How many values of $k $ , systeam of linear equations $\left( {k + 1} \right)x + 8y = 4k\;,\;kx + \left( {k + 3} \right)y$$ = 3k - 1$ has no solutions.
infinate
$1$
$2$
$3$
If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is
Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$x+y+z=1$ ; $2 x+N y+2 z=2$ ; $3 x+3 y+N z=3$
has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is
The system of equations $4x + y - 2z = 0\ ,\ x - 2y + z = 0$ ; $x + y - z =0 $ has
$\left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right| = $
The existence of the unique solution of the system $x + y + z = \lambda ,$ $5x - y + \mu z = 10$, $2x + 3y - z = 6$ depends on