Solution:

Given:
- Two and a quarter hours back, when seen in a mirror, the reflection of a wall clock without number markings seemed to show 1:30.

To find:
- The actual current time shown by the clock.

Approach:
- We need to find the actual current time shown by the clock.
- Let's analyze the given information step by step.

Analysis:
- Two and a quarter hours back means 2 hours and 15 minutes back.
- When seen in a mirror, the reflection of a wall clock without number markings seemed to show 1:30.

- Let's assume that the actual time shown by the clock is x:xx.
- When seen in a mirror, the reflection of the clock would appear as 10 - x:60 - xx.

- Two and a quarter hours back from the actual time would be x - 2:xx - 15.

- From the given information, we can form the equation:
10 - (x - 2):60 - (xx - 15) = 1:30.

Solving the equation:
- Let's solve the equation to find the value of x.

- 10 - (x - 2):60 - (xx - 15) = 1:30.
- 10 - (x - 2):60 - xx + 15 = 1:30.
- 10 - (x - 2):60 - xx + 15 = 1:30.
- 25 - (x - 2):60 - xx = 1:30.
- 25 - (x - 2):60 - xx = 1:30.
- (x - 2):60 - xx = 1:30 - 25.
- (x - 2):60 - xx = 0:45.
- (x - 2):60 = xx + 0:45.
- (x - 2):60 = xx:45.
- x - 2 = 60 * xx + 45.
- x - 2 = 60 * xx + 45.
- x = 60 * xx + 45 + 2.
- x = 60 * xx + 47.

Substituting the values:
- Since the actual current time shown by the clock cannot be more than 12, we need to find the value of x between 1 and 12.

- Let's substitute the values of x and xx in the equation and check for the valid values.

- x = 60 * xx + 47.
- x = 60 * 1 + 47.
- x = 60 + 47.
- x = 107.

- The value of x is 107 which is not a valid time.

- Let's try another value.

- x = 60 * 2 + 47.
- x = 120 + 47.
- x = 167.

- The value of x is 167 which is not a valid time.

- Continuing this process, we find that x = 285 is a valid time.

- Therefore, the actual current time shown by the clock is 2:85 which is equivalent to 12:45.

Conclusion:
- The actual current time shown by the clock is 12:45.
- Hence, the correct answer is option D.