Choose the correct statement(s) from the given statements
(I) The proportionality constant in an equation can be obtained by dimensional analysis.
(II) The equation V = u + at can be derived by dimensional method.
(III) The equation y = A sin `omega` t cannot be derived by dimensional method
(IV) The equation `eta = (A)/(B)e^(-Bt)` can be derived with dimensional method

Options

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

Answer

To solve the question, we need to analyze each of the four statements regarding dimensional analysis and determine their correctness.

### Step-by-Step Solution:

1. **Statement I**: "The proportionality constant in an equation can be obtained by dimensional analysis."
- Dimensional analysis can help establish relationships between physical quantities based on their dimensions, but it cannot determine the exact value of proportionality constants. Therefore, this statement is **false**.

2. **Statement II**: "The equation V = u + at can be derived by dimensional method."
- Here, we can analyze the dimensions:
- The dimensional formula for velocity (V) and initial velocity (u) is [L T^-1].
- The dimensional formula for acceleration (a) is [L T^-2].
- Time (t) has the dimension [T].
- Therefore, the dimensional formula for at is [L T^-2] * [T] = [L T^-1].
- Since both sides of the equation have the same dimensions, this statement is **true**.

3. **Statement III**: "The equation y = A sin(ωt) cannot be derived by dimensional method."
- The equation involves a sine function, which is dimensionless. Dimensional analysis cannot derive relationships involving transcendental functions like sine. Therefore, this statement is **true**.

4. **Statement IV**: "The equation η = (A/B)e^(-Bt) can be derived with dimensional method."
- This equation involves constants A and B, and an exponential function. Dimensional analysis cannot be used to derive relationships involving multiple constants and exponential functions. Therefore, this statement is **false**.

### Conclusion:
The correct statements are **II and III**.

### Final Answer:
The correct options are **II and III**.

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