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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-
$30 \sqrt{3}$
$75$
$15 \sqrt{15}$
$25 \sqrt{3}$
Solution
$z_1+z_2=5$
$z_1^3+z_2^3=20+15 i$
$z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right)$
$z_1^3+z_2^3=125-3 z_1 \cdot z_2(5)$
$\Rightarrow 20+15 i=125-15 z_1 z_2$
$\Rightarrow 3 z_1 z_2=25-4-3 i$
$\Rightarrow 3 z_1 z_2=21-3 i$
$\Rightarrow z_1 \cdot z_2=7-i$
$\Rightarrow\left(z_1+z_2\right)^2=25$
$\Rightarrow z_1^2+z_2^2=25-2(7-i)$
$\Rightarrow 11+2 i$
$\left(z_1^2+z_2^2\right)^2=121-4+44 i$
$\Rightarrow z_1^4+z_2^4+2(7-i)^2=117+44 i$
$\Rightarrow z_1^4+z_2^4=117+44 i-2(49-1-14 i)$
$\Rightarrow\left|z_1^4+z_2^4\right|=75$
Similar Questions
Let $z_k=\cos \left(\frac{2 k \pi}{10}\right)+ i \sin \left(\frac{2 k \pi}{10}\right) ; k =1,2, \ldots 9$.
| List $I$ | List $II$ |
| $P.$ For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j=1$ | $1.$ True |
| $Q.$ There exists a $k \in\{1,2, \ldots ., 9\}$ such that $z_{1 .} . z=z_k$ has no solution $z$ in the set of complex numbers. | $2.$ False |
| $R.$ $\frac{\left|1-z_1\right|\left|1-z_2\right| \ldots . .\left|1-z_9\right|}{10}$ equals | $3.$ $1$ |
| $S.$ $1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)$ equals | $4.$ $2$ |
Codes: $ \quad P \quad Q \quad R \quad S$
