The area (in sq. units) of the region described by

$$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$ and $$\left. {y \ge 4x - 1} \right\}$$ is :

Let $$y(x)$$ be the solution of the differential equation

$$\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).$$ Then $$y(e)$$ is equal to :

If $$12$$ different balls are to be placed in $$3$$ identical boxes, then the probability that one of the boxes contains exactly $$3$$ balls is :

Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of them are collinear and

$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\overrightarrow a .$$ If $$\theta $$ is the angle between vectors $$\overrightarrow b $$ and $${\overrightarrow c }$$ , then a value of sin $$\theta $$ is :