AP EAMCET · Maths · Complex Number
\(\omega\) is a complex cube root of unity and if \(Z\) is a complex number satisfying \(|\mathrm{Z}-1| \leq 2\) and \(\left|\omega^2 Z-1-\omega\right|=a\), then the set of possible values of \(a\) is
- A \(0 \leq a \leq 2\)
- B \(|\omega| \leq a \leq \frac{\sqrt{3}}{2}+2\)
- C \(\frac{1}{2} \leq a \leq \frac{\sqrt{3}}{2}\)
- D \(0 \leq a \leq 4\)
Answer & Solution
Correct Answer
(D) \(0 \leq a \leq 4\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {Given, }|Z-1| \leq 2 \text { and }\left|\omega^2 Z-1-\omega\right|=a \\ & \Rightarrow\left|\omega^2 Z+\omega^2\right|=a \qquad \left(\because 1+\omega+\omega^2=0\right) \\ & \Rightarrow\left|\omega^2\right||Z+1|=a \Rightarrow|Z-1+2|=a \\ &…
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