Let for some $\alpha \in \mathbb{R}, f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2 y^2+y+\alpha x y$ for all $x, y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum\limits_{n=1}^5(\alpha+f(n))$ is :

Let

$\mathrm{A}=\{(a, b, c): a, b, c$ are non-negative integers and $a+b+2 c=22\}$.

Then $n(\mathrm{~A})$ is equal to :

The area of the region bounded by the curves $x+3 y^2=0$ and $x+4 y^2=1$ is equal to :

Let $y=y(x)$ be the solution of the differential equation :

$$ \frac{d y}{d x}+\left(\frac{6 x^2+\left(3 x^2+2 x^3+4\right) e^{-2 x}}{\left(x^3+2\right)\left(2+e^{-2 x}\right)}\right) y=2+e^{-2 x} $$

$x \in(-1,2)$, satisfying $y(0)=\frac{3}{2}$. If $y(1)=\alpha\left(2+e^{-2}\right)$, then $\alpha$ is equal to :