$S$ denote the set of all real values of $\lambda$ such that the system of equations  $\lambda x + y + z =1$ ; $x +\lambda y + z =1$ ; $x + y +\lambda z =1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to

If $D(x) =$ $\left| {\begin{array}{*{20}{c}}{x - 1}&{{{(x - 1)}^2}}&{{x^3}}\\{x - 1}&{{x^2}}&{{{(x + 1)}^3}}\\x&{{{(x + 1)}^2}}&{{{(x + 1)}^3}}\end{array}} \right|$ then the coefficient of $x$ in $D(x)$ is

The solutions of the equation $\left| {\,\begin{array}{*{20}{c}}x&2&{ - 1}\\2&5&x\\{ - 1}&2&x\end{array}\,} \right| = 0$ are

Let $A =$ $\left[ {\begin{array}{*{20}{c}}1&{\sin \theta }&1\\{ - \sin \theta }&1&{\sin \theta }\\{ - 1}&{ - \sin \theta }&1\end{array}} \right]$, where $0 \le \theta < 2\pi$ , then

If the system of equations $2x + 3y - z = 0$, $x + ky - 2z = 0$ and  $2x - y + z = 0$ has a non -trivial solution $(x, y, z)$, then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

If $a \ne p,b \ne q,c \ne r$ and $\left| {\,\begin{array}{*{20}{c}}p&b&c\\{p + a}&{q + b}&{2c}\\a&b&r\end{array}\,} \right|$ =$ 0$, then $\frac{p}{{p - a}} + \frac{q}{{q - b}} + \frac{r}{{r - c}} = $