Mathematics 9709/12 — May/June 2022

Walkthrough

This problem asks us to find the value of a constant $a$ by equating two binomial coefficients. The key idea is to use the binomial theorem to expand each expression and extract the coefficient of the specified power of $x$.

First, we expand $(3 + x)^5$. The binomial theorem tells us the general term is $\binom{5}{k} 3^{5-k} x^k$. To find the $x^4$ term, we set $k = 4$, giving $\binom{5}{4} 3^1 x^4 = 5 \cdot 3 \, x^4 = 15x^4$. So the coefficient is 15.

Second, we expand $\left(2x + \frac{a}{x}\right)^6$. The general term is $\binom{6}{k} (2x)^{6-k} \left(\frac{a}{x}\right)^k$. Simplifying the powers of $x$, we get $x^{6-k} \cdot x^{-k} = x^{6-2k}$. To get the $x^2$ term, we need $6 - 2k = 2$, so $k = 2$. The term is $\binom{6}{2} 2^4 a^2 x^2 = 15 \cdot 16 \, a^2 x^2 = 240a^2 x^2$. So the coefficient is $240a^2$.

Finally, we equate the two coefficients: $240a^2 = 15$. Dividing both sides by 240 gives $a^2 = \frac{1}{16}$. Taking the square root gives $a = \pm \frac{1}{4}$. Since the question specifies $a$ is positive, we take $a = \frac{1}{4}$.

Key Takeaways

• The binomial theorem allows us to find the coefficient of any specific power of $x$ without expanding the whole expression.
• When the binomial has two terms involving $x$ (like $2x$ and $\frac{a}{x}$), the powers of $x$ combine, so we must carefully track the exponent of $x$ in the general term.
• When a problem asks for a "positive constant", remember to discard any negative solutions.

Common Mistakes

• Forgetting to include the factor $3^{5-k}$ when finding the coefficient in $(3 + x)^5$ — the coefficient is $5 \times 3 = 15$, not just $\binom{5}{4} = 5$.
• Incorrectly tracking the powers of $x$ in $\left(2x + \frac{a}{x}\right)^6$ — the exponent of $x$ in the general term is $6 - 2k$, not $6 - k$.
• Forgetting to include the factor $2^{6-k}$ when finding the coefficient — the coefficient is $15 \times 16 = 240$, not just $\binom{6}{2} = 15$.
• Accepting $a = -\frac{1}{4}$ as an answer when the question specifies $a$ is positive.

Things to Be Careful About

• The mark scheme says "Do not condone extra 'answer' of $-\frac{1}{4}$" — you must explicitly state that $a = \frac{1}{4}$ because $a$ is positive.
• The mark scheme also says not to allow $\sqrt{\frac{1}{16}}$ or similar — give the simplified final answer $\frac{1}{4}$.
• The mark scheme allows "condone inclusion of powers of $x$" when forming the equation, meaning you can write $240a^2x^2 = 15x^4$ as long as you then simplify to $240a^2 = 15$. But it's cleaner to extract coefficients directly.