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JEE Main 202229 JulyMorning Shift
Exam
JEE Main
Year
2022
Date
29 July
Shift
Morning Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2022 · 29 July · Morning Shift · Mathematics · Complex Number

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

If z=2+3iz = 2 + 3i, then ${z}^{5} + {\left(\overset{¯}{z}\right)}^{5} is equal to:

Options

  1. A244244
  2. B224224
  3. C245245
  4. D265265

Correct answer

A. 244244

Step-by-step solution

Given z=2+3iz = 2 + 3i,

So, z5+(z¯)5=(2+3i)5+(2−3i)5{z}^{5} + {\left(\overset{¯}{z}\right)}^{5} = {\left(2 + 3i\right)}^{5} + {\left(2 - 3i\right)}^{5}

Applying binomial expansion we get,

=2(C0525+C2523(3i)2+C4521(3i)4)= 2\left(C0 5 {2}^{5} + C2 5 {2}^{3}{\left(3i\right)}^{2} + C4 5 {2}^{1}{\left(3i\right)}^{4}\right)

=2(32+10×8(−9)+5×2×81)=244= 2\left(32 + 10 \times 8\left(- 9\right) + 5 \times 2 \times 81\right) = 244

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