Question

Let $z$ be an imaginary complex number satisfying $|z-1|=1.$ If $\alpha =2z,$ $\beta =2\alpha$ and $\gamma =2\beta ,$ then the value of $|z{|}^2+|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2+|z-2{|}^2+|\alpha -4{|}^2+|\beta -8{|}^2+|\gamma -16{|}^2$ is

• A. $100$
• B. $320$
• C. $340$
• D. $400$

Answer 1

The correct option is C $340$


As we can see,
$|z{|}^2+|z-2{|}^2=4\cdots \left(1\right)$
Similarly,
$|\alpha {|}^2+|\alpha -4{|}^2=16\cdots \left(2\right)$
$|\beta {|}^2+|\beta -8{|}^2=64\cdots \left(3\right)$
$|\gamma {|}^2+|\gamma -16{|}^2=256\cdots \left(4\right)$

Adding all the above equations $\left(1\right),\left(2\right),\left(3\right)$ and $\left(4\right)$,
$|z{|}^2+|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2+|z-2{|}^2+|\alpha -4{|}^2+|\beta -8{|}^2+|\gamma -16{|}^2=340$