If the straight lines $2x - y + 1 = 0$,$4x + y + 2 = 0$,and $x + y - k = 0$ are concurrent,then $k$ equals

Let $C$ be the centroid of the triangle with vertices $(3, -1), (1, 3),$ and $(2, 4).$ Let $P$ be the point of intersection of the lines $x + 3y - 1 = 0$ and $3x - y + 1 = 0.$ Then the line passing through the points $C$ and $P$ also passes through the point

If the three points $A(1, 6)$, $B(3, -4)$, and $C(x, y)$ are collinear, then the equation satisfied by $x$ and $y$ is:

The equation of a line passing through the point of intersection of the lines $x + 5y + 7 = 0$ and $3x + 2y - 5 = 0$,and perpendicular to the line $7x + 2y - 5 = 0$,is given by

The number of values of $a$ for which the system of equations $a^2 x + (2 - a) y = 4 + a^2$ and $a x + (2 a - 1) y = a^5 - 2$ possesses no solution is:

The value of $\lambda$ for which the lines $3x + 4y = 5,$ $5x + 4y = 4,$ and $\lambda x + 4y = 6$ meet at a point is