If z = 2 + 3 i , then z 5 + z ¯ 5 is equal to:

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Question: Answered & Verified by Expert

If

z = 2 + 3 i

, then

z^{5} + {(\bar{z})}^{5}

is equal to:

MathematicsComplex NumberJEE MainJEE Main 2022 (29 Jul Shift 1)

Options:

A $244$
B $224$
C $245$
D $265$

Solution:

1498 Upvotes Verified Answer

The correct answer is:

244

Given $z = 2 + 3 i$ ,
So, $z^{5} + {(\bar{z})}^{5} = {(2 + 3 i)}^{5} + {(2 - 3 i)}^{5}$
Applying binomial expansion we get,
$= 2 ({}^{5}C_{0} 2^{5} + {}^{5}C_{2} 2^{3} {(3 i)}^{2} + {}^{5}C_{4} 2^{1} {(3 i)}^{4})$
$= 2 (32 + 10 \times 8 (- 9) + 5 \times 2 \times 81) = 244$

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