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प्रश्न
Select the correct answer from the given alternatives:
If z = r(cos θ + i sin θ), then the value of `"z"/bar("z") + bar("z")/"z"`
विकल्प
cos 2θ
2 cos 2θ
2 cos θ
2 sin θ
उत्तर
2 cos 2θ
Explanation:
`"z"/bar"z" = ("r"(costheta + "i"sintheta))/("r"(costheta - "i" sintheta))`
= `(costheta + "i" sintheta)/(cos(-theta) + "i" sin(-theta))`
= (cos θ + i sin θ) (cos (– θ) + i sin (– θ))–1
= (cos θ + i sin θ) (cos θ + i sin θ) ...(De Movire’s Theorem)
= (cos θ + i sin θ)2
= cos 2θ + i sin 2θ ...(De Movire’s Theorem)
∵ `bar(("z"/bar"z")) = bar("z")/"z"`
∴ `"z"/bar("z") + bar("z")/"z" = "d" + bar("d") ...("Where d" = "z"/bar("z"))`
= cos 2θ + i sin 2θ + cos 2θ – i sin 2θ
= 2 cos 2θ
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