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 :

The sum of all values of $\alpha$, for which the shortest distance between the lines $\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha}$ and $\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2 \alpha}$ is $\sqrt{2}$, is

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 $\mathrm{X}=\{x \in \mathrm{~N}: 1 \leq x \leq 19\}$ and for some $a, b \in \mathbb{R}, \mathrm{Y}=\{a x+b: x \in \mathrm{X}\}$. If the mean and variance of the elements of Y are 30 and 750 , respectively, then the sum of all possible values of $b$ is