UGC NET Questions (Paper – 1)

Statements A, B, C and D together summarise the standard relationships between mutually exclusive and independent events. A correctly defines mutual exclusivity as events that cannot happen simultaneously. B properly describes independence in terms of unchanged probabilities. C follows because if two non-trivial events are mutually exclusive, their joint probability is zero and cannot equal the product of positive probabilities, so they are not independent. D restates the formal definition of independence using the product rule. E is false since mutual exclusivity and independence are distinct concepts, so the correct set is A, B, C and D only.

Option A:
Option A is incomplete because it omits D and thus fails to express the formal product definition of independence, even though A, B and C are true. The absence of D makes the description of independence less precise.

Option B:
Option B is correct as it combines the basic definitions and implications about mutual exclusivity and independence and excludes E, which wrongly equates the two. It matches the conceptual treatment of events found in NET-level probability questions.

Option C:
Option C is incorrect because it includes E, the false claim that mutual exclusivity and independence mean the same thing, and leaves out B, which is a core statement about independence. This mixture of error and omission invalidates the option.

Option D:
Option D is wrong as it drops A and includes only B, C and D, leaving out the explicit definition of mutual exclusivity while still not rejecting E. It therefore does not provide the complete and correct set of statements.

Statements A and B are fundamental facts about complements, D describes the multiplication rule for independent events and E is true because working with complements, like “at least one” via “1 − P(none)”, often simplifies calculations. Statement C is false because for mutually exclusive events, P(A and B) = 0, not P(A) + P(B), which is instead P(A or B) for such events. Therefore, the combination A, B, D and E only is correct.

Option A:
Option A is incomplete since it omits E and thus does not state explicitly the strategic advantage of using complementary events to simplify some NET probability questions.

Option B:
Option B is also incomplete because it leaves out D, thereby ignoring the key rule for independent events, which is central to many probability computations. Without D, the treatment of joint events is missing.

Option C:
Option C is wrong because it excludes A and consequently fails to express the basic identity P(A) + P(not A) = 1, and it still does not remove C, the incorrect equality about mutually exclusive events.

Option D:
Option D is correct as it collects all true statements and rejects C, which confuses “and” with “or” probabilities for mutually exclusive events. It represents how complementary and independent events are used in exam-level problems.

Statements A and B correctly capture fundamental logarithm values, D states the power rule and E refers to the change of base formula where one logarithm can be expressed as a constant multiple of another base’s logarithm. All of these are true. Statement C is false because the product rule for logarithms is logₐ(xy) = logₐ(x) + logₐ(y), not the difference. Therefore, the combination that includes A, B, D and E while excluding C is the only correct option.

Option A:
Option A is incomplete because although A, B and D are true, it leaves out E, which is an important fact about changing bases and is useful in more complex aptitude questions involving logarithms. The omission makes the set incomplete.

Option B:
Option B is also incomplete as it omits D, thereby ignoring the power rule, which is frequently used to simplify expressions like logₐ(xᵏ) in exam problems. Without D, the description of key log properties is not full.

Option C:
Option C is wrong because it excludes A and thereby omits the basic fact that logₐ(1) = 0, while still including the true D and E. Although some statements are correct, the absence of A means the option does not gather all true statements.

Option D:
Option D is correct because it retains all valid properties and discards the incorrect product rule in C, aligning with the standard set of logarithm rules that a NET candidate is expected to know.