The following system of linear equations $2 x+3 y+2 z=9$ ; $3 x+2 y+2 z=9$ ;$x-y+4 z=8$
has a solution $(\alpha, \beta, \gamma)$ satisfying $\alpha+\beta^{2}+\gamma^{3}=12$
has infinitely many solutions
does not have any solution
has a unique solution
The area of a triangle is $5$ and two of its vertices are $A(2, 1), B(3, -2)$. The third vertex which lies on line $y = x + 3$ is-
If the lines $ax + y + 1 = 0$, $x + by + 1 = 0$ and $x + y + c = 0$ (where $a, b$ and $c$ are distinct and different from $1$ ) are concurrent, then the value of $\frac{1}{{1 - a}} + \frac{1}{{1 - b}} + \frac{1}{{1 - c}} =$
$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2} - bc}\\1&b&{{b^2} - ac}\\1&c&{{c^2} - ab}\end{array}\,} \right| = $
In a $\Delta ABC,$ if $\left| {\,\begin{array}{*{20}{c}}1&a&b\\1&c&a\\1&b&c\end{array}\,} \right| = 0$, then ${\sin ^2}A + {\sin ^2}B + {\sin ^2}C = $
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 :