Let $A$  and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

If the system of equations

$x+y+z=6$

$2x+5y+\alpha z=\beta$

$x+2y+3z=14$

has infinitely many solutions, then $\alpha +\beta$ is equal to

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations

$$\begin{gathered} x+y+z=1 \\ 2x+Ny+2z=2 \\ 3x+3y+Nz=3 \end{gathered}$$

has unique solution is $\frac{k}{6},$ then the sum of value of $k$ and all possible values of $N$ is

Let $\alpha$ be a root of the equation $\left(a-c\right)x^2+\left(b-a\right)x+\left(c-b\right)=0$ where $a, b, c$ are distinct real numbers such that the matrix $\left[\begin{array}{ccc}{\alpha }^2& \alpha & 1\\ 1& 1& 1\\ a& b& c\end{array}\right]$ is singular. Then the value of  $\frac{{\left(a-c\right)}^2}{\left(b-a\right)\left(c-b\right)}+\frac{{\displaystyle {\left(b-a\right)}^2}}{\left(a-c\right){\displaystyle \left(c-b\right)}}+\frac{{\displaystyle {\left(c-b\right)}^2}}{\left(a-c\right){\displaystyle \left(b-a\right)}}$ is