Let $$A = \{2, 3, 6, 8, 9, 11\}$$ and $$B = \{1, 4, 5, 10, 15\}$$. Let $$R$$ be a relation on $$A \times B$$ defined by $$(a, b)R(c, d)$$ if and only if $$3ad - 7bc$$ is an even integer. Then the relation $$R$$ is :

First recall the sets

$$A = \{2,\,3,\,6,\,8,\,9,\,11\}, \quad B = \{1,\,4,\,5,\,10,\,15\}$$

The Cartesian product $$A \times B$$ consists of all ordered pairs $$(a,b)$$ with $$a \in A$$ and $$b \in B$$.
A relation $$R$$ on $$A \times B$$ is defined by

$$(a,b)\,R\,(c,d) \;\;\Longleftrightarrow\;\; 3ad - 7bc \text{ is even}$$

We test the three standard properties one by one.

Reflexivity

For any $$(a,b) \in A \times B$$ we must check $$(a,b)R(a,b)$$.

Put $$(c,d)=(a,b)$$ in the defining condition:

$$3ad - 7bc \;=\; 3ab - 7ba \;=\; (3-7)ab \;=\; -4ab$$

Since $$-4ab$$ is a multiple of $$4$$, it is always even. Hence every ordered pair is related to itself and $$R$$ is reflexive.

Symmetry

Assume $$(a,b)R(c,d)$$. Then

$$3ad - 7bc \text{ is even} \;\,\Longrightarrow\,\; ad - bc \equiv 0 \pmod 2$$ because $$3 \equiv 1 \pmod 2$$ and $$7 \equiv 1 \pmod 2$$.

Now consider $$(c,d)R(a,b)$$. We must examine

$$3cb - 7da = 3bc - 7ad$$

Modulo $$2$$ this becomes $$bc - ad$$ which is just the negative of $$ad - bc$$.
If $$ad - bc$$ is even, so is its negative, therefore $$3cb - 7da$$ is even.
Thus $$(c,d)R(a,b)$$ holds whenever $$(a,b)R(c,d)$$ holds, proving that $$R$$ is symmetric.

Transitivity (fails)

We need an explicit counter-example where

$$(a,b)R(c,d) \text{ and } (c,d)R(e,f) \text{ but } (a,b) \not R (e,f).$$

Choose the following ordered pairs from $$A \times B$$:

$$(a,b) = (3,5), \quad (c,d) = (2,4), \quad (e,f) = (3,4)$$

1. Check $$(a,b)R(c,d):$$

$$3ad - 7bc = 3(3)(4) - 7(5)(2) = 36 - 70 = -34,$$ $$-34$$ is even, so $$(3,5)R(2,4).$$

2. Check $$(c,d)R(e,f):$$

$$3cf - 7de = 3(2)(4) - 7(4)(3) = 24 - 84 = -60,$$ $$-60$$ is even, so $$(2,4)R(3,4).$$

3. Check $$(a,b)R(e,f):$$

$$3af - 7be = 3(3)(4) - 7(5)(3) = 36 - 105 = -69,$$ $$-69$$ is odd, so $$(3,5) \not R (3,4).$$

We have found two related pairs whose relation does not extend transitively to the third. Hence $$R$$ is not transitive.

Conclusion

The relation $$R$$ is reflexive and symmetric but not transitive.
Therefore the correct choice is Option B.