CAT Exam 2021 Slot 2 Solved Paper

Given a, b, c are rational numbers.
Hence a, b, c are three numbers that can be written in the form of $\frac{\mathit{p}}{\mathit{q}}$.
Hence if one both the root is $2=\sqrt{3}$ and considering the other root to be $\mathit{x}$.
The sum of the roots and the product of the two roots must be rational numbers.
For this to happen the other root must be the conjugate $2=\sqrt{3}$ of so the product and the sum of the roots are rational numbers which are represented by: $\frac{\mathrm{b}}{\mathrm{a}},\frac{\mathrm{c}}{\mathrm{a}}$
Hence the sum of the roots is $2+\sqrt{3}+2-\sqrt{3}=4$
The product of the roots is $\left(2+\sqrt{3}\right).\left(2-\sqrt{3}\right)=1$
$\frac{\mathit{b}}{\mathit{a}}=4,\frac{\mathit{c}}{\mathit{a}}=1$
Since $\mathit{b}={\mathit{c}}^3$
$4\times \mathit{a}={\mathit{a}}^3$
${\mathit{a}}^2=4$
$\mathit{a}=2$ or $-2$
$\left|\mathit{a}\right|=2$