The total number of distinct x ∈ R for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|
Step-by-Step Solution
Key Concept: Use properties of determinants to split the third column into two separate determinants. Factor out common terms from rows and columns to simplify the expression.
The given determinant is D = |x x^2 1+x^3; 2x 4x^2 1+8x^3; 3x 9x^2 1+27x^3|. This can be written as the sum of two determinants: D = |x x^2 1; 2x 4x^2 1; 3x 9x^2 1| + |x x^2 x^3; 2x 4x^2 8x^3; 3x 9x^2 27x^3|. Factoring out x from each row of the second determinant gives D = |x x^2 1; 2x 4x^2 1; 3x 9x^2 1| + x^3 |1 1 1; 2 4 8; 3 9 27|. The first determinant is 0 because the columns are proportional (or by direct calculation). The second determinant is x^3 * (1(108-72) - 1(54-24) + 1(18-12)) = x^3 * (36 - 30 + 6) = 12x^3. Setting 12x^3 = 10 gives x^3 = 10/12 = 5/6. This equation has only one real solution for x.
Correct Answer: 2