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

Answer

A

Explanation

`(a)` Differntiating given equation w.r.t.x, we get
`|{:(f'(x),g'(x),h'(x)),(a,b,c),(p,q,r):}|=4mx^(3)+3nx^(2)+2rx+5`...........`(1)`
Again differentiating w.r.t.x, we get
`|{:(f''(x),g''(x),h''(x)),(a,b,c),(p,q,r):}|=12mx^(2)+6nx+2x`...........`(2)`
Again differentiating w.r.t.x, we get
`|{:(f'''(x),g'''(x),h'''(x)),(a,b,c),(p,q,r):}|=24mx+6n`...........`(3)`
Putting `x=0` in `(2)`, we get
`2r=`|{:(f''(0),g''(0),h''(0)),(a,b,c),(p,q,r):}|`............`(4)`
Putting `x=0` in `(3)`, we get
`6n=|{:(f'''(0),g'''(0),h'''(0)),(a,b,c),(p,q,r):}|`.............`(5)`
From `(5)` and `(4)`, we get
`|{:(f'''(0)-f''(0),g'''(0)-g''(0),h'''(0)-h''(0)),(a,b,c),(p,q,r):}|=2(3n-r)`