Three Dimensional Geometry — NDA Mathematics PYQs

The locus of a point $P(x,y,z)$ which moves in such a way that $z=7$ is a

Consider the following statements :

1. A line in space can have infinitely many direction ratios.
2. It is possible for certain line that the sum of the squares of direction cosines can be equal to sum of its direction cosines.

Which of the above statements is/are correct?

The $xy$-plane divides the line segment joining the points $(-1,3,4)$ and $(2,-5,6)$

The number of spheres of radius $r$ touching the coordinate axes is

$ABCDEFGH$ is a cuboid with base $ABCD$. Let $A(0,0,0)$, $B(12,0,0)$, $C(12,6,0)$ and $G(12,6,4)$ be the vertices. If $\alpha$ is the angle between $AB$ and $AG$; $\beta$ is the angle between $AC$ and $AG$, then what is the value of $\cos 2\alpha +\cos 2\beta$?