If p : switch $\mathrm{S}_1$ is closed, q : switch $\mathrm{S}_2$ is closed then correct interpretation from the following circuit is

The solution of $\log \left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=2 x-5 y, y(0)=0$ is

The integrating factor of the differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y \log x=x \cdot \mathrm{e}^x x^{-\frac{1}{2}} \log x(x>0)$ is

ABCD is a quadrilateral with $\overline{\mathrm{AB}}=\overline{\mathrm{a}}, \overline{\mathrm{AD}}=\overline{\mathrm{b}}$ and $\overline{\mathrm{AC}}=2 \overline{\mathrm{a}}+3 \overline{\mathrm{~b}}$. If its area is $\alpha$ times the area of the parallelogram with $\mathrm{AB}, \mathrm{AD}$ as adjacent sides, then the value of $\alpha$ is