If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are distinct vectors such that
$$\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d \,.$$ Prove that
$$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) \ne 0\,\,i.e.\,\,\,\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c \ne \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $$

Find the centre and radius of circle given by $$\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$$

where, $${\rm{z = x + iy, }}\alpha {\rm{ = }}\,{\alpha _1}{\rm{ + i}}{\alpha _2}{\rm{,}}\,\beta = {\beta _1}{\rm{ + i}}{\beta _2}{\rm{ }}$$

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.

A box contains $$12$$ red and $$6$$ white balls. Balls are drawn from the box one at a time without replacement. If in $$6$$ draws there are at least $$4$$ white balls, find the probability that exactly one white is drawn in the next two draws. (binomial coefficients can be left as such)