2025 SJI P1 Q5

1. Prove that \((\alpha+\beta)^2-2\alpha\beta=\alpha^2+\beta^2\).[1]
2. Prove that \((\alpha+\beta)^4-4\alpha\beta(\alpha+\beta)^2+2(\alpha\beta)^2=\alpha^4+\beta^4\).[3]

The equation \(x^2-3x-5=0\) has two roots, \(\alpha\) and \(\beta\).

Consider the equation \(ax^2+bx+1=0\), where \(a,b\in\mathbb Z\) and which has roots \(\dfrac1{\alpha^4}\) and \(\dfrac1{\beta^4}\).

3. Without solving \(x^2-3x-5=0\), find the value of \(a\) and of \(b\).[6]