Question

If the equations $x^2+2x+3\mathrm{\lambda }=0\mathrm{and}2x^2+3x+5\mathrm{\lambda }=0$ have a non-zero common roots, then λ =
(a) 1
(b) −1
(c) 3
(d) none of these.

Answer 1

(b) −1

Let $\mathrm{\alpha }$ be the common roots of the equations, $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$

Therefore,

${\mathrm{\alpha }}^2+2\mathrm{\alpha }+3\lambda =0$ ... (1)

$2{\mathrm{\alpha }}^2+3\mathrm{\alpha }+5\lambda =0$ ... (2)

Solving (1) and (2) by cross multiplication, we get

$$\begin{gathered} \frac{{\mathrm{\alpha }}^2}{10\mathrm{\lambda }-9\mathrm{\lambda }}=\frac{\mathrm{\alpha }}{6\mathrm{\lambda }-5\mathrm{\lambda }}=\frac{1}{3-4} \\ \Rightarrow {\mathrm{\alpha }}^2=-\mathrm{\lambda },\mathrm{\alpha }=-\mathrm{\lambda } \\ \Rightarrow -\mathrm{\lambda }={\mathrm{\lambda }}^2 \\ \Rightarrow \mathrm{\lambda }=-1 \end{gathered}$$