There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the event of selecting ith urn, $$i=1,2,3........,n$$ and $$w$$ the event of getting a white ball.

If $$P\left( {{u_i}} \right) = c,$$ (a constant) then $$P\left( {{u_n}/w} \right) = $$

Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i + \widehat j - \widehat k.$$ A vector in the plane of $$\overrightarrow a $$ and $$\overrightarrow b $$ whose projection on $$\overrightarrow c $$ is $${1 \over {\sqrt 3 }},$$ is

Let $$\theta \in \left( {0,{\pi \over 4}} \right)$$ and $${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }},\,\,\,\,{t_2} = \,\,{\left( {\tan \theta } \right)^{\cot \theta }}$$, $${t_3}\, = \,\,{\left( {\cot \theta } \right)^{\tan \theta }}$$ and $${t_4}\, = \,\,{\left( {\cot \theta } \right)^{\cot \theta }},$$then

Let $${\overrightarrow A }$$ be vector parallel to line of intersection of planes $${P_1}$$ and $${P_2}.$$ Planes $${P_1}$$ is parallel to the vectors $$2\widehat j + 3\widehat k$$ and $$4\widehat j - 3\widehat k$$ and that $${P_2}$$ is parallel to $$\widehat j - \widehat k$$ and $$3\widehat i + 3\widehat j,$$ then the angle between vector $${\overrightarrow A }$$ and a given vector $$2\widehat i + \widehat j - 2\widehat k$$ is