Question

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

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Answer 1

Answer: (B)

Applying $C_1\to C_1+C_2+C_3$, we get
$f(x)=\left|\begin{array}{ccc}1& \left(1+b^2\right)x& \left(1+c^2\right)x\\ 1& \left(1+b^{2}x\right)& \left(1+c^2\right)x\\ 1& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right|$
$\left(\because a^2+b^2+c^2+2=0\right)$
Again, applying $R_2\to R_2-R_1,R_3\to R_3-R_1$,
we get
$=\left|\begin{array}{ccc}1& \left(1+b^2\right)x& \left(1+c^2\right)x\\ 0& 1-x& 0\\ 0& 0& 1-x\end{array}\right|=(1-x{)}^2$
Hence, degree of $f(x)=2$