<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Explain why the binomial expansion of \(\left(ax + \frac{2}{x}\right)^6\) contains only even powers of \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph">Write down, and simplify, the first three terms in the expansion of \(\left(1 - \frac{x}{a}\right)^3\) in ascending powers of \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-0-item-2" data-list-item-path="imported-item-block-0-item-2"><div class="list-item-paragraph">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 \(a\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-0-item-3" data-list-item-path="imported-item-block-0-item-3"><div class="list-item-paragraph"><p>Using the value of \(a\) found in <strong>(c)</strong>, show that there is no term independent of \(x\) for&nbsp;\(\left(1 - \frac{x}{a}\right)^3\left(1 + \frac{1}{x}\right)\).</p><span class="inline-marks">[2]</span></div></li></ol>

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

Write down, and simplify, the first three terms in the expansion of \(\left(1 - \frac{x}{a}\right)^3\) in ascending powers of \(x\).

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 \(a\).

<p>Using the value of \(a\) found in <strong>(c)</strong>, show that there is no term independent of \(x\) for&nbsp;\(\left(1 - \frac{x}{a}\right)^3\left(1 + \frac{1}{x}\right)\).</p>