If a line makes angles of measure $\frac{\pi}{6}$ and $\frac{\pi}{3}$ with $X$ and $Y$ axes respectively,then the angle made by the line with $Z$ axis is

Which of the following conditions is satisfied by a point lying on the $z$-axis?

If the direction cosines of two lines satisfy the equations $2l+m-n=0$ and $l^2-2m^2+n^2=0$, and $\theta$ is the angle between the lines, then $\cos \theta=$

If the relation between the direction ratios of two lines in $\mathbb{R}^3$ are given by $l+m+n=0$ and $2lm+2mn-ln=0$, then the angle between the lines is ($l, m, n$ have their usual meaning).

The angle between the lines whose direction cosines are $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)$ and $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)$ is

If a variable line in two adjacent positions has direction cosines $l, m, n$ and $l+\delta l, m+\delta m, n+\delta n,$ show that the small angle $\delta \theta$ between the two positions is given by $\delta \theta^{2}=\delta l^{2}+\delta m^{2}+\delta n^{2}$.