JEE Main 2022 (Online) 27th June Evening ShiftQuestion 1
Let for some real numbers $\alpha$ and $\beta$, $a=\alpha-i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)x+\bar{a}y=0$ has more than one solution, then $\frac{\alpha}{\beta}$ is equal to
Determinant
Solution
Non-trivial solution $\Rightarrow4i\bar a-(1+i)\cdot8\left(-\tfrac12+\tfrac{\sqrt3}{2}i\right)=0$, with $\bar a=\alpha+i\beta$. $(1+i)(-\tfrac12+\tfrac{\sqrt3}2i)=-\tfrac{1+\sqrt3}{2}+\tfrac{\sqrt3-1}{2}i$, so $4i\bar a=-4(1+\sqrt3)+4(\sqrt3-1)i$. $4i(\alpha+i\beta)=-4\beta+4\alpha i\Rightarrow\beta=1+\sqrt3,\ \alpha=\sqrt3-1$. $\dfrac\alpha\beta=\dfrac{\sqrt3-1}{\sqrt3+1}=\dfrac{(\sqrt3-1)^2}{2}=2-\sqrt3$. Option (B).