Question

If $\text{p,}\text{q,}\text{r,}\text{s},\in \text{R,}$ then equation $\left({\text{x}}^{\text{2}}\text{+ px + 3q}\right)\left(\text{-}{\text{x}}^{\text{2}}\text{+ sx - 2q}\right)\text{= 0}$ has

• A. 6 real roots
• B. at least two real roots
• C. 2 real and 4 imaginary roots
• D. 4 real and 2 imaginary roots

Answer 1

The correct option is B at least two real roots

$Let$
$x^2+px+3q$
$d_1=p^2-12q$
$-x^2+rx+q$
$d_2=r^2+4q$
$-x^2+sx-2q$
$d_3=s^2-8q$
$case1$
$ifq>0$
$d_1\ge 0or\le 0$
$d_2\ge 0$
$d_3\ge 0or\le 0$
$case2$
$q<0$
$d_1\ge 0$
$d_2\ge 0or\le 0$
$d_3\ge 0$
$Hencethegivenequationhasatleasttworealroots$