Question

If $A,G$ and $H$ are $A.M.,G.M.$ and $H.M$. respectively between two unequal positive integers. Then the equation $A{x}^2-Gx-2H=0$ has

• A. both roots fraction
• B. one negative fraction root
• C. exactly one positive root
• D. no root greater than $2$

Answer 1

Answer: (C), (D)

The correct options are
C exactly one positive root
D no root greater than $2$

Let, the two positive numbers be $a$ and $b$

The roots of the equation $A{x}^2-Gx-2H=0$ are

$x=\frac{G\pm \sqrt{G^2+8HA}}{2A}$

$\Delta =G^2+8HA$

$\Rightarrow \Delta =ab+8\times \frac{a+b}{2}\times \frac{2ab}{a+b}$

$\Rightarrow \Delta =9ab=9G^2$

$x=\frac{G\pm 3G}{2A}$

$x_1=\frac{-G}{A}$ and $x_2=\frac{2G}{A}$

$G=\sqrt{ab}$ Depending on the values of $a,b$, G will be rational or irrational. As there is insufficient information in the question, we cannot conclude if $x$ will be rational or irrational.

$x_1$ is negative and $x_2$ is positive.

A.M is greater than G.M . Hence, $x_2<2$

Hence, options 'C' and 'D' are correct.