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4-1.Complex numbers
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Let $z,w$be complex numbers such that $\overline z + i\overline w = 0$and $arg\,\,zw = \pi $. Then arg z equals
A
$5\pi /4$
B
$\pi /2$
C
$3\pi /4$
D
$\pi /4$
(AIEEE-2004)
Solution
(c)Given that arg zw =$\pi $ …..$(i)$
$\bar z + i\bar \omega = 0 \Rightarrow \bar z = – i\bar \omega $$ \Rightarrow z = i\omega $$ \Rightarrow \omega = – iz$
From $(i)$, arg$( – i{z^2}) = \pi $
$arg\;( – i) + 2arg(z) = \pi $ ;
$\frac{{ – \pi }}{2} + 2\;arg(z) = \pi $
$2\,arg\,(z) = \frac{{3\pi }}{2}$;
$a\,rg(z) = \frac{{3\pi }}{4}$
Standard 11
Mathematics
Similar Questions
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
| List-$I$ | List-$II$ |
| ($P$) $|z|^2$ is equal to | ($1$) $12$ |
| ($Q$) $|z-\bar{z}|^2$ is equal to | ($2$) $4$ |
| ($R$) $|z|^2+|z+\bar{z}|^2$ is equal to | ($3$) $8$ |
| ($S$) $|z+1|^2$ is equal to | ($4$) $10$ |
| ($5$) $7$ |
The correct option is:
