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Q.
If $\alpha, \beta \in R$ are such that $1-2 i$ (here $\left. i ^{2}=-1\right)$ is a root of $z^{2}+\alpha z+\beta=0,$ then $(\alpha-\beta)$ is equal to :
JEE MainJEE Main 2021Complex Numbers and Quadratic Equations
Solution:
$\because \alpha, \beta \in R$
$ \Rightarrow $ other root is $1+2 i$
$\alpha=-($ sum of roots $)=-(1-2 i +1+2 i )=-2$
$\beta=$ product of roots $=(1-2 i )(1+2 i )=5$
$\therefore \alpha-\beta=-7$