2024 DHS Promo Q9

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.

1. 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.[6]
2. 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.[3]
3. A student, Adam, claims that the approximation in part (b) is accurate. Without performing any further calculations, justify whether Adam's claim is valid.[1]
4. Suggest one way to improve the accuracy of the approximated value obtained in part (b).[1]