Correct Answer - Option 2 :
\(\frac{1}{81}\)
Concept:
Nature of roots of a quadratic equation:
The roots of a quadratic function are given by, \(x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\)
We call the term b2 – 4ac as discriminant (Δ).
Case 1: If Δ = 0, then the roots are real and equal.
Case 2: If Δ > 0, then the roots are real and unequal.
Case 3: If Δ < 0, then the roots are imaginary.
Calculation:
Given that roots of the equation x2 - 4x - log3 a = 0 are real.
Comparing the given equation with the standard form of quadratic equation: f(x) = ax2 + bx + c = 0
⇒ a = 1
⇒ b = -4
⇒ c = -log3 a
For roots to be real, Δ ≥ 0
⇒ Δ = b2 – 4ac = [(- 4)]2 – (4× 1× (- log3 a)
⇒ Δ = 16 + (4 × log3 a)
⇒ 16 + (4 × log3 a) ≥ 0
⇒ 4log3 a ≥ -16
⇒log3 a ≥ -4
Applying “ANTILOG” on both sides
⇒ a ≥ 3-4
⇒ a ≥ 1/81
The minimum value of a for roots to be real is 1/81.