Question

If $a^2+b^2+c^2=-2$ and $f(x)=\left|\begin{array}{ccc}1+a^{2}x& \left(1+b^2\right)x& \left(1+c^2\right)x\\ \left(1+a^2\right)x& 1+b^{2}x& \left(1+c^2\right)x\\ \left(1+a^2\right)x& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right|$ then $f(x)$ is a polynomial of degree

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

$C_1\to C_1+C_2+C_3$
$f(x)=\left|\begin{array}{ccc}1+2x+x\left(a^2+b^2+c^2\right)& \left(1+b^2\right)x& \left(1+c^2\right)x\\ 1+2x+x\left(a^2+b^2+c^2\right)& 1+b^{2}x& \left(1+c^2\right)x\\ 1+2x+x\left(a^2+b^2+c^2\right)& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right|($ as $a^2+b^2+c^2=-2)$
$\left|\begin{array}{ccc}1& \left(1+b^2\right)x& \left(1+c^2\right)x\\ 1& 1+b^{2}x& \left(1+c^2\right)x\\ 1& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right||$
$R_2\to R_2-R_1\&R_3\to R_3-R_1$
$$\begin{gathered} \left|\begin{array}{ccc}1& \left(1+b^2\right)x& \left(1+c^2\right)x\\ 0& 1-x& 0\\ 0& 1-x& 1-x\end{array}\right| \\ f(x)=(1-x{)}^2=1-2x+x^2\Rightarrow (C) \end{gathered}$$