The point of intersection of the two lines $\dfrac{x-3}{3} = \dfrac{y-3}{-1}, z - 1 = 0$ and $\dfrac{x-6}{2} = \dfrac{z-1}{3}, y - 2 = 0$ is....

If the centroid of a tetrahedron $OABC$ is $(1, 2, -1)$, where O is the origin, $A(a, 2, 3), B(1, b, 2), C(2, 1, c)$ are the other vertices, then the distance of the point $P(a, b, c)$ from the origin is...

If the lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-3}{1} = \dfrac{y-c}{2} = \dfrac{z}{1}$ intersect, then the radius of circle $x^2 + y^2 - 4x + 10y + c = 0$ is

The corner points of the feasible region determined by a system of linear constraints are $(0, 3), (1, 1)$ and $(3, 0)$. If the objective function is $z = px + qy$, where $p, q > 0$, then the condition on p and q such that the minimum of z occurs at both $(3, 0)$ and $(1, 1)$ is _____