Question

If \(z |z| = 24 + 7i\), where \(z\) is a complex number, then the value of \(|z|\) is equal to

• A. 5
• B. 7
• C. 12
• D. 15
• E. 25

Hint:

Recognizing Pythagorean triplets can save calculation time. \(7, 24, 25\) is a standard triplet, so \(\sqrt{24^2 + 7^2} = 25\) immediately.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are given an equation involving a complex number \(z\) and its modulus \(|z|\).
Taking the modulus of both sides will convert the complex equation into a real equation in terms of \(|z|\).

Step 2: Key Formula or Approach:
1. Property of modulus: \(|z_1 z_2| = |z_1| \cdot |z_2|\).
2. Modulus of a constant real number is the number itself (if positive). So, \(||z|| = |z|\).
3. Magnitude of \(a + bi\) is \(\sqrt{a^2 + b^2}\).

Step 3: Detailed Explanation:
Given: \(z |z| = 24 + 7i\).
Take modulus on both sides:
\[ |z \cdot |z|| = |24 + 7i| \]
Using modulus properties:
\[ |z| \cdot ||z|| = \sqrt{24^2 + 7^2} \]
\[ |z| \cdot |z| = \sqrt{576 + 49} \]
\[ |z|^2 = \sqrt{625} \]
\[ |z|^2 = 25 \]
Solving for \(|z|\) (noting that modulus must be positive):
\[ |z| = 5 \]

Step 4: Final Answer:
The value of \(|z|\) is 5.