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  4. Product of roots of equation |1+2x&1&1-x 2-x&2+x&3+x x&1+x&1-x2|=0 is .

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Q. Product of roots of equation $\begin{vmatrix}1+2x&1&1-x\\ 2-x&2+x&3+x\\ x&1+x&1-x^{2}\end{vmatrix}=0$ is ____.

Determinants

Solution:

$f \left(x\right)\begin{vmatrix}1+2x&1&1-x\\ 2-x&2+x&3+x\\ x&1+x&1-x^{2}\end{vmatrix}=0$
Constant term $=f \left(0\right)=\begin{vmatrix}1&1&1\\ 2&2&3\\ 0&1&1\end{vmatrix}$
$= 2 + 2 - 2 -3 = - 1$
Also coefficient of $x^{4}$ is '-2'
Hence product of roots is $p=1 / 2$