Question

If $z$ is a complex number such that the minimum value of $|z|+|z-1|+|2z-7|$ is $\lambda$ and if $y=2[x]+3=3[x-\lambda ]$, then find the value of $\left(\frac{[x+y]}{11}\right)$.
[Note: $[k]$ denotes greatest integer function less than or equal to $k$.

Answer 1

Answer: 6

$|z|+|z-1|+|2z-7|$
$|z|+|z-1|+|7-2z|\ge |2z-1+7-2z|=6$
$\ge 6$
$|z|+|z-1|+|2z-7|\ge 6$
$\therefore \lambda =6$
$y=2[x]+3=3[x-6]$
$2[x]+3=3[x]-18$
$[x]=21$
$\therefore y=2[x]+3=42+3=45$
$[x+y]=[x]+y=21+45=66$
$\therefore \left(\frac{[x+y]}{11}\right)=6$