Question

If f(x), g(x), h(x) are polynomials in x of degree 2 and F(x)=$\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ f"& g"& h"\end{array}\right|,$ , then F(x) is equal to

• A. 1
• B. 0
• C. -1
• D. $f\left(x\right).g\left(x\right).h\left(x\right)$

Answer 1

The correct option is B 0

$\begin{array}{c}Let,\\ f\left(x\right)=a{x}^2+bx+c\\ f^{\prime }\left(x\right)=2ax+b\\ f^{\prime \prime }\left(x\right)=2a\\ g\left(x\right)=a_1{x}^2+b_{1}x+c_1\\ g^{\prime }\left(x\right)=2a_{1}x+b_1\\ g^{\prime \prime }\left(x\right)=2a_1\\ and,\\ h\left(x\right)=a_2{x}^2+b_{2}x+c_2\\ h^{\prime }\left(x\right)=2a_{2}x+b_2\\ f\left(x\right)=\left|\begin{array}{ccc}a{x}^2+bx+c& a_1{x}^2+b_{1}x+c_1& a_2{x}^2+b_{2}x+c_2\\ 2ax+b& 2a_{1}x+b_1& 2a_{2}x+b_2\\ 2a& 2a_1& 2a_2\end{array}\right|\\ f^{\prime }\left(x\right)=\left|\begin{array}{ccc}2ax+b& 2a_{1}x+b_1& 2a_{2}x+b_2\\ 2ax+b& 2a_{1}x+b_1& 2a_{2}x+b_2\\ 2a& 2a_1& 2a_2\end{array}\right|+0+0\\ =0\end{array}$

Hence, the option $B$ is the correct answer.