UGC NET Questions (Paper – 1)

The equation (x − 3)(x + 5) = 0 is satisfied when either factor is zero. Setting x − 3 = 0 gives x = 3, and setting x + 5 = 0 gives x = −5. The sum of the roots is therefore 3 + (−5) = −2. This also agrees with the general property that for x² + bx + c = 0, the sum of the roots is −b, which is consistent if the equation is expanded.

Option A:
Option A correctly adds the two root values, 3 and −5, to obtain −2. It captures the essence of the relationship between factorized form and the numerical roots.

Option B:
Option B, 2, would arise if one mistakenly subtracted −5 from 3 or ignored the sign of one root, but that does not reflect the actual arithmetic.

Option C:
Option C, −8, is the product of the roots, not their sum. It reflects a different relation in quadratics and confuses two separate concepts.

Option D:
Option D, 8, would require both roots to be positive or both negative with different magnitudes, which is not the case for this factorization.

For a quadratic equation ax² + bx + c = 0, the sum of the roots is given by −b/a. In the given equation, a = 1 and b = −5, so the sum of the roots is −(−5)/1 = 5. Thus, the correct sum of the roots is 5.

Option A:
A value of 6 is actually the product of the roots for this quadratic, given by c/a = 6/1 = 6. This represents a different relationship between coefficients and roots. Since the question specifically asks for the sum of the roots, 6 is not the correct answer.

Option B:
4 does not correspond to either the sum or product derived from the standard formulas for this equation. It might result from a miscalculation when solving the quadratic. Therefore, 4 cannot be justified as the sum of the roots.

Option C:
5 is correct because it directly results from applying the formula −b/a with b = −5 and a = 1. If we factor the equation as (x−2)(x−3) = 0, the roots are 2 and 3, whose sum is also 5, confirming the calculation. This dual reasoning strengthens the validity of option C.

Option D:
2 is one of the roots of the equation but not the sum of both roots. Confusing a single root with the sum of all roots leads to this incorrect choice. Thus, 2 cannot satisfy the requirement of adding both roots together.

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.