Modulus of a complex number — A/L Combined Mathematics

How do I practise Modulus of a complex number?

1 question with a worked solution

Check before you answer: Triangle Inequality Proof: Proving |z₁ + z₂|² = (z₁ + z₂)(z̄₁ + z̄₂) = |z₁|² + |z₂|² + 2 Re(z₁ z̄₂) ≤ |z₁|² + |z₂|² + 2|z₁||z₂| = (|z₁| + |z₂|)².

1. What is the modulus of the complex number z = (3 + 4i) / (1 + i)?
   1. A) 25 / 2
   2. B) 5 / √2 (or 5√2 / 2)
   3. C) 7 / 2
   4. D) 5 / 2
   Check my answer

   Correct answer: B) 5 / √2 (or 5√2 / 2)

   Using the modulus quotient property |z₁ / z₂| = |z₁| / |z₂|: |3 + 4i| = √(3² + 4²) = √25 = 5. |1 + i| = √(1² + 1²) = √2. Thus, |z| = 5 / √2 = (5√2) / 2.

   Show worked solution

   Step 1: |3 + 4i| = √(9 + 16) = 5. Step 2: |1 + i| = √(1 + 1) = √2. Step 3: |z| = 5 / √2. Answer: 5 / √2 (or 5√2 / 2).
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Modulus of a complex number fit in the syllabus?

Year 2 · Term 1