Question

Assertion :The harmonic mean of the roots of the equation $(\sqrt{5}+2)x^2-(4+\sqrt{5})x+(8+2\sqrt{5})=0$ is 8 Reason: $H.M.$ of $a$ & $b$ is $\frac{2ab}{a+b}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Let $a$ and $b$ are roots of
$(\sqrt{5}+2)x^2-(4+\sqrt{5})x+(8+2\sqrt{5})=0$
$\therefore$ sum of roots $=a+b=\frac{4+\sqrt{5}}{2+\sqrt{5}}$
Product of roots $=ab=\frac{8+2\sqrt{5}}{2+\sqrt{5}}$
Now $H.M$ of roots $=\frac{2(8+2\sqrt{5})}{4+\sqrt{5}}$
$=\frac{4(4+\sqrt{5})}{4+\sqrt{5}}\Rightarrow 4\ne 8$
$\therefore$ Assertion (A) is false but Reason (R) is true
Ans: D