Question

If $|z_1-1|<1,|z_2-2|<2,|z_3-3|<3$, then $|z_1+z_2+z_3|$

• A. Is less than $6$
• B. Is more than $3$
• C. Is less than $12$
• D. Lies between $6$ and $12$

Answer 1

Answer: (C)

Is less than $12$

$\because |z_1-1|<1$
$\Rightarrow \left|z_1\right|-\left|1\right|<1$ (By triangle inequality)
$\Rightarrow \left|z_1\right|<\mathrm{2...}\left(i\right)$
Similarly
$|z_2-2|<2$
$\Rightarrow \left|z_2\right|-\left|2\right|<2$
$\Rightarrow \left|z_2\right|<\mathrm{4...}\left(ii\right)$
and
$|z_3-3|<3$
$\Rightarrow \left|z_3\right|-\left|3\right|<3\Rightarrow \left|z_3\right|<\mathrm{6...}\left(iii\right)$
On adding Eqs. (i),(ii),(iii) we get
$\left|z_1\right|+\left|z_2\right|+\left|z_3\right|<2+4+6$
$\Rightarrow \left|z_1\right|+\left|z_2\right|+\left|z_3\right|<\mathrm{12...}\left(iv\right)$
By using triangular inequality
$|z_1+z_2+z_3|\le \left|z_1\right|+\left|z_2\right|+\left|z_3\right|$
$\Rightarrow |z_1+z_2+z_3|<12$ (From Eq. (iv))