Let $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix}$,$B = \begin{bmatrix} 6 \\ 11 \\ 0 \end{bmatrix}$ and $X = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$. If $AX = B$,then the value of $2a + b + 2c$ is:

If the system of equations $2x - y + z = 4$,$5x + \lambda y + 3z = 12$,and $100x - 47y + \mu z = 212$ has infinitely many solutions,then $\mu - 2\lambda$ is equal to

Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x - y - z = 0$,$-3x + z = 0$,$-3x + 2y + z = 0$.
Then the number of such points for which $x^2 + y^2 + z^2 \leq 100$ is:

The ordered pair $(a, b)$,for which the system of linear equations $3x - 2y + z = b$,$5x - 8y + 9z = 3$,and $2x + y + az = -1$ has no solution,is

Let $M$ be a $3 \times 3$ matrix such that $M \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$,$M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}$ and $M \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$. If $M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}$,then $x + y + z$ equals :

If the system of linear equations $8x + y + 4z = -2$,$x + y + z = 0$,and $\lambda x - 3y = \mu$ has infinitely many solutions,then the distance of the point $\left(\lambda, \mu, -\frac{1}{2}\right)$ from the plane $8x + y + 4z + 2 = 0$ is: