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 simplify the expression. The determinant can be split into two determinants, one of which will be zero due to proportional columns, and the other can be factored.
The 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|. The first determinant is x(2x)(3x) * |1 1 1; 2 2 1; 3 3 1| = 0 (since columns 1 and 2 are proportional). The second determinant is x(2x)(3x) * |1 1 1; 1 2 4; 1 3 9| = 6x^3 * (2-1)(4-1)(3-1) = 6x^3 * 1 * 3 * 2 = 36x^3. Setting 36x^3 = 10 gives x^3 = 10/36 = 5/18. This equation has only one real solution for x.
Correct Answer: 2