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 =

विकल्प

• 1

• -1

• 3

• none of these.

Answer 1

-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

\[\frac{\alpha^2}{10\lambda - 9\lambda} = \frac{\alpha}{6\lambda - 5\lambda} = \frac{1}{3 - 4}\]

\[ \Rightarrow \alpha^2 = - \lambda, \alpha = - \lambda\]

\[ \Rightarrow - \lambda = \lambda^2 \]

\[ \Rightarrow \lambda = - 1\]