Question

A quadratic equation whose roots are ${\left(\frac{\gamma }{\alpha }\right)}^2$ and ${\left(\frac{\beta }{\alpha }\right)}^2$, where $\alpha ,\beta ,\gamma$ are the roots of $x^3+27=0$ is

• A. $x^2-x+1=0$
• B. $x^2+3x+9=0$
• C. $x^2+x+1=0$
• D. $x^2-3x+9=0$

Answer 1

The correct option is C $x^2+x+1=0$

$\alpha ,\beta ,\gamma$ are the roots of $x^3+27=0$
$\Rightarrow \alpha =-3,\beta =-3\omega ,\gamma =-3{\omega }^2$
$\therefore {\left(\frac{\gamma }{\alpha }\right)}^2={\left(\frac{-3{\omega }^2}{-3}\right)}^2=\omega$
$\therefore {\left(\frac{\beta }{\alpha }\right)}^2={\left(\frac{-3\omega }{-3}\right)}^2={\omega }^2$
$\therefore$ Equation whose roots are $\omega ,{\omega }^2$ is $x^2+x+1=0$
Hence, option C.