IP3 Class Test (May 2025) Q3

The diagram shows a rectangle \(ABCD\) with side \(AB = (2\sqrt{2} + \sqrt{3})\) cm.

The diagram shows a rectangle \(ABCD\) in which \(AB = (2\sqrt{2} + \sqrt{3})\) cm. Given that the area of the rectangle is \((9 + \sqrt{6})\) cm\(^2\), find, without using a calculator,

1. the exact value of \(BC\) in the form \((\sqrt{a} - \sqrt{b})\) cm, where \(a\) and \(b\) are integers.[4]
2. the exact value of \((AC)^2\) in the form \((c - \sqrt{d})\) cm\(^2\), where \(c\) and \(d\) are integers.[3]