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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
The value of $\left|\frac{1+i \sqrt{3}}{\left(1+\frac{1}{i+1}\right)^2}\right|$ is
  • A
    20
  • B
    9
  • C
    $\frac{5}{4}$
  • ✓
    $\frac{4}{5}$
✓

Answer

Correct option: D.
$\frac{4}{5}$
(D)
$|1+ i \sqrt{3}|=\sqrt{1+3}=2$
$1+\frac{1}{i+1}=1+\frac{i-1}{i^2-1}=1+\frac{(i-1)}{-2}=\frac{3}{2}-\frac{i}{2}$
$\therefore\left|1+\frac{1}{i+1}\right|=\sqrt{\left(\frac{3}{2}\right)^2+\left(-\frac{1}{2}\right)^2}=\sqrt{\frac{9}{4}+\frac{1}{4}}=\sqrt{\frac{10}{4}}$
$\therefore\left|\frac{1+i \sqrt{3}}{\left(1+\frac{1}{i+1}\right)^2}\right|=\frac{2}{\frac{10}{4}}=\frac{4}{5}$

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