If $x=a, y=b, z=c$ is the solution of the system of simultaneous linear equations $x+y+z=4$, $x-y+z=2$, and $x+2y+2z=1$, then $ab+bc+ca=$

If $\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}2 \\ 4\end{array}\right]$,then the values of $x$ and $y$ respectively are

Let $a$ be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b = \lim_{x \rightarrow 0} \left( \frac{\int_0^x \frac{\ln(1+t)}{t^{2024}+1} dt}{x^2} \right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root,where $c, d, e \in \mathbb{R}$,then $d:c:e$ equals

If $S$ is the set of distinct values of $b$ for which the following system of linear equations $x + y + z = 1$,$x + ay + z = 1$,and $ax + by + z = 0$ has no solution,then $S$ is:

If the system of equations $2x + 3y - z = 5$,$x + \alpha y + 3z = -4$,and $3x - y + \beta z = 7$ has infinitely many solutions,then $13\alpha\beta$ is equal to

For the system of linear equations:
$x - 2y = 1, x - y + kz = -2, ky + 4z = 6, k \in R$
Consider the following statements:
$(A)$ The system has a unique solution if $k \neq 2, k \neq -2$.
$(B)$ The system has a unique solution if $k = -2$.
$(C)$ The system has a unique solution if $k = 2$.
$(D)$ The system has no solution if $k = 2$.
$(E)$ The system has an infinite number of solutions if $k \neq -2$.
Which of the following statements are correct?