Correct Answer: E
Explanation:
If x is a positive integer, which of the following must be an even integer?
Option A: X + 2 ⇒ If x is odd, then x + 2 is also odd and if x is even, then x + 2 is even.
Option B: 2x + 1 ⇒ 2x is even number and 2x + 1 is odd number.
Option C: 3x + 1 ⇒ 3x may be even or odd based on x, so 3x + 1 is may also be odd or even based on x.
Option D: X2 + x + 1 ⇒ Square of x may or may not be even, similarly, x + 1 may or may not be even based on x.
Option E: X2 + x + 2 ⇒ Square of even number is even and of odd number is odd. Now two cases arises:
Case I: x is odd ⇒ square of odd number is odd. Sum of x2 + x is the sum of two odd numbers which is an even number. Even number + 2 is also an even number. X2 + x + 2 is even in this case.
Case II: x is even ⇒ square of even number is even. Sum of x2 + x is the sum of two even numbers which is an even number. Even number + 2 is also an even. X2 + x + 2 is even in this case.
So Option E is an even number.