Complex Number Question 12
Question 12 - 2024 (30 Jan Shift 2)
If $\mathrm{z}$ is a complex number, then the number of common roots of the equation $z^{1985}+z^{100}+1=0$ and $z^{3}+2 z^{2}+2 z+1=0$, is equal to :
(1) 1
(2) 2
(3) 0
(4) 3
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Answer (2)
Solution
Sol. $z^{1985}+z^{100}+1=0 & z^{3}+2 z^{2}+2 z+1=0$
$$ \begin{aligned} & (z+1)\left(z^{2}-z+1\right)+2 z(z+1)=0 \ & (z+1)\left(z^{2}+z+1\right)=0 \end{aligned} $$
$$ \Rightarrow \quad z=-1, \quad z=w, w^{2} $$
Now putting $z=-1$ not satisfy
Now put $\mathrm{z}=\mathrm{w}$
$$ \begin{aligned} & \Rightarrow \quad \mathrm{w}^{1985}+\mathrm{w}^{100}+1 \ & \Rightarrow \quad \mathrm{w}^{2}+\mathrm{w}+1=0 \end{aligned} $$
Also, $\mathrm{z}=\mathrm{w}^{2}$
$$ \begin{aligned} & \Rightarrow \quad \mathrm{w}^{3970}+\mathrm{w}^{200}+1 \ & \Rightarrow \quad \mathrm{w}+\mathrm{w}^{2}+1=0 \end{aligned} $$
Two common root
