The number of solutions of the equation z3+3(z)2|z|=0 (where , Z is a complex number) are

Z3+3(Z¯)2|Z|=0 ⇒Z5+3(Z¯)2 Z2|Z|=0 ⇒z5+3z2z=0 ⇒z5+3z=0z=r(cos⁡θ+isin⁡θ)⇒r=3 θ=π5+2k5k=0,1,2,3,4

The number of solutions of the equation z3+3(z)2|z|=0 (where , Z is a complex number) are

Z3+3(Z¯)2|Z|=0 ⇒Z5+3(Z¯)2 Z2|Z|=0 ⇒z5+3z2z=0 ⇒z5+3z=0z=r(cos⁡θ+isin⁡θ)⇒r=3 θ=π5+2k5k=0,1,2,3,4

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The number of solutions of the equation $z^{3} + \frac{3 (z)^{2}}{| z |} = 0$ (where , Z is a complex number) are

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answer is D.

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Detailed Solution

$Z^{3} + \frac{3 (\bar{Z})^{2}}{| Z |} = 0 \Rightarrow Z^{5} + \frac{3 (\bar{Z})^{2} Z^{2}}{| Z |} = 0$
$\Rightarrow z^{5} + \frac{3 {|z|}^{2}}{|z|} = 0 \Rightarrow z^{5} + 3 |z| = 0$
$z = r (\cos θ + isin θ) \Rightarrow r = \sqrt{3} θ = \frac{π}{5} + \frac{2 k}{5} k = 0, 1, 2, 3, 4$