Quadratic Equations
Positive roots with parameter
nta_pyq_2025_apr
Grade 12
Question:
If the set of all a$\i_n$R - {1}$, for which the roots of the equation ($1 - a)x + 2$(a - 3)$x + 9 = 0$are positive is 2 (-$$\infty, -$$\alpha$]$\cup$[$\beta,$$\gamma), then 2$$\alpha +$$\beta +$$\gamma is equal to _______.$
Step-by-Step Solution
Key Concept: Apply$positive-root$conditions: product$>0$, sum$>0$, and discriminant$\ge0$for the parameterized quadratic.
Both the roots are positive$(7)$D$\ge 0 2 4($a - 3)$- 4$$\times$ 9($1 - a)$$\ge 0 2$a - 6a + 9 - 9 + 9$a$$\ge 0 2$a + 3$a$$\ge 0 a($a + 3$)$$\ge 0 a$$\i_n (-$$\infty, -3$]$\cup$[0,$\infty)...(i) b -$> 0 2a 2($a - 3) > 0$2($a - 1)$a$\i_n (-$$\infty, 1)$$\cup (3,$$\infty)...(ii) f$(0)$= 9$> 0 Equation (i)$\cap (ii) a$$\i_n (-$$\infty, -3$]$\cup$[0, 1) 2$\alpha +$$\beta +$$\$gamma - 6 + 0 + 1$= 7$
Correct Answer: 7