If $f(x) = \frac{x}{x - 1} = \frac{1}{y}$,then $f(y) = $

Let $f(x + y) = f(x) + f(y)$ for all $x, y \in R.$ Then:

Let $f: R \rightarrow R$ be a twice differentiable function such that $(\sin x \cos y)(f(2x+2y) - f(2x-2y)) = (\cos x \sin y)(f(2x+2y) + f(2x-2y))$ for all $x, y \in R$. If $f'(0) = \frac{1}{2}$,then the value of $24f''\left(\frac{5\pi}{3}\right)$ is:

If $f: R \setminus \{0\} \rightarrow R$ is such that $2 f(x) + f\left(\frac{1}{x}\right) = 4x$ and $S = \{x \in R : f(x) = f(-x)\}$, then the number of elements in $S$ is

If a function $f$ satisfies $f(m+n) = f(m) + f(n)$ for all $m, n \in \mathbb{N}$ and $f(1) = 1$,then the largest natural number $\lambda$ such that $\sum_{k=1}^{2022} f(\lambda+k) \leq (2022)^2$ is equal to ..........

Let $f: R^{+} \rightarrow R^{+}$ be a function satisfying $f(x) - x = \lambda$ (constant),$\forall x \in R^{+}$ and $f(x f(y)) = f(x y) + x, \forall x, y \in R^{+}$. Then $\lim _{x \rightarrow 0} \frac{(f(x))^{\frac{1}{3}} - 1}{(f(x))^{\frac{1}{2}} - 1} =$