A dimensionless quantity y is represented by the formula `y=(a-bc)/(d+e)`. Which of following is/arecorrect?

Options

A. Dimensions of d and e are same
B. abc and de have same dimensions
C. `"bc"/"ae+d"` is dimensionless
D. de+dc is not meaningful

Answer

To solve the problem, we need to analyze the given expression for the dimensionless quantity \( y \):

\[ y = \frac{a - bc}{d + e} \]

Since \( y \) is dimensionless, we can conclude that the dimensions of the numerator must equal the dimensions of the denominator. This leads us to the following steps:

### Step 1: Understand the implications of \( y \) being dimensionless
Since \( y \) is dimensionless, we have:

\[ [a - bc] = [d + e] \]

This means that the dimensions of \( a \) and \( bc \) must be the same as the dimensions of \( d \) and \( e \).

### Step 2: Analyze the terms
1. **Dimensions of \( a \)**: Let the dimension of \( a \) be \( [A] \).
2. **Dimensions of \( b \)**: Let the dimension of \( b \) be \( [B] \).
3. **Dimensions of \( c \)**: Let the dimension of \( c \) be \( [C] \).
4. **Dimensions of \( d \)**: Let the dimension of \( d \) be \( [D] \).
5. **Dimensions of \( e \)**: Let the dimension of \( e \) be \( [E] \).

From the equation \( [a - bc] = [d + e] \), we can deduce:

- Since \( bc \) is a product, its dimension is \( [B][C] \).
- Therefore, we have \( [A] = [B][C] \) and \( [D] = [E] \).

### Step 3: Evaluate the options
1. **Option 1**: Dimensions of \( d \) and \( e \) are the same.
- Since we have \( [D] = [E] \), this statement is **correct**.

2. **Option 2**: \( a, b, c, \) and \( d \) have the same dimensions.
- We have \( [A] = [B][C] \) and \( [D] = [E] \). There is no direct relationship that states all four have the same dimensions. Thus, this statement is **incorrect**.

3. **Option 3**: \( \frac{bc}{a + d} \) is dimensionless.
- Since \( [bc] = [B][C] \) and \( [a + d] \) has dimensions \( [A] \) (which is not equal to \( [B][C] \)), this statement is **incorrect**.

4. **Option 4**: \( d + bc \) is not meaningful.
- Since \( [D] \) (dimension of \( d \)) is not the same as \( [B][C] \) (dimension of \( bc \)), we cannot add these two terms. Thus, this statement is **correct**.

### Conclusion
From the analysis, we find:
- Option 1: Correct
- Option 2: Incorrect
- Option 3: Incorrect
- Option 4: Correct

### Final Answer
The correct options are 1 and 4.

---