Is integer x positive?

You can use inequalities to solve this question or you can also solve it easily if you have a good understanding of the relation between x, x^2 and x^3 in various ranges. In fact, add \(\sqrt{x}\) to the list as well (though it is not needed in this question).

Method 1: Inequalities

(1) x > x^3
x^3 - x < 0
x(x - 1)(x + 1) < 0
-1, 0 and 1 are the transition points. Inequality will hold when 0 < x <1 or x < -1. So x cannot be integer and positive both. Sufficient.

(2) x < x^2
x^2 - x > 0
x(x - 1) > 0
0 and 1 are the transition points. Inequality will hold when x > 1 or x < 0. x can be an integer and either positive or negative. Not sufficient.

Answer (A)

Method 2: It is easy to see that A is enough if you understand the relation between x, x^2 and x^3. Look at the number lines below.



The green regions show the areas for which the relations hold. You can see that x^3 < x only for negative integer values (and some positive but non integer values) of x. Hence statement 1 is enough.
Answer (A)