JEE Main 2024 — 31 January, Shift 2 — Question 28

Determine the elements of $R$ The condition is $2x = 3y$, which implies $y = \frac{2x}{3}$. For $y$ to be an integer in the set $A$, $x$ must be a multiple of 3. The possible values for $x$ are:

$$x \in \{3, 6, 9, \dots, 99\}$$

The number of elements in $R$ is:

$$|R| = \frac{99}{3} = 33$$

Check for reflexive/symmetric elements in $R$ An element $(x, y)$ is the same as its symmetric counterpart $(y, x)$ only if $x = y$. Substituting $x = y$ into $2x = 3y$:

$$2x = 3x \implies x = 0$$

Since $0 \notin A$, there are no pairs in $R$ where $x = y$. Thus, for every $(x, y) \in R$, $x \neq y$.

Apply the Symmetry property For $R_1$ to be symmetric and contain $R$:

$$(x, y) \in R \implies (y, x) \in R_1$$

We already established that if $(x, y) \in R$, then $(y, x) \notin R$ (because $2y = 3x$ and $2x = 3y$ only intersect at $x = 0$). Therefore, for every pair in $R$, we must add a unique new pair to $R_1$ to satisfy symmetry.

Calculate $n$ The minimum number of elements $n$ is the sum of the elements in $R$ and their symmetric reflections:

$$n = |R| + |R_{reflected}|$$ $$n = 33 + 33 = 66$$

The minimum value of $n$ is $66.$