JEE Main 2024 — Quadratic Equation Question with Solution

Given,

$x^2-70x+\lambda =0$

Sum of the roots will be $\alpha +\beta =70$

Product of roots will be $\alpha \beta =\lambda$

$\Rightarrow \alpha (70-\alpha )=\lambda$

$\Rightarrow \frac{d\lambda }{d\alpha }=70-2\alpha$

Now, $\lambda$ is increasing for all $\alpha \le 35$

And $2$ and $3$ does not divide $\lambda$ so taking  $\alpha =5$ we get, $\beta =70-\alpha =65$ 

$\text{As} \text{for} \alpha =1, \beta =69 \& \text{for} \alpha =4, \beta =66$ which is not possible as they are multiple of $2 \& 3$

Hence, $\alpha =5, \beta =65, \lambda =325$

Now solving,

$\frac{\left(\sqrt{\alpha -1}+\sqrt{\beta -1}\right)\left(\lambda +35\right)}{|\alpha -\beta |}$

$=\frac{\left(\sqrt{5-1}+\sqrt{65-1}\right)\left(325+35\right)}{|5-65|}$

$=\frac{\left(2+8\right)·360}{60}$

$=60$