The solution of the linear system of equations $\begin{bmatrix} 2 & 2 & 3 \\ 7 & 1 & 1 \\ 0 & 6 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 y + 11 \\ 6 z - 1 \\ 5 y + 11 \end{bmatrix} + \begin{bmatrix} x \\ x \\ 4 z \end{bmatrix} + \begin{bmatrix} z \\ 3 x \\ 4 y \end{bmatrix}$ is

For what value of $k$ does the following system of equations possess a non-trivial solution?
$x + ky + 3z = 0$
$3x + ky - 2z = 0$
$2x + 3y - 4z = 0$

$A$ 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 $B - 2A - C =$

Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations:
$a x + 2 y = \lambda$
$3 x - 2 y = \mu$
Which of the following statement$(s)$ is(are) correct?
$(A)$ If $a = -3$,then the system has infinitely many solutions for all values of $\lambda$ and $\mu$.
$(B)$ If $a \neq -3$,then the system has a unique solution for all values of $\lambda$ and $\mu$.
$(C)$ If $\lambda + \mu = 0$,then the system has infinitely many solutions for $a = -3$.
$(D)$ If $\lambda + \mu \neq 0$,then the system has no solution for $a = -3$.

The system of linear equations $x+y+z=6, x+2y+3z=10$ and $x+2y+az=b$ has no solutions when

The system of equations $\lambda x - y + (\cos\theta) z = 0$,$3x + y + 2z = 0$,and $(\cos\theta) x + y + 2z = 0$ for $0 < \theta < 2\pi$ has non-trivial solution$(s)$: