Complex Number Question 14
Question 14 - 2024 (31 Jan Shift 2)
Let $z _1$ and $z _2$ be two complex number such that $z _1+z _2=5$ and $z _1^{3}+z _2^{3}=20+15 i$. Then $\left|z _1^{4}+z _2^{4}\right|$ equals-
(1) $30 \sqrt{3}$
(2) 75
(3) $15 \sqrt{15}$
(4) $25 \sqrt{3}$
Show Answer
Answer (2)
Solution
$$ \begin{aligned} & z _1+z _2=5 \\ & z _1^{3}+z _2^{3}=20+15 i \\ & z _1^{3}+z _2^{3}=\left(z _1+z _2\right)^{3}-3 z _1 z _2\left(z _1+z _2\right) \\ & z _1^{3}+z _2^{3}=125-3 z _1 \cdot z _2(5) \\ & \Rightarrow 20+15 i=125-15 z _1 z _2 \\ & \Rightarrow 3 z _1 z _2=25-4-3 i \\ & \Rightarrow 3 z _1 z _2=21-3 i \\ & \Rightarrow z _1 \cdot z _2=7-i \\ & \Rightarrow\left(z _1+z _2\right)^{2}=25 \\ & \Rightarrow z _1^{2}+z _2^{2}=25-2(7-i) \\ & \Rightarrow 11+2 i \\ & \left(z _1^{2}+z _2^{2}\right)^{2}=121-4+44 i \\ & \Rightarrow z _1^{4}+z _2^{4}+2(7-i)^{2}=117+44 i \\ & \Rightarrow z _1^{4}+z _2^{4}=117+44 i-2(49-1-14 i) \\ & \Rightarrow\left|z _1^{4}+z _2^{4}\right|=75 \end{aligned} $$
