1. Define events

Let:

• $A$ = event that Ajay appears in the exam
• $V$ = event that Vijay appears in the exam

Given:

• Probability that Ajay does not appear: $P(A') = \frac{2}{7}$
• Probability that both Ajay and Vijay appear: $P(A \cap V) = \frac{1}{5}$

We need to find: $P(A \cap V')$ that is, probability that Ajay appears and Vijay does not appear.

2. Find probability that Ajay appears

Since: $P(A) = 1 - P(A')$ we get $P(A) = 1 - \frac{2}{7} = \frac{5}{7}$

3. Use partition of event $A$

If Ajay appears, then either Vijay appears or Vijay does not appear. So: $P(A) = P(A \cap V) + P(A \cap V')$

Therefore, $P(A \cap V') = P(A) - P(A \cap V)$

Substitute the values: $P(A \cap V') = \frac{5}{7} - \frac{1}{5}$

Take LCM $35$: $P(A \cap V') = \frac{25}{35} - \frac{7}{35} = \frac{18}{35}$

4. Match with options

Thus the required probability is: $\boxed{\frac{18}{35}}$

So the correct option is D.