If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30^o and the angle of depression of reflection of the cloud in the lake from P be 60^o, then the height of the cloud (in meters) from the surface of the lake is :

If an angle between the line, $${{x + 1} \over 2} = {{y - 2} \over 1} = {{z - 3} \over { - 2}}$$ and the plane, $$x - 2y - kz = 3$$ is $${\cos ^{ - 1}}\left( {{{2\sqrt 2 } \over 3}} \right),$$ then a value of k is :

The tangent to the curve y = x² – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point :

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors, out of which vectors $$\overrightarrow b $$ and $$\overrightarrow c $$ are non-parallel. If $$\alpha $$ and $$\beta $$ are the angles which vector $$\overrightarrow a $$ makes with vectors $$\overrightarrow b $$ and $$\overrightarrow c $$ respectively and $$\overrightarrow a $$ $$ \times $$ ($$\overrightarrow b $$ $$ \times $$ $$\overrightarrow c $$) = $${1 \over 2}\overrightarrow b $$, then $$\left| {\alpha - \beta } \right|$$ is equal to :