What is the product of roots of equation \(\begin{vmatrix}1+2x & 1 & 1-x \\2-x & 2+x & 3+x \\ x & 1+x & 1-x^2 \end{vmatrix}\) = 0?

\(\frac{1}{2}\)

f(x) = \(\begin{vmatrix}1+2x & 1 & 1-x \\2-x & 2+x & 3+x \\ x & 1+x & 1-x^2 \end{vmatrix}\) = 0

Constant term = f(0) = \(\begin{vmatrix}1 & 1 & 1 \\2 & 2 & 3 \\ 0 & 1 & 1 \end{vmatrix}\) = 0 = 2+2-2-3 = -1

Also coefficient of x4 is -2

hence product of roots is \(\frac{1}{2}\)