UGC NET Questions (Paper – 1)

The terms can be generated by the formula aₙ = n³−n+2 with n starting from 1. For n = 1, 2, 3, 4 and 5 we have 1³−1+2 = 2, 2³−2+2 = 8, 3³−3+2 = 26, 4³−4+2 = 62 and 5³−5+2 = 122, which matches the sequence. For n = 6 we get 6³−6+2 = 216−6+2 = 212. Thus 212 is the unique next term that preserves this cubic-minus-linear pattern.

Option A:
Option A, 202, is 10 less than the value predicted by the formula for n = 6. It is not equal to 6³−6+2 and therefore cannot be justified by the same rule. Choosing 202 would disrupt the precise algebraic structure of the sequence.

Option B:
Option B, 206, also differs from 212 and does not satisfy the relationship aₙ = n³−n+2 for the next index. It would amount to an arbitrary adjustment with no support from the earlier terms. Hence 206 is not the correct continuation.

Option C:
Option C, 220, is larger than the correct cubic value and similarly fails to equal 6³−6+2. Using 220 would break the pattern that neatly connects all the preceding terms to their positions. For this reason, 220 is not a valid choice.

Option D:
Option D, 212, is exactly the result of applying the formula with n = 6. It continues the same rule without any modification and keeps the entire series logically coherent. Therefore this option is the correct next term.

The differences between consecutive terms are 10, 18, 26 and 34. These differences themselves form an arithmetic progression where each new difference is 8 more than the previous one. Therefore the next difference should be 42. Adding 42 to 91 gives 133 as the next term. This keeps the pattern at both the series and the difference levels consistent.

Option A:
Option A introduces a difference of 42, extending the sequence of gaps to 10, 18, 26, 34, 42. The increments between these differences are 8, 8, 8 and 8, preserving the arithmetic pattern. Hence 133 fits naturally as the next term.

Option B:
Option B would make the next difference 38, giving 10, 18, 26, 34, 38. The final increment from 34 to 38 is only 4, which breaks the consistent step of 8 seen earlier. Therefore 129 is not consistent with the rule.

Option C:
Option C corresponds to a difference of 44 from 91, creating the gaps 10, 18, 26, 34, 44. The increase from 34 to 44 is 10, not 8, so the progression among differences becomes irregular. Hence 135 cannot be the correct continuation.

Option D:
Option D gives a difference of 46, which results in an increment of 12 from the previous difference 34. This diverges even further from the constant +8 pattern and makes 137 an unsuitable next term.

Statements A, C and D all describe standard facts from elementary number theory. Every prime greater than 2 is odd because 2 is the only even prime. Composite numbers must have at least one additional divisor besides 1 and themselves, giving at least three distinct positive divisors. There are infinitely many primes, a result known since Euclid’s proof. Statements B and E are false because 1 is neither prime nor composite and many even numbers greater than 2, such as 4, 6 and 8, are not prime, so A, C and D only are correct.

Option A:
Option A is correct because it picks exactly those statements that are true and excludes the common misconceptions in B and E. It recognises that 1 is excluded from the definition of primes and that even numbers beyond 2 are usually composite. By including C, it also captures a key structural property of composite numbers.

Option B:
Option B is incorrect as it omits D, thereby ignoring the important theorem that there are infinitely many primes. Even though it includes two true statements, it fails to include all the correct ones.

Option C:
Option C is wrong because it brings in E, which incorrectly claims that every even number greater than 2 is prime. Introducing a clearly false statement alongside A and D makes the overall combination incorrect.

Option D:
Option D is incomplete since it includes only C and D and leaves out A. While C and D are true, this option fails to capture the basic fact that primes greater than 2 are odd, so it does not list the full set of correct statements.

