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
$R$
$R -\{-11,13\}$
$R -\{13\}$
$R -\{-11,11\}$
If the system of equations $x + y+ z = 5$ ; $x + 2y + 3z = 9$ ; $x + 3y + \alpha z = \beta $ has infinitely many solutions, then $\beta - \alpha $ equals
If $ 5$ is one root of the equation $\left| {\,\begin{array}{*{20}{c}}x&3&7\\2&x&{ - 2}\\7&8&x\end{array}\,} \right| = 0$, then other two roots of the equation are
If $A=\left[\begin{array}{lll}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right],$ find $|A|$.
Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3 x+2 y+5 z=3$ $9 x+4 y+(28+[\lambda]) z=[\lambda]$ has a solution is:
The number of solutions of equations $x + y - z = 0$, $3x - y - z = 0, \,x - 3y + z = 0$ is