Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade 11
Question:
If $\alpha = \cos\frac{2\pi}{7} + i\sin\frac{2\pi}{7}$ and $\sum_{r=0}^{n-1}\alpha^r\Big|^2 = 32$ then $n$ is:
$8$
$4$
$\log_e 8$
$-4\sum_{k=1}^{7}e^{\frac{2\pi k}{7}}$
Step-by-Step Solution
Key Concept: Prime factorization of a number allows systematic calculation of derived quantities by selecting and combining factors.
Given $p = 2520 = 2^3 \times 3^2 \times 5 \times 7$, we find $x = 3 \times 2 \times 2 \times 2 = 24$ and $y = 4 \times 1 \times 2 \times 2 = 16$ by multiplying specific subsets of prime factors according to the problem constraints.
Correct Answer: 2,4