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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Complex Number

NDA 2023 · Mathematics · Complex Number

If z= 1+i 3 1-i 3 where i= -1 , then what is the argument of z ?

If z=1+i31−i3z=\frac{1+\mathrm{i} \sqrt{3}}{1-\mathrm{i} \sqrt{3}} where i=−1i=\sqrt{-1}, then what is the argument of zz ?

Options

  1. Aπ3\frac{\pi}{3}
  2. B2π3\frac{2 \pi}{3}
  3. C4π3\frac{4 \pi}{3}
  4. D5π6\frac{5 \pi}{6}

Correct answer

B. 2π3\frac{2 \pi}{3}

Step-by-step solution

z=1+3i1−3i=1+3i1−3i×1+3i1+3i=(1+3i)21+3z=1−3+23i4z=−12+32i=cos⁡2π3+isin⁡2π3arg⁡(z)=2π3z=\frac{1+\sqrt{3} i}{1-\sqrt{3} i}=\frac{1+\sqrt{3} i}{1-\sqrt{3} i} \times \frac{1+\sqrt{3} i}{1+\sqrt{3} i}=\frac{(1+\sqrt{3} i)^2}{1+3}z=\frac{1-3+2 \sqrt{3} i}{4}z=\frac{-1}{2}+\frac{\sqrt{3}}{2} i=\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\arg (z)=\frac{2 \pi}{3}
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