JEE Main 2022 · 26 Jun · Shift 2 · Mathematics Q23

← JEE Main 2022 · 26 Jun · Shift 2

MathematicsInteger+4 / -1Question 23 of 90

If $z^2+z+1=0$ , $z\in\mathbb C$ , then $\left|\displaystyle\sum_{n=1}^{15}\left(z^n+(-1)^n\frac1{z^n}\right)^2\right|$ is equal to ______.

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Answer and worked solution

Official answer: 2

$z=\omega$ (a cube root of unity), $z^{-n}=\omega^{2n}$ .

Even $n$ . $(\omega^n+\omega^{2n})^2$ is $1$ if $3\nmid n$ and $4$ if $3\mid n$ . Odd $n$ . $(\omega^n-\omega^{2n})^2=\omega^{2n}+\omega^n-2$ is $-3$ if $3\nmid n$ and $0$ if $3\mid n$ .

Count for $n=1$ to 15. $5(1)+2(4)+5(-3)+3(0)=-2$ ; the modulus is 2.

Answer: 2

Topics:Roots of unity

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JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 23