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Reviewed CSCA Mathematics question · Hard

Let z = 2 + 3i and w = 1 - i. What is the modulus of z * w?

  1. sqrt(26)
  2. sqrt(13)
  3. 5
  4. sqrt(10)
Show the answer and explanation

Correct answer

A. sqrt(26)

Principle or equation

The modulus of a product of complex numbers equals the product of their moduli: |z * w| = |z| * |w|. For a complex number a + bi, the modulus is sqrt(a^2 + b^2).

Why this answer is correct

First, compute |z| = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13). Then |w| = sqrt(1^2 + (-1)^2) = sqrt(2). Therefore, |z * w| = sqrt(13) * sqrt(2) = sqrt(26).

Example

For z = 1 + i and w = 1 - i, |z| = sqrt(2), |w| = sqrt(2), so |z*w| = 2.

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