Question

If $z=(3+4i{)}^6+(3-4i{)}^6,$ where $i=\sqrt{-1},$ then $Im\left(z\right)$ equals to

• A. $-6$
• B. $0$
• C. $6$
• D. None of these

Answer 1

Answer: (B)

$0$

$z={(3+4i)}^6+{(3-4i)}^6$

$={\left[5\left(e^{i\theta }\right)\right]}^6+{\left[5\left(e^{-i\theta }\right)\right]}^6$ [where $\theta ={tan}^{-1}\left(4/3\right)$]

$=5^6[e^{i6\theta }+e^{-i6\theta }]$

$=5^6\left[2Re\left(e^{i6\theta }\right)\right]$

$\therefore$ In $\left(z\right)=0$ [B]