Note: In the following questions a statement of assertion followed by a statement of reason is given. Choose the correct answer out of the following choices.
Assertion : In allene `(C_3 H_4)` all carbon atoms are `sp^2` hybridised.
Reason : All the three carbon atoms are joined by double bond.

Answer

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

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that in allene (C₃H₄), all carbon atoms are sp² hybridized.
- To evaluate this, we need to consider the structure of allene. Allene has the formula C₃H₄, and its structure can be represented as H₂C=C=CH₂.

2. **Analyzing the Structure**:
- In the structure of allene, the central carbon atom is involved in two double bonds (one with each of the terminal carbon atoms).
- The terminal carbon atoms are each bonded to one hydrogen atom and one carbon atom through a double bond.

3. **Determining Hybridization**:
- For the terminal carbon atoms:
- Each terminal carbon has 3 sigma bonds (one with hydrogen and one with the central carbon) and no lone pairs.
- The steric number (number of sigma bonds + lone pairs) is 3, which corresponds to sp² hybridization.
- For the central carbon atom:
- The central carbon has 2 double bonds (which consist of 1 sigma bond each) and no lone pairs.
- The steric number is 2, which corresponds to sp hybridization.

4. **Conclusion on Assertion**:
- Since the central carbon atom is sp hybridized and not sp² hybridized, the assertion is **false**.

5. **Understanding the Reason**:
- The reason states that all three carbon atoms are joined by double bonds.
- In allene, the terminal carbons are indeed connected to the central carbon by double bonds, which is true.

6. **Conclusion on Reason**:
- The reason is **true** because there are double bonds between the carbon atoms.

### Final Evaluation:
- The assertion is **false**, while the reason is **true**. Therefore, the correct answer is option C: "Assertion is false, Reason is true."