Option A is correct.
Let us take x as 5 which is odd, then (x + 1)^2 is (5 + 1)^2 which is equal to 36, which is even. Hence I is true. Let us take x as 8 which is even, then (x - 1)^2 is (8 - 1)^2 which is equal to 49, which is odd. Hence II is also true. As I and II are true option A is correct.
Option B is not correct.
I is true as seen above. If x is 8 which is even, then √(8-1) = √7
is irrational, but If x is 10 which is even, then √(10-1) = √9 = 3 is rational, hence III is not true.
Option C is not correct.
II is true as seen above but III is not true as seen above.
Option D is not correct.
Only I is false, as both I and II are true.
Option E is not correct.
Only II is false, as both I and II are true.