The particular solution of the diffrential equation $$y(1+\log x)=\left(\log x^x\right) \frac{d y}{d x}$$, when $$y(e)=e^2$$ is

If statements $$\mathrm{p}$$ and $$\mathrm{q}$$ are true and $$\mathrm{r}$$ and $$\mathrm{s}$$ are false, then truth values of $$\sim(\mathrm{p} \rightarrow \mathrm{q}) \leftrightarrow(\mathrm{r} \wedge \mathrm{s})$$ and $$(\sim \mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{r} \leftrightarrow \mathrm{s})$$ are respectively.

Bismath has half life period of 5 days. A sample originally has a mass of 1000 mg, then the mass of Bismath after 30 days is

In $$\triangle A B C$$, with usual notations, $$2 a b \sin \frac{1}{2}(A+B-C)=$$