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NDA 2025 Maths · Mathematics · Complex Numbers and Quadratic Equations · medium

/are correct for non-zero complex numbers z, z1, z2? 1. zbar z=|z|^2 2. arg(z1z2)=arg z1+arg z2 for the principal arguments, always.

Single correct option

Which of the following statements is/are correct for non-zero complex numbers zz, z1z_1, z2z_2?
1. zzˉ=∣z∣2z\bar z=|z|^2
2. arg⁡(z1z2)=arg⁡z1+arg⁡z2\arg(z_1z_2)=\arg z_1+\arg z_2 for the principal arguments, always.
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
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Answer: (A) 1 only

Statement 1 is the basic identity zzˉ=x2+y2=∣z∣2z\bar z=x^2+y^2=|z|^2. Statement 2 holds only modulo 2π2\pi: for z1=z2=−1z_1=z_2=-1 the principal arguments are π\pi each, their sum 2π2\pi is not principal, while arg⁡(1)=0\arg(1)=0.
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Original question written in the NDA 2025 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake