Question

If $z$ is a complex number such that the minimum value of $|z|+|z-1|+|2 z-7|$ 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

$ |z|+|z-1|+|2 z-7| $
$|z|+|z-1|+|7-2 z| \geq|2 z-1+7-2 z|=6 $
$\geq 6$
$|z|+|z-1|+|2 z-7| \geq 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 $