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

The system of linear equations $x + y + z = 6$, $2x + 5y + az = 36$, and $x + 2y + 3z = b$ has:

If the system of equations
$x-2y+3z=9$
$2x+y+z=b$
$x-7y+az=24$
has infinitely many solutions,then $a-b$ is equal to

If the system of linear equations,$x+y+z = 6$,$x+2y+3z = 10$,and $3x+2y+\lambda z = \mu$ has more than two solutions,then $\mu-\lambda^{2}$ is equal to

Let $\alpha$ be a solution of $x^2+x+1=0$,and for some $a$ and $b$ in $\mathbb{R}$,$\begin{bmatrix} 4 & a & b \end{bmatrix} \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$. If $\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$,then $m + n$ is equal to

If $x=\alpha, y=\beta, z=\gamma$ is the unique solution of the system of linear equations $2x-3y+5z=12$, $5x+2y+3z=11$, and $x+2y-3z=-3$, then $2\alpha+5\beta+3\gamma=$