JEE Main 2026 — Apr 4, Shift 2 — Mathematics solution

JEE Main2026Apr 4, Shift 2Mathematics

If the quadratic equation $(λ + 2) x^{2} - 3 λ x + 4 λ = 0$ , $λ \neq = - 2$ , has two positive roots, then the number of possible integral values of $λ$ is:

Solution

Three conditions for two distinct positive roots of $a x^{2} + b x + c = 0$ (writing $a = λ + 2$ , $b = - 3 λ$ , $c = 4 λ$ ):

(i) Discriminant $\geq 0$ : $9 λ^{2} - 16 λ (λ + 2) \geq 0 \Rightarrow - 7 λ^{2} - 32 λ \geq 0 \Rightarrow λ (7 λ + 32) \leq 0 \Rightarrow λ \in [- \frac{32}{7}, 0]$ .

(ii) Sum of roots positive: $- b / a = 3 λ / (λ + 2) > 0$ , so $λ$ and $λ + 2$ have the same sign: $λ > 0$ or $\lambda < -2$.

(iii) Product of roots positive: $c / a = 4 λ / (λ + 2) > 0$ , same condition: $λ > 0$ or $\lambda < -2$.

Intersecting (i) with (ii)/(iii): $λ \in [- \frac{32}{7}, - 2) \approx [- 4.57, - 2)$ .

Integers in this interval: $λ = - 4, - 3$ . That's 2 values.

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