MCQ
જો $\lambda \in R$ એવો મળે કે જેથી દ્રીઘાત સમીકરણ $x^2 +(2 - \lambda ) x+ (10 - \lambda ) = 0$ ના ઉકેલોના ઘનનો સરવાળો ન્યૂનતમ થાય તો બંને ઉકેલોના તફાવતનો માનાંક મેળવો.
- A$20$
- ✓$2\sqrt 5 $
- C$2\sqrt 7 $
- D$4\sqrt 2 $
$x^{2}+(2-\lambda) x+(10-\lambda)=0$ are $\alpha$ and $\beta$.
Also roots of the given equation are
$\frac{\lambda-2 \pm \sqrt{4-4 \lambda+\lambda^{2}-40+4 \lambda}}{2}=\frac{\lambda-2 \pm \sqrt{\lambda^{2}-36}}{2}$
The magnitude of the difference of the roots is $|\sqrt{\lambda^{2}-36}|$
So, $\alpha^{3}+\beta^{3}=\frac{(\lambda-2)^{3}}{4}+\frac{3(\lambda-2)\left(\lambda^{2}-36\right)}{4}$
$=\frac{(\lambda-2)\left(4 \lambda^{2}-4 \lambda-104\right)}{4}$
$=(\lambda-2)\left(\lambda^{2}-\lambda-26\right)=\mathrm{f}(\lambda)$
As $f(\lambda)$ attains its minimum value at $\lambda=4$.
Therefore, the magintude of the difference of the roots is $|i \sqrt{20}|=2 \sqrt{5}$
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