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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Complex Number

NDA 2025 · Mathematics · Complex Number

If z ≠ 0 is a complex number, then what is amp(z) + amp( z ) equal...

If z≠0z \neq 0 is a complex number, then what is amp(z)+amp(zˉ)\text{amp}(z) + \text{amp}(\bar{z}) equal to?

Options

  1. A00
  2. Bπ2\dfrac{\pi}{2}
  3. Cπ\pi
  4. D2π2\pi

Correct answer

A. 00

Step-by-step solution

Let z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta) where r>0r > 0 and θ=amp(z)\theta = \text{amp}(z).

The conjugate of zz is given by zˉ=r(cos⁡θ−isin⁡θ)\bar{z} = r(\cos \theta - i\sin \theta).

This can be rewritten as zˉ=r(cos⁡(−θ)+isin⁡(−θ))\bar{z} = r(\cos(-\theta) + i\sin(-\theta)).

Thus, the amplitude of zˉ\bar{z} is amp(zˉ)=−θ\text{amp}(\bar{z}) = -\theta.

Therefore, amp(z)+amp(zˉ)=θ+(−θ)=0\text{amp}(z) + \text{amp}(\bar{z}) = \theta + (-\theta) = 0.

Answer: 00
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