Question

The set of values of $aϵR$ for which $x^2+i(a-1)x+5=0$ will have a pair of conjugate complex roots is

• A. $R$
• B. $\left\{1\right\}$
• C. $\{a|a^2-2a+2|>0\}$
• D. None of these

Answer 1

The correct option is B $\left\{1\right\}$

$x^2+i(a-1)x+5=0$ where $a\in R$
Let $\alpha$ & $\beta$ be the roots of the equation.
Since the equation have pair of complex conjugate roots.
$\therefore \alpha =\overline{\beta }\&\beta =\overline{\alpha }$
Sum of the roots of the equation= $\alpha +\beta =-i(a-1)$ ...(1)
By taking conjugate.
$\Rightarrow \overline{\alpha }+\overline{\beta }=i(a-1)$ ...(2)
Substracting (1) & (2)
$\Rightarrow \alpha +\beta -\overline{\alpha }-\overline{\beta }=-2i(a-1)$
$\therefore a=1$ ...{ $\because \alpha =\overline{\beta }\&\beta =\overline{\alpha }$}

Ans: B