Explain why the binomial expansion of $\left( ax + \frac{2}{x} \right)^6$ contains only even power

Question

Explain why the binomial expansion of $\left( ax + \frac{2}{x} \right)^6$ contains only even powers of $x$.[2] Write down, and simplify, the first three terms in the expansion of $\left( 1 - \frac{x}{a} \right)^3$ in ascending powers of $x$.[2] Given that the coefficient of $\frac{1}{x^4}$ in the expansion of $ \left(ax+\frac{2}{x} \right)^6$ is 1728 times the coefficient of $x^2$ in the expansion of $ \left(1-\frac{x}{a} \right)^3$ , find the value of $a$.[4] Using the value of $a$ found in (c), show that there is no term independent of $x$ for $ \left(1-\frac{x}{a} \right)^3 \left(1+\frac{1}{x} \right)$[2]

Accepted Answer

(b)$1+\frac{3x}{a}+\frac{3x^2}{a^2}+...$ (c) a=3

2025 SJI PRELIMS P2 Q6

2025 SJI PRELIMS P2 Q6