Consider the following statements:
${\mathrm{S}}_1:$ Number of points where $f\left(x\right)=\left|xsgn\left(1-x^2\right)\right|$ is non-differentiable is $3$
$S_2:$ Defined $f\left(x\right)=\left\{\begin{array}{ll}a\sin \left(\frac{\pi }{2}\left(x+1\right)\right)& , x\le 0\\ \frac{\tan x-\sin x}{x^3}& , x>0\end{array}\right.$, in order that, $f\left(x\right)$ is continuous,$\text{'}a\text{'}$should be equal to $\frac{1}{2}$

${\mathrm{S}}_3:$ The set of all points, where the function $\sqrt[3]{x^2\left|x\right|}$ is differentiable is $(-\infty ,0)\cup (0,\infty )$
${\mathrm{S}}_4:$ Number of points where $f\left(x\right)=\frac{1}{{\sin }^{-1}\left(\sin x\right)}$ is non-differentiable in the interval $(0,3\pi )$ is $3 .$ State, in order, whether $S_1,S_2,S_3,S_4$ are true or false

Consider the following statements:

${\mathrm{S}}_1:$ Let $f\left(x\right)=\frac{\sin \left(\pi \left[\mathrm{x}-\pi \right]\right)}{1+[\mathrm{x}{]}^2}$ where $[.]$ stands for the greatest integer function. Then $f\left(x\right)$ is discontinuous at $x=n+\pi , n\in I$.
${\mathrm{S}}_2:$ The function $f\left(x\right)=p[x+1]+q[x-1]$, where $[.]$ denotes the greatest integer function is continuous at $x=1$ if $p+q=0$.
${\mathrm{S}}_3:$ Let $f\left(x\right)=\left|\right[x] x|$ for $-1\le x\le 2,$ where $[.]$ is greatest integer function, then $f$ is not differentiable at $x=2$.
${\mathrm{S}}_4:$ If $f\left(x\right)$ takes only rational values for all real $x$ and is continuous, then $f^{\text{'}}\left(10\right)=10$.

Mark $\mathrm{F}$ if the statement is false and $\mathrm{T}$ if the statement is true.

Find the equation of tangent to the curve $y=\left\{\begin{array}{l}{x}^2\sin \left(\frac{1}{x}\right); x\ne 0\\ 0; x=0\end{array}\right.$ at $(0,0)$