Step-by-step Solution:

Statement I: Given the quadratic equation: \[ x^2 - 4x - \log_3 a = 0 \] For the roots to be real, the discriminant must be non-negative: \[ \Delta = b^2 - 4ac \geq 0 \] Here, \( a = 1 \), \( b = -4 \), and \( c = -\log_3 a \). So, \[ (-4)^2 - 4(1)(-\log_3 a) \geq 0 \] \[ 16 + 4\log_3 a \geq 0 \] \[ 4\log_3 a \geq -16 \] \[ \log_3 a \geq -4 \] \[ a \geq 3^{-4} \] \[ a \geq \frac{1}{81} \] Thus, the least value of \( a \) is \( \frac{1}{81} \). Statement I is true. Statement II: Given the quadratic equation: \[ (5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + (8 + 2\sqrt{5}) = 0 \] Let the roots be \( \alpha \) and \( \beta \). The harmonic mean (HM) of the roots is given by: \[ \text{HM} = \frac{2\alpha\beta}{\alpha + \beta} \] From the quadratic equation: \[ \alpha + \beta = \frac{4 + \sqrt{5}}{5 + \sqrt{2}} \] \[ \alpha\beta = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} \] Calculating the harmonic mean: \[ \text{HM} = \frac{2 \cdot \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}}}{\frac{4 + \sqrt{5}}{5 + \sqrt{2}}} = \frac{2(8 + 2\sqrt{5})}{4 + \sqrt{5}} \] Simplifying: \[ \text{HM} = \frac{16 + 4\sqrt{5}}{4 + \sqrt{5}} = 4 \] The statement claims the harmonic mean is 2, but our calculation shows it is 4. Statement II is false. Conclusion: - Statement I is true. - Statement II is false. Therefore, both statements are not correct. The correct answer is: C. Statement I is true but Statement II is false