Question

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

Answer 1

Hint: According to the question we have to choose the option which is not correct.
So, first of all we have to check each option to determine that it is correct or not but as given in the options Z is a complex number which is $(a + ib)$ where a, and b are the real numbers and i is the imaginary term. Now, we have determined the option is correct or not by checking each term of the options.
As given in the as we know that the value of z is $(a + ib)$ so, the value of $\overline z $ is $(a - ib)$ which is the inverse of z.

Complete step-by-step answer:
Step 1: First of all we have to determine the option (A) which is $\left| z \right| \geqslant 0$
As we know that z = $(a + ib)$
Hence,
\[\left| {(a + ib)} \right| \geqslant 0\] which is true because by placing values of a, and b which are real numbers the value obtained will always be greater than or equal to zero.
Step 2: Now, we have to determine the option (B) which is $\left| z \right| \geqslant \operatorname{Re} (z)$ so, for every real z as given in the question \[\left| {(a + ib)} \right| > \operatorname{Re} (a + ib)\] which is true because for all the negative values of z, $\left| z \right|$ will always be positive.
Step 3: Now, we have to determine the option (C) which is $\left| x \right| \geqslant \operatorname{Im} (z)$ which is definitely true because given that z is an imaginary term.
Step 4: Now, same as the previous steps we will determine the option (D) which is $z\overline z = {\left| z \right|^{ - 3}}$ so, as mentioned in the solution hint that the value of z is $(a + ib)$ so, the value of $\overline z $ is $(a - ib)$ which is the inverse of z.
Hence on solving the L.H.S. which is $z\overline z $ by placing the values of z and $\overline z $
$
   = (a + ib)(a - ib) \\
   = {a^2} - {(ib)^2} \\
   = {a^2} - {i^2}{b^2} \\
 $
Step 5: Now, on substituting the value of i which is -1 in the expression obtained in the step 4.
$ = {a^2} + {b^2}$
Which is not equal to ${\left| z \right|^{ - 3}}$

Hence, by solving all the options we have obtained the correct option which is (A) $\left| z \right| \geqslant 0$

Note: As mentioned in the question z is a complex number which is equal to $(a + ib)$ and the value of $\overline z $ is $(a - ib)$ which is the inverse of z.
On multiplying the imaginary term i with i we will obtain ${i^2}$ and the value of ${i^2}$ is equal to 1.