UGC NET Questions (Paper – 1)

A gives the defining properties: reflexive, symmetric and transitive. C (equality) and D (congruence modulo n) are classic examples satisfying these properties. B is false because reflexivity cannot be dropped. E is false because equivalence classes can contain multiple elements (e.g., congruence classes modulo n). Therefore A, C and D only are correct.

Option A:
Option A is correct since it includes the definition and two standard examples while excluding B and E, which are false.

Option B:
Option B is incomplete because it ignores D, leaving out an important arithmetic example.

Option C:
Option C is wrong because it includes E, which incorrectly claims all equivalence classes are singletons.

Option D:
Option D is incorrect because it omits A, which states the defining criteria of an equivalence relation.

Statement E is the only wrong statement in this list. A is correct because induction proofs typically start by verifying the statement for the first natural number in the domain. B is correct since the induction step assumes P(k) and proves P(k+1). C correctly states that if the base and step are valid, the result holds for all relevant n. D is true because induction is a method for proving infinitely many cases using a finite argument. E is false because without proving the base case, there is no guarantee that the chain of implications starts, so the induction proof is incomplete.

Option A:
Option A is incorrect since it claims A alone is wrong, but A accurately describes the base step of induction. Declaring it wrong contradicts standard textbook presentations.

Option B:
Option B is wrong because it treats both A and E as wrong. While E is indeed incorrect, A is a true description of the base case and so cannot be grouped with E as wrong.

Option C:
Option C is correct because it singles out E, which wrongly dismisses the need for a base case, as the only incorrect statement. The other statements together present a coherent picture of mathematical induction.

Option D:
Option D is incorrect because it adds A and C to E in the set of wrong statements. C is a true summary of the outcome of induction, and A correctly describes the base step, so this option misclassifies several true statements as wrong.

Statements A and B state the two standard De Morgan identities for sets. Statement C is correct because these laws explicitly relate complements of unions and intersections. Statement E is also true; in Venn diagrams one can see that the shaded regions for both sides of each law coincide. Statement D is false since De Morgan’s laws hold in general, not only for finite sets. Hence the correct set of statements is A, B, C and E only, corresponding to option A.

Option A:
Option A is correct because it collects the two algebraic forms of the laws, their conceptual role and their Venn-diagram visualisation, while rightly excluding D, which wrongly restricts them to finite sets.

Option B:
Option B is incomplete as it omits E and so fails to mention the important geometric interpretation in Venn diagrams, which is often tested in NET questions on sets.

Option C:
Option C is incorrect because it includes D, the false claim about failure for infinite sets, and so mixes correct principles with an incorrect limitation.

Option D:
Option D is wrong since it leaves out A and picks only B, C and E; without A, one of the two core algebraic identities of De Morgan’s laws is missing, so the description is not complete.

Statements A and B give the definition of a binary operation and associativity. D is also correct since there are operations that are associative but not commutative, such as matrix multiplication. C is false because commutativity must hold for all pairs, not just some, and E is false because commutativity does not imply associativity. Thus A, B and D only are correct.

Option A:
Option A is incomplete as it omits D, missing the important fact that associativity and commutativity are independent properties.

Option B:
Option B is wrong because it includes C, which weakens commutativity incorrectly by requiring it only for a single pair.

Option C:
Option C is incorrect since it omits A, the definition of a binary operation, which is one of the correct statements.

Option D:
Option D is correct because it gathers the proper definition, the associativity law and a fact about independence of associativity from commutativity, while excluding C and E, which are incorrect.

Statements A, B, D and E are correct standard number-theoretic facts: even integers are divisible by 2, primes greater than 2 are odd, divisibility by 6 implies divisibility by both 2 and 3 and divisibility tests accelerate factor checking in exams. Statement C is false because a composite number has more than two positive divisors, whereas having exactly two is the defining feature of a prime. Therefore, the combination containing A, B, D and E only is correct.

Option A:
Option A is incomplete because it omits E, thereby ignoring the important exam-oriented point that divisibility rules are employed to solve questions quickly. Without E, the applied angle of the topic is underrepresented.

Option B:
Option B is correct as it includes all true statements while leaving out C, which reverses the definition of composite and prime numbers. It aligns with the basic arithmetic knowledge expected at UGC NET level.

Option C:
Option C is wrong because it includes C, the incorrect statement about composites having exactly two divisors, and omits A, a correct definition of even numbers. This combination thus mixes error with omission.

