If (2,3,4) is the centroid of the tetrahedron for which  (2,3,−1),(3,0,−2),(−1,4,3)are three vertices , then the distance of the fourth vertex from the origin is

The centroid of the tetrahedron formed by the vertices  (x1,y1,z1),(x2,y2,z2),(x3,y3,z3),(x4,y4,z4) is Gx1+x2+x3+x44,y1+y2+y3+y44,z1+z2+z3+z44Hence, Fourth vertex is D=4G−A−B−C=(8−2−3+1,12−3−4,16+1+2−3)=(4,5,16)The distance between the fourth vertex and the centroid is DG=16+25+256=297Therefore, the required distance is 333