Question

$\sf if \: z = 2 + i.then \: (z - 1)( \overset - {z} - 5) + ( \overset - z - 1)(z - 5) \: is \: what$
a) 2
b) 7
c) -1
d) -4​

Answer 1

$\large \dag$ Question :-

$\text{If z = 2 + }i \text{ then Find value of } : -$

$\rm (z - 1)(\bar{z} - 5) + ( \bar{z} - 1)(z - 5)$

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf  \green{Value  \: of \:  Given  \: Expression \:  is \: - 4}} }$

$\large \dag$ Step by step Explanation :-

❒ Pre - Required Knowledge :-

▪ Conjugate of a Complex Number :

For any complex Number $\green{\text{z = a + }i \rm b}$ it's conjugate is $\red{\bar \text{z} = \text{a} - i \rm b }$

▪ Value of $\large {i}^{2}$ is $\large - 1$

❒ Let's Begin the Solution Now :-)

Here in the Given Question ;

$\large\text{ z = 2 + }i$

$\large\text{ So} \: \: \bar \text{z} = \text{2} - i \rm$

Now,

$\rm (z - 1)(\bar{z} - 5) + ( \bar{z} - 1)(z - 5)$

Putting Values of $\large\rm z$ and $\large \rm \bar z$

$= (2 + i - 1)(2 - i - 5) \\ +\: (2 - i - 1)( 2 + i - 5)$

$= (1 + i)( - 3 - i) + (1 - i)( - 3 + i)$

$= ( - 3 - i - 3i - {i}^{2} ) + ( - 3 + i + 3i - {i}^{2} )$

$= ( - 3 - 4i - {i}^{2} ) + ( - 3 + 4i - {i}^{2} )$

$= - 3 - \cancel{4i }- {i}^{2} + - 3 + \cancel{4i} - {i}^{2}$

⏩ $\text{Putting}\: {i}^{2} = - 1$

$= - 3 - ( - 1) - 3 - ( - 1)$

$= - 6 + 2$

$\Large \purple{ \bf= - 4}$

$\large\blue\dashrightarrow\underline{\underline{\sf  \red{The  \: Answer \:  is  \: - 4}} }$

Therefore,

$\large \pmb{ \red\bigstar \: \: \orange{ \underbrace{ \underline{  \frak{Option \:  (\text d)  \: is \:  Correct }}}}}$

Answer 2

Answer:

$\large \dag$ Question(1):-

$\sf if \: z = 2 + i .then \: (z - 1)( \overset - {z} - 5) + ( \overset - z - 1)(z - 5) \: is \: what$

1. 2
2. 7
3. -1
4. -4.

$\sf\tt\large{\green {\underline {\underline{⚘\;Answer:}}}}$

$\sf\tt\large{\blue {\underline {\underline{⚘\;For \;your \;Question \;the \;value \;of \;given \;expression \;is \;-4:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;Given:}}}}$

• z = 2 + i

$\sf\tt\large{\red {\underline {\underline{⚘\;To find:}}}}$

• The value of the given expression in the problem.

$\sf\tt\large{\red {\underline {\underline{⚘\;Solution:}}}}$

• Kindly refer the above given attachment for better understanding.
• And for more information.

$\sf\tt\large{\purple {\underline {\underline{⚘\;Option \;D \;is \;the \;Answer \;to \;your \;Question:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;Hope \;it \;helps \;you:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;Thank \;you:}}}}$