If $f(x)$ is a polynomial function satisfying the condition $f(x) \cdot f(1/x) = f(x) + f(1/x)$ and $f(2) = 9$,then:

Let $f : R \rightarrow R$ be a continuous function such that $f(3x) - f(x) = x$. If $f(8) = 7$,then $f(14)$ is equal to.

Let $f(x)$ be a polynomial of degree $3$ such that $f(k) = -\frac{2}{k}$ for $k = 2, 3, 4, 5$. Then the value of $52 - 10 f(10)$ is equal to:

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 $f\left( \frac{x + 8y}{9} \right) = \frac{f(x) + 8f(y)}{9}$ for all real $x$ and $y$. If $f'(0)$ exists and is equal to $2$,and $f(0) = -5$,then $f(7)$ is equal to:

Let $f(x)$ be defined for all $x > 0$ and be continuous. Let $f(x)$ satisfy $f\left( \frac{x}{y} \right) = f(x) - f(y)$ for all $x, y > 0$ and $f(e) = 1$. Then: