The values of $\lambda$ and $\mu$ for which the system of linear equations

$x+y+z=2$

$x+2 y+3 z=5$

$x+3 y+\lambda z=\mu$

has infinitely many solutions are, respectively

Let $m$ and $M$ be respectively the minimum and maximum values of

$\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$.

Then the ordered pair $( m , M )$ is equal to

The sum of distinct values of $\lambda$ for which the system of equations

$(\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0$

$(\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0$

$2 x+(3 \lambda+1) y+3(\lambda-1) z=0$

has non-zero solutions, is

Let $\alpha, \beta, \gamma$ be the real roots of the equation, $x ^{3}+ ax ^{2}+ bx + c =0,( a , b , c \in R$ and $a , b \neq 0)$ If the system of equations (in, $u,v,w$) given by $\alpha u+\beta v+\gamma w=0, \beta u+\gamma v+\alpha w=0$ $\gamma u +\alpha v +\beta w =0$ has non-trivial solution, then the value of $\frac{a^{2}}{b}$ is

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) = $. . . . .

$\left| {\,\begin{array}{*{20}{c}}{{{({a^x} + {a^{ - x}})}^2}}&{{{({a^x} - {a^{ - x}})}^2}}&1\\{{{({b^x} + {b^{ - x}})}^2}}&{{{({b^x} - {b^{ - x}})}^2}}&1\\{{{({c^x} + {c^{ - x}})}^2}}&{{{({c^x} - {c^{ - x}})}^2}}&1\end{array}\,} \right| = $