Question

For a complex number $Z$ , if $\left|Z - i\right|\leq 2$ and $Z_{1}=5+3i$ , then the maximum value of $\left|i Z + Z_{1}\right|$ is (where, $i^{2}=-1$ )

• A. $7+\sqrt{13}$
• B. $7+\sqrt{12}$
• C. $7$
• D. $\sqrt{34}-2$

Answer 1

Answer: (C)

$\left|i Z + Z_{1}\right|=\left|i \left(Z - i\right) + \left(Z_{1} - 1\right)\right|$
$\leq \left|i \left(Z - i\right)\right|+\left|Z_{1} - 1\right|$
$\leq \left|i\right|\left|Z - i\right|+\left|4 + 3 i\right|$
$\leq 2+\sqrt{4^{2} + 3^{2}}=7$