Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade None
Question:
Two triangles having vertices as $z_1, z_2, z_3$ and $a, b, c$ are similar. Then
$az_1 + bz_2 + cz_3 = 0$
$\frac{z_1}{a} + \frac{z_2}{b} + \frac{z_3}{c} = 0$
$z_1(b - c) + z_2(c - a) + z_3(a - b) = 0$
$a(z_2 - z_3) + b(z_3 - z_1) + c(z_1 - z_2) = 0$
Step-by-Step Solution
Key Concept: The total number of subsets of an n-element set is $2^n$, derived from summing all binomial coefficients.
Objects can be categorized by how many are identical: 0 identical and $n$ distinct, 1 identical and $n-1$ distinct, up to $n$ identical and 0 distinct. Summing all cases: $\sum_{r=0}^{n} ^nC_r = 2^n$. Additionally, $2^{n+1}(^nC_0 + 2^{n+1}C_1 + 2^{n+1}C_2 + \ldots + 2^{n+1}C_n) = 2^{2n}$, so all three statements (A), (B), and (C) are true.
Correct Answer: 3,4