B is the definition of independence: P(A ∩ B) = P(A)P(B). C is correct because mutually exclusive events with positive probabilities have P(A ∩ B)=0 but P(A)P(B)>0, so they cannot be independent. D is true because independence carries over to complements. A is false (addition relates to mutually exclusive unions, not independence). E is false because for independent events P(A|B) = P(A), not 0. Therefore B, C and D only are correct.

Option A:
Option A is incomplete because it includes only B and leaves out C and D, which are also correct statements.

Option B:
Option B is correct as it includes exactly the true statements and excludes A and E, which confuse addition/conditional probability with independence.

Option C:
Option C is wrong because it includes E, which misstates conditional probability under independence.

Option D:
Option D is incorrect because it includes A, which is not the criterion for independence.

Statement A is correct: arguments have premises and a conclusion. Statement B is true: indicator words help identify roles. Statement C is true: validity is about form, so with false premises the conclusion can be false even if the argument form is valid. Statement D is false because a true conclusion does not guarantee validity. Statement E is false since irrelevant information does not strengthen logical support. Therefore, A, B and C only are correct.

Option A:
Option A is incomplete because it omits C (validity vs truth nuance).

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

Option C:
Option C is correct as it includes all true statements and excludes D and E.

Option D:
Option D is incorrect because it includes E (false) and omits A.

For an implication "If P, then Q," the contrapositive is "If not Q, then not P." These two statements are logically equivalent, meaning they are either both true or both false in every situation. The contrapositive reverses the direction of the implication and negates both the hypothesis and the conclusion. Therefore, the given statement "If not Q, then not P" is the contrapositive of "If P, then Q."

Option A:
The converse of "If P, then Q" is "If Q, then P," which merely swaps the positions of P and Q without negating them. It is not logically equivalent to the original statement in general. Since the given statement includes negations, it is not the converse.

Option B:
The inverse of "If P, then Q" is "If not P, then not Q," which negates both parts but keeps the order the same. This is different from "If not Q, then not P," which also reverses the roles of P and Q. Thus, the inverse is not the statement described in the question.

Option C:
The contrapositive is correct because it both negates and reverses the components of the original implication, matching the structure "If not Q, then not P." In logical reasoning, proving the contrapositive is a common technique for establishing the truth of an implication. Recognising this form is important for analysing arguments and proofs.

Option D:
A biconditional statement is of the form "P if and only if Q," signifying that P implies Q and Q implies P simultaneously. It does not involve the pattern "If not Q, then not P" as a standalone statement. Therefore, biconditional is not the correct term here.

Let the numbers be 9x and 13x. After subtracting 12 from each, they become 9x − 12 and 13x − 12, and their ratio is 3:5. So (9x − 12)/(13x − 12) = 3/5. Cross-multiplying gives 5(9x − 12) = 3(13x − 12), which simplifies to 45x − 60 = 39x − 36 and then to 6x = 24, so x = 4. Hence the original numbers are 36 and 52, and the larger number is 52.

Option A:
Option A, 36, is the smaller of the two numbers for x = 4 and does not answer the question, which asks specifically for the larger number. Though 36 fits the ratio 9:13 with 52, choosing it would ignore the wording about which number is larger.

Option B:
Option B, 44, does not fit into the pair 9x and 13x when the transformation is imposed. If we tried to use 44 as the larger number, the corresponding smaller number from ratio 9:13 would not be an integer, and the new ratio after subtracting 12 would not become 3:5. Thus 44 does not satisfy the stated conditions.

Option C:
Option C, 48, might appear plausible as a mid-range value but cannot be expressed as 13x with integer x satisfying the ratio change. Any attempt to treat 48 as the larger number yields inconsistencies when we try to reach a ratio of 3:5 after subtracting 12. Therefore, 48 is incorrect.

Option D:
Option D is correct because with x = 4 we get 9x = 36 and 13x = 52, and 36:52 reduces to 9:13. Subtracting 12 from each gives 24 and 40, whose ratio is 24:40 = 3:5, precisely matching the requirement. This confirms 52 as the correct larger original number.

