Complex Numbers
True/False Statements
MMTS_Full_Test_06
Grade 12
Question:
Which must be true (T/F in order): I) $\arg(\bar{z})=-\arg(z)$ for any complex number $z$. II) If $|z-z_1|-|z-z_2|=k$ where $k<|z_1-z_2|$, then locus of $z$ is a hyperbola. III) $z_1,z_2,\ldots,z_n$ represent vertices of $n$-sided regular polygon with centre $z_0$, then $\sum z_i^2=nz_0^2$. IV) If $z_1^2+z_2^2+z_1z_2=0$, then $z_1,z_2$ and origin form equilateral triangle.
Step-by-Step Solution
Key Concept: Verify each statement
I) True. II) True (hyperbola definition). III) False ($\sum z_i^2\ne nz_0^2$ generally). IV) False ($z_1^2+z_1z_2+z_2^2=0$: equilateral needs $|z_1|=|z_2|$ and arg difference $\pi/3$... actually True). FFTT.
Correct Answer: 3