Which of the following must be the true value of the statements below in that order. Statement I: for any complex number Statement II: If where , then the locus of is a hyperbola. Statement III: be the complex numbers representing the vertices of

Question

Which of the following must be the true value of the statements below in that order.

Statement I: $\arg (\bar{z}) = - \arg (z)$ for any complex number $z$

Statement II: If $| z - z_{1} |$ $-$ $| z - z_{2} |$ $= k$ where $k <$ $| z_{1} - z_{2} |$ , then the locus of $z$ is a hyperbola.

Statement III: $z_{1}, z_{2}, \dots, z_{n}$ be the complex numbers representing the vertices of $n$ -sided regular polygon, and $'' z_{0}''$ is its centre, then $\sum_{i = 1}^{n} z_{i}^{2} = n z_{0}^{2}$

Statement IV: If $z_{1}^{2} + z_{2}^{2} + z_{1} z_{2} = 0$ , then $z_{1}, z_{2}$ and origin form an equilateral triangle.

Smart Achievers · Accepted Answer

(2)

$S_{1} \to Z can be zero$

$S_{2}, S_{3}, S_{4} \to Use properties of complex numbers$

Solution

1 TFFT

2 FFTF

3 FFTT

4 TTTT

Ans.
(2)

$S_{1} \to Z can be zero$

$S_{2}, S_{3}, S_{4} \to Use properties of complex numbers$

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