The number of distinct real values of $\lambda$ for which the lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{\lambda^2}$ and $\frac{x - 3}{1} = \frac{y - 2}{\lambda^2} = \frac{z - 1}{2}$ are coplanar is

Find the position vector of the point where the line $r = (i - j + k) + t(i + j + k)$ meets the plane $r \cdot (i + j - k) = 5$.

The equation of a plane containing the line $\frac{x + 1}{-3} = \frac{y - 3}{2} = \frac{z + 2}{1}$ and the point $(0, 7, -7)$ is

Let the coordinates of one vertex of $\triangle ABC$ be $A(0, 2, \alpha)$ and the other two vertices lie on the line $\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$. For $\alpha \in \mathbb{Z}$,if the area of $\triangle ABC$ is $21$ sq. units and the line segment $BC$ has length $2\sqrt{21}$ units,then $\alpha^2$ is equal to $...........$.

$A$ line $L$ passes through the points $(1, 2, -3)$ and $(3, 3, -1)$ and a plane $\pi$ passes through the points $(2, 1, -2), (-2, -3, 6)$,and $(0, 2, -1)$. If $\theta$ is the angle between the line $L$ and plane $\pi$,then $27 \cos^2 \theta = $

If the line $\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6}$ lies in the plane $3x-14y+6z+49=0$,then the value of $m$ is