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NDA 2016 Maths · Mathematics · Complex Numbers and Quadratic Equations · easy

Let z=x+iy be a non-zero complex number. 1. arg(bar z)=arg(z). 2. |z|=x+y.

Single correct option

Let z=x+iyz=x+iy be a non-zero complex number. Consider the following statements:
1. arg⁡(zˉ)=arg⁡(z)\arg(\bar z)=\arg(z).
2. ∣z∣=x+y|z|=x+y.
Which of the statements given above is/are correct?
  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: (D) Neither 1 nor 2

The conjugate reflects zz in the real axis, so arg⁡(zˉ)=−arg⁡(z)\arg(\bar z)=-\arg(z) (mod 2π2\pi); statement 1 holds only for real zz. The modulus is ∣z∣=x2+y2|z|=\sqrt{x^2+y^2}, which differs from x+yx+y (take z=3+4iz=3+4i: 5≠75\ne7). Both statements are false.
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Original question written in the NDA 2016 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