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Properties of argument of a complex number - (Concept)
i) $\arg \left(z_1 z_2\right)=\arg \left(z_1\right)+\arg \left(z_2\right)+2 k \pi, \quad k$ is an integer
Value of $k$ should be chosen in such a way that arg lies between (-?,?]
This formula for argument can be generalised for n number of complexes in a similar way
ii) $\arg \left(\frac{\mathrm{z}_1}{\mathrm{z}_2}\right)=\arg \left(\mathrm{z}_1\right)-\arg \left(\mathrm{z}_2\right)+2 \mathrm{k} \pi$ where $k$ is an integer
iii) $\arg (\bar{z})=-\arg (z)$
iv) $\arg \left(\mathrm{z}^{\mathrm{n}}\right)=\mathrm{n} \cdot \arg (\mathrm{z})+2 \mathrm{k} \pi$ k belongs to an integer
(v) $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right| \Rightarrow \arg \left(z_1\right)=\arg \left(z_2\right)$
(vi) $\left|z_1+z_2\right|=\left|\left|z_1\right|-\left|z_2\right|\right| \Rightarrow \arg \left(z_1\right)-\arg \left(z_2\right)=\pi$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
Let be complex numbers such that
and arg
Then arg
equals:
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Polar representation of complex number will be
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Let then in Euler's form z will be represented as
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Let then
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Let then arg
equals
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&
are complex numbers such that
then
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Principal value of arg equals
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Let and
be two complex number such that
and arg
. Then arg z2 is.
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Let $\arg (\mathrm{z})$ represents the principal argument of the complex number z . Then $|\mathrm{z}|=3$ and $\arg (z-1)-\arg (z+1)=\frac{\pi}{4} {\text { intersects }}$
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If are two complex numbers such that
and
, then
is.
(Here arg(z) denotes the principal argument of complex number z).
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Let $z_1$ and $z_2$ be two complex numbers such that arg $\left(z_1-z_2\right)=\frac{\pi}{4}$ and $z_1, z_2{ }_{\text {satisfy the equation }}|z-3|=\operatorname{Re}(z)$. Then the imaginary part of $z_1+z_2$ is equal to $\qquad$ .
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If $\left|z_2+i z_1\right|=\left|z_1\right|+\left|z_2\right|$ and $\left|z_1\right|=3 \quad \&\left|z_2\right|=4$ then area of $\triangle A B C$, if affix of $A, B$ and $C_{\text {are }}\left(z_1\right),\left(z_2\right)$ and $\left(\frac{z_2-i z_1}{1-i}\right)$ respectively , is
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Match the following
z $\operatorname{Arg}(z)$
(i) $1-i$
(p) $\frac{-2 \pi}{3}$
(ii) $2+2 \sqrt{3} i$
(q) $\frac{-\pi}{4}$
(iii) $-\sqrt{3}+i$
(r) $\frac{5 \pi}{6}$
(iv) $-1-\sqrt{3} i$
(s) $\frac{\pi}{3}$
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If $Z_1, Z_2, Z_3$ represents three vertices of an equilateral triangle, then the relation between $Z_1, Z_2$ and $Z_3$ is
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$z_1, z_2$ and $\left(\frac{z_2-i z_1}{1-i}\right)$ are vertices of $\triangle A B C$ such that $\left|z_2+i z_1\right|=\left|z_1\right|+\left|z_2\right|$ and $\left|z_1\right|=3,\left|z_2\right|=4$. The area of $\triangle A B C$ is
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If $z=\frac{-2}{1+i \sqrt{3}}$, then the value of $\arg (z)$ is: [Up to three decimal places]
[Here, $\pi =\frac{22}7$]
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