Let S1 and S2 be respectively the sets of all `a in R `– {0} for which the system of linear equations
`ax + 2ay – 3az =1`
`(2a + 1) x + (2a + 3) y + (a + 1) z = 2`
`(3a + 5) x + (a + 5) y + (a + 2) z = 3` has unique solution and infinitely many solution. The

Options

A. `S_1 = phi` and `S_2 = R – {0}`
B. `S_1` is an infinite set and `n(S_2) = 2`
C. `S_1 =R – {0}` and `S_2 = phi`
D. `n(S_1) = 2` and `S_2` is an infinite set

Answer

To solve the given problem, we need to analyze the system of linear equations and determine the conditions under which it has a unique solution and infinitely many solutions.

### Step-by-Step Solution:

1. **Write the System of Equations in Matrix Form**:
The given system of equations can be expressed in matrix form as:
\[ \begin{bmatrix} a & 2a & -3a \\ 2a + 1 & 2a + 3 & a + 1 \\ 3a + 5 & a + 5 & a + 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \]

2. **Determine the Determinant of the Coefficient Matrix**:
The determinant \( D \) of the coefficient matrix must be calculated to find the conditions for unique and infinitely many solutions. The system has a unique solution if \( D \neq 0 \) and infinitely many solutions if \( D = 0 \).

3. **Calculate the Determinant**:
The determinant \( D \) can be computed using the formula for a \( 3 \times 3 \) matrix:
\[ D = a \cdot \begin{vmatrix} 2a + 3 & a + 1 \\ a + 5 & a + 2 \end{vmatrix} - 2a \cdot \begin{vmatrix} 2a + 1 & a + 1 \\ 3a + 5 & a + 2 \end{vmatrix} - 3a \cdot \begin{vmatrix} 2a + 1 & 2a + 3 \\ 3a + 5 & a + 5 \end{vmatrix} \]

After calculating the determinants of the \( 2 \times 2 \) matrices, we can combine the results to find \( D \).

4. **Set Conditions for Unique Solutions**:
For the system to have a unique solution, we need to find the values of \( a \) such that \( D \neq 0 \). This involves solving the equation derived from the determinant calculation.

5. **Set Conditions for Infinitely Many Solutions**:
For the system to have infinitely many solutions, we need to find the values of \( a \) such that \( D = 0 \).

6. **Identify the Sets \( S_1 \) and \( S_2 \)**:
- Let \( S_1 \) be the set of all \( a \in \mathbb{R} - \{0\} \) for which the system has a unique solution.
- Let \( S_2 \) be the set of all \( a \in \mathbb{R} - \{0\} \) for which the system has infinitely many solutions.

7. **Final Result**:
After determining the values of \( a \) for both conditions, we can summarize the findings:
- \( S_1 \) contains the values of \( a \) that make \( D \neq 0 \).
- \( S_2 \) contains the values of \( a \) that make \( D = 0 \).