Assertion : In `CH_(2)=C=CH_(2)`, all the carbon atoms are `sp^(2)`-hybridized.
Reason : All the hydrogen atoms lie in one plane.

Options

A. If both assertion and reason are correct and reason is correct explanation for assertion.
B. If both assertion and reason are correct but reason is not correct explanation for assertion.
C. If assertion is correct but reason is incorrect.
D. If both assertion and reason are incorrect.

Explanation

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Analyze the Assertion
The assertion states that in \( CH_2=C=CH_2 \) (propadiene), all carbon atoms are \( sp^2 \)-hybridized.

1. **Identify the Structure**: The compound is propadiene, which has a structure with two double bonds.
2. **Determine Hybridization**:
- The terminal carbon atoms (the ones at either end) each form two sigma bonds (one with hydrogen and one with the central carbon) and participate in one pi bond (the double bond). Thus, they have three regions of electron density, leading to \( sp^2 \) hybridization.
- The central carbon atom forms two pi bonds (one with each terminal carbon) and has only two sigma bonds (one with each terminal carbon). Therefore, it has two regions of electron density, leading to \( sp \) hybridization.

**Conclusion for Assertion**: The assertion is **false** because not all carbon atoms are \( sp^2 \)-hybridized; the central carbon is \( sp \)-hybridized.

### Step 2: Analyze the Reason
The reason states that all hydrogen atoms lie in one plane.

1. **Consider the Geometry**: In the case of propadiene, the terminal carbon atoms are \( sp^2 \)-hybridized, which means the hydrogen atoms attached to them will lie in the same plane as the carbon atoms.
2. **Position of Central Carbon**: The central carbon is \( sp \)-hybridized, which means it has a linear arrangement with respect to the two terminal carbons. The hydrogen atoms attached to the terminal carbons will not all lie in the same plane due to the presence of the double bonds and the linear arrangement of the central carbon.

**Conclusion for Reason**: The reason is also **false** because the hydrogen atoms do not lie in one plane; they are arranged in a way that some are perpendicular to others.

### Final Conclusion
Both the assertion and the reason are false.