})]^{-1} = a \cdot (3a)^{-1}(1+\frac{x}{3a})^{-1} = \frac{1}{3}(1+\frac{x}{3a})^{-1}$

  $\frac{1}{3}(1+\frac{x}{3a})^{-1} = \frac{1}{3} \left[ 1 + (-1)(\frac{x}{3a}) + \frac{(-1)(-2)}{2}(\frac{x}{3a})^2 + \dots \right]$ $= \frac{1}{3} \left[ 1 - \frac{x}{3a} + \frac{x^2}{9a^2} + \dots \right]$ $= \frac{1}{3} - \frac{x}{9a} + \frac{x^2}{27a^2} + \dots$

  Now, add the two expansions: $f(x) = \left( 2 + \frac{4x}{a} + \frac{8x^2}{a^2} \right) + \left( \frac{1}{3} - \frac{x}{9a} + \frac{x^2}{27a^2} \right) + \dots$

  Collect the terms: Constant: $2 + \frac{1}{3} = \frac{7}{3}$ Term in $x$: $\frac{4x}{a} - \frac{x}{9a} = (\frac{36}{9a} - \frac{1}{9a})x = \frac{35x}{9a}$ Term in $x^2$: $\frac{8x^2}{a^2} + \frac{x^2}{27a^2} = (\frac{216}{27a^2} + \frac{1}{27a^2})x^2 = \frac{217x^2}{27a^2}$

  So, the expansion of $f(x)$ is: $f(x) = \frac{7}{3} + \frac{35}{9a}x + \frac{217}{27a^2}x^2 + \dots$

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  How the marks are awarded

  • M1 — Correctly rewriting one of the terms in the form $k(1+y)^{-1}$ and applying the binomial formula to get the first two terms. For example, expanding $2(1-2x/a)^{-1}$ to $2(1 + 2x/a + ...)$.
  • A1ft — Obtaining a fully correct expansion for one of the partial fractions up to the $x^2$ term, such as $2 + 4x/a + 8x^2/a^2$. The 'ft' allows for follow-through if the constant from part (a) was incorrect.
  • A1ft — Obtaining correct expansions for both terms and showing the intention to add them. For example, writing out $(2 + 4x/a + 8x^2/a^2) + (1/3 - x/9a + x^2/27a^2)$. This mark also allows follow-through.
  • A1 — Obtaining the final correct and simplified expansion $\frac{7}{3} + \frac{35}{9a}x + \frac{217}{27a^2}x^2$. No follow-through is allowed for this mark.

  Common mistakes

  • Sign errors in the expansion, especially for the $(1-2x/a)^{-1}$ term, leading to $1 - 2x/a$ instead of the correct $1 + 2x/a$.
  • Incorrectly factoring the constant from the denominator, for example writing $(a-2x)^{-1}$ as $a^{-1}(1-2x)^{-1}$ instead of $a^{-1}(1-2x/a)^{-1}$.
  • Forgetting to multiply the expansion by the constant from the partial fraction numerator, e.g. forgetting the '2' in $2(1-2x/a)^{-1}$.
  • Arithmetic errors when combining the coefficients of the two expansions, particularly when finding a common denominator for the $x$ and $x^2$ terms.

  Examiner tip: Master the two-step process for binomial expansions of the form $(a+bx)^n$: first, factor out the constant 'a' to create the form $a^n(1+\frac{b}{a}x)^n$, then apply the standard $(1+y)^n$ formula.

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