1.

Number of solutions of the equation `z^3+[3(barz)^2]/|z|=0` where z is a complex number is

Answer» `Z^3+(3(overlineZ)^2)/|Z|=0`
`r^3e^(i3theta)+(3r^2e^(i(-2theta)))/r=0`
`r^3e^(i3theta)+3re^(i(-2theta)=0`
`r/e^(i2theta)(r^2e^(i5theta)+3)=0`
`r!=0`
`|r^2e^(i5theta)|=|-3|`
`r^2=3`
`r=sqrt3`
`e^(i5theta)=-1`
`cos5theta+isintheta=-1`
`cos5theta=-1`
`sin5theta=0`
`theta in(-pi,pi]`
`-5pi<5theta<=5pi`
`sin5theta=0`
the correct answer is 5.


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