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Question
If the equations \[x^2 + 2x + 3\lambda = 0 \text { and } 2 x^2 + 3x + 5\lambda = 0\] have a non-zero common roots, then λ =
Options
1
-1
3
none of these.
Solution
-1
Let \[\alpha\] be the common roots of the equations, \[x^2 + 2x + 3\lambda = 0\] and \[2 x^2 + 3x + 5\lambda = 0\]
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
\[\frac{\alpha^2}{10\lambda - 9\lambda} = \frac{\alpha}{6\lambda - 5\lambda} = \frac{1}{3 - 4}\]
\[ \Rightarrow \alpha^2 = - \lambda, \alpha = - \lambda\]
\[ \Rightarrow - \lambda = \lambda^2 \]
\[ \Rightarrow \lambda = - 1\]
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