यदि \(z\ne0\) एक सम्मिश्र संख्या है, तो किसके बराबर है?

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  1. 0
  2. π/2
  3. π
  4. 2π

Answer (Detailed Solution Below)

Option 1 : 0
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संप्रत्यय:

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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