Question

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

• A. $R$
• B. $\left(1\right)$
• C. $(a:a^2-2a+21>0)$
• D. $\left(0\right)$

Answer 1

The correct option is B $\left(1\right)$

$x^2+\left(a-1\right)ix+5=0$

$D=b^2-4ac$

$={\left\{\left(a-1\right)i\right\}}^2-4\times 1\times 5$

$={\left(a-1\right)}^2{i}^2-20$

$=-{\left(a-1\right)}^2-20$

$=-\left[{\left(a-1\right)}^2+20\right]$

$x=\frac{-b\pm \sqrt{D}}{2a}$

$=\frac{-\left(a-1\right)i\pm \sqrt{-\left\{{\left(a-1\right)}^2+20\right\}}}{2}$

$=\frac{-\left(a-1\right)i\pm \sqrt{-{\left(a-1\right)}^2+20}}{2}$

For conjugate pair of imaginary roots

$a-1=0$

$a=1$

there the required set is $\left\{1\right\}$