Aspire Faculty ID #16659 · Topic: CUET 2023 · Just now
CUET 2023

Given below are two statements:

Statement I: If the roots of the quadratic equation
$x^2 - 4x - \log_3 a = 0$ are real, then the least value of $a$ is $\dfrac{1}{81}$.

Statement II: The harmonic mean of the roots of the equation
$(5+\sqrt{2})x^2 - (4+\sqrt{5})x + (8+2\sqrt{5}) = 0$ is $2$.

In the light of the above statements, choose the correct answer from the options given below:

Solution

For Statement I:
For real roots, discriminant $\ge 0$
$(-4)^2 - 4(1)(-\log_3 a) \ge 0$
$16 + 4\log_3 a \ge 0$
$\log_3 a \ge -4$
$a \ge 3^{-4} = \dfrac{1}{81}$
So Statement I is true.

For Statement II:
Harmonic mean of roots $= \dfrac{2\alpha\beta}{\alpha+\beta}$
Here,
$\alpha+\beta = \dfrac{4+\sqrt{5}}{5+\sqrt{2}}$
$\alpha\beta = \dfrac{8+2\sqrt{5}}{5+\sqrt{2}}$

So,
HM $= \dfrac{2(8+2\sqrt{5})}{4+\sqrt{5}} = 4 \ne 2$

So Statement II is false.
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