If $Z_1=4 \mathrm{i}^{40}-5 \mathrm{i}^{35}+6 \mathrm{i}^{17}+2, \mathrm{Z}_2=-1+\mathrm{i}$, where…

If $Z_1=4 \mathrm{i}^{40}-5 \mathrm{i}^{35}+6 \mathrm{i}^{17}+2, \mathrm{Z}_2=-1+\mathrm{i}$, where $\mathrm{i}=\sqrt{-1}$, then $\left|\mathrm{Z}_1+\mathrm{Z}_2\right|=$
  1. 5
  2. 13
  3. 12
  4. 15

Solution

$\begin{aligned} & \mathrm{Z}_1=4 \mathrm{i}^{40}-5 \mathrm{i}^{35}+6 \mathrm{i}^{17}+2 \\ & \mathrm{Z}_1=6+11 \mathrm{i} \\ & \mathrm{Z}_2=-1+\mathrm{i} \\ \therefore \quad \mathrm{Z}_1 & +\mathrm{Z}_2=6+11 \mathrm{i}-1+\mathrm{i} \\ & =5+12 \mathrm{i} \\ \therefore \quad & \left|\mathrm{Z}_1+\mathrm{Z}_2\right|=\sqrt{(5)^2+(12)^2}=\sqrt{169}=13\end{aligned}$

Asked in: MHT CET 2023 (09 May Shift 1)

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