If x and y are integers, and w = x^2y + x + 3y, which of the following

Actually I and II can be derived from each other if it were a CR question..
I. If w is even, then x must be even..... MEANS... If x is NOT even, w is NOT even ... MEANS if x is odd, w is odd :- SAME as II
so our answer must have BOTH I and Ii.. ONLY C and E are left

But let's solve ..

Ofcourse many ways to solve but just to understand the logic behind it....

\(w = x^2y + x + 3y\)
there are TWO terms, x and 3y , which contain only one of the two variable AND 1 term containing both variable and that too as a product

1) Therefore, if BOTH the variables x and y are opposite that is one odd and one even
a) the term containing both variables will be EVEN as it will be product of E*O
b) the oher terms will be one E and other O
so SUM = w = E+O+E = O

2) if BOTH the variables x and y are SAME
a) ALL the terms will be EVEN if both are even so SUM= w = E+E+E = E
b) ALL the terms will be ODD if both are odd so SUM = w = O+O+O = O

conclusion
if both x and y are EVEN, w is EVEN or, else, w is ODD

let's see the choices...

I. If w is even, then x must be even.
Refer 2(a), w is even only when both x and y are even....TRUE

II. If x is odd, then w must be odd.
refer 1 and 2(b), when x is odd, w is ODD....TRUE

III. If y is odd, then w must be odd.
Again refer the conclusion or 1 and 2(b), y is ODD means w is ODD....TRUE

E all three true