To find the LCM, we factor each number into primes. The factorization of 6 is 2 × 3, of 8 is 2³ and of 9 is 3². The LCM must contain the highest powers of each prime present, so it is 2³ × 3² = 8 × 9 = 72. This is the smallest number divisible by 6, 8 and 9 without remainder.

Option A:
Option A, 36, is divisible by 6 and 9 but not by 8, because 36 ÷ 8 is not an integer. Thus, it fails the requirement of being a common multiple of all three numbers.

Option B:
Option B correctly uses the highest powers of primes appearing in the factorization of each number. By combining 2³ and 3², we ensure divisibility by 6, 8 and 9, and 72 is the least such number.

Option C:
Option C, 48, is divisible by 6 and 8 but not by 9, as 48 ÷ 9 is not an integer. Therefore, it cannot be the LCM.

Option D:
Option D, 54, is divisible by 6 and 9 but not by 8, since 54 ÷ 8 is not a whole number. It is a common multiple of only two of the three numbers.

Statements A, B, C and E correctly describe the logic behind classification questions. Such items ask for the outlier, use a clear basis such as number properties or meaning, rely on a single consistent rule for the odd item and are best solved by first identifying what the majority have in common. Statement D is false because well-designed questions typically have exactly one odd item, not two, so that there is a unique correct answer. Hence, the combination that includes A, B, C and E but excludes D is the correct one.

Option A:
Option A is correct since it contains all of the true statements and omits D, which introduces ambiguity by suggesting two odd items. This option best reflects how classification questions are set in UGC NET and similar exams.

Option B:
Option B is incomplete because it omits E, failing to highlight the crucial strategy of first spotting the common feature shared by most items. Without E, the guidance on how to approach such questions is not fully conveyed.

Option C:
Option C is incorrect as it leaves out A and therefore does not explicitly mention that the task is to identify the non-member of the group, which is the central idea of classification items. Even though B, C and E are true, the set is not complete.

Option D:
Option D is wrong because it includes D, the false claim about having two odd items, and omits B, which correctly indicates different possible bases of grouping. By combining a wrong statement with omissions, this option cannot be accepted.

Statements A and B correctly define tautology and contradiction, D states that truth tables are a systematic method to classify statements and E rightly explains that contingent statements are sometimes true and sometimes false. Statement C is false because it mistakenly assigns the property of tautology to contingent statements by saying they are true in all situations. Consequently, A, B, D and E together form the only correct combination.

Option A:
Option A is correct because it lists exactly the statements that match standard logical definitions and excludes C, which confuses contingency with tautology. It reflects the approach used in UGC NET when analysing compound statements.

Option B:
Option B is incomplete since it omits E and therefore does not describe how contingent statements behave across different truth assignments. Without E, the classification of logical status is less complete.

Option C:
Option C is wrong because it leaves out A and thereby fails to define tautology explicitly, while still not correcting the misstatement about contingency. Although some included statements are true, the set is not complete.

Option D:
Option D is incorrect because it includes C, the false description of contingent statements, and so mixes an error with otherwise true statements. That mixture makes the option logically inconsistent.

Statements A, B and C cover the main patterns tested in cause–effect questions: one being cause of the other, both being independent causes of a common outcome and both being effects of a hidden common cause. Statement D is false because mere temporal closeness does not guarantee causation. Statement E is also false since some questions classify the pair as unrelated. Therefore, A, B and C only form the correct set of statements.

Option A:
Option A is incomplete because it omits C, which correctly recognises cases where two statements are joint effects of a single cause.

Option B:
Option B is incorrect as it adds D, which conflates temporal sequence with causation, and still lacks C.

Option C:
Option C is correct because it brings together the three valid patterns of interpretation while excluding D and E.

Option D:
Option D is wrong since it includes E (false) and omits B (independent-cause case).

