UPSC NDA I 2025 Mathematics Paper

The quadratic equation is x2−kx+k=0.
One root exceeds the other by 2√‌3.
⇒α−β=2√‌3.
Also,
Sum of roots: α+β=k
Product of roots: α×β=k
Calculation:
We know the following identity
‌(α+β)2=(α−β)2−4αβ
‌⇒k2=(2√‌3)2−4k
‌⇒k2−12−4k=0
‌⇒k2−6k+2k−12=0
‌⇒k(k−6)+2(k−6)=0
‌⇒(k−6)(k+2)=0
‌⇒k=6‌ and ‌k=−2
Thus, the possible values of k are 6 and -2