Question

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

Answer: (B)

For real roots,
$p^{2}-12q\geq 0$ ...(i)
$r^{2}+4q\geq 0$ ...(ii)
$s^{2}-8q\geq 0$ ...(iii)
If $q>0$ , then (ii) is always true.
If $q < 0$ , then (i) and (iii) are always true.
So, the equation has atleast two real roots.