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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
$\left(\frac{1+\cos \phi+i \sin \phi}{1+\cos \phi-i \sin \phi}\right)^n=$
  • A
    $\cos n \phi-i \sin n \phi$
  • ✓
    $\cos n \phi+i \sin n \phi$
  • C
    $\sin n \phi+i \cos n \phi$
  • D
    $\sin n \phi-i \cos n \phi$
✓

Answer

Correct option: B.
$\cos n \phi+i \sin n \phi$
(B)
L.H.S.
$=\left[\frac{2 \cos ^2(\phi / 2)+2 i \sin (\phi / 2) \cos (\phi / 2)}{2 \cos ^2(\phi / 2)-2 i \sin (\phi / 2) \cos (\phi / 2)}\right]^{ n }$
$=\left[\frac{\cos (\phi / 2)+ i \sin (\phi / 2)}{\cos (\phi / 2)- i \sin (\phi / 2)}\right]^{ n }$
$=\left[\frac{ e ^{ i (\phi / 2)}}{ e ^{- i (\phi / 2)}}\right]^{ n }$
$=\left( e ^{ i \phi}\right)^{ n }$
$=\cos n \phi+i \sin n \phi$

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