Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$, and $\lim\limits_{t \rightarrow x}\left(\frac{t^2 y(x)-x^2 y(t)}{x-t}\right)=3$ for each $x > 0$. Then $2 y(2)$ is equal to :

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function

$$ f(x)=\left\{\begin{array}{cl} b^2 \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x) \cos x\right]\right), & x<0 \\ \frac{\sin x-\frac{1}{2} \sin 2 x}{x^3} & , x>0 \\ a & , x=0 \end{array}\right. $$

is continuous at $x=0$, then $a^2+b^2$ is equal to :

Let $\alpha, \beta \in \mathbb{R}$ be such that the function $f(x)= \begin{cases}2 \alpha\left(x^2-2\right)+2 \beta x & , x<1 \\ (\alpha+3) x+(\alpha-\beta) & , x \geq 1\end{cases}$ be differentiable at all $x \in \mathbb{R}$. Then $34(\alpha+\beta)$ is equal to

If the function $f(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+\log _e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to