QJEEM24Ma049f: JEE Main 2024 Mathematics - Class 11 JEEM24Ma049f Solution: Let the relations and on the set be given by and . If and be…

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

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

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

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

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

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

For symmetric we need to add 4 element as

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