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JEE Main 20252 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
2 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2025 · 2 April · Evening Shift · Mathematics · Quadratic Equation

If the set of all a ∈ R-\ 1\ , for which the roots of the equation...

If the set of all a∈R−{1}\mathrm{a} \in \mathrm{R}-\{1\}, for which the roots of the equation (1−a)x2+2(a−3)x+9=0(1-a) x^2+2(a-3) x+9=0 are positive is (−∞,−α]∪[β,γ)(-\infty,-\alpha] \cup[\beta, \gamma), then 2α+β+γ2 \alpha+\beta+\gamma is equal to _______ .

Correct answer

0

Step-by-step solution

Both the roots are positive
<br/>D≥04(a−3)2−4×9(1−a)≥0a2−6a+9−9+9a≥0a2+3a≥0a(a+3)≥0<br/>\begin{aligned}<br/>& 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<br/>\end{aligned}
a∈(−∞,−3]∪[0,∞)\mathrm{a} \in(-\infty,-3] \cup[0, \infty) ...(i)
−b2a>02(a−3)2(a−1)>0\begin{aligned} & -\frac{\mathrm{b}}{2 \mathrm{a}} \gt 0 \\ & \frac{2(\mathrm{a}-3)}{2(\mathrm{a}-1)} \gt 0\end{aligned}
a∈(−∞,1)∪(3,∞)a \in(-\infty, 1) \cup(3, \infty) ...(ii)
f(0)=9>0f(0)=9 \gt 0
Equation (i) ∩\cap (ii)
<br/>a∈(−∞,−3]∪[0,1)2α+β+γ−6+0+1=7<br/>\begin{aligned}<br/>& a \in(-\infty,-3] \cup[0,1) \\ & 2 \alpha+\beta+\gamma-6+0+1=7<br/>\end{aligned}
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