The system of equations : $2x\, \cos^2\theta + y\, \sin2\theta - 2\sin\theta = 0$ $x\, \sin2\theta + 2y\, \sin^2\theta = - 2\, \cos\theta$ $x\, \sin\theta - y \cos\theta = 0$ , for all values of $\theta$ , can

If $\left| {\begin{array}{*{20}{c}}{x - 4}&{2x}&{2x}\\{2x}&{x - 4}&{2x}\\{2x}&{2x}&{x - 4}\end{array}} \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2},$ then the ordered pair $\left( {A,B} \right) = $. . . . .

Let $\lambda $ be a real number for which the system of linear equations $x + y + z = 6$
 ; $4x + \lambda y - \lambda z = \lambda - 2$ ; $3x + 2y -4z = -5$ Has indefinitely many solutions. Then $\lambda $ is a root of the quadratic equation

If $S$ is the set of distinct values of $'b'$ for which the following system of linear equations $x + y + z = 1;x + ay + z = 1;ax + by + z = 0$ has no solution , then $S$ is :

If the system of equations, $x + 2y -3z = 1, (k + 3) z = 3, (2k + 1)x + z = 0$ is inconsistent, then the value of $k$ is :-

Consider the system of equations

$ x-2 y+3 z=-1 $ ; $ -x+y-2 z=k $ ; $ x-3 y+4 z=1$

$STATEMENT -1$ : The system of equations has no solution for $\mathrm{k} \neq 3$. and

$STATEMENT - 2$ : The determinant $\left|\begin{array}{ccc}1 & 3 & -1 \\ -1 & -2 & \mathrm{k} \\ 1 & 4 & 1\end{array}\right| \neq 0$, for $\mathrm{k} \neq 3$.