Option D:
Option D is incorrect as it accepts C along with true statements and omits B, failing to state that primes greater than 2 are odd. The presence of C again makes the option unacceptable.

C and E are wrong because they contradict basic properties and uses of ratios. Ratios can indeed be simplified by dividing both terms by the same non-zero number, in the same way that fractions are simplified, so C is false. E is also false because ratio and proportion are central to many aptitude problems, especially those involving sharing, mixtures and scaling. A, B and D all correctly describe comparison, equality in proportion and sharing applications, so the wrong statements are exactly C and E.

Option A:
Option A is incorrect because it identifies only C as wrong and overlooks E, which dismisses the importance of ratios in aptitude, a claim that clearly conflicts with the UGC NET syllabus. Leaving out E means not all wrong statements are captured.

Option B:
Option B is also incorrect since it treats only E as wrong and ignores C, which incorrectly states that ratios cannot be simplified. A proper understanding of ratio includes the idea of simplification.

Option C:
Option C is correct because it brings together both C and E, the two statements that misrepresent ratio properties and their examination relevance. It implicitly acknowledges that A, B and D provide accurate descriptions used in aptitude questions.

Option D:
Option D is wrong because it adds B, which is true about cross products in a proportion, to the set of supposed wrong statements. Including B makes the option inconsistent with the mathematical definition of proportion.

Statements A, B, C and E correctly describe the behaviour of clock hands and the nature of related aptitude questions: the minute and hour hands have known revolution times, their relative speed determines meeting or opposite positions and questions hinge on both angular calculations and relative motion. Statement D is false because the hour also affects the angle; for example, 3:00 and 4:00 have different angles even though the minutes are zero. Hence, the combination A, B, C and E only is correct.

Option A:
Option A is correct because it includes all the true statements while excluding D, which oversimplifies angular dependence by ignoring the hour. It reflects the standard method used to model clock questions in NET exams.

Option B:
Option B is incomplete since it omits E and so does not mention that these problems blend geometry and speed concepts. Without E, the description of the skill set tested is less comprehensive.

Option C:
Option C is wrong because it includes D, which incorrectly denies the effect of the hour hand on the angle, and it also omits A, failing to mention the minute hand’s revolution time. This mix of omission and error makes the option invalid.

Option D:
Option D is incorrect as it leaves out B, and so does not state that the hour hand takes 12 hours to complete a revolution, which is important for computing relative motion. The combination is thus incomplete even though some statements in it are true.

Statements A, B and C give the standard definitions and formulas for GPs. Statement E is also true because phenomena like compound interest and population growth are often modelled using GPs. Statement D is false since a GP does not have constant differences (so it is not an AP). Thus A, B, C and E only are correct.

Option A:
Option A is incomplete since it omits E (applications).

Option B:
Option B is wrong because it includes D (false).

Option C:
Option C is correct because it keeps all true statements and excludes D, which confuses multiplicative and additive patterns.

Option D:
Option D is wrong because it includes D (false) and omits A (definition).

A is correct because each element of the domain must be assigned exactly one image. B is true as many-one mappings allow different domain elements to share a common image. C correctly describes surjective functions where every codomain element is hit by at least one domain element, and D is correct that a mapping both one-one and onto is a bijection. E is false because that condition describes surjectivity; a function in general does not require every codomain element to have a pre-image. Hence A, B, C and D only are correct.

Option A:
Option A omits D and so fails to name bijection explicitly, even though the other statements are true. Because it does not list all correct statements, it is incomplete.

Option B:
Option B leaves out A and therefore does not include the basic definition of a function as a mapping with single-valued images. This omission makes the combination insufficient.

Option C:
Option C drops B and thus ignores the many-one possibility, which is an important non-injective behaviour, even though A, C and D are true. The set is therefore not fully representative.

Option D:
Option D is correct in collecting all accurate statements about functions, surjections and bijections, while excluding E, which confuses general functions with onto functions. It matches the functional concepts tested in NET.

Statements A, B and D describe core properties of common binary operations. A correctly defines a binary operation as a rule from ordered pairs in S back to S, capturing closure. B is true because addition of reals satisfies both commutativity and associativity. D is also correct since matrix multiplication, though associative, is in general not commutative. Statement C is false because subtraction is neither commutative nor associative, and E is false since having an identity element does not ensure that every element has an inverse unless additional conditions such as group structure hold. Hence the correct combination is A, B and D only.

