Let α,β∈N be roots of the equation x2−70x+λ=0 , where λ2,λ3∉N. If λ assumes the minimum possible value, then (α−1+β−1)(λ+35)|α−β| is equal to :

Given x2−70x+λ=0 ⇒ Let roots be α and β ⇒β=70−αλ=α(70−α)λ is not divisible by 2 and 3⇒α=5,β=65∴λ=5×65=325∴(α−1+β−1)(λ+35)|α−β|=1060×360=60

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

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Detailed Solution

Given $x^{2} - 70 x + λ = 0$
$\Rightarrow$ Let roots be $α a n d β$
$\Rightarrow β = 70 - α$
$λ = α (70 - α)$
$λ$ is not divisible by 2 and 3
$\begin{array}{l} \Rightarrow α = 5, β = 65 \\ ∴ λ = 5 \times 65 = 325 \end{array}$
$∴ \frac{(\sqrt{α - 1} + \sqrt{β - 1}) (λ + 35)}{| α - β |} = \frac{10}{60} \times 360 = 60$