JEE Main 2022 · 27 Jun · Shift 2 · Mathematics Q3

← JEE Main 2022 · 27 Jun · Shift 2

MathematicsSingle correct+4 / -1Question 3 of 90

Let for some real numbers $\alpha$ and $\beta$ , $a=\alpha-i\beta$ . If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3\right)x+\bar ay=0$ has more than one solution, then $\dfrac\alpha\beta$ is equal to

-2+\sqrt3

2-\sqrt3

2+\sqrt3

-2-\sqrt3

Answer and worked solution

Official answer: B

More than one solution needs the determinant to vanish: $4i\,\bar a=8(1+i)\omega$ , with $\omega=e^{2\pi i/3}$ .

Solve. $\bar a=\alpha+i\beta=2(1-i)\omega=2\sqrt2\,e^{i(2\pi/3-\pi/4)}=2\sqrt2\,e^{5\pi i/12}$ .

Ratio. $\dfrac\alpha\beta=\cot\dfrac{5\pi}{12}=2-\sqrt3$ .

(B)

Topics:Polar and Euler form Systems of linear equations

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JEE Main 2022 · 27 Jun · Shift 2, Mathematics question 3