Let Ajay will not appear in JEE exam with probability $$p = \frac{2}{7}$$, while both Ajay and Vijay will appear in the exam with probability $$q = \frac{1}{5}$$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:

Let $$P(\text{Ajay not appear}) = p = \frac{2}{7}$$.

So $$P(\text{Ajay appears}) = 1 - \frac{2}{7} = \frac{5}{7}$$.

$$P(\text{Both Ajay and Vijay appear}) = q = \frac{1}{5}$$.

We need: $$P(\text{Ajay appears and Vijay does not appear})$$.

$$P(\text{Ajay appears and Vijay does not}) = P(\text{Ajay appears}) - P(\text{Both appear})$$

$$= \frac{5}{7} - \frac{1}{5} = \frac{25 - 7}{35} = \frac{18}{35}$$

The answer is $$\boxed{\dfrac{18}{35}}$$, which corresponds to Option (2).