The value of $k$ for which the following lines are concurrent is

$x-y-1=0 \newline 2x+3y-12=0 \newline 2x-3y+k=0$

Two points on a ground are 1 m apart. If a cow moves in the field in such a way that its distance from the two points is always in ratio $3: 2$ then

The x-intercept of the line that passes through the intersection of the lines $x + 2y = 4$ and $2x + 3y = 6$, and is perpendicular to the line $3x - y = 2$ is

If one of the lines given by the equation $2x^2 + axy + 3y^2 = 0$ coincides with one of those given by $2x^2 + bxy - 3y^2 = 0$ and the other lines represented by them are perpendicular then $a^2+b^2 =$

The equation of the straight line passing through the point $M (-5,4)$, such that the portion of it between the axes is divided by the point $M$ into two equal halves, is

Let $A(1,3)$ and $B(5,1)$ be two points. If a line with slope $m$ intersects $AB$ at an angle of $45°$, then the possible values of $m$ are

Consider the polynomials $f(x) = ax^2 + bx + c$, where $a > 0, b, c$ are real, $g(x) = -2x$. If $f(x)$ cuts the x-axis at $(-2, 0)$ and $g(x)$ passes through $(a, b)$, then the minimum value of $f(x) + 9a + 1$ is