Question

If $z(2 - i) = 3 + i$ then ${z^{20}} = $ :
A. $ - 1024$
B. $1 - i$
C. $1 + i$
D. $1024$

Answer 1

Hint: First try to find out the $z$ after getting rationalise it and try to convert it as $z = \left| z \right|\left( {\cos \theta + i\sin \theta } \right)$ because this can be also written as $z = \left| z \right|{e^{i\theta }}$ Now we have to find out the ${z^{20}}$ so take $20$ on both sides and solve according to that .

Complete step-by-step answer:
In this question first try to find out the z for this ,
$z(2 - i) = 3 + i$
Now transfer $2 - i$ to the RHS , or in denominator of RHS ,
we get
$\Rightarrow$$z = \dfrac{{3 + i}}{{2 - i}}$
Now multiply and divide $2 + i$ in both numerator and denominator ,
$\Rightarrow$$z = \dfrac{{3 + i}}{{2 - i}} \times \dfrac{{2 + i}}{{2 + i}}$
On solving we get ,
$\Rightarrow$$z = \dfrac{{6 + 3i + 2i + {i^2}}}{{{2^2} - {i^2}}}$
We know that the value of ${i^2} = - 1$ hence put it on equation
$\Rightarrow$$z = \dfrac{{5 + 5i}}{5}$
or $z = 1 + i$
Now try to convert $z = \left| z \right|\left( {\cos \theta + i\sin \theta } \right)$ because this can be also written as $z = \left| z \right|{e^{i\theta }}$ so for this take common $\sqrt 2 $ from the equation $z = 1 + i$
$\Rightarrow$$z = \sqrt 2 \left( {\dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }}i} \right)$
As we know that the $\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }} = \sin \dfrac{\pi }{4}$ hence
$\Rightarrow$$z = \sqrt 2 \left( {\cos \dfrac{\pi }{4} + i\sin \dfrac{\pi }{4}} \right)$
It can also be written as $z = \sqrt 2 {e^{i\dfrac{\pi }{4}}}$
Now we have to find out the ${z^{20}}$ so ,
$\Rightarrow$${z^{20}} = {\left( {\sqrt 2 {e^{i\dfrac{\pi }{4}}}} \right)^{20}}$
$\Rightarrow$${z^{20}} = \left( {{{\left( {\sqrt 2 } \right)}^{20}}{e^{i\dfrac{\pi }{4} \times 20}}} \right)$
$\Rightarrow$${z^{20}} = \left( {{2^{10}}{e^{i5\pi }}} \right)$
So ${2^{10}} = 1024$ and we can write ${e^{i5\pi }} = \cos 5\pi + i\sin 5\pi $
$\Rightarrow$${z^{20}} = 1024\left( {\cos 5\pi + i\sin 5\pi } \right)$
And as we know that the value of $\cos 5\pi = - 1$ and $\sin 5\pi = 0$ therefore ,
$\Rightarrow$${z^{20}} = - 1024$

Hence option A is the correct answer.

Note: De Moivre’s Theorem for integral index state that If n is a integer, then ${\left( {\cos \theta + i\sin \theta } \right)^n} = \cos n\theta + i\sin n\theta $ we will use this proof in solving the question .
In general, if n be a positive integer then, where $\omega $ is the cube root of unity
$\
  {\omega ^{3n}} = {({\omega ^3})^n} = {1^n} = 1 \\
  {\omega ^{3n + 1}} = {\omega ^{3n}}.\omega = 1.\omega = \omega \\
  {\omega ^{3n + 2}} = {\omega ^{3n}}.{\omega ^2} = 1.{\omega ^2} = {\omega ^2} \\
\ $