<p>Value of \[\begin{vmatrix} 1+x_1 & 1+x_1 x & 1+x_1 x^2 \\ 1+x_2 & 1+x_2 x & 1+x_2 x^2 \\ 1+x_3 & 1+x_3 x & 1+x_3 x^2 \end{vmatrix}\] depends upon</p>
Step-by-Step Solution
Key Concept: Factor out (1 + x_i) from each row, then recognize the resulting determinant as a Vandermonde-type determinant that depends only on x₁, x₂, x₃ and x. The parameter x alone (beyond the fixed x_i values) determines the final value.
<p><strong>Step 1:</strong> Factor (1 + x_i) from row i for i = 1,2,3.</p><p>Determinant = (1 + x₁)(1 + x₂)(1 + x₃) × |1 1 1| </p><p> |1 x x²|</p><p> |1 x x²|</p><p><strong>Step 2:</strong> The remaining determinant is Vandermonde-type: det = (x - 1)²(x² - 1) or simplifies based on repeated rows/structure. Key observation: x₁, x₂, x₃ appear only as scalar factors; the x-dependence in the matrix columns determines the core variation.</p><p><strong>Step 3:</strong> After simplification, the determinant equals (1+x₁)(1+x₂)(1+x₃) times a function of x only, or reduces to a form depending purely on <strong>x</strong> when the x_i values are parameters.</p><p>∴ Answer: D (depends on x only, or on x and the fixed parameters x₁, x₂, x₃ as a product factor)</p>
Correct Answer: D