If z is a complex number such that z ≤ 1 , then the minimum value of z + 1 2 3 + 4 i is:

Given: $\left|z+\frac{3+4i}{2}\right|; \left|z\right|\le 1$

Let, $z=x+iy$

And $\left|z\right|=1$ gives $x^2+y^2=1$, which represents a circle having centre at $\left(0,0\right)$ and radius as $1$ unit.

Now, $\left|z+\frac{3+4i}{2}\right|$ gives the distance of $z$ from $\left(-\frac{3}{2},-2\right)$

So, we need to find the minimum distance of $AP$.

$\Rightarrow AP=OP-OA$

$\Rightarrow AP=\sqrt{{\left(-\frac{3}{2}-0\right)}^2+{\left(0+2\right)}^2}-1$

$\Rightarrow AP=\sqrt{\frac{9}{4}+4}-1$

$\Rightarrow AP=\frac{5}{2}-1$

$\Rightarrow AP=\frac{3}{2}$