In a $\Delta ABC$, angle A, B, C are in A.P. and $b:c=\sqrt{3}:\sqrt{2,}$ then angle A is:

Question

Accepted Answer

To solve the problem, we need to determine the measure of angle A in triangle $\Delta ABC$ given that angles A, B, and C are in arithmetic progression (A.P.) and the ratio of sides $b:c = \sqrt{3}:\sqrt{2}$.

### Explanation:

1. **Angles in Arithmetic Progression:**
   - If angles A, B, and C are in A.P., then we can express them as:
     $$
     A = x - d, \quad B = x, \quad C = x + d
     $$
   - The sum of angles in a triangle is $180^\circ$:
     $$
     (x - d) + x + (x + d) = 180^\circ
     $$
     Simplifying, we get:
     $$
     3x = 180^\circ \quad \Rightarrow \quad x = 60^\circ
     $$
   - Therefore, the angles are:
     $$
     A = 60^\circ - d, \quad B = 60^\circ, \quad C = 60^\circ + d
     $$

2. **Using the Ratio of Sides:**
   - Given $b:c = \sqrt{3}:\sqrt{2}$, we use the sine rule:
     $$
     \frac{b}{c} = \frac{\sin B}{\sin C} = \frac{\sqrt{3}}{\sqrt{2}}
     $$
   - Since $B = 60^\circ$, $\sin B = \sin 60^\circ = \frac{\sqrt{3}}{2}$.
   - Therefore:
     $$
     \frac{\sqrt{3}/2}{\sin C} = \frac{\sqrt{3}}{\sqrt{2}}
     $$
     Solving for $\sin C$:
     $$
     \sin C = \frac{\sqrt{2}}{2}
     $$
   - $\sin C = \frac{\sqrt{2}}{2}$ corresponds to $C = 45^\circ$.

3. **Determining Angle A:**
   - Since $C = 45^\circ$ and $C = 60^\circ + d$, we have:
     $$
     60^\circ + d = 45^\circ \quad \Rightarrow \quad d = -15^\circ
     $$
   - Therefore, angle $A = 60^\circ - d = 60^\circ + 15^\circ = 75^\circ$.

### Conclusion:

- **Correct Answer: Option 1 ($75^\circ$):** This is the correct measure of angle A, as derived from the conditions given in the problem.
- **Why Other Options are Incorrect:**
  - **Option 2 ($45^\circ$):** This would imply a different configuration of angles not consistent with the given A.P. and side ratio.
  - **Option 3 ($60^\circ$):** This would imply all angles are equal, which contradicts the A.P. condition.
  - **Option 4 ($30^\circ$):** This would not satisfy the A.P. condition or the given side ratio.

Thus, the correct answer is indeed Option 1, where angle A is $75^\circ$.