Which of the following reactions are disproportionation reactions?
`(A)" " Cu^(+)rarrCu^(2+)+Cu`
`(B)" "3 MnO_(4)+4H^(+)rarr2MnO_(4)^(-)+MnO_(2)+2H_(2)O`
`(C )" "2KMnO_(4)rarrK_(2)MnO_(4)+MnO_(2)+O_(2)`
`(D)" "2MnO_(4)^(-)+3Mn^(2+)+2H_(2)O rarr5MnO_(2)+4H^(+)`

Options

A. a,b
B. a,b,c
C. b,c,d
D. a,d

Explanation

To determine which of the given reactions are disproportionation reactions, we need to analyze each reaction to see if the same species undergoes both oxidation and reduction. A disproportionation reaction is characterized by a single species being oxidized and reduced simultaneously.

### Step-by-Step Solution

1. **Identify the Reactions**:
- (A) \( \text{Cu}^+ \rightarrow \text{Cu}^{2+} + \text{Cu} \)
- (B) \( 3 \text{MnO}_4 + 4 \text{H}^+ \rightarrow 2 \text{MnO}_4^- + \text{MnO}_2 + 2 \text{H}_2\text{O} \)
- (C) \( 2 \text{KMnO}_4 \rightarrow \text{K}_2\text{MnO}_4 + \text{MnO}_2 + \text{O}_2 \)
- (D) \( 2 \text{MnO}_4^- + 3 \text{Mn}^{2+} + 2 \text{H}_2\text{O} \rightarrow 5 \text{MnO}_2 + 4 \text{H}^+ \)

2. **Analyze Reaction (A)**:
- In this reaction, \( \text{Cu}^+ \) is oxidized to \( \text{Cu}^{2+} \) (oxidation state increases from +1 to +2).
- Simultaneously, \( \text{Cu}^+ \) is reduced to elemental \( \text{Cu} \) (oxidation state decreases from +1 to 0).
- Since the same species \( \text{Cu}^+ \) is both oxidized and reduced, **this is a disproportionation reaction**.

3. **Analyze Reaction (B)**:
- Here, \( \text{Mn} \) in \( \text{MnO}_4 \) has an oxidation state of +7 and is reduced to \( \text{MnO}_2 \) with an oxidation state of +4.
- The \( \text{MnO}_4^- \) is also converted to \( \text{MnO}_4^- \) (remains at +7).
- Since there is no simultaneous oxidation and reduction of the same species, **this is not a disproportionation reaction**.

4. **Analyze Reaction (C)**:
- In this reaction, \( \text{KMnO}_4 \) (Mn at +7) is converted to \( \text{K}_2\text{MnO}_4 \) (Mn at +6) and \( \text{MnO}_2 \) (Mn at +4).
- Here, all manganese species are undergoing reduction; there is no oxidation occurring.
- Therefore, **this is not a disproportionation reaction**.

5. **Analyze Reaction (D)**:
- In this reaction, \( \text{MnO}_4^- \) (Mn at +7) is reduced to \( \text{MnO}_2 \) (Mn at +4), while \( \text{Mn}^{2+} \) (Mn at +2) does not undergo oxidation.
- Since different manganese species are involved and there is no simultaneous oxidation and reduction of the same species, **this is not a disproportionation reaction**.

### Conclusion:
Only reaction (A) is a disproportionation reaction.

### Final Answer:
**The only disproportionation reaction is (A) \( \text{Cu}^+ \rightarrow \text{Cu}^{2+} + \text{Cu} \)**.