Consider a seven-digit number 735x6y4, divisible by 44, where the two digits x and y are unknown.
Consider the following two additional pieces of information:
I. x and y are even numbers
II. x and y are equal
To determine the values of x and y UNIQUELY, which of the above pieces of information is/are MINIMALLY SUFFICIENT?

A seven-digit number 735x6y4 is divisible by 44. It means that the number is also divisible by 4 and 11. 

Divisibility rule of 4 = Last two digits of the number is divisible by 4. 

Divisibility rule of 11 = Sum of odd digits starting from left - Sum of even digits starting from left

For the given number to be divisible by 4, y4 should be divisible by 4. 

So, the possible values of y = 0,2,4,6,8

For the number to be divisible by 11, (7+5+6+4)-(3+x+y) = 19-(x+y) should be divisible by 11. 

Case 1 : Both x and y are even numbers. 

If y = 0, x = 8 

If y = 2, x = 6

If y = 4, x = 4

If y = 6, x = 2

If y = 8, x = 0

In this case, we got 5 different combinations of values. 

Case 2 : Both x and y are equal. 

If y = 4, x = 4

In this case, we got a unique solution. 

Hence, we only need condition 2 to uniquely determine the values of x and y. 

$$\therefore\ $$ The required answer is C.