Derivatives

Q. Answer the following by appropriately matching the lists based on the information given in Column $\text{I}$ and Column $\text{II}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a.}\text{If the function}y=e^{4x}+2e^{-x}\text{is a solution of}& \\ \text{the differential equation}\frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=K,& \\ \text{then the value of}\frac{K}{3}\text{is}& \text{p.}3\\ \text{b.}\text{Number of straight lines which satisfy the}& \\ \text{differential equation}\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2-y=0& \\ \text{is}& \text{q.}4\\ \text{c.}\text{If real value of}m\text{for which the substitution,}& \\ y=u^m\text{will transform the differential equat-}& \\ \text{ion,}2x^{4}y\frac{dy}{dx}+y^4=4x^6\text{into a homogeneous}& \\ \text{equation, then the value of}2m\text{is}& \text{r.}2\\ \text{d.}\text{If the solution of differential equation}& \\ x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}=12y\text{is}y=A{x}^m+B{x}^{-n},& \\ \text{then}|m+n|\text{is}& \text{s.}7\end{array}$

Then which of the following is correct ?

1. $a\to s,b\to r,c\to p,d\to q$
2. $a\to q,b\to r,c\to p,d\to s$
3. $a\to q,b\to p,c\to r,d\to s$
4. $a\to s,b\to q,c\to p,d\to s$

Q. If $f\left(x\right),g\left(x\right)$ be twice differential functions on $[0,2]$ satisfying $f^{\prime \prime }\left(x\right)=g^{\prime \prime }\left(x\right)$, $f^{\prime }\left(1\right)=2g^{\prime }\left(1\right)=4$ and $f\left(2\right)=3g\left(2\right)=9$, then

1. $f\left(4\right)-g\left(4\right)=10$
   
2. $|f\left(x\right)-g\left(x\right)|<2\Rightarrow -2<x<0$
   
3. $f\left(2\right)=g\left(2\right)\Rightarrow x=-1$
   
4. $f\left(x\right)-g\left(x\right)=2x$ has real root