Test Preparation on Mathematics- 3D geometry-1

Let Ā be vector parallel to the line of intersection of planes P₁ ​ and P₂ ​ through the origin. P₁ ​ is parallel to the vectors 2j^{^}​ +3 k^{^} and 4 j^{^}-3 k^{^} and P₂ ​ is parallel to j^{^}-k^{^} and 3i+3j^{^} ​ , then the angle between vectors Ā and 2i^{^}+ j^{^}-2k^{^} is

The value of k such that x-4/1=y-2/1=z-k/2 lies in the plane 2x-4y+z=7 is

If the angles of a triangle are in the ratio 4: 1: 1, then the ratio of the longest side to the perimeter is

If lines x-1/2 = y+1/3 = z-1/4 and x-3/1 = y-k/2 = z/1 intersect, then find the value of k is

A variable plane at a distance of 1 unit from the origin cuts the axes at A, B, and C. If centroid D(x,y,z) of ΔABC satisfies the relation 1/x² ​+ 1/y² + 1/z² ​=k, then the value of k is