UPSC CDS 2 2021 Math Paper

Given equation,
x2−4x−log10N=0
Roots of the equation will be real only, when the discriminant (D)≥0.
For ax2+bx+c=0,D=b2−4ac
For x2−4x−log10N=0
a=1,b=−4 and c=−log10N
∴b2−4ac≥0[∵ roots are real ( given )]
⇒(−4)2−4(1)(−log10N)≥0
⇒16+4log10N≥0
⇒‌‌4log10N≥−16
⇒‌‌log10N≥−4
⇒‌‌N≥(10)−4‌‌ [on ‌‌ taking ⇒‌‌ anti-logarithm]
⇒‌‌N≥‌

1

10000

⇒‌‌N≥0.0001
Hence, the minimum value of N is 0.0001.