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Jee Main 2024
31-01-2024 (S2)
Question
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-
Select the correct option:
A
$30 \sqrt{3}$
B
75
C
$15 \sqrt{15}$
D
$25 \sqrt{3}$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
z 1 + z 2 = 5 z 1 3 + z 2 3 = 20 + 15 i z 13 + z 23 = ( z 1 + z 2 ) 3 − 3 z 1 z 2 ( z 1 + z 2 ) z 1 3 + z 2 3 = 125 − 3 z 1 ⋅ z 2 ( 5 ) ⇒ 20 + 15 i = 125 − 15 z 1 z 2 ⇒ 3 z 1 z 2 = 25 − 4 − 3 i ⇒ 3 z 1 z 2 = 21 − 3 i ⇒ z 1 ⋅ z 2 = 27 = − 25 i ⇒ z 1 + z 2 ⇒ z 1 2 + z 2 2 = 25 − 2 ( 7 − i ) ⇒ 11 + 2 i z 1 2 + z 2 2 2 = 121 − 4 + 44 i ⇒ z 1 4 + z 2 4 + 2 ( 7 − i ) 2 = 117 + 44 i ⇒ z 1 4 + z 2 4 = 117 + 44 i − 2 ( 49 − 1 − 14 i ) ⇒∣ z 1 4 + z 2 4 = 75
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