Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If $z$ is a complex number such that $\arg\left(\frac{z-3\sqrt{3}}{z+3\sqrt{3}}\right) = \frac{\pi}{3}$, then the locus of $z$ is:
$|z - 3i| = 6$
$|z - 3i| = 6, \text{Im} z > 0$
$|z - 3i| = 6, \text{Im} z 0)$
Step-by-Step Solution
Key Concept: The number of arithmetic progressions of length 3 depends on whether $n$ is even or odd, determined by counting valid common differences.
For $n = 2m$, triplets with common difference $d$ have count $2m - 2d$. Summing from $d=1$ to $d=m-1$ gives $2+4+6+\ldots+(2m-2) = m(m-1) = \frac{n(n-2)}{4}$. For $n = 2m+1$, triplets sum to $1+3+5+\ldots+(2m-1) = m^2 = \left(\frac{n-1}{2}\right)^2$.
Correct Answer: 2,4