Question

$Iff\left(x\right)=a{x}^2+bx+ca,b,cϵ$ R and the equation $f\left(x\right)-x=0$ has imaginary roots $\alpha$ and $\beta$ and $\gamma$ and$\delta$ be the roots of $f\left(f\right(x\left)\right)-x=0$, then$\left|\begin{array}{ccc}2& \alpha & \delta \\ \beta & 0& \alpha \\ \gamma & \beta & 1\end{array}\right|is$

• A. purely real
• B. purely imaginary
• C. none of these

Answer 1

The correct option is A purely real

$f\left(x\right)-x>0or,f\left(x\right)-x<0\forall xϵRf\left(f\right(x\left)\right)-f\left(x\right)>0orf\left(f\right(x\left)\right)-f\left(x\right)<0Adding,f\left(f\right(x\left)\right)-x>0orf\left(f\right(x\left)\right)-x<0\Rightarrow rootsoff\left(f\right(x\left)\right)-x=0areimaginary$
Let $z=\left|\begin{array}{ccc}2& \alpha & \delta \\ \beta & 0& \alpha \\ \gamma & \beta & 1\end{array}\right|\overline{z}=\left|\begin{array}{ccc}2& \overline{\alpha }& \overline{\delta }\\ \overline{\beta }& 0& \overline{\alpha }\\ \overline{\gamma }& \overline{\beta }& 1\end{array}\right|=\left|\begin{array}{ccc}2& \beta & \gamma \\ \alpha & 0& \beta \\ \delta & \alpha & 1\end{array}\right|=z$