Consider the following linear equations $$ax+by+cz=0;$$ $$\,\,\,$$ $$bx+cy+az=0;$$ $$\,\,\,$$ $$cx+ay+bz=0$$

Match the conditions/expressions in Column $$I$$ with statements in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS.$$

$$\,\,\,$$ Column $$I$$
(A)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$
(B)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(C)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(D)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$

$$\,\,\,$$ Column $$II$$
(p)$$\,\,\,$$ the equations represents planes meeting only at asingle point
(q)$$\,\,\,$$ the equations represents the line $$x=y=z.$$
(r)$$\,\,\,$$ the equations represent identical planes.
(s) $$\,\,\,$$ the equations represents the whole of the three dimensional space.

Find the equation of the plane containing the line $$2x-y+z-3=0,3x+y+z=5$$ and at a distance of $${1 \over {\sqrt 6 }}$$ from the point $$(2, 1, -1).$$

Find the equation of plane passing through $$(1, 1, 1)$$ & parallel to the lines $${L_1},{L_2}$$ having direction ratios $$(1,0,-1),(1,-1,0).$$ Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.

$${P_1}$$ and $${P_2}$$ are planes passing through origin. $${L_1}$$ and $${L_2}$$ are two line on $${P_1}$$ and $${P_2}$$ respectively such that their intersection is origin. Show that there exists points $$A, B, C,$$ whose permutation $$A',B',C'$$ can be chosen such that (i) $$A$$ is on $${L_1},$$ $$B$$ on $${P_1}$$ but not on $${L_1}$$ and $$C$$ not on $${P_1}$$ (ii) $$A'$$ is on $${L_2},$$ $$B'$$ on $${P_2}$$ but not on $${L_2}$$ and $$C'$$ not on $${P_2}$$