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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Complex Number

NDA 2024 · Mathematics · Complex Number

Paragraph: Let Z_1 and Z_2 be any two complex numbers such that...

Paragraph: Let Z1Z_1 and Z2Z_2 be any two complex numbers such that Z12+Z22+Z1Z2=0Z_1^2+Z_2^2+Z_1 Z_2=0
Question: What is the value of 12+Re⁡(Z1Z2)\frac{1}{2}+\operatorname{Re} \left(\frac{\mathrm{Z}_1}{\mathrm{Z}_2}\right)?

Options

  1. A−1-1
  2. B00
  3. C11
  4. D22

Correct answer

B. 00

Step-by-step solution

Given that, Z12+Z22+Z1Z2=0⇒Z1=w and  Z_1^2+Z_2^2+Z_1 Z_2=0 \Rightarrow Z_1=w \text{\text{ and } } Z_2=w^2 Now, 12+Re⁡(Z1Z2)=12+Re⁡(ωω2)=12+Re⁡(ω2)=12+Re⁡(−12−32i)=12+(−12)=0=0\begin{aligned} & \frac{1}{2}+\operatorname{Re}\left(\frac{Z_1}{Z_2}\right)=\frac{1}{2}+\operatorname{Re}\left(\frac{\omega}{\omega^2}\right)=\frac{1}{2}+\operatorname{Re}\left(\omega^2\right) \\ & =\frac{1}{2}+\operatorname{Re}\left(-\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right)=\frac{1}{2}+\left(-\frac{1}{2}\right)=0 \\ & =0 \end{aligned}
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