Correction note for part (a): the stated integer-coefficient representation does not exist. Comparing coefficients gives \(a^2+2b^2=17\) and \(ab=-4\); integer factor pairs give \(a^2+2b^2\in\{12,18,33\}\), never \(17\). The requested quartic follows conditionally from the printed assumption, but that assumption is inconsistent. The printed numbers are retained; part (b) is unaffected.
A square has an area \( (17 - 8\sqrt{2}) \text{ cm}^2 \). The length of each side of the square can be expressed in the form \( (a + b\sqrt{2}) \text{ cm} \), where \(a\) and \(b\) are integers. Show that \(2b^4 - 17b^2 + 16 = 0\).[4]
[The area of a sector is \(\frac{1}{2}r^2\theta\) and the arc length of a sector is \(r\theta\).] The sector of a circle with radius, \(r\), has an arc length of \((\sqrt{15} - \sqrt{3})\) cm and an area of \((3\sqrt{3} - \sqrt{15})\) cm\(^2\). Show that \(r = \frac{6\sqrt{3} - 2\sqrt{15}}{\sqrt{15} - \sqrt{3}}\) and hence express \(r\) in the form \((p + q\sqrt{5})\) cm, where \(p\) and \(q\) are integers.[5]