CUET Mathematics 2022 4 Aug Shift 1 PYQs

Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$

Then at the origin, which one of the following is true?

If $e^y(x + 1) = 1$, then

If $x^y = e^{x-y}$, then $\frac{dy}{dx} =$