If the set of all $\mathrm{a} \in \mathrm{R}-\{1\}$, for which the roots of the equation $(1-a) x^2+2(a-3) x+9=0$ are positive is $(-\infty,-\alpha] \cup[\beta, \gamma)$, then $2 \alpha+\beta+\gamma$ is equal to _______ .

Question

Aicharya · Accepted Answer

Both the roots are positive
$\begin{aligned}
& D \geq 0 \\ & 4(a-3)^2-4 \times 9(1-a) \geq 0 \\ & a^2-6 a+9-9+9 a \geq 0 \\ & a^2+3 a \geq 0 \\ & a(a+3) \geq 0
\end{aligned}$
$\mathrm{a} \in(-\infty,-3] \cup[0, \infty)$ ...(i)
$\begin{aligned} & -\frac{\mathrm{b}}{2 \mathrm{a}} \gt 0 \\ & \frac{2(\mathrm{a}-3)}{2(\mathrm{a}-1)} \gt 0\end{aligned}$
$a \in(-\infty, 1) \cup(3, \infty)$ ...(ii)
$f(0)=9 \gt 0$
Equation (i) $\cap$ (ii)
$\begin{aligned}
& a \in(-\infty,-3] \cup[0,1) \\ & 2 \alpha+\beta+\gamma-6+0+1=7
\end{aligned}$

If the set of all $\mathrm{a} \in \mathrm{R}-\{1\}$, for which the roots of the equation $(1-a) x^2+2(a-3)…

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