Question

If $\alpha ,\beta$ be unequal real roots of the equation $a{x}^2+bx+c=0$ where $a,b,c$ are real and $\gamma$ is the solution of $2ax+b=0$ then

• A. $\gamma >\alpha ,\gamma >\beta$
• B. $\gamma >\alpha ,\gamma <\beta$
• C. $\alpha <\gamma <\beta$ or $\beta <\gamma <\alpha$
• D. $noneofthese$

Answer 1

Answer: (C)

$\alpha <\gamma <\beta$ or $\beta <\gamma <\alpha$

$\alpha$ & $\beta$ are the roots of the equation $f\left(x\right)=a{x}^2+bx+c=0$
for $\alpha >\beta$
Since, roots are real
There lies a $\gamma$ such that $f^{{}^{\prime }}\left(\gamma \right)=0$ for $\beta <\gamma <\alpha$
$\Rightarrow 2a\gamma +b=0$
Similarly for $\alpha <\beta$, we get $\alpha <\gamma <\beta$
Ans: C