The GATE Grind

GATE 2023 CH – Question 13

Engineering Mathematics · Complex Variables: Complex numbers and polar form · 1 mark · Multiple choice

Given that $F = \frac{|z_1 + z_2|}{|z_1| + |z_2|}$, where $z_1 = 2 + 3i$ and $z_2 = -2 + 3i$ with $i = \sqrt{-1}$, which one of the following options is CORRECT?

  1. $F < 0$
  2. $F < 1$
  3. $F > 1$
  4. $F = 1$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $F < 1$

Explanation

$z_1 + z_2 = 6i$, so $|z_1 + z_2| = 6$. $|z_1| = |z_2| = \sqrt{13}$, so the denominator is $2\sqrt{13} = 7.21$. $F = 0.83$, which is less than 1 (as it must be by the triangle inequality when the numbers do not have the same argument).