JEE Advanced 2024 Paper 2 · Mathematics · Complex Numbers · medium

Let ω=e^2π i/5. The value of prodk=1^4(2-ω^k(2-ω^2k) is

Numerical answer

Let

\omega=e^{2\pi i/5}

. The value of

\displaystyle\prod_{k=1}^{4}\left(2-\omega^{k}\right)\left(2-\omega^{2k}\right)

Show answer and solution

Answer: 961

The fifth roots of unity satisfy

x^5-1=\prod_{k=0}^{4}(x-\omega^k)

. Putting

x=2

\prod_{k=1}^{4}(2-\omega^k)=\dfrac{2^5-1}{2-1}=31

. Since

\gcd(2,5)=1

, as

k

runs over

1,2,3,4

the numbers

\omega^{2k}

run over the same set

\{\omega,\omega^2,\omega^3,\omega^4\}

in a different order, so

\prod_{k=1}^4(2-\omega^{2k})=31

as well. The required product is

31\times31=961

, confirmed by direct complex-number multiplication.

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