For the equation `x=ACsin (Bt)+D e^((BCt))`, where x and t represent position and time respectively, which of the following is/are CORRECT :-

Options

A. Dimension of AC is `LT^(-1)`
B. Dimension of B is `T^(-1)`
C. Dimension of AC and D are same
D. Dimension of C is `T^(-1)`

Answer

To solve the equation \( x = AC \sin(Bt) + D e^{(BCt)} \) and determine the correctness of the options regarding the dimensions of \( AC \), \( B \), \( D \), and \( C \), we will analyze each term step by step.

### Step 1: Identify the dimensions of \( x \) and \( t \)
- \( x \) represents position, so its dimension is \( [x] = L \) (length).
- \( t \) represents time, so its dimension is \( [t] = T \) (time).

**Hint:** Remember that dimensions of physical quantities are represented by symbols like \( L \) for length and \( T \) for time.

### Step 2: Analyze the term \( AC \sin(Bt) \)
- The term \( \sin(Bt) \) is a dimensionless quantity. For \( Bt \) to be dimensionless, the dimensions of \( B \) must be such that when multiplied by the dimensions of \( t \), the result is dimensionless.
- Since \( [t] = T \), we have:
\[ [B] \cdot [t] = [B] \cdot T = 1 \quad \Rightarrow \quad [B] = T^{-1} \]

**Hint:** A dimensionless quantity means that it has no units, which implies that the product of its dimensions must equal 1.

### Step 3: Determine the dimension of \( AC \)
- Since \( AC \sin(Bt) \) must have the same dimension as \( x \), we can write:
\[ [AC] \cdot [\sin(Bt)] = [x] \quad \Rightarrow \quad [AC] \cdot 1 = L \quad \Rightarrow \quad [AC] = L \]

**Hint:** The sine function does not change the dimensionality of its argument; it remains dimensionless.

### Step 4: Analyze the term \( D e^{(BCt)} \)
- The term \( e^{(BCt)} \) must also be dimensionless. Therefore, the product \( BCt \) must be dimensionless:
\[ [B] \cdot [C] \cdot [t] = 1 \quad \Rightarrow \quad [B] = T^{-1}, \quad [t] = T \quad \Rightarrow \quad [C] = 1 \quad (\text{dimensionless}) \]

**Hint:** Exponential functions require their exponent to be dimensionless.

### Step 5: Determine the dimension of \( D \)
- Since \( D e^{(BCt)} \) must also have the dimension of \( x \):
\[ [D] \cdot [e^{(BCt)}] = [x] \quad \Rightarrow \quad [D] \cdot 1 = L \quad \Rightarrow \quad [D] = L \]

**Hint:** The dimension of \( D \) is determined by the requirement that the entire term has the same dimension as \( x \).

### Summary of Dimensions
- \( [AC] = L \)
- \( [B] = T^{-1} \)
- \( [D] = L \)
- \( [C] = 1 \) (dimensionless)

### Conclusion
- **Option A:** Dimension of \( AC \) is \( LT^{-1} \) - **Incorrect** (it should be \( L \)).
- **Option B:** Dimension of \( B \) is \( T^{-1} \) - **Correct**.
- **Option C:** Dimensions of \( AC \) and \( D \) are the same - **Correct**.
- **Option D:** Dimension of \( C \) is \( T^{-1} \) - **Incorrect** (it is dimensionless).

### Correct Options
- **Option B** and **Option C** are correct.