Question

The equation $x^2-6x+8+\lambda (x^2-4x+3)=0,\lambda \in R$, has

• A. real and unequal roots for all $\lambda$
• B. real roots for $\lambda <0$ only
• C. real roots for $\lambda >0$ only
• D. real and unequl roots for $\lambda =0$ only

Answer 1

The correct option is A real and unequal roots for all $\lambda$

$x^2-6x+8+\lambda (x^2-4x+3)=0$
$\Rightarrow (1+\lambda )x^2-(6+4\lambda )x+8+3\lambda =0$
Now, $D=4[{(3+2\lambda )}^2-(1+\lambda )(8+3\lambda )]=4({\lambda }^2+\lambda +1)$
Therefore, $D=4({\lambda }^2+\lambda +1)>0$ for all $\lambda \in R$.
Therefore, given equation has real and distinct roots.
Ans: A