For particular integral of differential equation which of the following options are correct ?
A. $\frac{1}{f\left(D^{2}\right)} \sin ax = \frac{\sin ax}{f\left(-a^{2}\right)}$
B. If $f\left(-a^{2}\right) = 0$, then $\frac{1}{f\left(D^{2}\right)} \sin ax = x \frac{\sin ax}{f'\left(-a^{2}\right)}$
C. If $f\left(-a^{2}\right) = 0$, then $\frac{1}{f\left(D^{2}\right)} \sin ax = x \frac{\cos ax}{f'\left(-a^{2}\right)}$
D. If $f'\left(-a^{2}\right) = 0$, then $\frac{1}{f\left(D^{2}\right)} \sin ax = x^{2} \frac{\sin ax}{f''\left(-a^{2}\right)}$
Choose the correct answer from the options given below :

Match List - I with List - II.

List - I	List - II
A. $\vec{\nabla}\cdot\vec{A}=0$	I. $\vec{A}$ is conservative
B. $\oint \vec{A}\cdot d\vec{r}=0$	II. $\vec{A}$ is irrotational
C. $\vec{A} = \vec{\nabla}\phi$	III. $\vec{A}$ is solenoidal
D. $\vec{\nabla}\times\vec{A}=0$	IV. $\vec{A}$ points in the direction of maximum increase of the function $\phi$

Choose the correct answer from the options given below :