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BITSAT 2021
Exam
BITSAT
Year
2021
Subject
Mathematics
Chapter
Complex Number

BITSAT 2021 · Mathematics · Complex Number

If z_ 1 = 2 [ (π)/(4)+i (π)/(4)] and z_ 2 = 3 [ (π)/(3)+ i (π)/(3)]...

\text{ If } z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right] \text{andand} z_{2}=\sqrt{3} \left[\cos \frac{\pi}{3}+isin⁡i \sin\frac{\pi}{3}\right]then∣z1z2∣isequaltothen \left| z_{1} z_{2}\right|is equal to\sqrt{m}. Value ofmm is

Options

  1. A6
  2. B3
  3. C2
  4. D5

Correct answer

A. 6

Step-by-step solution

z1=2[cos⁡π4+isin⁡π4]=2[12+i12]=1+i∣z1∣=2z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right]=\sqrt{2}\left[\frac{1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}\right]=1+i\left|z_{1}\right|=\sqrt{2} and z2=3[cos⁡π3+isin⁡π3]=3[12+i32]∣z2∣=34+94=3∣z1z2∣=∣z1∣∣z2∣=2⋅3=6z_{2}=\sqrt{3}\left[\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right]=\sqrt{3}\left[\frac{1}{2}+i \frac{\sqrt{3}}{2}\right]\left|z_{2}\right|=\sqrt{\frac{3}{4}+\frac{9}{4}}=\sqrt{3}\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|=\sqrt{2} \cdot \sqrt{3}=6
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