<p>It is given that&nbsp;\(y = \cos^{-1}(2x) \sin^{-1}(2x)\), where&nbsp;\(-0.5 \le x \le 0.5\), and \(\cos^{-1}(2x)\) and \(\sin^{-1}(2x)\) denote their principal values.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\((1-4x^2)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 4x\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right) + 8 = 0\). Hence find the Maclaurin series for&nbsp;\(y\) up to and including the term in \(x^3\), giving the coefficients in exact form.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Use your expansion from part (a) and integration to find an approximate value for&nbsp;\(\int_{0.1}^{0.2} \left(\frac{2}{x}\right)^3 \cos^{-1}(2x) \sin^{-1}(2x) \, \mathrm{d}x\), correct to 4 decimal places.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">A student, Adam, claims that the approximation in part <strong>(b)</strong> is accurate. Without performing any further calculations, justify whether Adam's claim is valid.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Suggest one way to improve the accuracy of the approximated value obtained in part <strong>(b)</strong>.<span class="inline-marks">[1]</span></div></li></ol>

It is given that \(y = \cos^{-1}(2x) \sin^{-1}(2x)\), where \(-0.5 \le x \le 0.5\), and \(\cos^{-1}(2x)\) and \(\sin^{-1}(2x)\) denote their principal values.

Show that \((1-4x^2)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 4x\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right) + 8 = 0\). Hence find the Maclaurin series for \(y\) up to and including the term in \(x^3\), giving the coefficients in exact form.

Use your expansion from part (a) and integration to find an approximate value for \(\int_{0.1}^{0.2} \left(\frac{2}{x}\right)^3 \cos^{-1}(2x) \sin^{-1}(2x) \, \mathrm{d}x\), correct to 4 decimal places.

A student, Adam, claims that the approximation in part **(b)** is accurate. Without performing any further calculations, justify whether Adam's claim is valid.

Suggest one way to improve the accuracy of the approximated value obtained in part **(b)**.

2024 DHS Promo Q9

2024 DHS Promo Q9