Question

If $f (x), g(x)$and $h (x)$ are three polynomials of degree 2 and $\Delta \left(x\right) = \begin{vmatrix}f\left(x\right)&g\left(x\right)&h\left(x\right)\\ f'\left(x\right)&g'\left(x\right)&h'\left(x\right)\\ f''\left(x\right)&g''\left(x\right)&h''\left(x\right)\end{vmatrix}$ then $ \Delta\left(x\right) $ is a polynomial of degree

• A. 2
• B. 3
• C. 0
• D. atmost 3

Answer 1

Answer: (C)

Since $f(x),\, g(x)$ and $h(x)$ are the polynomials of degree $2$, therefore
$f '''\left(x\right)= g '''\left(x\right) = h'''\left(x\right) = 0 $
Now, $\Delta\left(x\right)=\begin{vmatrix}f\left(x\right)&g\left(x\right)&h\left(x\right)\\ f'\left(x\right)&g'\left(x\right)&h'\left(x\right)\\ f''\left(x\right)&g''\left(x\right)&h''\left(x\right)\end{vmatrix} $
$ \Rightarrow \; \Delta '(x) = \begin{vmatrix} f'\left(x\right)&g'\left(x\right)&h'\left(x\right)\\ f'\left(x\right)&g'\left(x\right)&h'\left(x\right)\\ f''\left(x\right)&g''\left(x\right)&h''\left(x\right)\end{vmatrix} $
$ + \begin{vmatrix} f\left(x\right)&g\left(x\right)&h\left(x\right) \\ f''\left(x\right)&g''\left(x\right)&h''\left(x\right)\\ f''\left(x\right)&g''\left(x\right)&h''\left(x\right)\end{vmatrix} $
$ + \begin{vmatrix} f''\left(x\right)&g''\left(x\right)&h''\left(x\right) \\ f'\left(x\right)&g'\left(x\right)&h'\left(x\right)\\ f'''\left(x\right)&g'''\left(x\right)&h'''\left(x\right)\end{vmatrix} $
$ \Rightarrow \; \Delta '(x) = 0 + 0 + 0 = 0$
$ \Rightarrow \; \Delta (x) =$ constant.
Thus, $\Delta (x)$ is the polynomial of degree zero.