$A$ line makes an angle $\theta$ with the $X$ and $Z$ axes and an angle $\beta$ with the $Y$ axis. If $\sin^2 \beta = 3 \sin^2 \theta$,then $\cos^2 \theta = \dots$

Let $ABC$ be a triangle with vertices at points $A(2, 3, 5)$,$B(-1, 3, 2)$ and $C(\lambda, 5, \mu)$ in three-dimensional space. If the median through $A$ is equally inclined with the axes,then $(\lambda, \mu)$ is equal to

The direction ratios of the line joining the points $(4, 3, -5)$ and $(-2, 1, -8)$ are

If $l, m, n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $1, 2, -1$ and $1, -2, 1$,then $(l+m+n)^2=$

The direction cosines of the line passing through $P(2, 3, -1)$ and the origin are

If the angle between the lines whose direction cosines are $\left(-\frac{2}{\sqrt{21}}, \frac{C}{\sqrt{21}}, \frac{1}{\sqrt{21}}\right)$ and $\left(\frac{3}{\sqrt{54}}, \frac{3}{\sqrt{54}}, -\frac{6}{\sqrt{54}}\right)$ is $\frac{\pi}{2}$, then the value of $C$ is