If a2+b2+c2=-2 and f(x)=1+a2x(1+b2)x(1+c2)x(1+a2)x1+b2x(1+c2)x(1+a2)x(1+b2)x1+c2x then f(x) is a polynomial of degree

C1→C1+C2+C3 =1+2xa2+b2+c2x(1+b2)x(1+c2)x1+2x+a2+b2+c2x1+b2x(1+c2)x1+2x+a2+b2+c2x(1+b2)x1+c2x =11+b2x1+c2x11+b2x1+c2x11+b2x1+c2x ∴ a2+b2+c2=-2 =11+b2x1+c2x01-x0001-x R2→R2-R1, R3→R3-R1 =1-x2∴ f(x) is a polynomial of degree '2'

If a2+b2+c2=-2 and f(x)=1+a2x(1+b2)x(1+c2)x(1+a2)x1+b2x(1+c2)x(1+a2)x(1+b2)x1+c2x then f(x) is a polynomial of degree

C1→C1+C2+C3 =1+2xa2+b2+c2x(1+b2)x(1+c2)x1+2x+a2+b2+c2x1+b2x(1+c2)x1+2x+a2+b2+c2x(1+b2)x1+c2x =11+b2x1+c2x11+b2x1+c2x11+b2x1+c2x ∴ a2+b2+c2=-2 =11+b2x1+c2x01-x0001-x R2→R2-R1, R3→R3-R1 =1-x2∴ f(x) is a polynomial of degree '2'

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

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Detailed Solution

$\begin{array}{rcl} C_{1} & \to & C_{1} + C_{2} + C_{3} \\ = & |\begin{matrix} 1 + 2 x (a^{2} + b^{2} + c^{2}) x & (1 + b^{2}) x & (1 + c^{2}) x \\ 1 + 2 x + (a^{2} + b^{2} + c^{2}) x & 1 + b^{2} x & (1 + c^{2}) x \\ 1 + 2 x + (a^{2} + b^{2} + c^{2}) x & (1 + b^{2}) x & 1 + c^{2} x \end{matrix}| \\ = & |\begin{matrix} 1 & (1 + b^{2}) x & (1 + c^{2}) x \\ 1 & 1 + b^{2} x & (1 + c^{2}) x \\ 1 & (1 + b^{2}) x & 1 + c^{2} x \end{matrix}| (∴ a^{2} + b^{2} + c^{2} = - 2) \\ = & |\begin{matrix} 1 & (1 + b^{2}) x & (1 + c^{2}) x \\ 0 & 1 - x & 0 \\ 0 & 0 & 1 - x \end{matrix}| (R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}) \\ = & {(1 - x)}^{2} \end{array}$