Question:
Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2+\frac{1}{ z ^2}$ are integers, then which of the following is/are possible value(s) of $|z|$?
Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2+\frac{1}{ z ^2}$ are integers, then which of the following is/are possible value(s) of $|z|$?
Updated On: Apr 23, 2024
\(\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}\)
\(\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}\)
\(\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}\)
\(\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}\)
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The Correct Option is A
Solution and Explanation
The correct option is (A): \(\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}\)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.