The total number of distinct `x in R` for which `|[x, x^2, 1+x^3] , [2x,4x^2,1+8x^3] , [3x, 9x^2,1+27x^3]|=10` is (A) 0 (B) 1 (C) 2 (D) 3

Explanation

To solve the problem, we need to evaluate the determinant of the given matrices and set it equal to 10. Here’s the step-by-step solution:

### Step 1: Set up the determinant
We have the determinant:
\[ D = \begin{vmatrix} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{vmatrix} \]
We want to find the values of \( x \) such that \( D = 10 \).

### Step 2: Factor out common terms
We can factor out \( x \) from the first column and \( x^2 \) from the second column:
\[ D = x \cdot x^2 \begin{vmatrix} 1 & x & 1 + x^3 \\ 2 & 4 & 1 + 8x^3 \\ 3 & 9 & 1 + 27x^3 \end{vmatrix} = x^3 \begin{vmatrix} 1 & x & 1 + x^3 \\ 2 & 4 & 1 + 8x^3 \\ 3 & 9 & 1 + 27x^3 \end{vmatrix} \]

### Step 3: Simplify the determinant
Now we need to simplify the determinant:
\[ D = x^3 \begin{vmatrix} 1 & x & 1 + x^3 \\ 2 & 4 & 1 + 8x^3 \\ 3 & 9 & 1 + 27x^3 \end{vmatrix} \]

### Step 4: Perform row operations
We can perform row operations to simplify the determinant. Subtract the first row from the second and third rows:
\[ D = x^3 \begin{vmatrix} 1 & x & 1 + x^3 \\ 1 & 3 & 7x^3 \\ 2 & 8 & 26x^3 \end{vmatrix} \]

### Step 5: Expand the determinant
We can expand the determinant using the first row:
\[ D = x^3 \left[ 1 \cdot \begin{vmatrix} 3 & 7x^3 \\ 8 & 26x^3 \end{vmatrix} - x \cdot \begin{vmatrix} 1 & 7x^3 \\ 2 & 26x^3 \end{vmatrix} + (1 + x^3) \cdot \begin{vmatrix} 1 & 3 \\ 2 & 8 \end{vmatrix} \right] \]

### Step 6: Calculate the 2x2 determinants
Calculating the 2x2 determinants:
1. \(\begin{vmatrix} 3 & 7x^3 \\ 8 & 26x^3 \end{vmatrix} = 3(26x^3) - 7x^3(8) = 78x^3 - 56x^3 = 22x^3\)
2. \(\begin{vmatrix} 1 & 7x^3 \\ 2 & 26x^3 \end{vmatrix} = 1(26x^3) - 7x^3(2) = 26x^3 - 14x^3 = 12x^3\)
3. \(\begin{vmatrix} 1 & 3 \\ 2 & 8 \end{vmatrix} = 1(8) - 3(2) = 8 - 6 = 2\)

### Step 7: Substitute back into the determinant
Substituting these back into the determinant:
\[ D = x^3 \left[ 22x^3 - 12x^4 + 2(1 + x^3) \right] \]
This simplifies to:
\[ D = x^3 \left[ 22x^3 - 12x^4 + 2 + 2x^3 \right] = x^3 \left[ -12x^4 + 24x^3 + 2 \right] \]

### Step 8: Set the determinant equal to 10
Now we set the determinant equal to 10:
\[ x^3 (-12x^4 + 24x^3 + 2) = 10 \]

### Step 9: Rearranging the equation
Rearranging gives us:
\[ -12x^7 + 24x^6 + 2x^3 - 10 = 0 \]

### Step 10: Analyze the polynomial
This is a polynomial equation of degree 7. We can use numerical methods or graphing to find the number of distinct real roots.

### Conclusion
After analyzing the polynomial, we find that there are 2 distinct real roots.

### Final Answer
The total number of distinct \( x \in \mathbb{R} \) for which the determinant equals 10 is **2** (Option C).

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