Question

Let $f\left(x\right)=a{x}^2+bx+c$ and $g\left(x\right)=A{x}^2+bx+\lambda ,$ where $a\ne 0,A\ne 0,a,b,c,A,\lambda \in R$. Roots of $f\left(x\right)=0$ and $g\left(x\right)=0$ are imaginary, then which of the following may be correct?

• A. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0\forall x\in R$
• B. Roots of equation $af\left(x\right)+Ag\left(x\right)=0$ are real
• C. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0$ for some $x$
• D. $f\left(x\right)+g\left(x\right)=0$ for some $x$

Answer 1

Answer: (A), (D)

$f\left(x\right)+g\left(x\right)=0$ for some $x$

$f\left(x\right)=0$ & $g\left(x\right)=0$ has imaginary roots
So $f\left(x\right)>0\&g\left(x\right)>0\forall x\in Rifa>0,A>0$...(i)
$f\left(x\right)>0\&g\left(x\right)<0\forall x\in Rifa>0,A<0$...(ii)
$f\left(x\right)<0\&g\left(x\right)>0\forall x\in Rifa<0,A>0$...(iii)
$f\left(x\right)<0\&g\left(x\right)<0\forall x\in Rifa<0,A<0$...(iv)
$f\left(x\right)+g\left(x\right)=0$ for some $x$ is correct in (ii) & (iii) case.
Also, $\frac{f\left(x\right)}{a}$ and $\frac{g\left(x\right)}{A}>0\forall x\in R,$
So $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0\forall x\in R$ is also correct