Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If an equilateral triangle $ABC$ with vertices at $z_1, z_2$ and $z_3$ be inscribed in the circle $|z|=2$ and again a circle is inscribed in the triangle $ABC$ touching the sides $AB, BC$ and $CA$ at $D(z_4), E(z_5)$ and $F(z_5)$ respectively:
Step-by-Step Solution
Key Concept: Harmonic alternating series can be evaluated using telescoping and combinatorial manipulation.
Option (A): $[7 - (1 - \frac{1}{1} + \frac{1}{2} - \frac{1}{3} + \cdots - \frac{1}{7})] = 1854$. The expression telescopes using properties of harmonic series and alternating sums, combined with combinatorial identities for factorials.
Correct Answer: [A – q ] [B-p] [C-r] [D-s]