JEE Main 2024 — 31 January, Shift 2 — Question 3

$\mathrm{z}_{1}+\mathrm{z}_{2}=5$

$z_{1}^{3}+z_{2}^{3}=20+15 i$

$\mathrm{z}_{1}^{3}+\mathrm{z}_{2}^{3}=\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)^{3}-3 \mathrm{z}_{1} \mathrm{z}_{2}\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)$

$\mathrm{z}_{1}^{3}+\mathrm{z}_{2}^{3}=125-3 \mathrm{z}_{1} \cdot \mathrm{z}_{2}(5)$

$\Rightarrow 20+15 \mathrm{i}=125-15 \mathrm{z}_{1} \mathrm{z}_{2}$

$\Rightarrow 3 \mathrm{z}_{1} \mathrm{z}_{2}=25-4-3 \mathrm{i}$

$\Rightarrow 3 \mathrm{z}_{1} \mathrm{z}_{2}=21-3 \mathrm{i}$

$\Rightarrow \mathrm{z}_{1} \cdot \mathrm{z}_{2}=7-\mathrm{i}$

$\Rightarrow\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)^{2}=25$

$\Rightarrow \mathrm{z}_{1}^{2}+\mathrm{z}_{2}^{2}=25-2(7-\mathrm{i})$

$\Rightarrow 11+2 \mathrm{i}$

$\left(\mathrm{z}_{1}^{2}+\mathrm{z}_{2}^{2}\right)^{2}=121-4+44 \mathrm{i}$

$\Rightarrow \mathrm{z}_{1}^{4}+\mathrm{z}_{2}^{4}+2(7-\mathrm{i})^{2}=117+44 \mathrm{i}$

$\Rightarrow \mathrm{z}_{1}^{4}+\mathrm{z}_{2}^{4}=117+44 \mathrm{i}-2(49-1-14 \mathrm{i})$

$\Rightarrow\left|z_{1}^{4}+z_{2}^{4}\right|=75$