Question
Download Solution PDF\(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}\) का मापांक (मॉड्यूलस) क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFप्रयुक्त सूत्र:
सम्मिश्र संख्या z = x + iy के लिए
|zn| = |z|n
जहाँ, i = √-1 और
|z| = √(x2 + y2)
गणना:
माना कि, z = \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right) \)
This can be written as:
\(z=\left(\frac{{i\sqrt 3}}{2}-\frac{1}{2}\right)\)
⇒ |z200| = \(|\left(\frac{{i\sqrt 3}}{2}-\frac{1}{2}\right)^{200}|\)
मापांक के गुण का प्रयोग करने पर
⇒ |z|200 = \(|\left(\frac{{i\sqrt 3}}{2}-\frac{1}{2}\right)^{200}|\)
हम जानते हैं कि |z| = √(x2 + y2)
⇒ |z|200 = \(\left(\sqrt{(\frac{{\sqrt3}}{2})^2+(\frac{-1}{2})^2}\right)^{200}\)
⇒ |z|200 = \((\sqrt {\frac{3}{4}+\frac{1}{4}})^{200} = 1^{200}\)
∴ \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200} =1\)
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