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NDA 2018 Maths · Mathematics · Complex Numbers and Quadratic Equations · medium

|z1+z2|≤|z1|+|z2| for all complex numbers z1,z2. 2. If z≠0 and z+1/z is real, then |z|=1.

Single correct option

Consider the following statements: 1. ∣z1+z2∣≤∣z1∣+∣z2∣|z_1+z_2|\le|z_1|+|z_2| for all complex numbers z1,z2z_1,z_2. 2. If z≠0z\ne0 and z+1zz+\dfrac1z is real, then ∣z∣=1|z|=1. 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
Show answer and solution

Answer: (A) 1 only

Statement 1 is the triangle inequality and always holds. For statement 2, take z=2z=2: z+1/z=2.5z+1/z=2.5 is real but ∣z∣=2≠1|z|=2\ne1. In fact z+1/zz+1/z is real when either zz is real or ∣z∣=1|z|=1, so the claim is false. Only statement 1 is correct.
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Original question written in the NDA 2018 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