Question

Minimum possible number of positive roots of the equation $x^2-(\lambda +1)x+(\lambda -2)=0$ where $\lambda \in R$ is

• A. 0
• B. 1
• C. 2
• D. Can't be determined

Answer 1

Answer: (B)

$D=(\lambda +1{)}^2-4(\lambda -2)$
$D={\lambda }^2-2\lambda +9=(\lambda -1{)}^2+8>0$
Hence roots are distinct
Case-1: If $\lambda <2$, then $f(0)<0\Rightarrow 1$ root positive and 1 root negative
Case-2: If $\lambda >2$, then $f(0)>0$ and $\frac{-b}{2a}>0$. Hence both roots are positive and distinct.
Case-3: If $\lambda =2$, equation is $x^2-3x=0\Rightarrow x=0,3$
$\therefore$ Minimum number of positive roots of equation is 1.$