Let the system of equations: $2x + 3y + 5z = 9$,$7x + 3y - 2z = 8$,$12x + 3y - (4 + \lambda)z = 16 - \mu$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4x = 3y$ is

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 system of equations $x+2y+3z=6$,$x+3y+5z=9$,and $2x+5y+az=12$ has no solution when $a=$

If the point $P(\alpha, \beta, \gamma)$ lies on the plane $2x + y + z = 1$ and $\begin{bmatrix} \alpha & \beta & \gamma \end{bmatrix} \begin{bmatrix} 1 & 9 & 1 \\ 8 & 2 & 1 \\ 7 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$,then $\alpha^2 + \beta^2 + \gamma^2 = $

The values of $\lambda$ and $\mu$ for which the system of linear equations $x+y+z=2$,$x+2y+3z=5$,and $x+3y+\lambda z=\mu$ has infinitely many solutions are,respectively:

Consider the following system of equations in matrix form $\begin{bmatrix} 1 \\ 2 \\ \lambda \end{bmatrix} \begin{bmatrix} 1 & 2 & \lambda \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = 0$. Then which one of the following statements is true?