If $$\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)$$ are given vectors, then a vector $$B$$ satifying the equations $$\overrightarrow A \times \overrightarrow B = \overrightarrow {\,C} $$ and $$\overrightarrow A .\overrightarrow B = \overrightarrow {3\,} $$ ..........

A box contains $$100$$ tickets numbered $$1, 2, ....., 100.$$ Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than $$10.$$ The minimum number on them is $$5$$ with probability ........

From the origin chords are drawn to the circle $${(x - 1)^2} + {y^2} = 1$$. The equation of the locus of the mid-points of these chords is.............

If $$a,\,b,\,c$$ and $$u,\,v,\,w$$ are complex numbers representing the vertics of two triangles such that $$c = \left( {1 - r} \right)a + rb$$ and $$w = \left( {1 - r} \right)u + rv,$$ where $$w = \left( {1 - r} \right)u + rv,$$ is a complex number, then the two triangles