If $z$ is a complex number, then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^3+2 z^2+2 z+1=0$, is equal to
Solution
<p>$$\begin{array}{ll}
\text { } & z^{1985}+z^{100}+1=0 \& z^3+2 z^2+2 z+1=0 \\
& (z+1)\left(z^2-z+1\right)+2 z(z+1)=0 \\
& (z+1)\left(z^2+z+1\right)=0 \\
\Rightarrow \quad & z=-1, \quad z=w, w^2 \\
& \text { Now putting } z=-1 \text { not satisfy } \\
& \text { Now put } z=w \\
\Rightarrow \quad & w^{1985}+w^{100}+1 \\
\Rightarrow \quad & w^2+w+1=0 \\
\Rightarrow \quad & \text { Also, } z=w^2 \\
\Rightarrow \quad & w^{3970}+w^{200}+1 \\
\Rightarrow & w+w^2+1=0
\end{array}$$</p>
<p>Two common root</p>
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
This question is part of PrepWiser's free JEE Main question bank. 223 more solved questions on Complex Numbers and Quadratic Equations are available — start with the harder ones if your accuracy is >70%.