If `Delta=|(x^2-5x+3,2x-5,3),(3x^2+x+4,6x+1,9),(7x^2-6x+9,14x-6,21)| = ax^3+bx^2+cx+d` then `Delta` i.e. `d/(dx)(Delta)` =

Options

A. 6
B. 5
C. 4
D. 0

Answer

To solve the problem, we need to evaluate the determinant \( \Delta \) given by:

\[ \Delta = \begin{vmatrix} x^2 - 5x + 3 & 2x - 5 & 3 \\ 3x^2 + x + 4 & 6x + 1 & 9 \\ 7x^2 - 6x + 9 & 14x - 6 & 21 \end{vmatrix} \]

and then find the derivative \( \frac{d}{dx}(\Delta) \).

### Step 1: Set up the determinant

We have the determinant as follows:

\[ \Delta = \begin{vmatrix} x^2 - 5x + 3 & 2x - 5 & 3 \\ 3x^2 + x + 4 & 6x + 1 & 9 \\ 7x^2 - 6x + 9 & 14x - 6 & 21 \end{vmatrix} \]

### Step 2: Perform row operations

To simplify the calculation, we can perform row operations. We will subtract appropriate multiples of the first row from the second and third rows.

1. For the second row \( R_2 \), we will perform the operation \( R_2 - 3R_1 \):
\[ R_2 = (3x^2 + x + 4) - 3(x^2 - 5x + 3) \]
This simplifies to:
\[ R_2 = (3x^2 + x + 4 - 3x^2 + 15x - 9) = (16x - 5) \]
The second element becomes \( 6x + 1 - 3(2x - 5) = 6x + 1 - 6x + 15 = 16 \), and the last element becomes \( 9 - 9 = 0 \).

2. For the third row \( R_3 \), we perform the operation \( R_3 - 7R_1 \):
\[ R_3 = (7x^2 - 6x + 9) - 7(x^2 - 5x + 3) \]
This simplifies to:
\[ R_3 = (7x^2 - 6x + 9 - 7x^2 + 35x - 21) = (29x - 12) \]
The second element becomes \( 14x - 6 - 7(2x - 5) = 14x - 6 - 14x + 35 = 29 \), and the last element becomes \( 21 - 21 = 0 \).

Now the determinant becomes:

\[ \Delta = \begin{vmatrix} x^2 - 5x + 3 & 2x - 5 & 3 \\ 16x - 5 & 16 & 0 \\ 29x - 12 & 29 & 0 \end{vmatrix} \]

### Step 3: Expand the determinant

Since the last column consists of zeros, we can expand along the third column:

\[ \Delta = 3 \cdot \begin{vmatrix} 16x - 5 & 16 \\ 29x - 12 & 29 \end{vmatrix} \]

Calculating the 2x2 determinant:

\[ = 3 \cdot ((16x - 5) \cdot 29 - (29x - 12) \cdot 16) \]

Expanding this gives:

\[ = 3 \cdot (464x - 145 - 464x + 192) = 3 \cdot (192 - 145) = 3 \cdot 47 = 141 \]

### Step 4: Find the derivative

Now we need to find \( \frac{d}{dx}(\Delta) \):

Since \( \Delta = 141 \) is a constant, its derivative is:

\[ \frac{d}{dx}(\Delta) = 0 \]

### Final Answer

Thus, the final answer is:

\[ \frac{d}{dx}(\Delta) = 0 \]