A lot contains $$50$$ defective and $$50$$ non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events $$A, B, C$$ are defined as
$$A=$$ (the first bulbs is defective)
$$B=$$ (the second bulbs is non-defective)
$$C=$$ (the two bulbs are both defective or both non defective)
Determine whether
(i) $$\,\,\,\,\,$$ $$A, B, C$$ are pairwise independent
(ii)$$\,\,\,\,\,$$ $$A, B, C$$ are independent

A unit vector coplanar with $$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$ and $$\overrightarrow i + 2\overrightarrow j + \overrightarrow k $$ and perpendicular to $$\overrightarrow i + \overrightarrow j + \overrightarrow k $$ is ...........

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points $$0,$$ $$1$$ and $$2$$ are $$0.45, 0.05$$ and $$0.50$$ respectively. Assuming that the outcomes are independent, the probability of India getting at least $$7$$ points is

Let a circle be given by 2x (x - a) + y (2y - b) = 0, $$(a\, \ne \,0,\,\,b\, \ne 0)$$. Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from $$\left( {a,\,\,{b \over 2}} \right)$$.