Question

Which of the following is not correct?
(A) \[\left| z \right|\ge 0\]
(B) \[\left| z \right|\ge \operatorname{Re}\left( z \right)\]
 (C) \[\left| z \right|\ge \operatorname{Im}\left( z \right)\]
(D) \[z\overline{z}={{\left| z \right|}^{-3}}\]

Answer 1

Hint: Assume a complex number \[z\] such that \[z=a+ib\] . Here, \[a\] is the real part of the complex number \[z\] whereas \[b\] is the imaginary part of the complex number \[z\] i.e., \[\operatorname{Re}\left( z \right)=a\] and \[\operatorname{Im}\left( z \right)=b\] . Since \[\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\] and \[z\] is the magnitude which cannot be negative so, \[z\] must be always greater than or equal to zero, \[\left| z \right|\ge 0\] . Now, compare the real part of the complex number \[z\] with the magnitude of the complex number \[z\] . Similarly, compare the imaginary part of the complex number \[z\] with the magnitude of the complex number \[z\] . Now, identify all the correct options and pick the incorrect option as an answer.

Complete step by step answer:
According to the question, we are given four options and we are asked to select that one option which is not correct.
First of all, let us assume that the complex number \[z\] is equal to \[a+ib\] .
\[z=a+ib\] ………………………………………(1)
In the above equation, \[a\] is the real part of the complex number \[z\] whereas \[b\] is the imaginary part of the complex number \[z\] .
\[\operatorname{Re}\left( z \right)=a\] ………………………………….(2)
\[\operatorname{Im}\left( z \right)=b\] ……………………………………(3)
We also know the formula for magnitude of any complex number, Magnitude =
\[\sqrt{{{\left( real\,part \right)}^{2}}+\left( imaginary\,part \right)}\] ………………………………………………(4)
Now, using the formula shown in equation (4), we get
Magnitude of the complex number \[z\] = \[\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\] ……………………………….(5)
Now, on comparing equation (2) and equation (5), we can say that \[a\] will be always smaller than \[\sqrt{{{a}^{2}}+{{b}^{2}}}\] .
So, \[\left| z \right|\ge \operatorname{Re}\left( z \right)\] ……………………………………….(6)
Similarly, on comparing equation (3) and equation (5), we can say that \[b\] will be always smaller than \[\sqrt{{{a}^{2}}+{{b}^{2}}}\] .
So, \[\left| z \right|\ge \operatorname{Im}\left( z \right)\] …………………………………(7)
Since \[\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}\] and \[z\] is the magnitude which cannot be negative so, \[z\] must be always greater than or equal to zero, \[\left| z \right|\ge 0\] ……………………………….(8)
Now, from equation (6), equation (7), and equation (8), we have
\[\left| z \right|\ge \operatorname{Re}\left( z \right)\] , \[\left| z \right|\ge \operatorname{Im}\left( z \right)\] , and \[\left| z \right|\ge 0\] .
So, option (A), option (B), and option (C) are correct.
Therefore, only option (D) is not correct.

Note:
For this type of question, where we have to comment on real or imaginary part of the complex number. Always approach this type of question by assuming the complex number \[z\] as \[z=a+ib\] . At last, take the formula for finding magnitude of the complex number \[z\] into consideration, Magnitude of a complex number = \[\sqrt{{{\left( real\,part \right)}^{2}}+\left( imaginary\,part \right)}\] .