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JEE Advanced 2021 Paper 2 · Mathematics · Complex Numbers · medium

Let α=e^2π i/5.

One or more options correct

Let α=e2πi/5\alpha=e^{2\pi i/5}. Which of the following statements is/are correct?
  1. A. ∏k=14(1−αk)=5\displaystyle\prod_{k=1}^{4}\left(1-\alpha^{k}\right)=5
  2. B. ∑k=1411−αk=52\displaystyle\sum_{k=1}^{4}\frac{1}{1-\alpha^{k}}=\frac52
  3. C. ∏k=14(2−αk)=32\displaystyle\prod_{k=1}^{4}\left(2-\alpha^{k}\right)=32
  4. D. ∑k=14(1−αk)2=6\displaystyle\sum_{k=1}^{4}\left(1-\alpha^{k}\right)^2=6
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Answer: A

The numbers αk\alpha^k (k=1,…,4k=1,\dots,4) are the roots of Φ(z)=1+z+z2+z3+z4=∏(z−αk)\Phi(z)=1+z+z^2+z^3+z^4=\prod(z-\alpha^k). Hence ∏(1−αk)=Φ(1)=5\prod(1-\alpha^k)=\Phi(1)=5, so (A) is true. For (B), ∑11−αk=42=2\sum\frac1{1-\alpha^k}=\frac{4}{2}=2 because 11−αk+11−α−k=1\frac1{1-\alpha^k}+\frac1{1-\alpha^{-k}}=1 (pairing kk with 5−k5-k), so 5/25/2 is wrong. For (C), ∏(2−αk)=Φ(2)=1+2+4+8+16=31≠32\prod(2-\alpha^k)=\Phi(2)=1+2+4+8+16=31\neq32. For (D), ∑(1−αk)2=4−2∑αk+∑α2k=4+2−1=5\sum(1-\alpha^k)^2=4-2\sum\alpha^k+\sum\alpha^{2k}=4+2-1=5, not 66. Only (A) is correct.
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