Which of the following statement(s) is / are correct ?

Options

A. `[Co(NH_(3))_(6)]^(3+), [Co(CN)_(6)]^(3-) and [Co(NO_(2))_(6)]^(3-)` are diamagnetic involving `d^(2)sp^(3)` hybridisation.
B. `[Zn(NH_(3))_(4)]^(2+),[FeCl_(4)]^(-) and [Ni(CO)_(4)]` are diamagnetic involving `sp^(3)` hybridisation.
C. The magnetic moment of `[Fe(H_(2)O)_(6)]^(3+)` is 5.92 BM and that of `[Fe(CN)_(6)]^(3-) ` is 1.73
D. The magnetic moment of `K_(4)[MnF_(6)] and K_(3)[FeF_(6)]` are same .

Explanation

To determine which of the given statements regarding coordination compounds are correct, we will analyze each statement step by step.

### Step 1: Analyze the First Statement
The first statement claims that the complexes \( \text{Co(NH}_3\text{)}_6^{3+} \), \( \text{Co(CN)}_6^{3-} \), and \( \text{Co(NH}_2\text{)}_2^{3-} \) are all diamagnetic and involve the hybridization \( d^2sp^3 \).

1. **Determine Oxidation State**:
- All complexes have cobalt in the +3 oxidation state.

2. **Electron Configuration**:
- Cobalt in +3 state has a \( d^6 \) configuration.

3. **Hybridization**:
- For \( d^6 \) configuration, the hybridization can be \( d^2sp^3 \) in an octahedral field.

4. **Magnetic Properties**:
- Since \( d^6 \) in a strong field ligand (like CN) leads to pairing of electrons, these complexes are indeed diamagnetic.

**Conclusion**: The first statement is **correct**.

### Step 2: Analyze the Second Statement
The second statement claims that the complexes \( \text{Zn(NH}_3\text{)}_4^{2+} \), \( \text{FeCl}_4^{-} \), and \( \text{Ni(CO)}_4 \) are all diamagnetic and involve \( sp^3 \) hybridization.

1. **Zinc Complex**:
- \( \text{Zn(NH}_3\text{)}_4^{2+} \) has a \( d^{10} \) configuration and is indeed diamagnetic with \( sp^3 \) hybridization.

2. **Iron Complex**:
- \( \text{FeCl}_4^{-} \) has a \( d^5 \) configuration and is **paramagnetic** due to the presence of 5 unpaired electrons. This makes the statement incorrect.

3. **Nickel Complex**:
- \( \text{Ni(CO)}_4 \) has a \( d^{10} \) configuration and is diamagnetic with \( sp^3 \) hybridization.

**Conclusion**: The second statement is **incorrect** due to the iron complex being paramagnetic.

### Step 3: Analyze the Third Statement
The third statement claims that the magnetic moment of \( \text{Fe(H}_2\text{O)}_6^{3+} \) is 5.92 Bohr magnetons and that of \( \text{Fe(CN)}_6^{3-} \) is 1.73 Bohr magnetons.

1. **Iron Aqua Complex**:
- \( \text{Fe(H}_2\text{O)}_6^{3+} \) has 5 unpaired electrons, leading to a magnetic moment of:
\[ \mu = \sqrt{n(n+2)} = \sqrt{5(5+2)} = \sqrt{35} \approx 5.92 \text{ Bohr magnetons} \]

2. **Iron Cyanide Complex**:
- \( \text{Fe(CN)}_6^{3-} \) has 1 unpaired electron, leading to a magnetic moment of:
\[ \mu = \sqrt{1(1+2)} = \sqrt{3} \approx 1.73 \text{ Bohr magnetons} \]

**Conclusion**: The third statement is **correct**.

### Step 4: Analyze the Fourth Statement
The fourth statement claims that the magnetic moments of \( \text{K}_3\text{FeF}_6 \) and \( \text{K}_4\text{MnF}_6 \) are the same.

1. **Iron Fluoride Complex**:
- \( \text{K}_3\text{FeF}_6 \) has a \( d^5 \) configuration and 5 unpaired electrons, leading to a magnetic moment of:
\[ \mu = \sqrt{5(5+2)} = \sqrt{35} \approx 5.92 \text{ Bohr magnetons} \]

2. **Manganese Fluoride Complex**:
- \( \text{K}_4\text{MnF}_6 \) has a \( d^3 \) configuration and 3 unpaired electrons, leading to a magnetic moment of:
\[ \mu = \sqrt{3(3+2)} = \sqrt{15} \approx 3.87 \text{ Bohr magnetons} \]

**Conclusion**: The fourth statement is **incorrect** since the magnetic moments are not the same.

### Final Conclusion
- The correct statements are:
- **First Statement**: Correct
- **Second Statement**: Incorrect
- **Third Statement**: Correct
- **Fourth Statement**: Incorrect

### Summary of Correct Statements
- The first and third statements are correct, while the second and fourth statements are incorrect.