Question
Download Solution PDFयदि \(z\ne0\) एक सम्मिश्र संख्या है, तो किसके बराबर है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंप्रत्यय:
1. एक सम्मिश्र संख्या \( z = r(\cos\theta + i\sin\theta) \) का आयाम (या कोणांक) \( \text{amp}(z) = \theta \) द्वारा दिया जाता है।
2. \( z \) के संयुग्मी, जिसे \( \overline{z} \) द्वारा दर्शाया गया है, का आयाम \( \text{amp}(\overline{z}) = -\theta \) होता है, क्योंकि सम्मिश्र संख्या का संयुग्मी ज्ञात करने पर आर्गैंड समतल में इसे वास्तविक अक्ष के परितः परावर्तित किया जाता है।
प्रयुक्त सूत्र:
\( \text{amp}(z) + \text{amp}(\overline{z}) = \theta + (-\theta) = 0 \).
गणना:
\( z = r(\cos\theta + i\sin\theta) \)
\( \overline{z} = r(\cos\theta - i\sin\theta) \)
\( \text{amp}(z) + \text{amp}(\overline{z}) = \theta + (-\theta) = 0 \)
निष्कर्ष:
\( \therefore \text{amp}(z) + \text{amp}(\overline{z}) = 0 \).
इसलिए, सही उत्तर विकल्प 1 है।
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