The direction cosines of a line which is perpendicular to the lines $\dfrac{x-7}{2} = \dfrac{y+17}{-3} = \dfrac{z-6}{1}$ and $\dfrac{x+5}{1} = \dfrac{y+3}{2} = \dfrac{z-6}{-2}$ are...

The equation of the plane containing the lines $\dfrac{x-1}{2} = \dfrac{y+1}{\lambda} = \dfrac{z}{2}$ and $\dfrac{x+1}{5} = \dfrac{y+1}{2} = \dfrac{z}{\lambda}$ is

If a unit vector makes angles $\dfrac{\pi}{4}$ with $\hat{i}$, $\dfrac{\pi}{3}$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$, then a value of $\theta$ is equal to...

The minimum value of $Z = 3x + y$, subject to the constraints $2x + 3y \leq 6, x + y \geq 1, x \geq 0, y \geq 0$ is....