Integers \(a\) and \(b\) are such that \((a+3 \sqrt{5})^{2}+a-b \sqrt{5}=51\). Find the possible values of \(a\) and the corresponding values of \(b\).

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2016 Oct Nov Paper 21 & 22 Q2

Without using a calculator, find the integers \(a\) and \(b\) such that \(\frac{a}{\sqrt{3}+1}+\frac{b}{\sqrt{3}-1}=\sqrt{3}-3\).

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2016 May June Paper 21 & 23

Do not use a calculator in this question. (b) Given that \(28+p \sqrt{3}=(q+2 \sqrt{3})^{2}\), where \(p\) and \(q\) are integers, find the values of \(p\) and of \(q\).