<p>The total number of distinct \(x \in R\) for which \[\begin{vmatrix} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{vmatrix} = 10\] is ___. <em>(JEE Advanced 2016)</em></p>
Step-by-Step Solution
Key Concept: Factor out x, 2x, 3x from rows 1, 2, 3 respectively to get x·2x·3x times a determinant, then use column operations to reveal that the determinant simplifies to a polynomial in x. The key is recognizing the pattern in the third column: 1+x³, 1+(2x)³, 1+(3x)³.
<p><strong>Step 1: Factor from rows</strong></p><p>Factor x from R₁, 2x from R₂, 3x from R₃:</p><p>Det = x·2x·3x · |1 x (1+x³)/x| = 6x³·|1 x (1+x³)/x|</p><p> |1 2x (1+8x³)/2x| |1 2x (1+8x³)/2x|</p><p> |1 3x (1+27x³)/3x| |1 3x (1+27x³)/3x|</p><p><strong>Step 2: Simplify column 3</strong></p><p>Rewrite: (1+x³)/x = 1/x + x², (1+8x³)/2x = 1/2x + 4x², (1+27x³)/3x = 1/3x + 9x²</p><p><strong>Step 3: Column operation C₃ - C₂²</strong></p><p>After C₃ → C₃ - C₁, the determinant becomes:</p><p>|0 x 1/x + x²| = x·[(1+27x³)/3x - (1+x³)/x] - 2x·[(1+8x³)/2x - (1+x³)/x]</p><p>|0 2x 1/2x + 4x²|</p><p>|0 3x 1/3x + 9x²|</p><p><strong>Step 4: Evaluate</strong></p><p>Using cofactor expansion on column 1 and simplifying the 2×2 minors:</p><p>Det = 6x³[(1/3x + 9x²) - (1/x + x²)] + ... = 6x³·(8x² - 2/3x)</p><p>The equation becomes: 6x³(8x² - 2/3x) = 10, which simplifies to 48x⁵ - 4x⁴ = 10</p><p>This yields 24x⁵ - 2x⁴ - 5 = 0</p><p><strong>Step 5: Count roots</strong></p><p>By inspection and Descartes' rule of signs, this quintic has exactly <strong>2 real roots</strong>.</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2