Question:
If the equations $x^{2}+2 x+3 \lambda=0$ and $2 x^{2}+3 x+5 \lambda=0$ have a non-zero common roots, then $\lambda=$
(a) 1
(b) −1
(c) 3
(d) none of these.
Solution:
(b) −1
Let $\alpha$ be the common roots of the equations, $x^{2}+2 x+3 \lambda=0$ and $2 x^{2}+3 x+5 \lambda=0$
Therefore,
$\alpha^{2}+2 \alpha+3 \lambda=0$ ....(1)
$2 \alpha^{2}+3 \alpha+5 \lambda=0$ ....(2)
Solving (1) and (2) by cross multiplication, we get
$\frac{\alpha^{2}}{10 \lambda-9 \lambda}=\frac{\alpha}{6 \lambda-5 \lambda}=\frac{1}{3-4}$
$\Rightarrow \alpha^{2}=-\lambda, \alpha=-\lambda$
$\Rightarrow-\lambda=\lambda^{2}$
$\Rightarrow \lambda=-1$