Question

Consider the equation $a{z}^2+z+1=0$ having purely imaginary root where $a=\cos \theta +i\sin \theta ,i=\sqrt{-1}$ and function $f\left(x\right)=x^3-3x^2+3(1+\cos \theta )x+5$, then answer the following questions

$ii)$ Which of the following is true ?

• A. $f\left(x\right)=0$ has three but not distinct roots
• B. $f\left(x\right)=0$ has one positive real root
• C. $f\left(x\right)=0$ has one negative real root
• D. $f\left(x\right)=0$ has three real distinct roots

Answer 1

The correct option is C $f\left(x\right)=0$ has one negative real root

$f\left(x\right)=x^3-3x^2+3(1+\cos \theta )x+5$ (Given)
$f\left(x\right)$ is increasing $\forall x\in R$
(From the previous result)
$f\left(0\right)=(0{)}^3-3(0{)}^2+3\left(0\right)+5=5$
Since, $f\left(x\right)$ is increasing function and $f\left(0\right)=5$,
There will always be a point on negative $x-$axis where $f\left(x\right)$ will be $0$.
Hence, $f\left(x\right)=0$ has one negative real root.