If the equations 2x^2 + 3x + 5lambda = 0 and | vedclass

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Question

(C) Let $\alpha$ be the common root of the equations $2x^2 + 3x + 5\lambda = 0$ and $x^2 + 2x + 3\lambda = 0$.Then,$2\alpha^2 + 3\alpha + 5\lambda = 0

Vedclass · Accepted Answer

$0, -1$

Vedclass · Answer

(C) Let $\alpha$ be the common root of the equations $2x^2 + 3x + 5\lambda = 0$ and $x^2 + 2x + 3\lambda = 0$.Then,$2\alpha^2 + 3\alpha + 5\lambda = 0$  --- $(1)$And $\alpha^2 + 2\alpha + 3\lambda = 0$  --- $(2)$Multiply equation $(2)$ by $2$: $2\alpha^2 + 4\alpha + 6\lambda = 0$ --- $(3)$Subtract equation $(1)$ from equation $(3)$:$(2\alpha^2 + 4\alpha + 6\lambda) - (2\alpha^2 + 3\alpha + 5\lambda) = 0$$\alpha + \lambda = 0 \implies \alpha = -\lambda$Substitute $\alpha = -\lambda$ into equation $(2)$:$(-\lambda)^2 + 2(-\lambda) + 3\lambda = 0$$\lambda^2 - 2\lambda + 3\lambda = 0$$\lambda^2 + \lambda = 0$$\lambda(\lambda + 1) = 0$Therefore,$\lambda = 0$ or $\lambda = -1$.

If the equations $2x^2 + 3x + 5\lambda = 0$ and $x^2 + 2x + 3\lambda = 0$ have a common root,then $\lambda = $

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