If $$A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$$ and $$A_{i j}$$ is a cofactor of $$a_{i j}$$ then the value of $$a_{21} A_{21}+a_{22} A_{22}+a_{23} A_{23}$$ is

Rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant and it is found that the number doubles in 6 hours. The number of bacteria becomes ________ times at the end of 18 hours.

The slope of the normal to the curve $$x=\sqrt{t}$$ and $$y=t-\frac{1}{\sqrt{t}}$$ at $$t=4$$ is

If $$3 \mathrm{f}(x)-\mathrm{f}\left(\frac{1}{x}\right)=8 \log _2 x^3, x>0$$, then $$\mathrm{f}(2), \mathrm{f}(4)$$, $$f(8)$$ are in