if z(2-i) = 3+i, find z²⁰. a. 2¹⁰ b. -2¹⁰ c. 2²⁰ d. -2²⁰ Hope a correct and quick answer from all u genius

Question

if z(2-i) = 3+i, find z²⁰.
a. 2¹⁰ b. -2¹⁰
c. 2²⁰ d. -2²⁰
Hope a correct and quick answer from all u geniuses…​

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Ruby 1 month 2021-08-17T14:38:26+00:00 1 Answer 0 views 0

Answers ( )

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    2021-08-17T14:39:37+00:00

    Given :

    • z(2-i) = 3 + i

    To Find :

    • \rm{ {z}^{20} }

    Solution :

    \rm{\implies z (2-i) = 3+i }

    \rm{\implies z = \dfrac{ 3+i }{2-i}}

    \rm{\implies z = \dfrac{ 3+i }{2-i} \times \dfrac{2+i}{2+i}}

    \rm{\implies z = \dfrac{6+5i + {i}^{2}}{ 4- {i}^{2}}}

    = 1 .

    \rm{\implies z = \dfrac{6+5i -1}{ 4.+1 }}

    \rm{\implies z = \dfrac{5+5i }{ 5}}

    • \rm{\implies z = 1+ i }

    In polar form :

    \rm{\implies z = \sqrt{2} [\dfrac{1}{\sqrt{2} } + \dfrac{1}{\sqrt{2}}i  ] }

    \rm{\implies z = \sqrt{2} [ \cos \dfrac{\pi }{4 } +i \sin \dfrac{\pi}{4} ] }

    •  \rm{ \implies z \:  =  \sqrt{2}  {e}^{i \frac{\pi}{4} } }

    \rm{ {z }^{20}\:  =  (\sqrt{2}  {e}^{i \frac{\pi}{4} } }  {)}^{20}

     \rm{ \implies  {z}^{20}  =  {2}^{10} . {e}^{i \frac{\pi}{4} \times 20 } }

    \rm{ \implies  {z}^{20}  =  {2}^{10} . {e}^{i. 5 \pi} }

    \rm{\implies{ z}^{20} = {2}^{10} [ \cos 5 \pi +i \sin 5 \pi] }

    Cos 5π = 1 and sin 5π = 0

    \rm{\implies {z}^{20} = {2}^{10} [ -1 +0.i  ] }

    • \rm{\implies {z}^{20} = - {2}^{10} }

    Option B .

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