If area of the parallelogram with $$\bar{a}$$ and $$\bar{b}$$ as two adjacent sides is 20 square units, then the area of the parallelogram having $$3 \overline{\mathrm{a}}+\overline{\mathrm{b}}$$ and $$2 \overline{\mathrm{a}}+3 \overline{\mathrm{b}}$$ as two adjacent sides in square units is

If $$f(x) = {{{4^{x - \pi }} + {4^{x - \pi }} - 2} \over {{{(x - \pi )}^2}}}$$, for $$x \ne \pi $$, is continuous at $$x=\pi$$, then k =

The equation of tangent to the circle $$x^2+y^2=64$$ at the point $$\mathrm{P\left(\frac{2\pi}{3}\right)}$$ is

If $$2 f(x)-3 f\left(\frac{1}{x}\right)=x$$, then $$\int_\limits1^e f(x) d x=$$