Question

If \(z=\frac{(3+i)(7-i)^2}{3-i}\), then the value of |z| is equal to

• A. 48
• B. √50
• C. 50
• D. √500
• E. √48

Answer 1

The Correct Option is C

We are given the complex number $z = \frac{(3 + i)(7 - i)}{3 - i}$. To find the magnitude $|z|$, we follow these steps: 1. Simplify the expression for $z$: \[ z = \frac{(3 + i)(7 - i)}{3 - i} \] 2. Multiply the numerator $(3 + i)(7 - i)$: \[ (3 + i)(7 - i) = 3 \cdot 7 - 3 \cdot i + i \cdot 7 - i \cdot i = 21 - 3i + 7i + 1 = 22 + 4i \] 3. Divide by the denominator $(3 - i)$. Multiply both the numerator and denominator by the complex conjugate of the denominator, $3 + i$: \[ \frac{22 + 4i}{3 - i} \cdot \frac{3 + i}{3 + i} = \frac{(22 + 4i)(3 + i)}{(3 - i)(3 + i)} \] 4. Simplify the denominator: \[ (3 - i)(3 + i) = 3^2 - i^2 = 9 + 1 = 10 \] 5. Simplify the numerator: \[ (22 + 4i)(3 + i) = 22 \cdot 3 + 22 \cdot i + 4i \cdot 3 + 4i \cdot i = 66 + 22i + 12i + 4(-1) = 66 + 34i - 4 = 62 + 34i \] 6. Therefore, we have: \[ z = \frac{62 + 34i}{10} = 6.2 + 3.4i \] 7. Find the magnitude of $z$, which is $|z|$: \[ |z| = \sqrt{(6.2)^2 + (3.4)^2} = \sqrt{38.44 + 11.56} = \sqrt{50} \]

The correct option is (C) : \(\sqrt {50}\)

Answer 2

We are given the complex number \(z = \frac{(3+i)(7-i)^2}{3-i}\). We want to find the value of |z|.

We can use the property that \( |z| = \frac{|z_1||z_2|^2}{|z_3|} \), where \(z_1 = 3+i\), \(z_2 = 7-i\), and \(z_3 = 3-i\).

First, let's find the magnitudes:

• \(|3+i| = \sqrt{3^2 + 1^2} = \sqrt{9+1} = \sqrt{10}\)
• \(|7-i| = \sqrt{7^2 + (-1)^2} = \sqrt{49+1} = \sqrt{50}\)
• \(|3-i| = \sqrt{3^2 + (-1)^2} = \sqrt{9+1} = \sqrt{10}\)

Now, we can find |z|:

\(|z| = \frac{|3+i||7-i|^2}{|3-i|} = \frac{\sqrt{10} (\sqrt{50})^2}{\sqrt{10}} = \frac{\sqrt{10} (50)}{\sqrt{10}} = 50\)

Therefore, the value of |z| is equal to 50.