The maximum value of $z=4 x+2 y$, subject to the constraints $3 x+4 y \geqslant 12, x+y \leqslant 5, x, y \geqslant 0$ is

The value of $\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}$ is equal to

If $\mathrm{p} \rightarrow(\sim \mathrm{p} \vee \sim \mathrm{q})$ is false, then the truth values of p and q are respectively

Let $\alpha, \beta$ be the roots of the equation $x^2-\mathrm{p} x+\mathrm{r}=0$ and $\frac{\alpha}{2}, 2 \beta$ be the roots of the equation $x^2-q x+r=0$. Then the value of r is