Let denote the complex conjugate of a complex number and let . In the set of complex numbers, the number of distinct roots of the equation is _______ . [2022]

Question

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

Smart Achievers · Accepted Answer

(4)

$Given, \bar{z} - z^{2} = i (\bar{z} + z^{2})$

$It can be written as \bar{z} (1 - i) = z^{2} (1 + i)$

$So | \bar{z} | | 1 - i | = | z |^{2} | 1 + i |$

$| z | = | z |^{2} \Rightarrow | z | = 0 or | z | = 1$

$Let \arg (z) = α . So from (i), we get$

$2 n π - α - \frac{π}{4} = 2 α + \frac{π}{4}$

$\Rightarrow α = \frac{1}{3} (\frac{4 n - 1}{2}) π = \frac{(4 n - 1) π}{6}$

So we will get 3 distinct values of $α$ . Hence there will be total 4 possible values of complex number $z$ .

Solution

Q.
Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

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Solution

Q. Let z¯ denote the complex conjugate of a complex number z and let i=-1. In the set of complex numbers, the number of distinct roots of the equation z¯-z2=i(z¯+z2)is _______ . [2022]

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Q.
Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]