A-Level Mathematics May/June 2025 Q2(a): Expand (6-x)(1-2x)¯½ in ascending powers of x, up to and including the term in x², simp…

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  To expand $(6-x)(1-2x)^{-\frac{3}{2}}$, we first find the expansion of $(1-2x)^{-\frac{3}{2}}$ up to the term in $x^2$.

  Using the binomial expansion formula $(1+y)^n = 1 + ny + \frac{n(n-1)}{2!}y^2 + \dots$

  Let $y = -2x$ and $n = -\frac{3}{2}$.

  $(1-2x)^{-\frac{3}{2}} = 1 + (-\frac{3}{2})(-2x) + \frac{(-\frac{3}{2})(-\frac{3}{2}-1)}{2!}(-2x)^2 + \dots$

  $= 1 + 3x + \frac{(-\frac{3}{2})(-\frac{5}{2})}{2}(4x^2) + \dots$

  $= 1 + 3x + \frac{\frac{15}{4}}{2}(4x^2) + \dots$

  $= 1 + 3x + \frac{15}{8}(4x^2) + \dots$

  $= 1 + 3x + \frac{15}{2}x^2 + \dots$

  Now, we multiply this expansion by $(6-x)$:

  $(6-x)(1 + 3x + \frac{15}{2}x^2 + \dots) = 6(1 + 3x + \frac{15}{2}x^2) - x(1 + 3x + \frac{15}{2}x^2) + \dots$

  We only need terms up to $x^2$:

  $= (6 + 18x + 6(\frac{15}{2})x^2) - (x + 3x^2) + \dots$

  $= (6 + 18x + 45x^2) - (x + 3x^2) + \dots$

  Now, collect the like terms:

  Constant term: $6$

  Term in $x$: $18x - x = 17x$

  Term in $x^2$: $45x^2 - 3x^2 = 42x^2$

  So, the expansion is $6 + 17x + 42x^2$.

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  How the marks are awarded

  • B1 — Correctly finding the first two terms of the expansion of $(1-2x)^{-3/2}$, which are $1 + 3x$.
  • B1 — Obtaining the correct third term, $\frac{15}{2}x^2$, from the correct formula and simplification: $\frac{(-3/2)(-5/2)}{2}(-2x)^2$.
  • M1 — Multiplying the three-term expansion $(1 + 3x + \frac{15}{2}x^2)$ by $(6-x)$ and attempting to collect terms up to $x^2$.
  • A1 — Obtaining the final, fully simplified correct answer of $6 + 17x + 42x^2$.

  Common mistakes

  • Sign errors, particularly in calculating $(n-1)$ as $(-\frac{3}{2}-1) = -\frac{5}{2}$ or in squaring the term $(-2x)^2 = 4x^2$, not $-4x^2$.
  • Arithmetic errors when simplifying the coefficient of the $x^2$ term, for example calculating $\frac{15}{8} \times 4$ incorrectly.
  • Errors in the final multiplication, such as forgetting to multiply the second part of the bracket, $-x$, with the expansion, or only multiplying corresponding terms (e.g. $6 \times 1$ and $-x \times 3x$).
  • Using an incorrect value for 'n', such as $n = -1/2$ from the question text typo, which would lead to an entirely different expansion and final answer.

  Examiner tip: For multi-part expressions, always complete the binomial expansion to the required power first before systematically multiplying out the terms and collecting them.

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