A-Level Mathematics May/June 2024 Q1: Find the coefficient of x² in the expansion of (2-5x)(1+3x)¹⁰.

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  To find the coefficient of $x^2$ in the expansion of $(2-5x)(1+3x)^{10}$, we first need to find the terms up to $x^2$ in the expansion of $(1+3x)^{10}$.

  Using the binomial expansion formula, $(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ...$

  For $(1+3x)^{10}$, we have $a=1$, $b=3x$, and $n=10$.

  The expansion is: $(1+3x)^{10} = \binom{10}{0}(1)^{10} + \binom{10}{1}(1)^9(3x)^1 + \binom{10}{2}(1)^8(3x)^2 + ...$

  $= 1 + (10)(1)(3x) + \frac{10 \times 9}{2 \times 1}(1)(9x^2) + ...$

  $= 1 + 30x + 45(9x^2) + ...$

  $= 1 + 30x + 405x^2 + ...$

  Now, we multiply this by $(2-5x)$ and collect the terms in $x^2$: $(2-5x)(1 + 30x + 405x^2 + ...)$

  The terms that produce $x^2$ are:

  1. The constant term from $(2-5x)$ multiplied by the $x^2$ term from $(1+3x)^{10}$. $2 \times 405x^2 = 810x^2$

  2. The $x$ term from $(2-5x)$ multiplied by the $x$ term from $(1+3x)^{10}$. $-5x \times 30x = -150x^2$

  Adding these terms together: Total $x^2$ term = $810x^2 - 150x^2 = 660x^2$.

  Therefore, the coefficient of $x^2$ is 660.

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  How the marks are awarded

  • B1 — Correctly finding the term '30x' in the expansion of (1+3x)¹⁰. This is shown in the line '$= 1 + 30x + 405x^2 + ...$'.
  • B1 — Correctly finding the term '+405x²' in the expansion of (1+3x)¹⁰. This is shown by the calculation '$45(9x^2)$' which simplifies to '$405x^2$'.
  • M1 — Demonstrating the correct method to find the two terms that result in x² by multiplying the relevant parts of (2-5x) and the expansion of (1+3x)¹⁰. This is shown by the calculations '$2 \times 405x^2$' and '$-5x \times 30x$'.
  • A1 — Obtaining the correct final coefficient of 660. This is clearly stated as the final answer after combining the two x² terms.

  Common mistakes

  • Forgetting to square the '3' in (3x)², leading to a calculation of $\binom{10}{2}(3x^2) = 45(3x^2) = 135x^2$ instead of $405x^2$.
  • Only considering one of the ways to make an x² term, for example, only calculating $2 \times 405x^2$ and ignoring the term from $-5x \times 30x$.
  • Making a sign error when multiplying, for example calculating $-5x \times 30x$ as $+150x^2$, leading to a final answer of 960.
  • Expanding $(1+3x)^{10}$ but only finding the $x^2$ term ($405x^2$) and then multiplying it by $(2-5x)$ to get $810x^2 - 2025x^3$, incorrectly concluding the coefficient is 810.

  Examiner tip: When dealing with a product of expansions, identify which pairs of terms will multiply to give the required power before you start calculating.

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