Question:
If $z(2-i)=(3+i)$, then $z^{20}$ is equal to
If $z(2-i)=(3+i)$, then $z^{20}$ is equal to
Updated On: Sep 24, 2024
- $2^{10}$
- $-2^{10}$
- $2^{20}$
- $-2^{20}$
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The Correct Option is B
Solution and Explanation
We have, $z\left(2-i\right)=\left(3+i\right)$
$\Rightarrow\, z=\left(\frac{3+i}{2-i}\right)\times\left(\frac{2+i}{2+i}\right)$
$=\frac{5+5i}{5}$
$\Rightarrow\, z=1+i$
$\Rightarrow\, z^{2}=2i$
$\Rightarrow\, z^{20}$
$=-2^{10}$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
