Consider the fallowing statement :
(I) `CH_3OC^(o+)H_2` is more stable that `CH_3CH_2^(o+)`
(II) `Me_3C^(o+)` is more stable than `CH_3CH_2C^(o+)H_2`
(III) `CH_2=CH-C^(o+)H_2` is more stable than `CH_3CH_2C^(o+)H_2`
(IV) `CH_2=C^(o+)H` is more stable thn `CH_3C^(o+)H_2` of these statement:

Options

A. I and II are correct
B. III and IV are correct
C. I,II and III are correct
D. All are correct

Answer

To analyze the stability of the given carbocations in the statements, we will evaluate each statement one by one based on the principles of carbocation stability, such as hyperconjugation, inductive effect, and resonance.

### Step-by-Step Solution:

**Statement I: `CH3OCH2^+` is more stable than `CH3CH2^+`**

1. **Identify the carbocations**:
- `CH3OCH2^+` (methoxycarbocation) has a positive charge on the carbon adjacent to an oxygen atom.
- `CH3CH2^+` (ethyl carbocation) has a positive charge on a primary carbon.

2. **Evaluate stability**:
- The methoxy group (OCH3) can donate electron density through resonance (mesomeric effect) due to the lone pairs on oxygen.
- The ethyl carbocation does not have any such electron-donating group and is less stable due to being a primary carbocation.

3. **Conclusion**: The first statement is correct; `CH3OCH2^+` is indeed more stable than `CH3CH2^+`.

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**Statement II: `Me3C^+` is more stable than `CH3CH2C^+H2`**

1. **Identify the carbocations**:
- `Me3C^+` (tertiary carbocation) has three methyl groups attached to the positively charged carbon.
- `CH3CH2C^+H2` (secondary carbocation) has two alkyl groups.

2. **Evaluate stability**:
- Tertiary carbocations are more stable than secondary carbocations due to hyperconjugation; the presence of three alkyl groups allows for more hyperconjugative structures.
- Secondary carbocations have fewer hyperconjugative interactions.

3. **Conclusion**: The second statement is correct; `Me3C^+` is more stable than `CH3CH2C^+H2`.

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**Statement III: `CH2=CH-C^+H2` is more stable than `CH3CH2C^+H2`**

1. **Identify the carbocations**:
- `CH2=CH-C^+H2` (allylic carbocation) has a positive charge on a carbon adjacent to a double bond.
- `CH3CH2C^+H2` (secondary carbocation) has a positive charge on a secondary carbon.

2. **Evaluate stability**:
- The allylic carbocation can be stabilized by resonance; the positive charge can be delocalized over the double bond.
- The secondary carbocation does not have such resonance stabilization.

3. **Conclusion**: The third statement is correct; `CH2=CH-C^+H2` is more stable than `CH3CH2C^+H2`.

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**Statement IV: `CH2=C^+H` is more stable than `CH3C^+H2`**

1. **Identify the carbocations**:
- `CH2=C^+H` (vinyl carbocation) has a positive charge on a carbon that is part of a double bond.
- `CH3C^+H2` (secondary carbocation) has a positive charge on a secondary carbon.

2. **Evaluate stability**:
- Vinyl carbocations are generally less stable than secondary carbocations because they are sp-hybridized and have less ability to stabilize the positive charge through hyperconjugation or resonance.
- Secondary carbocations benefit from hyperconjugation from adjacent alkyl groups.

3. **Conclusion**: The fourth statement is incorrect; `CH2=C^+H` is less stable than `CH3C^+H2`.

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### Final Conclusion:
- Statements I, II, and III are correct.
- Statement IV is incorrect.
- Therefore, the correct answer is that statements I, II, and III are true.