If `f(x), h(x)` are polynomials of degree 4 and `|(f(x), g(x),h(x)),(a, b, c),(p,q,r)|``=mx^4+nx^3+rx^2+sx+r` be an identity in x, then `|(f'''(0) - f''(0),g'''(0) - g''(0),h'''(0) -h''(0)),(a,b,c),(p,q,r)|` is

Options

A. `2(3n-r)`
B. `2(2n-3r)`
C. `3(n-2r)`
D. none of these

Explanation

To solve the given problem step by step, we need to analyze the determinants and their derivatives as described in the video transcript.

### Step 1: Understand the Given Determinant
We are given the determinant:
\[ D = \begin{vmatrix} f(x) & g(x) & h(x) \\ a & b & c \\ p & q & r \end{vmatrix} \]
This determinant is stated to be equal to:
\[ mx^4 + nx^3 + rx^2 + sx + r \]
This is an identity in \(x\), meaning it holds for all values of \(x\).

### Step 2: Differentiate the Determinant
We differentiate \(D\) with respect to \(x\):
\[ D' = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ a & b & c \\ p & q & r \end{vmatrix} \]
The right-hand side differentiates to:
\[ D' = 4mx^3 + 3nx^2 + 2rx + s \]

### Step 3: Differentiate Again
We differentiate \(D'\) with respect to \(x\):
\[ D'' = \begin{vmatrix} f''(x) & g''(x) & h''(x) \\ a & b & c \\ p & q & r \end{vmatrix} \]
The right-hand side differentiates to:
\[ D'' = 12mx^2 + 6nx + 2r \]

### Step 4: Differentiate Once More
We differentiate \(D''\) with respect to \(x\):
\[ D''' = \begin{vmatrix} f'''(x) & g'''(x) & h'''(x) \\ a & b & c \\ p & q & r \end{vmatrix} \]
The right-hand side differentiates to:
\[ D''' = 24mx + 6n \]

### Step 5: Subtract Equations
Now we subtract \(D''\) from \(D'''\):
\[ D''' - D'' = \begin{vmatrix} f'''(x) - f''(x) & g'''(x) - g''(x) & h'''(x) - h''(x) \\ a & b & c \\ p & q & r \end{vmatrix} \]
This gives us:
\[ D''' - D'' = (24mx + 6n) - (12mx^2 + 6nx + 2r) \]
Simplifying the right-hand side:
\[ = 24mx + 6n - 12mx^2 - 6nx - 2r \]
\[ = -12mx^2 + (24m - 6n)x + 6n - 2r \]

### Step 6: Evaluate at \(x = 0\)
Now we set \(x = 0\):
\[ D'''(0) - D''(0) = \begin{vmatrix} f'''(0) - f''(0) & g'''(0) - g''(0) & h'''(0) - h''(0) \\ a & b & c \\ p & q & r \end{vmatrix} \]
The left-hand side becomes:
\[ = 0 + 0 + 6n - 2r = 6n - 2r \]

### Step 7: Final Result
Thus, we have:
\[ \begin{vmatrix} f'''(0) - f''(0) & g'''(0) - g''(0) & h'''(0) - h''(0) \\ a & b & c \\ p & q & r \end{vmatrix} = 2(3n - r) \]

### Conclusion
The value of the determinant is:
\[ 2(3n - r) \]