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Question

In the argand plane, the distinct roots of $$1+z+{ z }^{ 3 }+{ z }^{ 4 }=0$$ ($$z$$ is a complex number) represent vertices of


A
A square
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B
An equilateral triangle
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C
A rhombus
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D
A rectangle
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Solution

The correct option is B An equilateral triangle
Given equation is
$$1+z+{ z }^{ 3 }+{ z }^{ 4 }=0$$
$$\Rightarrow \left( 1+z \right) +{ z }^{ 3 }\left( 1+z \right) =0$$
$$\Rightarrow \left( 1+z \right) \left( 1+{ z }^{ 3 } \right) =1$$
$$\Rightarrow z=-1$$
$$\Rightarrow { z }^{ 3 }=-1$$
$$\Rightarrow z=-1,z=-1,-\omega ,-{ \omega  }^{ 2 }$$ 
Hence, roots are in cubic roots of unity.
Hence, these roots are the vertices of an equilateral triangle.

Maths

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