There are several ways to approach this question. Here's one approach:

First recognize that, if mn + 2m + n + 1 is even, then mn + 2m + n + 2 must be ODD
Now recognize that we can factor mn + 2m + n + 2 to get: mn + 2m + n + 2 = (m + 1)(n + 2)
So, if mn + 2m + n + 2 is ODD, then it must be true that (m + 1)(n + 2) is ODD

If (m + 1)(n + 2) is ODD, then we know that (m + 1) is ODD, AND (n + 2) is ODD
If (m + 1) is ODD, then m must be EVEN
If (n + 2) is ODD, then n must be ODD

Now check the 3 statements:
i) (2n + m)² is even.
(2n + m)² = [2(ODD) + EVEN)]²
= [EVEN + EVEN]²
= [EVEN]²
= EVEN
So, statement i is TRUE


ii) n² + 2n – 11 is even
n² + 2n – 11 = (ODD)² + 2(ODD) – ODD
= ODD + EVEN - ODD
= ODD - ODD
= EVEN
So, statement ii is TRUE


iii) m² – 2mn + n² is odd
m² – 2mn + n² = (EVEN)² – 2(EVEN)(ODD) + (ODD)²
= EVEN - EVEN + ODD
= EVEN + ODD
= ODD
So, statement iii is TRUE

Answer: E
IMPORTANT: Let's say you didn't see that mn + 2m + n + 2 factors nicely into (m + 1)(n + 2) [most students will NOT see that]
No problem. In the next solution, you'll see another way to handle this question.

Cheers,
Brent
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Here's a different solution:

Since m and n can each be either even or odd, there are 4 possible cases to consider:

case a) m is EVEN and n is EVEN
case b) m is ODD and n is EVEN
case c) m is EVEN and n is ODD
case d) m is ODD and n is ODD

Now let's test each case as we examine \(mn + 2m + n + 1\)
To make things super easy, let's plug in 0 as a nice EVEN number, and we'll plug in 1 as a nice ODD number.

case a) m is EVEN and n is EVEN
mn + 2m + n + 1 = (0)(0) + 2(0) + (0) + 1
= 1 (an ODD number)
We're told that mn + 2m + n + 1 is EVEN, so case a is NOT POSSIBLE

case b) m is ODD and n is EVEN
mn + 2m + n + 1 = (1)(0) + 2(1) + (0) + 1
= 3 (an ODD number)
We're told that mn + 2m + n + 1 is EVEN, so case b is NOT POSSIBLE

case c) m is EVEN and n is ODD
mn + 2m + n + 1 = (0)(1) + 2(0) + (1) + 1
= 2 (an EVEN number)
We're told that mn + 2m + n + 1 is EVEN, so case c IS POSSIBLE

case d) m is ODD and n is ODD
mn + 2m + n + 1 = (1)(1) + 2(1) + (1) + 1
= 5 (an ODD number)
We're told that mn + 2m + n + 1 is EVEN, so case d is NOT POSSIBLE

Since case c is the ONLY possible case, we know that m is EVEN and n is ODD

Now check the 3 statements (using the same strategy that we applied above):

i) (2n + m)² is even.
(2n + m)² = [2(ODD) + EVEN)]²
= [EVEN + EVEN]²
= [EVEN]²
= EVEN
So, statement i is TRUE


ii) n² + 2n – 11 is even
n² + 2n – 11 = (ODD)² + 2(ODD) – ODD
= ODD + EVEN - ODD
= ODD - ODD
= EVEN
So, statement ii is TRUE


iii) m² – 2mn + n² is odd
m² – 2mn + n² = (EVEN)² – 2(EVEN)(ODD) + (ODD)²
= EVEN - EVEN + ODD
= EVEN + ODD
= ODD
So, statement iii is TRUE

Answer: E

RELATED VIDEO

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Here is a faster way. Create a table, there are only 2 variables so it shouldn't be too hard.

Let E be even and O be odd.

m|n|m^2 + 2m + n + 1
---------------------------
E|E| E^2 + 2 E + E + 1 = E + E + E + 1 = O
E|O| EO + E + O + 1 = E
O|E|OE + 2O + E + 1 = O
O|O|O^2 + 2O + O + 1 = O

Where we used the rules
odd +- odd = even
odd+- even = odd
even+-even = even
odd*odd = odd
odd*even = even
even*even = even

You don't need to memorize these, just keep 1 and 2 as the specific cases if you forget, i.e. 1*2 = 2 so odd*even = even.

Now, notice that the only one that satisfies the criteria that mn + 2m + n + 1 is the second option. Therefore m is even and n is odd.

We check all three cases.
i) (2n + m)² is even
ii) n² + 2n – 11 is even
iii) m² – 2mn + n² is odd

i) (2O + e)^2 = even
ii) O^2 + 2O - 1 = even
iii) E^2 - 2EO +O^2 = odd

So all three check, therefore E is the correct answer.