If the equations x^2+2x+3 =0 and 2x^2+3x+5 =0 have a non-zero com… - Vidyadip

MCQ

If the equations $\text{x}^2+2\text{x}+3\lambda=0$ and $2\text{x}^2+3\text{x}+5\lambda=0$ have a non$-$zero common roots, then $\lambda=$

A
$1$
✓
$-1$
C
$3$
D
None of these.

✓

Answer

Correct option: B.

$-1$

Let a be the common roots of the equations $\text{x}^2+2\text{x}+3\lambda=0$ and $2\text{x}^2+3\text{x}+5\lambda=0$
Therefore
$\alpha^2+2\text{a}+3\lambda=0\ ...(1)$
$2\alpha^2+3\alpha+5\lambda=0\ ...(2)$
Solving $(1)$ and $(2)$ by cross multiplication, we get
$\frac{\alpha^2}{10\lambda-9\lambda}=\frac{\alpha}{6\lambda-5\lambda}=\frac{1}{3-4}$
$\Rightarrow\text{a}^2=-\lambda,\alpha=-\lambda$
$\Rightarrow-\lambda=\lambda^2$
$\Rightarrow\lambda=-1$

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