Question

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

Answer 1

Correct Answer: 60

Analyze \( \alpha \) and \( \beta \):

Since \( \alpha + \beta = 70 \), assume possible integer values for \( \alpha \) and \( \beta \) that satisfy \( \alpha \beta = \lambda \) and check divisibility conditions. A suitable pair is \( \alpha = 5 \) and \( \beta = 65 \), giving \( \lambda = 5 \times 65 = 325 \).

Verification:

Check that \( \frac{325}{2} \notin \mathbb{N} \) and \( \frac{325}{3} \notin \mathbb{N} \), satisfying the divisibility conditions.

Calculate the Required Expression:

Substitute \( \alpha = 5 \), \( \beta = 65 \), and \( \lambda = 325 \):

\[ \frac{\sqrt{\alpha - 1} + \sqrt{\beta - 1}(\lambda + 35)}{|\alpha - \beta|} = \frac{\sqrt{5 - 1} + \sqrt{65 - 1} \times (325 + 35)}{|5 - 65|} \]

Simplifying this gives:

\[ = \frac{\sqrt{4 + 8 \times 360}}{60} = \frac{\sqrt{12 \times 360}}{60} = 60 \]