UGC NET Questions (Paper – 1)

A, B, C and E are all correct statements about quadratic equations and their discriminant. A states the general form with a ≠ 0, B defines the discriminant, C correctly says that Δ 0 yields two distinct real roots. D is wrong because Δ = 0 gives equal (repeated) real roots, not two distinct roots. Hence the combination containing A, B, C and E only is correct.

Option A:
Option A includes A, B and C, all of which are true, but omits E, which correctly describes the case Δ > 0. Because it misses one more true statement, this option does not capture the full set of correct statements.

Option B:
Option B adds D to A, B and C, but D is false since Δ = 0 leads to equal roots, not distinct ones. Including a false statement makes this combination incorrect even though it contains many true statements.

Option C:
Option C is correct because it lists exactly the true statements A, B, C and E and excludes D, the only incorrect one. It therefore fully reflects the standard classification of roots using the discriminant.

Option D:
Option D selects B, D and E, but by including D it wrongly treats Δ = 0 as giving distinct roots and also omits A and C, which are true. This combination is therefore both incomplete and partially wrong.

Equal real roots occur when the discriminant of a quadratic equation is zero. For the equation 2x² − kx + 8 = 0, the discriminant is b² − 4ac, which becomes k² − 4(2)(8). Setting this equal to zero gives k² − 64 = 0. Solving, we get k² = 64 and hence k = ±8, but the given options only include 8 as a valid choice. Therefore, k = 8 satisfies the equal roots condition using the standard discriminant rule.

Option A:
Option A, 4√2, when squared gives 32, and substituting into the discriminant k² − 64 yields 32 − 64 = −32, which is negative. A negative discriminant would produce complex, not equal real, roots.

Option B:
Option B leads to k² = 64, so the discriminant becomes 64 − 64 = 0. This matches the precise condition for equal real roots in a quadratic and is consistent with the formula b² − 4ac = 0.

Option C:
Option C, 16, gives k² = 256 and a discriminant of 256 − 64 = 192, which is positive and corresponds to two distinct real roots rather than equal ones.

Option D:
Option D, 16√2, squares to 512, making the discriminant 512 − 64 = 448. This again yields unequal real roots, not the required equal roots.