Is b even? (1) a/b is even (2) (a+1)b is even

This is a bit of a trick question as we're not explicitly told that a,b are integers.
As such, since we're only given general information about the product and quotient of a and b, (E) will be our answer.
For example in (1) if we set b=1/2 than a/b = 2a which is even for any integer a and in (2) we can keep using fractions such as a=2 and b=2/3. Similarly, when combining, we can find fractions that work in both cases such as a=3 and b=1/2

If we assume that a,b are integers, then a more standard solution would be:

As we're asked about number-properties (even/odd) we'll look for a theoretical solution (one without calculation).
This is a Logical approach.

(1) If a/b is even then it is divisible by 2. Since a/b is a factor of a, this means that a is divisible by 2. It tells us nothing about b, however. (For example, consider a = 20, b=2 or a=20, b=5)
Insufficient.

(2) If a product is even, then at least one of its factors is even. So, at least one of (a+1) and b are even. As we don't know which one, we cannot answer.
Insufficient.

Combined: From (1) we know that a is even meaning that a+1 is odd. Therefore, by (2), b must be even.
Sufficient!

In this case (C) is our answer.