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BITSAT 2022
Exam
BITSAT
Year
2022
Subject
Mathematics
Chapter
Complex Number

BITSAT 2022 · Mathematics · Complex Number

If |w|=2 , then the set of points z=w-(1)/(w) is contained in or...

If ∣w∣=2|w|=2, then the set of points z=w−1wz=w-\frac{1}{w} is contained in or equal to the set of points zz satisfying

Options

  1. AIm⁡(z)=0\operatorname{Im}(z)=0
  2. B∣Im⁡(z)∣≤1|\operatorname{Im}(z)| \leq 1
  3. C∣Re⁡(z)∣≤2|\operatorname{Re}(z)| \leq 2
  4. D∣z∣≤3|z| \leq 3

Correct answer

D. ∣z∣≤3|z| \leq 3

Step-by-step solution

We have, ∣w∣=2⇒w=2(cos⁡θ+isin⁡θ)∴z=w−1w=2(cos⁡θ+isin⁡θ)−12(cos⁡θ−isin⁡θ)⇒z=32cos⁡θ+52isin⁡θ⇒∣z∣=94+254=172<3|w|=2 \Rightarrow w=2(\cos \theta+i \sin \theta)\therefore \quad z=w-\frac{1}{w}=2(\cos \theta+i \sin \theta)-\frac{1}{2}(\cos \theta-i \sin \theta)\Rightarrow \quad z=\frac{3}{2} \cos \theta+\frac{5}{2} i \sin \theta\Rightarrow \quad|z|=\sqrt{\frac{9}{4}+\frac{25}{4}}=\frac{\sqrt{17}}{2} < 3 Hence, option (d) is correct.
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