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 $\frac{1}{{1 - a}} + \frac{1}{{1 - b}} + \frac{1}{{1 - c}} =$
$-1$
$1$
$0$
$2$
The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is
The value of $k$ for which the set of equations $x + ky + 3z = 0,$ $3x + ky - 2z = 0,$ $2x + 3y - 4z = 0$ has a non trivial solution over the set of rationals is
If the system of equations $x +y + z = 6$ ; $x + 2y + 3z= 10$ ; $x + 2y + \lambda z = 0$ has a unique solution, then $\lambda $ is not equal to
The value of $k \in R$, for which the following system of linear equations
$3 x-y+4 z=3$
$x+2 y-3 x=-2$
$6 x+5 y+k z=-3$
has infinitely many solutions, is:
The value of a for which the system of equations ; $a^3x + (a +1)^3 y + (a + 2)^3 \, z = 0$ ,$ax + (a + 1) y + (a + 2)\, z = 0$ & $x + y + z = 0$ has a non-zero solution is :