Question:
What is the modulus of \(\left(\dfrac{\sqrt{-3}}{2} – \dfrac{1}{2}\right)^{200}\)?
\(\left(\dfrac{\sqrt{-3}}{2} – \dfrac{1}{2}\right)^{200}\) का मापांक क्या है?
Options:
- \(\dfrac{1}{2}\)
- 1
- \(\dfrac{1}{4}\)
- \(2^{200}\)
- \(\dfrac{1}{2}\)
- 1
- \(\dfrac{1}{4}\)
- \(2^{200}\)
Correct Answer: b)
Explanation:
\(\dfrac{\sqrt{-3}}{2} – \dfrac{1}{2} = -\dfrac{1}{2} + i\dfrac{\sqrt{3}}{2} = e^{i 2\pi/3} = \omega\).
Modulus of \(z = \omega^{200}\) is \(|z| = |\omega|^{200} = 1^{200} = 1\).
\(\dfrac{\sqrt{-3}}{2} – \dfrac{1}{2} = -\dfrac{1}{2} + i\dfrac{\sqrt{3}}{2} = \omega\)।
चूँकि \(|\omega| = 1\) होता है, अतः \(|\omega^{200}| = |\omega|^{200} = 1^{200} = 1\)।