A-Level Mathematics May/June 2024 Q1: Expand (3+x)(1-2x) in ascending powers of x, up to and including the term in x², simpli…

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  The question is to expand $(3+x)(1-2x)^{1/2}$ in ascending powers of $x$, up to and including the term in $x^2$.

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

  Here, $y = -2x$ and $n = \frac{1}{2}$.

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

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

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

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

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

  Now, multiply this expansion by $(3+x)$:

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

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

  To find terms up to $x^2$, we only need to consider products that result in powers of $x$ less than or equal to 2:

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

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

  Now, collect the like terms:

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

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

  $= 3 - 2x - \frac{5}{2}x^2$

  Final Answer: $3 - 2x - \frac{5}{2}x^2$

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  How the marks are awarded

  • B1 — Stating the correct unsimplified first two terms of the binomial expansion. This is shown in the first line of the calculation: $1 + (\frac{1}{2})(-2x)$.
  • B1 — Stating the correct unsimplified term in $x^2$ for the binomial expansion. This is shown in the working as $\frac{(\frac{1}{2})(-\frac{1}{2})}{2}(-2x)^2$.
  • M1 — Multiplying the three-term expansion of $(1-2x)^{1/2}$ by $(3+x)$ and attempting to collect terms up to $x^2$. This is demonstrated by the step: $3(1 - x - \frac{1}{2}x^2) + x(1 - x) + \dots$.
  • A1 — Obtaining the final, correct, and simplified answer of $3 - 2x - \frac{5}{2}x^2$.

  Common mistakes

  • Sign errors, particularly when squaring the $(-2x)$ term, for example calculating $(-2x)^2$ as $-4x^2$ instead of $4x^2$.
  • Using an incorrect value for 'n' in the binomial formula, such as $n=-1/2$ or $n=2$, instead of the correct $n=1/2$.
  • Errors in the multiplication stage, such as forgetting to multiply all the required terms, for instance omitting the product of $x$ and $-x$.
  • Arithmetic mistakes when simplifying the coefficients, especially when adding or subtracting the fractions for the $x^2$ term.

  Examiner tip: Master the two-stage process of first applying the binomial formula correctly and then systematically multiplying out and collecting only the required terms to save time and avoid errors.

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