Question

If $Z = \dfrac{{1 + 7i}}{{{{\left( {2 - i} \right)}^2}}}$, then the polar form of Z is
(A) $\sqrt 2 \left( {\cos \left( {\dfrac{{3\pi }}{4}} \right) + i\sin \left( {\dfrac{{3\pi }}{4}} \right)} \right)$
(B) $\sqrt 2 \left( {\cos \left( {\dfrac{\pi }{4}} \right) + i\sin \left( {\dfrac{\pi }{4}} \right)} \right)$
(C) $\sqrt 2 \left( {\cos \left( {\dfrac{{7\pi }}{4}} \right) + i\sin \left( {\dfrac{{7\pi }}{4}} \right)} \right)$
(D) $\sqrt 2 \left( {\cos \left( {\dfrac{{5\pi }}{4}} \right) + i\sin \left( {\dfrac{{5\pi }}{4}} \right)} \right)$

Answer 1

Hint: In the given problem, we are required to simplify an expression involving a complex number and find its modulus and argument. For simplifying the given expression, we need to have a thorough knowledge of complex number sets and its applications in such questions. Algebraic rules and properties also play a significant role in simplification of such expressions. We must know the polar representation of the complex number to answer the problem.

Complete step by step answer:
In the question, we are given an expression which first needs to be simplified using the knowledge of complex number sets.
$Z = \dfrac{{1 + 7i}}{{{{\left( {2 - i} \right)}^2}}}$
Simplifying the denominator, we get,
$ \Rightarrow Z = \dfrac{{1 + 7i}}{{\left( {{2^2} + {i^2} - 2\left( 2 \right)i} \right)}}$
$ \Rightarrow Z = \dfrac{{1 + 7i}}{{\left( {4 + \left( { - 1} \right) - 4i} \right)}}$
$ \Rightarrow Z = \dfrac{{1 + 7i}}{{\left( {3 - 4i} \right)}}$
For simplifying the given expression involving complex numbers, we need to multiply the numerator and denominator with the conjugate of the complex number present in the denominator so as to obtain a real number in the denominator.
So, $Z = \dfrac{{1 + 7i}}{{\left( {3 - 4i} \right)}} \times \dfrac{{\left( {3 + 4i} \right)}}{{\left( {3 + 4i} \right)}}$
Using the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$,
$ \Rightarrow Z = \dfrac{{\left( {1 + 7i} \right)\left( {3 + 4i} \right)}}{{{{\left( 3 \right)}^2} - {{\left( {4i} \right)}^2}}}$
$ \Rightarrow Z = \dfrac{{\left( {3 + 4i} \right) + 7i\left( {3 + 4i} \right)}}{{9 - \left( { - 16} \right)}}$
We know that ${i^2} = - 1$. Hence, substituting ${i^2}$ as $ - 1$, we get,
$ \Rightarrow Z = \dfrac{{3 + 4i + 21i + 28{i^2}}}{{9 - \left( { - 16} \right)}}$
Opening brackets and simplifying further, we get,
$ \Rightarrow Z = \dfrac{{3 + 4i + 21i + 28\left( { - 1} \right)}}{{9 + 16}}$
Adding up the like terms, we get,
$ \Rightarrow Z = \dfrac{{25i - 25}}{{25}}$
$ \Rightarrow Z = - 1 + i$
Therefore, the complex number $Z = \dfrac{{1 + 7i}}{{{{\left( {2 - i} \right)}^2}}}$ can be simplified as $Z = - 1 + i$ .
Now, we need to find the modulus and argument of the complex number.
The modulus of a complex number $Z = x + iy$ is given by $\left| Z \right|$ and it is calculated as: $\left| Z \right| = \sqrt {{x^2} + {y^2}} $
Thus, putting in the values of x and y, we get the modulus of given complex number as:
$\left| {{Z_1}} \right| = \sqrt {{{( - 1)}^2} + {{(1)}^2}} = \sqrt 2 $
So, the modulus of the given complex number $Z = \dfrac{{1 + 7i}}{{{{\left( {2 - i} \right)}^2}}}$ is $\left( {\sqrt 2 } \right)$.
Now, we know that a complex number can be represented in polar form as $r\left( {\cos \theta + i\sin \theta } \right)$, where r is the modulus of complex numbers and $\theta $ is the argument. So, we have,
$Z = \left( { - 1 + i} \right) = \sqrt 2 \left( {\cos \theta + \sin \theta } \right)$
Comparing both sides, we get,
$ \Rightarrow \cos \theta = - \dfrac{1}{{\sqrt 2 }}$ and $\sin \theta = \dfrac{1}{{\sqrt 2 }}$.
Since sine is positive and cosine is negative. So, the argument of complex numbers is in the second quadrant. Also, we know that values of $\sin \left( {\dfrac{{3\pi }}{4}} \right)$ and $\cos \left( {\dfrac{{3\pi }}{4}} \right)$ are $\dfrac{1}{{\sqrt 2 }}$ and $ - \dfrac{1}{{\sqrt 2 }}$ respectively.
Hence, the argument of complex number $Z = \dfrac{{1 + 7i}}{{{{\left( {2 - i} \right)}^2}}}$ is $\left( {\dfrac{{3\pi }}{4}} \right)$.
Therefore, the polar form of the complex number is $\sqrt 2 \left( {\cos \left( {\dfrac{{3\pi }}{4}} \right) + i\sin \left( {\dfrac{{3\pi }}{4}} \right)} \right)$. Hence, option (A) is the correct answer.

Note:
Algebraic rules and properties also play a significant role in simplification of such expressions and we also need to have a thorough knowledge of complex number sets and their applications in such questions. We should know the process of finding the argument and modulus of a given complex number. We must take care while doing the calculations so as to be sure of the final answer. The question tells us about the wide ranging significance of the complex number set and its properties.