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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Complex Number

NDA 2023 · Mathematics · Complex Number

If ω is a non-real cube root of 1 , then what is the value of...

If ω\omega is a non-real cube root of 1 , then what is the value of ∣1−ωω+ω2∣?\left|\frac{1-\omega}{\omega+\omega^2}\right| ?

Options

  1. A3\sqrt{3}
  2. B2\sqrt{2}
  3. C11
  4. D43\frac{4}{\sqrt{3}}

Correct answer

A. 3\sqrt{3}

Step-by-step solution

∣1−ww+w2∣=∣1−w−1∣=∣w−1∣[∵1+w+w2=0]=∣−1+3i2−1∣=∣−1+3i−22∣=∣−32+32i∣=(−32)2+(32)2=94+34=3\left|\frac{1-w}{w+w^2}\right|=\left|\frac{1-w}{-1}\right|=|w-1| \quad\left[\because 1+w+w^2=0\right]=\left|\frac{-1+\sqrt{3} i}{2}-1\right|=\left|\frac{-1+\sqrt{3} i-2}{2}\right|=\left|\frac{-3}{2}+\frac{\sqrt{3}}{2} i\right|=\sqrt{\left(\frac{-3}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}=\sqrt{\frac{9}{4}+\frac{3}{4}}=\sqrt{3}
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