UPSC NDA I 2025 Mathematics Paper

1. The amplitude (or argument) of a complex number z=r(cos‌θ+isin‌θ) is given by amp(z)=θ.
2. The conjugate of z , denoted as Z, has an amplitude amp(z)=−θ, because conjugating a complex number reflects it across the real axis in the Argand plane.
Formula Used:
amp(z)+amp(z)=θ+(−θ)=0.
Calculation:
‌z=r(cos‌θ+isin‌θ)
‌z=r(cos‌θ−isin‌θ)
‌amp(z)+amp(z)=θ+(−θ)=0
Conclusion:
∴amp(z)+amp(z)=0.