If for some $m \in \mathbb{R}$ the lines $L_1 : \dfrac{x+1}{m} = \dfrac{y-m}{-1} = \dfrac{z-1}{1}$ and $L_2 : \dfrac{x+2}{-4} = \dfrac{y+1}{9} = \dfrac{z+1}{1}$ are coplanar, then line $L_1$ passes through the point

If the lines $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(q\hat{i} - 2\hat{j} + \hat{k})$ and $\vec{r} = p\hat{i} - 3\hat{j} + 2\hat{k} + \mu(\hat{i} - 2\hat{j} + 2\hat{k})$ intersect each other and $q\hat{i} - 2\hat{j} + \hat{k}$ is collinear to $4\hat{i} - 4\hat{j} + 2\hat{k}$, then the values of $p$ and $q$ are

The angle between a diagonal and one of its edges of a cube is ...................

For the linear programming problem, $x + 2y \leq 10,\ 3x + y \leq 12,\ x, y \geq 0$, the maximum value of $z = 5x + 10y$ occurs at every point on the line segment joining the points..