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 $\left(0 , 0\right), \, \left(0 , 1\right), \, \left(1 , 1\right) \, \& \, \left(\omega , \, \left(\omega \right)^{2}\right)$