Statements A, B, C and D summarise key facts for calendar questions. A correctly notes that 365 days leave 1 odd day beyond 52 weeks. B is true for leap years with 366 days and 2 odd days. C is correct because adding multiples of 7 days cycles the week back to the same weekday. D correctly defines odd days as leftovers after full weeks are removed. E is false since adding 7 days returns to the same weekday, not the next one. Therefore, A, B, C and D only are correct.

Option A:
Option A is incomplete as it omits D and thus fails to define odd days explicitly, even though that concept is central to computing weekday shifts.

Option B:
Option B is incomplete because it omits B and/or C and does not capture the full set of rules needed for weekday calculations.

Option C:
Option C is correct because it combines facts about year lengths, odd days and the effect of multiples of seven on weekdays, while excluding E, which misstates how adding 7 days affects the day of the week.

Option D:
Option D is wrong since it includes E (false) and omits A (true), so it both adds an error and loses a key fact.

This option is correct because simplifying a ratio involves dividing both terms by their greatest common divisor. The greatest common divisor of 8 and 12 is 4. Dividing both by 4 gives 8:12 = 2:3. Therefore, the simplest form of the ratio is 2:3.

Option A:
The ratio 3:2 reverses the order of the terms and does not preserve the original relationship. It corresponds to 12:8 after scaling, which is not equal to 8:12. Thus, this option is incorrect.

Option B:
The ratio 4:6 is equivalent to 8:12 but not in simplest form. Both terms still have a common factor of 2. Since we can reduce further to 2:3, 4:6 is not fully simplified.

Option C:
The ratio 2:3 is obtained by dividing both 8 and 12 by 4. It has no further common factor greater than 1, so it is in simplest form. This matches the standard requirement for simplified ratios.

Option D:
The ratio 1:3 would correspond to 4:12 when scaled up, which is different from 8:12. Therefore, 1:3 does not represent the same relationship and is not correct.

This option is correct because when two ratios are equal, we say they are in proportion. The notation a:b = c:d indicates that a/b equals c/d. Such an equality is called a proportion in standard arithmetic terminology. Therefore, the situation described in the question defines a proportion.

Option A:
Variation describes how one quantity changes when another changes, such as direct or inverse variation. It does not specifically refer to the equality of two ratios. Hence, variation is not the term used for a:b = c:d.

Option B:
A proportion is an equation that states two ratios are equal. In the example a:b = c:d, the four terms are said to be in proportion. This matches exactly the relationship described by the question, so proportion is the correct name.

Option C:
Commission is a fee or percentage earned on sales or transactions. It belongs to commercial arithmetic rather than ratio equality. Since the question is about two ratios being equal, commission is irrelevant.

Option D:
Percentage is a way to express a number per hundred, such as 30%. Although percentages are related to ratios, the specific term used when two ratios are equal is proportion, not percentage. Therefore, this option is not correct.

The positions of the letters are Z(26), X(24), U(21) and Q(17). The decreases between them are −2, −3 and −4, which form an increasing sequence of negative steps in magnitude. The next step should be −5, giving 17 − 5 = 12, which is the letter L. Hence L uniquely satisfies the pattern of successively larger backward jumps.

Option A:
Option A, O, lies at position 15, which would give a decrease of only 2 from Q. This contradicts the trend of increasing the size of the backward step. Since the differences must grow by 1 each time, O cannot fit the rule.

Option B:
Option B, N, is at position 14, implying a decrease of 3 from 17. This repeats a previous step size instead of extending it to −5. Therefore N does not reflect the evolving pattern of differences.

Option C:
Option C, K, corresponds to position 11, giving a drop of 6 from Q. This overextends the next decrement beyond what the sequence suggests. Because the model clearly shows steps of −2, −3, −4, then −5, moving to −6 breaks the logic.

Option D:
Option D is correct because L, at position 12, produces the required decrease of 5 from 17. It aligns with the systematic growth of the backward interval. By respecting the exact structure of the differences, L is the only consistent next term.