`z_(1) and z_(2)` are two complex numbers with different moduli, then `| sqrt(3) z_(1) + i sqrt(2) z_(2) |^(2) + | sqrt(2) z_(1) + i sqrt(3) z_(2) |^(2)` is equal to

Options

A. 0
B. `2 |z_(1)|^(2) + 3 |z_(2)|^(2)`
C. ` lt 5 (|z_(1)|^(2) + |z_(2)|^(2))`
D. ` gt 10 |z_(1) z_(2)|`

Answer

To solve the problem, we need to evaluate the expression:

\[ | \sqrt{3} z_1 + i \sqrt{2} z_2 |^2 + | \sqrt{2} z_1 + i \sqrt{3} z_2 |^2 \]

### Step 1: Apply the Modulus Squared Formula
Recall that for any complex number \( z \), the modulus squared is given by \( |z|^2 = z \cdot \overline{z} \). We will apply this property to both terms in our expression.

### Step 2: Calculate the First Term
Let \( z_1 \) and \( z_2 \) be complex numbers. The first term can be calculated as follows:

\[ | \sqrt{3} z_1 + i \sqrt{2} z_2 |^2 = (\sqrt{3} z_1 + i \sqrt{2} z_2)(\sqrt{3} \overline{z_1} - i \sqrt{2} \overline{z_2}) \]

Expanding this product:

\[ = 3 |z_1|^2 + 2 |z_2|^2 + \sqrt{6} i (z_1 \overline{z_2} - \overline{z_1} z_2) \]

### Step 3: Calculate the Second Term
Now, we calculate the second term:

\[ | \sqrt{2} z_1 + i \sqrt{3} z_2 |^2 = (\sqrt{2} z_1 + i \sqrt{3} z_2)(\sqrt{2} \overline{z_1} - i \sqrt{3} \overline{z_2}) \]

Expanding this product:

\[ = 2 |z_1|^2 + 3 |z_2|^2 + \sqrt{6} i (z_1 \overline{z_2} - \overline{z_1} z_2) \]

### Step 4: Combine the Two Terms
Now, we combine the results from Step 2 and Step 3:

\[ | \sqrt{3} z_1 + i \sqrt{2} z_2 |^2 + | \sqrt{2} z_1 + i \sqrt{3} z_2 |^2 = (3 |z_1|^2 + 2 |z_2|^2 + 2 |z_1|^2 + 3 |z_2|^2) + (\sqrt{6} i (z_1 \overline{z_2} - \overline{z_1} z_2) + \sqrt{6} i (z_1 \overline{z_2} - \overline{z_1} z_2)) \]

This simplifies to:

\[ = 5 |z_1|^2 + 5 |z_2|^2 + 0 \]

### Final Result
Thus, we have:

\[ | \sqrt{3} z_1 + i \sqrt{2} z_2 |^2 + | \sqrt{2} z_1 + i \sqrt{3} z_2 |^2 = 5 |z_1|^2 + 5 |z_2|^2 \]

### Conclusion
The final answer is:

\[ 5 (|z_1|^2 + |z_2|^2) \]