A-Level Mathematics October/November 2024 Q1: In the expansion of (kx+2/x)^4, where k is a positive constant, the term independent of…

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  The general term in the expansion of $(kx+\frac{2}{x})^4$ is given by the binomial theorem: $T_{r+1} = \binom{4}{r} (kx)^{4-r} \left(\frac{2}{x}\right)^r$ $T_{r+1} = \binom{4}{r} k^{4-r} x^{4-r} \cdot 2^r x^{-r} = \binom{4}{r} k^{4-r} 2^r x^{4-2r}$

  Part 1: Find the value of k

  For the term independent of x, the power of x must be 0. $4 - 2r = 0 \implies 2r = 4 \implies r = 2$

  Substitute $r=2$ into the general term formula: $\text{Term} = \binom{4}{2} (kx)^{4-2} \left(\frac{2}{x}\right)^2$ $= 6 (k^2x^2) \left(\frac{4}{x^2}\right)$ $= 24k^2$

  We are given that this term is equal to 150. $24k^2 = 150$ $k^2 = \frac{150}{24} = \frac{25}{4}$ Since k is a positive constant: $k = \sqrt{\frac{25}{4}} = \frac{5}{2}$

  Part 2: Determine the coefficient of x²

  For the term in $x^2$, the power of x must be 2. $4 - 2r = 2 \implies 2r = 2 \implies r = 1$

  Substitute $r=1$ into the general term formula: $\text{Term} = \binom{4}{1} (kx)^{4-1} \left(\frac{2}{x}\right)^1$ $= 4 (kx)^3 \left(\frac{2}{x}\right)$ $= 4 (k^3x^3) \left(\frac{2}{x}\right) = 8k^3x^2$

  The coefficient of $x^2$ is $8k^3$.

  Substitute the value of $k = \frac{5}{2}$: $\text{Coefficient} = 8 \left(\frac{5}{2}\right)^3 = 8 \left(\frac{125}{8}\right)$ $\text{Coefficient} = 125$

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  How the marks are awarded

  • M1 — Correctly identifying the term independent of x (the $r=2$ term) and writing it as $\binom{4}{2}(kx)^2(2/x)^2$ or the simplified form $6(kx)^2(4/x^2)$.
  • A1 — Correctly equating the constant term expression $24k^2$ to 150 and solving to find the exact value $k = 5/2$.
  • M1 — Correctly identifying the term in $x^2$ (the $r=1$ term) as $\binom{4}{1}(kx)^3(2/x)^1$ and attempting to find the coefficient using their found value of k.
  • A1 — Correctly calculating the final coefficient as 125, following a correct method.

  Common mistakes

  • Forgetting to square the constants in the terms, for example writing $(2/x)^2$ as $2/x^2$ or $(kx)^2$ as $kx^2$, leading to an incorrect equation for k.
  • Making an error with the binomial coefficient, for example calculating $\binom{4}{2}$ as 4 instead of 6.
  • Incorrectly identifying the term for $x^2$. For example, solving $4-2r=1$ instead of $4-2r=2$, leading to a non-integer value for r and confusion.
  • Making a calculation error when substituting $k=5/2$ into the expression for the coefficient of $x^2$, for example 8(5/2)^3 = 8(25/4) = 50.

  Examiner tip: Master using the general term formula, $\binom{n}{r}a^{n-r}b^r$, to find any specific term in an expansion by setting up and solving an equation for the required power of x.

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