Which of the following is incorrect

Options

A. All derived quantities may be represented dimensionally in terms of the base quantities
B. A base quantity cannot be represented dimensionally in terms of other base quantities
C. The dimension of a derived quantity is never zero in any base quantity
D. The dimension of a base quantity in other base quantities is always zero.

Answer

To solve the question of identifying the incorrect statement among the given options, we will analyze each statement step by step.

### Step 1: Analyze Statement A
**Statement A:** "Derived quantity may be represented dimensionally in terms of base quantity."

- **Explanation:** Derived quantities are formed from base quantities. For example, velocity is a derived quantity that can be expressed as displacement (length) divided by time. In dimensional terms, this is represented as:
\[ \text{Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{L}{T} \implies [V] = [L][T]^{-1} \]
- **Conclusion:** This statement is true.

### Step 2: Analyze Statement B
**Statement B:** "A base quantity cannot be represented dimensionally in terms of other base quantities."

- **Explanation:** Base quantities are fundamental and cannot be expressed in terms of other quantities. For example, mass (M), length (L), and time (T) are base quantities. Mass cannot be expressed in terms of length or time.
- **Conclusion:** This statement is also true.

### Step 3: Analyze Statement C
**Statement C:** "The dimension of a derived quantity is never 0 in any base quantity."

- **Explanation:** The dimension of a derived quantity can indeed be zero in terms of some base quantities. For example, the dimension of velocity is:
\[ [V] = [M^0 L^1 T^{-1}] \]
Here, the exponent of mass (M) is zero, indicating that the dimension of mass in the context of velocity is zero.
- **Conclusion:** This statement is false.

### Step 4: Analyze Statement D
**Statement D:** "The dimension of a base quantity in terms of other base quantities is always zero."

- **Explanation:** This statement is true because a base quantity is independent and cannot be expressed in terms of other base quantities. For instance, the dimension of mass (M) in terms of length (L) and time (T) is indeed zero.
- **Conclusion:** This statement is true.

### Final Conclusion
After analyzing all the statements, we find that:
- Statement A: True
- Statement B: True
- Statement C: False (this is the incorrect statement)
- Statement D: True

Thus, the incorrect statement is **Statement C**.