Question

The number of quadratic equations that are unchanged by squaring their roots is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (B)

As per the given conditions-
$\alpha +\beta ={\alpha }^2+{\beta }^2\&\alpha \beta ={\alpha }^2{\beta }^2$
$\Rightarrow \alpha \beta =0$ or $1$ .
If $\alpha \beta =0,$ then let, $\alpha =0\Rightarrow \beta =0$ or $1$ .
If $\beta =\frac{1}{\alpha }$ ,
$\alpha +\frac{1}{\alpha }={\left(\alpha \right)}^2+\frac{1}{{\left(\alpha \right)}^2}={\left(\alpha +\frac{1}{\alpha }\right)}^2-2$
$\Rightarrow {\left(\alpha +\frac{1}{\alpha }\right)}^2-\left(\alpha +\frac{1}{\alpha }\right)-2=0\Rightarrow \alpha +\frac{1}{\alpha }=2$ or $-1\Rightarrow \alpha =1$ or $\omega ,{\omega }^2$
Hence number of such equations are four $\left(0,0\right),\left(0,1\right),\left(1,1\right)\&\left(\omega ,{\left(\omega \right)}^2\right)$