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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 triangleExplanation for the correct option:Finding the distinct root of $⇒1+z+{z}^{3}+{z}^{4}=0\phantom{\rule{0ex}{0ex}}⇒\left(1+z\right)+{z}^{3}\left(1+z\right)=0\phantom{\rule{0ex}{0ex}}⇒\left(1+z\right)\left(1+{z}^{3}\right)=0\phantom{\rule{0ex}{0ex}}⇒z=-1or{z}^{3}=-1\phantom{\rule{0ex}{0ex}}⇒z=-1,-\omega ,-{\omega }^{2}$From the above values, we know that the roots are the cube root of unity.So, these roots are the vertices of an equilateral triangleHence, option (B) is the correct answer.

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