The existence of a unique solution for the system of equations $x+y+z=\beta$,$5x-y+\alpha z=10$,and $2x+3y-z=6$ depends on:

If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to:

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 the system of linear equations $x - 2y + kz = 1$,$2x + y + z = 2$,and $3x - y - kz = 3$ has a non-zero solution $(x, y, z) \neq 0$,then $(x, y)$ lies on the straight line whose equation is

The system of equations $x + 2y = 3$ and $2x + 3y = 3$ has

$A$ and $C$ lie in $\left[0, \frac{\pi}{2}\right)$ and $B$ lies in $[0, 2\pi]$. If $\tan A + 3 \cos B + 6 \sin C = 1$; $3 \tan A + \cos B + 4 \sin C = 4$; $5 \tan A + 3 \cos B - 8 \sin C = -2$, then $B - 2A - C =$