UPSC CDS 1 2022 Math Paper

Given:
Equation is x2−k2x+30kx−161x−64=0
Sum of roots =0
Concept Used:
For quadratic equation ax2+bx+c=0
Sum of roots (α+β)=−b∕a
Product of roots (α.β)=c∕a
(α−β)2=(α+β)2−4α⋅β
Calculation:
We have x2−k2x+30kx−161x−64=0
⇒x2−(k2−30k+161)x−64=0
α+β=(k2−30k+161)∕1=0
And α⋅β=−64∕1=−64
Now, We have to find α−β
So, Using formula
(α−β)2=(α+β)2−4α⋅β
⇒(α−β)2=(0)2−4×−64
⇒(α−β)2=4×64
⇒α−β=√‌(4×64)
⇒α−β=±16
∴ The difference between the roots is 16 .