UGC NET Questions (Paper – 1)

A is true because absolute value is never negative, so |x| ≥ 0 for all real x. B is true: |x| 0) means x lies within distance a of 0, i.e., −a < x a means x is outside that interval, i.e., x a. D is false because |x| > −2 is true for every real x (since |x| ≥ 0), so it does not force x to be positive. Therefore A, B and C only are correct.

A, B and C are correct standard facts: absolute value measures distance, ∣x∣ < 3 implies −3 < x 4 implies x 4. D is false because absolute value cannot be negative, so ∣x∣ = −2 has no real solution. E is false because ∣x∣ = 5 has two real solutions, x = 5 and x = −5, not exactly one. Therefore, the correct set is A, B and C only.

Option A:
Option A is incomplete because it omits C, which is also a correct rule for solving strict “greater than” absolute value inequalities.

Option B:
Option B is incorrect because it includes E, which is false since ∣x∣ = 5 has two solutions, not one.

Option C:
Option C is correct because it lists exactly the true statements (A, B, C) and excludes both false statements (D, E).

Option D:
Option D is incorrect because it includes D (which is impossible) and E (which is also false), so it mixes false statements with true ones.