Question

The conjugate of the complex number \[\dfrac{{{{(i + 1)}^2}}}{{(1 - i)}}\] is
A.\[1 - i\]
B.\[1 + i\]
C.\[ - 1 + i\]
D.\[ - 1 - i\]

Answer 1

Hint: We have to find the complex number of simplifications of the given expression \[\dfrac{{{{(i + 1)}^2}}}{{(1 - i)}}\] . We solve this question using the concept of complex numbers and various properties of complex numbers . We firstly expand the term using the formula of sum of two terms and then simplifying the expression , we get the simplified complex number .

Complete step-by-step answer:
Given :
The given complex number is \[\dfrac{{{{(i + 1)}^2}}}{{(1 - i)}}\]
Let us consider that \[z = \dfrac{{{{(i + 1)}^2}}}{{(1 - i)}}\]
Now ,
Firstly expanding the term using the formula given as :
\[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
Using the formula , we get
\[z = \dfrac{{({i^2} + {1^2} + 2i)}}{{(1 - i)}}\]
\[z = \dfrac{{({i^2} + 1 + 2i)}}{{(1 - i)}}\]
We also know that the value of \[i = \sqrt { - 1} \]
So , we get the value of \[{i^2} = - 1\]
Putting the value of i^2 , we get
\[z = \dfrac{{( - 1 + 1 + 2i)}}{{(1 - i)}}\]
On further simplifying , we get
\[z = \dfrac{{2i}}{{(1 - i)}}\]
Now rationalising the term
Multiplying numerator and denominator by \[1 + i\]
\[z = \dfrac{{2i \times (1 + i)}}{{(1 - i) \times \left( {1 + i} \right)}}\]
Using the formula \[(a - b)(a + b) = {a^2} - {b^2}\] and using the value of \[{i^2}\] , we get
\[z = \dfrac{{2i \times (1 + i)}}{2}\]
Cancelling the terms , we get
\[z = i \times (1 + i)\]
Expanding the term and putting the value of \[{i^2}\] , we get
\[z = i - 1\]
We get the simplified term of the expression as \[z = i - 1\] .
Now , we have to find the conjugate of \[z\] .
Conjugate of \[z = - 1 - i\]
Thus , The conjugate of the complex number \[\dfrac{{{{(i + 1)}^2}}}{{(1 - i)}}\] is \[ - 1 - i\] .
Hence , the correct option is \[(4)\] .
So, the correct answer is “Option 4”.

Note: let us consider that there is a complex number \[z\] such that \[z = a + ib\] . Then the conjugate of the complex number \[z\] is given as \[a - ib\] . A number of the form \[a + ib\] , where \[a\] and \[b\] are real numbers , is called a complex number , \[a\] is called the real part and \[b\] is called the imaginary part of the complex number .
Every real number can be represented in terms of complex numbers but the converse is not true .
Since \[{b^2} - 4ac\] determines whether the quadratic equation \[a{x^2} + bx + c = 0\]
If \[{b^2} - 4ac < 0\] then the equation has imaginary roots .