The area enclosed between the curves $$y = a{x^2}$$ and
$$x = a{y^2}\left( {a > 0} \right)$$ is $$1$$ sq. unit, then the value of $$a$$ is

If $$y=y(x)$$ and $${{2 + \sin x} \over {y + 1}}\left( {{{dy} \over {dx}}} \right) = - \cos x,y\left( 0 \right) = 1,$$
then $$y\left( {{\pi \over 2}} \right)$$ equals

If three distinct numbers are chosen randomly from the first $$100$$ natural numbers, then the probability that all three of them are divisible by both $$2$$ and $$3$$ is

The unit vector which is orthogonal to the vector $$3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $$ and is coplanar with the vectors $$\,2\widehat i + \widehat j + \widehat k$$ and $$\,\widehat i - \widehat j + \widehat k$$$$\,\,\,$$ is