If the system of linear equations $x + y + z = 5$ ; $x = 2y + 2z = 6$ ; $x + 3y + \lambda z = u (\lambda \, \mu \in R)$, has infinitely many solutions then the value of $\lambda  + \mu $ is

One of the roots of the given equation $\left| {\,\begin{array}{*{20}{c}}{x + a}&b&c\\b&{x + c}&a\\c&a&{x + b}\end{array}\,} \right| = 0$ is

The values of $\mathrm{m}, \mathrm{n}$, for which the system of equations

$ x+y+z=4 $

$ 2 x+5 y+5 z=17 $

$ x+2 y+m z=n$

has infinitely many solutions, satisfy the equation :

The value of $a$ for which the system of equations

$a^3x + ( a + 1)^3y + (a + 2)^3z = 0$ ; $ax + (a + 1) y + ( a + 2) z = 0$ ; $x + y + z = 0$, has a non zero solution is

Show that points $A(a, b+c), B(b, c+a), C(c, a+b)$ are collinear

The system of linear equations $x + y + z = 2$, $2x + y - z = 3,$ $3x + 2y + kz = 4$has a unique solution if