- Home
- Standard 11
- Mathematics
4-1.Complex numbers
hard
જો $\mathrm{z}_1$ અને $\mathrm{z}_2$ બે સંકર સંખ્યા માટે $\mathrm{z}_1+\mathrm{z}_2=5$ અને $z_1^3+z_2^3=20+15 i$ છે. તો $\left|z_1^4+z_2^4\right|=$__________.
A
$30 \sqrt{3}$
B
$75$
C
$15 \sqrt{15}$
D
$25 \sqrt{3}$
(JEE MAIN-2024)
Solution
$z_1+z_2=5$
$z_1^3+z_2^3=20+15 i$
$z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right)$
$z_1^3+z_2^3=125-3 z_1 \cdot z_2(5)$
$\Rightarrow 20+15 i=125-15 z_1 z_2$
$\Rightarrow 3 z_1 z_2=25-4-3 i$
$\Rightarrow 3 z_1 z_2=21-3 i$
$\Rightarrow z_1 \cdot z_2=7-i$
$\Rightarrow\left(z_1+z_2\right)^2=25$
$\Rightarrow z_1^2+z_2^2=25-2(7-i)$
$\Rightarrow 11+2 i$
$\left(z_1^2+z_2^2\right)^2=121-4+44 i$
$\Rightarrow z_1^4+z_2^4+2(7-i)^2=117+44 i$
$\Rightarrow z_1^4+z_2^4=117+44 i-2(49-1-14 i)$
$\Rightarrow\left|z_1^4+z_2^4\right|=75$
Standard 11
Mathematics
