Question

If $a^2+b^2+c^2=-2$ and $f(x)=\begin{vmatrix} 1+a^2x &(1+b^2)x & (1+c^2)x \\[0.3em] (1+a^2)x& 1+b^2x &(1+c^2)x \\[0.3em] (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}$ then f(x) is a polynomial of degree

• A. 2
• B. 3
• C. 0
• D. 1

Answer 1

Answer: (A)

Operate $C_1 + C_2 + C_3$, we get
$ f(x) = \begin{vmatrix} 1+a^2x +b^2x + x c^2 x &(1+b^2)x & (1+c^2)x \\[0.3em] x+a^2 x+1 + b^2 x +x +c^2 x& 1+b^2x &(1+c^2)x \\[0.3em] x+a^2x+x +b^2x +1 +c^2 x & (1+b^2)x & 1+c^2x \end{vmatrix}$
= $ (1+2x + (a^2 + b^2 + c2)x)$
$\begin{vmatrix} 1&(1+b^2)x & (1+c^2)x \\[0.3em] 1& 1+b^2x &(1+c^2)x \\[0.3em] 1 & (1+b^2)x & 1+c^2x \end{vmatrix}$
$\begin{vmatrix} 1 &(1+b^2)x & (1+c^2)x \\[0.3em] 0& 1-x &0 \\[0.3em] 0 & 0& 1-x \end{vmatrix}$
[$\because \, a^2 + b^2 +c^2 = - 2 \, \therefore \, 1 +2x + (ca^2 + b^2 +c^2 ) x =1 + 2x -2x =1$]
= $(1 - x)^2$ which is a polynomial of degree 2.