There are certain 2-digit numbers. The difference between the number and the one obtained on reversing it is always How many such maximum 2-digit numbers are there?

Let the 2-digit number be represented as 10a + b, where 'a' is the tens digit (a is from 1 to 9) and 'b' is the units digit (b is from 0 to 9). The number obtained on reversing it is 10b + a.

The difference between the number and the one obtained on reversing it is: (10a + b) - (10b + a) = 10a + b - 10b - a = 9a - 9b = 9(a - b).

The question states that this difference is "always" a certain value for "certain 2-digit numbers". This means we are looking for a set of 2-digit numbers for which the difference 9(a - b) is a constant. For 9(a - b) to be constant, the value of (a - b) must be constant.

We need to find the maximum number of 2-digit numbers that can share a common constant difference. This means we need to find which constant value of (a - b) allows for the most possible 2-digit numbers.

Let's examine the possible values for (a - b) and the number of corresponding 2-digit numbers:

1. If (a - b) = 0: This means a = b. The numbers are 11, 22, 33, 44, 55, 66, 77, 88, 99. There are 9 such numbers. The constant difference is 9 * 0 = 0.

2. If (a - b) = 1: The numbers are 21, 32, 43, 54, 65, 76, 87, 98. There are 8 such numbers. The constant difference is 9 * 1 = 9.

3. If (a - b) = -1: (i.e., b - a = 1) The numbers are 12, 23, 34, 45, 56, 67, 78, 89. There are 8 such numbers. The constant difference is 9 * (-1) = -9.

For other values of (a - b), the number of possible 2-digit numbers will be less than 9 or 8. For example:

• If (a - b) = 2: Numbers are 20, 31, 42, 53, 64, 75, 86, 97 (8 numbers).
• If (a - b) = 9: Number is 90 (1 number).
• If (a - b) = -8: Number is 19 (1 number).

The maximum number of 2-digit numbers for which the difference between the number and its reverse is always a constant value occurs when (a - b) = 0. In this case, there are 9 such numbers (11, 22, ..., 99), and their difference with their reverse is always 0.

Since the maximum count we found is 9, and 9 is not among the options A (3), B (4), or C (5), the correct answer is D) NONE.

The final answer is D.