Option A:
Option A is incorrect as it lists only A and B and omits D, which conveys an important contrast between addition and matrix multiplication. Without D, the option does not fully represent the intended range of binary operation examples. It is therefore not the best match for the correct set.

Option B:
Option B is wrong because it includes only B and D, dropping the general definition of a binary operation in A. While B and D are true, the absence of A means the option does not capture the foundational concept being tested. This makes it incomplete.

Option C:
Option C is correct since it includes the definition in A, the familiar properties of addition in B and the non-commutativity of matrix multiplication in D, while excluding C and E, both of which misstate important algebraic facts. This aligns with the structure of binary operations as tested in NET mathematics.

Option D:
Option D is incorrect because it incorporates C, which wrongly claims that subtraction is commutative, along with true statements. Once a false statement is added to the combination, it cannot represent the exact set of correct statements.

B is the definition of a GP (constant common ratio). C is true for positive terms when r > 1. D is true for positive terms when 0 < r < 1. A is false (that describes an AP), and E is false (a GP is not generally an AP). Therefore B, C and D only are correct. Option A: Option A is incomplete because it omits D, the key decreasing-case when 0 < r < 1. Option B: Option B is incorrect since it includes E, which falsely claims every GP is an AP. Option C: Option C is correct because it includes exactly the true statements (B, C, D) and excludes A and E. Option D: Option D is wrong because it includes A, which incorrectly describes a GP as having a fixed additive difference.

Statement A is correct for linear permutations. Statement B is true because circular permutations fix rotation, giving (n − 1)!. Statement C is true via the block method for adjacency. Statement E is true because extra constraints reduce valid arrangements. Statement D is false because in permutations, order matters. Thus A, B, C and E only are correct.

Option A:
Option A is correct because it includes all true statements and excludes D (false).

Option B:
Option B is incomplete as it omits E (constraints).

Option C:
Option C is wrong because it includes D (false) and omits A.

Option D:
Option D is incorrect because it includes D (false) and omits B (circular formula).

A is the standard inclusion–exclusion formula for two sets. B is a special case when A and B are disjoint (intersection is empty). C is true because an intersection cannot have more elements than either set. D is true for subset relations. E is false because inclusion–exclusion is commonly extended to three (and more) sets. Therefore, A, B, C and D only are correct.

Option A:
Option A is incomplete because it omits D, which is also a true statement about subset relations with union and intersection.

Option B:
Option B is correct because it includes all true statements (A, B, C, D) and excludes E, which is false.

Option C:
Option C is incomplete because it omits A, which is a key identity for set cardinality.

Option D:
Option D is wrong because it includes E (false), so it cannot be the correct set.

Statements A and B correctly distinguish permutations and combinations in terms of the role of order. D is true because permutations can be obtained from combinations by arranging the chosen r objects in r! ways. E is also correct since there is only one way to choose all n objects from n. Statement C is false because nPr generally exceeds nCr when 0 < r < n, so permutations and combinations are not equal in count. Thus the correct combination is A, B, D and E only. Option A: Option A is incomplete as it omits D and E, and hence does not show the quantitative relationship between permutations and combinations or the special case when all objects are chosen. It therefore does not capture the full set of correct statements. Option B: Option B is incorrect because it leaves out E and includes only A, B and D, so it does not mention the important special case nCr = 1 when r = n. This omission makes the combination incomplete. Option C: Option C is correct because it gathers the true distinctions and the exact formula relating nPr and nCr and excludes C, which wrongly claims equality of counts. It therefore matches the combinatorial facts emphasised in NET aptitude. Option D: Option D is wrong as it drops A and includes B, D and E only, failing to state explicitly that permutations take order into account while still not rejecting C. As a result, it does not provide the precise set of correct statements requested.

Statements A and B are the precise symbolic forms of De Morgan’s laws for sets. E is also correct because it broadly describes the role of De Morgan’s laws in relating complements to unions and intersections. C and D are false since they swap union/intersection incorrectly: the complement of a union is an intersection of complements, and the complement of an intersection is a union of complements. Hence A, B and E only are correct.

Option A:
Option A is incomplete because it leaves out E and therefore omits the general conceptual summary of what De Morgan’s laws do.

Option B:
Option B is wrong because it adds C, which contradicts De Morgan’s law for complements of unions.

Option C:
Option C is correct as it selects exactly those statements that align with De Morgan’s laws and excludes C and D, which misuse union and intersection in the context of complements.

Option D:
Option D is incorrect since it omits A (a true De Morgan law), so it cannot be correct.