2025 PHS PRELIMS P1 Q9

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.

1. 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]

2. \([\text{The area of a sector is}\hspace{0.5em} \frac{1}{2}r^2\theta \text{ and the arc length of a sector is}\hspace{0.5em} r\theta.]\)

   The sector of a circle with radius, \(r\), has an arc length of \((\sqrt{15} - \sqrt{3})\text{ cm}\) and an area of \((3\sqrt{3} - \sqrt{15})\text{ cm}^2\). Show that \(r = \frac{6\sqrt{3} - 2\sqrt{15}}{\sqrt{15} - \sqrt{3}}\text{ and hence express}\hspace{0.5em} r \text{ in the form}\hspace{0.5em} (p + q\sqrt{5})\text{ cm}\), where \(p\) and \(q\) are integers.

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