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NDA2026MathematicsComplex NumberActual

Let Z₁ and Z₂ be complex numbers such that 3Z₁ 4Z₂ is purely imaginary. What is | Z₁+Z₂ Z₁-Z₂ | equal to ?

Options

  1. A2
  2. B3 2
  3. C5 4
  4. D1

Correct answer

D. 1

Step-by-step solution

Given 3Z₁ 4Z₂ is purely imaginary. Let 3Z₁ 4Z₂ = i k , where k R . Z₁ Z₂ = 4 3 i k Let 4 3 k = y , so Z₁ Z₂ = i y , where y R . We need to evaluate | Z₁+Z₂ Z₁-Z₂ | . Dividing the numerator and the denominator by Z₂ , we get: | Z₁ Z₂ + 1 Z₁ Z₂ - 1 | Substituting Z₁ Z₂ = i y : | i y + 1 i y - 1 | = |1 + i y| |-1 + i y| Since |1 + i y| = 1^2 + y^2 and |-1 + i y| = (-1)^2 + y^2 = 1 + y^2 , we have: 1 + y^2 1 + y^2 = 1 Answer: 1

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