<p>An ordered pair \((\alpha, \beta)\) for which the system of linear equations<br>\((1 + \alpha)x + \beta y + z = 2\)<br>\(\alpha x + (1 + \beta)y + z = 3\)<br>\(\alpha x + \beta y + 2z = 2\)<br>has a unique solution, is:</p>
Step-by-Step Solution
Key Concept: A system has a unique solution if and only if the coefficient matrix is non-singular (determinant ≠ 0). Find when det(A) = 0 to identify the exceptional pairs, then choose any pair NOT satisfying that condition.
<p><strong>Step 1:</strong> Write the coefficient matrix $A$:</p>
<p>$$A = \begin{bmatrix} 1+\alpha & \beta & 1 \\ \alpha & 1+\beta & 1 \\ \alpha & \beta & 2 \end{bmatrix}$$</p>
<p><strong>Step 2:</strong> For a unique solution, $\det(A) \neq 0$. Find when $\det(A) = 0$:</p>
<p>Expanding along the first row:</p>
<p>$$\det(A) = (1+\alpha)[(1+\beta) \cdot 2 - \beta] - \beta[\alpha \cdot 2 - \alpha] + 1[\alpha \beta - \alpha(1+\beta)]$$</p>
<p>$$= (1+\alpha)[2 + 2\beta - \beta] - \beta \cdot \alpha + \alpha\beta - \alpha - \alpha\beta$$</p>
<p>$$= (1+\alpha)(2 + \beta) - \alpha\beta - \alpha + \alpha\beta - \alpha$$</p>
<p>$$= 2 + \beta + 2\alpha + \alpha\beta - \alpha\beta - 2\alpha$$</p>
<p>$$= 2 + \beta$$</p>
<p><strong>Step 3:</strong> $\det(A) = 0$ when $\beta = -2$. Therefore, a unique solution exists when $\beta \neq -2$ ($\alpha$ can be any value).</p>
<p><strong>Step 4:</strong> Any pair $(\alpha, \beta)$ with $\beta \neq -2$ is valid. Checking the given options for $\beta \neq -2$:</p>
<p>Option (A): $(2, 4)$ has $\beta = 4 \neq -2$ ✓</p>
<p>Option (B): $(-3, 1)$ has $\beta = 1 \neq -2$ ✓</p>
<p>Option (C): $(-4, 2)$ has $\beta = 2 \neq -2$ ✓</p>
<p>Option (D): $(1, -3)$ has $\beta = -3 \neq -2$ ✓</p>
<p>All options satisfy the condition. The answer depends on additional constraints not specified in the problem.</p>
Correct Answer: B