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Complex Numbers Part 2

Section: Mathematics

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Question : 99 of 106

Marks: +1, -0

The real part of the complex number

\;\frac{(1+2 i)^8 \cdot (1-2 i)^2}{(3+2 i) \cdot \overline{(4-6 i)}}

[30-Jun-2022-Shift-1]

$\;\frac{500}{13}$
$\;\frac{110}{13}$
$\;\frac{55}{6}$
$\;\frac{550}{13}$

Solution: 👈: Video Solution

Given,

\;\frac{(1+2 i)^8 \cdot (1-2 i)^2}{(3+2 i) \cdot \overline{(4-6 i)}}

=\;\frac{(1+2 i)^2(1-2 i)^2(1+2 i)^6}{(3+2 i)(4+6 i)}

=\;\frac{(1-4 i^2)^2(1+2 i)^6}{12+18 i+8 i+12 i^2}

=\;\frac{(1+5)^2 [(1+2 i)^2]^3}{12+26 i-12}

=\;\frac{25 (1+4 i^2+4 i)^3}{26 i}

=\;\frac{25(1-4+4 i)^3}{26 i}

=\;\frac{25(-3+4 i)^3}{26 i}

=\;\frac{25}{26 i} [(-3)^3+(4 i)^3+3 \cdot (-3)^2 \cdot 4 i+3(-3) \cdot (4 i)^2]

=\;\frac{25}{26 i}(-27-64 i+108 i+144)

=\;\frac{25}{26 i}(117+44 i)

=\;\frac{25 i}{26 i^2}(117+44 i)

=\;\frac{25 i}{-26}(117+44 i)

=\;\frac{25 \times 117 i}{-26}-\;\frac{25 \times 44 i^2}{26}

=\;\frac{25 \times 117 i}{-26}+\;\frac{22 \times 25}{13}

=\;\frac{25 \times 117 i}{-26}+\;\frac{550}{13}

\therefore\;\text{ Real part }\;=\;\frac{550}{13}

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