Comprehension

मान लीजिए z =  जहां i =  है

z का मापांक क्या है?

This question was previously asked in
NDA 02/2022 Mathematics Official Paper (Held On 04 Sep 2022)
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  1. 1
  2. 1 + sin2 θ

Answer (Detailed Solution Below)

Option 1 : 1
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NDA 01/2025: English Subject Test
30 Qs. 120 Marks 30 Mins

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संकल्पना:

यदि z = x + iy है

⇒z का मापांक = 

गणना:

दिया गया है,

z =  जहाँ i = 

(1 + i sin θ) से अंश और हर को गुणा करने पर,

,

\(⇒ |z| =\sqrt{(\frac{ \cos^2\theta }{1+\sin^2 θ})^2 +(\frac{2\sin θ}{1+\sin^2 θ})^2}\)

\(⇒ |z| ={1 \over 1+ \sin^2 \theta }\sqrt{\cos^4 \theta + 4\sin^2 \theta}\)

\(⇒ |z| = {1 \over 1+ \sin^2 \theta }\sqrt{(1 -\sin^2 \theta)^2 + 4\sin^2 \theta}\)

\(⇒ |z| ={1 \over 1+ \sin^2 \theta }\sqrt{1 -2\sin^2 \theta + \sin^4 \theta + 4\sin^2 \theta}\)

\(⇒ |z| ={1 \over 1+ \sin^2 \theta }\sqrt{1 +2\sin^2 \theta + \sin^4 \theta }\)

\(⇒ |z| ={1+ \sin^2 \theta \over 1+ \sin^2 \theta } = 1\)

∴ सही उत्तर विकल्प (1) है।

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