Let $|Z|=|W|=r$ $\Rightarrow Z=r e^{i \theta}, W=r e^{i \phi}$ where $\theta+\phi=\pi$ $\therefore \bar{W}=r e^{-i \phi}$ $\mathrm{Now}, Z

Statement $1$ is true, Statement $2$ is false.

Let $|Z|=|W|=r$ $\Rightarrow Z=r e^{i \theta}, W=r e^{i \phi}$ where $\theta+\phi=\pi$ $\therefore \bar{W}=r e^{-i \phi}$ $\mathrm{Now}, Z=r e^{i(\pi-\phi)}=r e^{i \pi} \times e^{-i \phi}=-r e^{-i \phi}$ $=-\vec{W}$ Thus, statement $-\,1$ is true but statement $-\,2$ is false

4-1.Complex numbersDifficult

Let $Z$ and $W$ be complex numbers such that $\left| Z \right| = \left| W \right|,$ and arg $Z$ denotes the principal argument of $Z.$

Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.

Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$

[AIEEE 2012]

A
Statement $1$ is true, Statement $2$ is false.
B
Statement $1$ is true, Statement $2$ is true, Statement $2$ is a correct explanation for Statement $1$
C
Statement $1$ is true, Statement $2$ is true, Statement $2$ is not a correct explanation for Statement $1.$
D
Statement $1$ is false, Statement $2$ is true

Std 11
Mathematics

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[JEE MAIN 2024]

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$z_1$ and $z_2$ are two complex numbers such that $|z_1 + z_2|$ = $1$ and $\left| {z_1^2 + z_2^2} \right|$ = $25$ , then minimum value of $\left| {z_1^3 + z_2^3} \right|$ is

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Let $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that

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JEE Main

Similar Questions

If complex number $z = x + iy$ is taken such that the amplitude of fraction $\frac{{z - 1}}{{z + 1}}$ is always $\frac{\pi }{4}$, then

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