Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& \mathrm{f}(\mathrm{m}+\mathrm{n})=\mathrm{f}(\mathrm{m})+\mathrm{f}(\mathrm{n})\\ & \Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{kx}\\ & \Rightarrow \mathrm{f}(1)=1\\ & \Rightarrow \mathrm{k}=1\\ & \mathrm{f}(\mathrm{x})=\mathrm{x}\end{array}$
Now ${\sum }_{\mathrm{k}=1}^{2022}\mathrm{f}(\lambda +\mathrm{k})\le (2022{)}^2$ $\begin{array}{rl}& \Rightarrow \sum _{\mathrm{k}=1}^{2022}(\lambda +\mathrm{k})\le (2022{)}^2\\ & \Rightarrow 2022\lambda +\frac{2022\times 2023}{2}\le (2022{)}^2\end{array}$ $\begin{array}{rl}& \Rightarrow \lambda \le 2022-\frac{2023}{2}\\ & \Rightarrow \lambda \le 1010.5\end{array}$ $\therefore$ largest natural no. $\lambda$ is 1010 .