Let's analyze each statement step-by-step given that [tex]\( a \)[/tex] is an even positive integer:
### Statement A: [tex]\( 3a \)[/tex] is odd.
- An even number can be expressed as [tex]\( a = 2k \)[/tex] where [tex]\( k \)[/tex] is an integer.
- Therefore, [tex]\( 3a = 3 \times 2k = 6k \)[/tex].
- Since [tex]\( 6k \)[/tex] is divisible by 2 (as it is a multiple of 6), [tex]\( 3a \)[/tex] is even.
- This statement is false.
### Statement B: [tex]\( a^2 \)[/tex] is even.
- If [tex]\( a \)[/tex] is even, then [tex]\( a = 2k \)[/tex].
- Squaring both sides, [tex]\( a^2 = (2k)^2 = 4k^2 \)[/tex].
- Since [tex]\( 4k^2 \)[/tex] is divisible by 2 (as it is a multiple of 4), [tex]\( a^2 \)[/tex] is even.
- This statement is true.
### Statement C: [tex]\( a + 2 \)[/tex] is odd.
- If [tex]\( a \)[/tex] is an even number, adding another even number 2 to it gives: [tex]\( a + 2 = 2k + 2 = 2(k + 1) \)[/tex].
- Since [tex]\( 2(k + 1) \)[/tex] is still a multiple of 2, [tex]\( a + 2 \)[/tex] is even.
- This statement is false.
### Statement D: [tex]\( a - 3 \)[/tex] is even.
- As [tex]\( a \)[/tex] is even, subtracting 3 (an odd number) from it gives: [tex]\( a - 3 = 2k - 3 \)[/tex].
- [tex]\( 2k \)[/tex] is even and subtracting an odd number from an even number results in an odd number.
- Therefore, [tex]\( 2k - 3 \)[/tex] is odd.
- This statement is false.
Based on the analysis, the correct statement is:
[tex]\[ \boxed{B} \][/tex]
