Question

If $If p\lambda^{4} +q\lambda^{3} +r\lambda^{2} +s\lambda +t=\begin{vmatrix}\lambda^{2} +3\lambda&\lambda -1&\lambda +3\\ \lambda +1&2 -\lambda&\lambda -4\\ \lambda -3&\lambda +4&3\lambda\end{vmatrix}$ then the value of $t$ is

Answer 1

Since it is an indentity in$\lambda$ so it is satisfied by every value of $\lambda$. Now put $\lambda = 0$ in the given equation, we have
$t =\begin{vmatrix}0&-1&3\\ 1&2&-4\\ -3&4&0\end{vmatrix} = 12 + 30 =18$