Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
Let $z_i, i = 1, 2, \ldots, 6$ be the roots of $z^6 + z^4 = 2$ then $\sum_{i=1}^{6} |z_i|^4$ is equal to ____.
Step-by-Step Solution
Key Concept: The sum of binomial coefficients $\sum_{k=0}^{n} \binom{n}{k} = 2^n$ directly counts all selection possibilities.
Ramesh selects from $2n$ fruits where $n$ are distinct and $n$ are identical. Case (i): 0 distinct fruits selected gives $^nC_0 = 1$ way. Case (ii): 1 distinct fruit selected gives $^nC_1$ ways. Case (iii): 2 distinct fruits selected gives $^nC_2$ ways, continuing through case (n): all $n$ distinct fruits selected gives $^nC_n$ ways. Total ways: $^nC_0 + ^nC_1 + ^nC_2 + ... + ^nC_n = 2^n$. Setting $2^n = 16$ gives $n = 4$.
Correct Answer: 10