(A) : Plane angle has unit but no dimensional formula
(B) : All dimension less quantities are unit less

Options

A. Both A & B are tue
B. Both A & B are false
C. Only A is true
D. Only B is true

Answer

To solve the question, we need to analyze the two statements given:

**Statement A:** "Plane angle has a unit but no dimensional formula."
**Statement B:** "All dimensionless quantities are unitless."

### Step 1: Analyzing Statement A
1. **Understanding Plane Angle:**
- A plane angle is a measure of rotation between two intersecting lines.
- Common units for measuring angles include degrees, radians, and gradians.
- In SI units, the radian is the standard unit for measuring angles.

2. **Dimensional Formula of Plane Angle:**
- A plane angle is defined as the ratio of two lengths (arc length to radius).
- Since it is a ratio of two lengths, it is dimensionless.
- Therefore, while it has a unit (like radians), it does not have a dimensional formula.

**Conclusion for Statement A:**
- Statement A is **true** because a plane angle has units (like radians) but does not have a dimensional formula.

### Step 2: Analyzing Statement B
1. **Understanding Dimensionless Quantities:**
- Dimensionless quantities are those that do not have any physical dimensions (length, mass, time, etc.).
- Examples include ratios, such as the coefficient of friction, which is a ratio of two forces.

2. **Unitless vs. Dimensionless:**
- While all unitless quantities are dimensionless, not all dimensionless quantities are unitless.
- For instance, the mechanical equivalent of heat is a dimensionless quantity but has dimensions associated with it (it relates energy to temperature).

**Conclusion for Statement B:**
- Statement B is **false** because there are dimensionless quantities that still have dimensions, even if they do not have units.

### Final Answer:
- Based on the analysis, the correct option is that **only Statement A is true**. Therefore, the answer is option **3**.

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