If z  is a complex number such that  $\left|z\right|\ge 1,$ then the minimum value of $|z+\frac{1}{2}(3+4i)|$  is:

Given  $\left|z\right|\ge 1$
Consider graph of  z (argand planes)
$z\in$ outside part of circle whose radius is 1 & centre is (0, 0)
 $\frac{-3}{2}-\frac{4i}{z}\in z[\because |-\frac{3}{2}-\frac{4i}{2}|=\frac{5}{2}\ge 1]$
 
Then $|z+\frac{1}{2}(3+4i)|$  minimum value is zero for $=\frac{-3}{2}-\frac{4i}{2}$ 
$\therefore {|z+\frac{1}{2}(3+4i)|}_{\min }=0$