Test Index

Complex Numbers Part 1

Section: Mathematics

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Question : 93 of 100

Marks: +1, -0

Let

z

w

be complex numbers such that

\overline{z}+i \overline{w}=0

\arg z w = \pi

z

$\frac{5\pi}{4}$
$\frac{\pi}{2}$
$\frac{3\pi}{4}$
$\frac{\pi}{4}$

Solution:

Given

\overline{z}+i \overline{w}=0

\Rightarrow \overline{z}=-i \overline{w}

\Rightarrow \overline{\overline{z}} = -\overline{i \overline{w}}

\Rightarrow \overline{\overline{z}} = -\overline{i \overline{w}}

\Rightarrow z = -\overline{i} w

\Rightarrow z = -(-i) w

\Rightarrow z = i w

Now given that

\operatorname{Arg}(zW)=\pi

\Rightarrow \operatorname{Arg}\left(Z\times \frac{z}{i}\right)=\pi

\Rightarrow \operatorname{Arg}(z^2)-\operatorname{Arg}(i)=\pi

\Rightarrow 2\operatorname{Arg}(z)-\frac{\pi}{2}=\pi

[

i

complex number represent

(0,1)

point on imaginary axis and

\operatorname{Arg}(i)

means the angle made by the point

(0,1)

with real axis which is

\frac{\pi}{2}]

\Rightarrow 2\operatorname{Arg}(z)=\frac{3\pi}{2}

\Rightarrow \operatorname{Arg}(z)=\frac{3\pi}{4}

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