Let α, β ∈ N be roots of equation x^2 − 70x + λ = 0, where λ/2, λ/3 ∉ N.

Question

Let \(\alpha, \beta \in N\) be roots of equation \(x^{2}-70 x+\lambda=0\), where \(\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathrm{N}\). If \(\lambda\) assumes the minimum possible value, then \(\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}\) is equal to ______.

Answer 1

Correct answer: 60

\(x^{2}-70 x+\lambda=0\)

\(\alpha+\beta=70\)

\(\alpha \beta=\lambda \)

\(\therefore \alpha(70-\alpha)=\lambda\)

Since, 2 and 3 does not divide \(\lambda\)

\(\therefore \alpha=5, \beta=65, \lambda=325\)

By putting value of \(\alpha, \beta, \lambda\) we get the required value 60.