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4-1.Complex numbers
normal
If $z = x + iy\, (x, y \in R,\, x \neq \, -1/2)$ , the number of values of $z$ satisfying ${\left| z \right|^n}\, = \,{z^2}{\left| z \right|^{n - 2}}\, + \,z{\left| z \right|^{n - 2}}\, + \,1\,.\,\left( {n \in N,n > 1} \right)$ is
A
$0$
B
$1$
C
$2$
D
$3$
Solution
The given equation is $|z|^{n}=\left(z^{2}+z\right)|z|^{n-2}+1$
$\Rightarrow \quad z^{2}+z$ is real
$\Rightarrow \quad z^{2}+z=\bar{z}^{2}+\bar{z}$
$\Rightarrow \quad(z-\bar{z})(z+\bar{z}+1)=0$
$\Rightarrow \quad z=\bar{z}$ as $z+\bar{z}+1 \neq 0(x \neq-1 / 2)$
Hence, the given equation reduces to
$x^{n}=x^{n}+x|x|^{n-2}+1$
$\Rightarrow \quad x|x|^{n-2}=-1$
$\Rightarrow \quad x=-1$
Standard 11
Mathematics
