If [t] denotes the greatest integer $$\le \mathrm{t}$$, then the value of $${{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx} $$ is :

If $$\mathrm{P}(\mathrm{h}, \mathrm{k})$$ be a point on the parabola $$x=4 y^{2}$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the distance of $$\mathrm{P}$$ from the directrix of the parabola $$\quad y^{2}=4(x+y)$$ is equal to :

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are three non-zero vectors and $$\widehat n$$ is a unit vector perpendicular to $$\overrightarrow c $$ such that $$\overrightarrow a = \alpha \overrightarrow b - \widehat n,(\alpha \ne 0)$$ and $$\overrightarrow b \,.\overrightarrow c = 12$$, then $$\left| {\overrightarrow c \times (\overrightarrow a \times \overrightarrow b )} \right|$$ is equal to :

If the coefficient of $$x^{15}$$ in the expansion of $$\left(\mathrm{a} x^{3}+\frac{1}{\mathrm{~b} x^{1 / 3}}\right)^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $$\left(a x^{1 / 3}-\frac{1}{b x^{3}}\right)^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$(\mathrm{a}, \mathrm{b})$$ :