JEE Main 2022 · 28 Jul · Shift 2 · Mathematics Q88

← JEE Main 2022 · 28 Jul · Shift 2

MathematicsInteger+4 / -1Question 88 of 90

Let $z=a+ib$ , $b\ne0$ be complex numbers satisfying $z^2=\bar z\cdot2^{1-|z|}$ . Then the least value of $n\in\mathbf N$ , such that $z^n=(z+1)^n$ , is equal to ______.

Your answer

Answer and worked solution

Official answer: 6

Modulus. $|z|=2^{1-|z|}$ forces $|z|=1$ , so $z^2=\bar z$ , $z^3=1$ , $z=\omega$ .

Condition. $\left(\frac\omega{1+\omega}\right)^n=(-\omega^2)^n=1$ first at $n=6$ .

Answer: 6

Topics:Roots of unity Modulus and argument

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JEE Main 2022 · 28 Jul · Shift 2, Mathematics question 88