Determinants Matrices Question 50
Question: If $ f(x),g(x) $ and $ h(x) $ are three polynomials of degree 2 and $ \Delta (x)= \begin{vmatrix} f(x) & g(x) & h(x) \\ f’(x) & g’(x) & h’(x) \\ f’’(x) & g’’(x) & h’’(x) \\ \end{vmatrix}, $ then $ \Delta (x) $ is a polynomial of degree
Options:
A) 2
B) 3
C) At most 2
D) At most 3
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] Let $ f(x)=a_0x^{2}+a_1x+a_2 $
$ g(x)=b_0x^{2}+b_1x+b_2 $
$ h(x)=c_0x^{2}+c_1x+c_2 $ Then, $ \Delta (x) \begin{vmatrix} f(x) & g(x) & h(x) \\ 2a_0x+a_1 & 2b_0x+b_1 & 2c_0x+c_1 \\ 2a_0 & 2b_0 & 2c_0 \\ \end{vmatrix} $
$ =x \begin{vmatrix} f(x) & g(x) & h(x) \\ 2a_0 & 2b_0 & 2c_0 \\ 2a_0 & 2b_0 & 2c_0 \\ \end{vmatrix}+ \begin{vmatrix} f(x) & g(x) & h(x) \\ a_1 & b_1 & c_1 \\ 2a_0 & 2b_0 & 2c_0 \\ \end{vmatrix} $
$ =0+2 \begin{vmatrix} f(x) & g(x) & h(x) \\ a_1 & b_1 & c_1 \\ a_0 & b_0 & c_0 \\ \end{vmatrix} $
$ =2[(b_1c_0-b_0c_1]f(x)-(a_1c_0-a_0c_1)g(x) $
$ +(a_1b_0-a_0b_1)h(x)] $ Hence degree of $ \Delta (x)\le 2 $ .
BETA
