If $p{\lambda ^4} + q{\lambda ^3} + r{\lambda ^2} + s\lambda + t = $ $\left| {\,\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda }&{\lambda - 1}&{\lambda + 3}\\{\lambda + 1}&{2 - \lambda }&{\lambda - 4}\\{\lambda - 3}&{\lambda + 4}&{3\lambda }\end{array}\,} \right|,$ the value of $t$ is

If the system of equations $\mathrm{x}+4 \mathrm{y}-\mathrm{z}=\lambda$, $7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu .+3 \lambda)$ is equal to :

The value of $\left| {\begin{array}{*{20}{c}}
1&x&y\\
2&{\sin x + 2x}&{\sin y + 2y}\\
3&{\cos x + 3x}&{\cos y + 3y}
\end{array}} \right|$ is

If $A = \left| {\,\begin{array}{*{20}{c}}{ - 1}&2&4\\3&1&0\\{ - 2}&4&2\end{array}\,} \right|$and $B = \left| {\,\begin{array}{*{20}{c}}{ - 2}&4&2\\6&2&0\\{ - 2}&4&8\end{array}\,} \right|$, then $B$ is given by

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 :

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set