JEE Main 2024 — Sets And Relations Question with Solution

$$\begin{array}{rl}& R_1=\{(x,y):2x-3y=2\}\\ & R_2=\{(x,y):-5x+4y=0\}\\ & 2x-3y=2\end{array}$$

So $$2x$$ and $$3y$$ both has to be even or odd simultaneously and $$2x$$ can't be odd so $$2x$$ and $$3y$$ both will be even

$$R_1=\{(4,2),(7,4),(10,6),(13,8),(16,10),(19,12)\}$$

For symmetric we need to add 6 elements as

$$\begin{array}{rl}& (2,4),(4,7),(6,10),(8,13),(10,16),(12,19)\\ & M=6\end{array}$$

For $$R_2-5x+4y=0$$

$$5x$$ and $$4y$$ has to be equal $$4y$$ is always even so $$5x$$ will also be even

$$R_2=\{(4,5),(8,10),(12,15),(16,20)\}$$

For symmetric we need to add 4 element as

$$\begin{array}{rl}& (5,4)(10,8)(15,12)(20,16)\\ & N=4\\ & M+N=6+4=10\end{array}$$