UGC NET Questions (Paper – 1)

This sequence is defined by the rule (a_n = 3n^4 - n^2 + 2) with n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 3−1+2 = 4, 48−4+2 = 46, 243−9+2 = 236, 768−16+2 = 754 and 1875−25+2 = 1852. For n = 6 we calculate (3×6^4 - 6^2 + 2 = 3888 - 36 + 2 = 3854). Therefore 3854 is the correct next term.

Option A:
Option A, 3854, is exactly the value produced by the formula for n = 6. It preserves the quartic and quadratic contributions in the same way as for the earlier terms and keeps the pattern coherent. Thus 3854 is the correct continuation of the series.

Option B:
Option B, 3818, is 36 less than the correct value and does not equal (3×6^4 - 6^2 + 2). It suggests an unwarranted reduction that is not reflected in the previous behaviour. Hence 3818 is not valid.

Option C:
Option C, 3886, overshoots the computed value and again fails to satisfy the formula. Choosing 3886 would break the algebraic structure and introduce an arbitrary jump. Therefore 3886 is not correct.

Option D:
Option D, 3922, deviates even more from 3854 and has no basis in the generating expression. Using 3922 would destroy the close match between the rule and the data, so it is not the right answer.

Each term after the first is obtained by doubling the previous term and subtracting 1. We have 3×2−1 = 5, 5×2−1 = 9, 9×2−1 = 17 and 17×2−1 = 33, confirming the rule. Applying it again gives 33×2−1 = 66−1 = 65. Hence, 65 is the correct next term.

Option A:
Option A is 63, which would require subtracting 3 from 33×2, not 1. The series consistently subtracts 1, so altering that constant is unjustified. Therefore, 63 is incorrect.

Option B:
Option B is 64, implying that we simply double 33 and subtract 2. Such a modification has no support from the earlier steps. Thus, 64 cannot match the pattern.

Option C:
Option C equals 65 and follows the rule 33×2−1 exactly. The extended sequence 3, 5, 9, 17, 33, 65 is perfectly consistent with the defined operation. This makes 65 the valid continuation.

Option D:
Option D is 67, which would result from 33×2+1 and introduces addition instead of subtraction. Since all previous transitions subtracted 1, this change breaks the rule. Therefore, 67 is not appropriate.

Statements A and B describe systematic coding and how multiple examples reveal the pattern. Statement D is also correct because many codes use alphabet positions, sums or reversals. Statement C is false since one example can match many different rules, and E is false because solving requires pattern detection, not guessing. Hence A, B and D only are correct.

Option A:
Option A is incomplete because it omits D (common operations), so it does not include all true statements.

Option B:
Option B is correct as it includes all true statements and excludes C and E, both of which are false.

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

Option D:
Option D is wrong because it includes C (false) and omits B (need multiple examples), so it cannot be correct.

Statements A and B correctly define closure and commutativity for binary operations, D notes that exams introduce custom operations to test rule-following rather than standard arithmetic and E emphasises the need to use the given definition faithfully. Statement C is false because associativity means (a * b) * c = a * (b * c), not that the results are always different. Hence, the combination A, B, D and E only is correct.

Option A:
Option A is incomplete as it omits E and thus fails to stress that candidates must not revert to usual addition or multiplication when a custom rule is specified. This omission makes the description less helpful for exam strategy.

Option B:
Option B is also incomplete because it leaves out A, so it does not explicitly mention closure, the idea that combining elements of S stays within S, which is often implied when operations are defined.

Option C:
Option C is wrong since it includes C, which reverses the meaning of associativity by saying results are always different, and omits B, thereby ignoring the commutative law. This mixture of an error and an omission makes the option invalid.

Option D:
Option D is correct because it captures closure, commutativity, exam usage of custom operations and the necessity of following the definition, while excluding C, which misstates associativity. It reflects the way NET questions check conceptual clarity about operations.

This series is generated by the rule aₙ = 3n⁵ + n² − 2 with n starting at 1. Substituting n = 1, 2, 3, 4 and 5 yields 2, 98, 736, 3086 and 9398, which correspond exactly to the given terms. For n = 6 the same expression produces 23362. Therefore 23362 is the unique value that continues this high-degree polynomial pattern.

Option A:
Option A, 23278, is 84 less than the value obtained from aₙ = 3n⁵ + n² − 2 at n = 6. Choosing 23278 would require a large downward adjustment of the correct result only at this stage. Since earlier terms match the formula exactly, this option does not conform to the established rule and is incorrect.

Option B:
Option B, 23362, matches exactly the number produced by the expression when n = 6. It preserves the dominance of the fifth-power term together with the smaller quadratic and constant components. Because it fits perfectly into the same pattern that explains all previous terms, 23362 is the correct answer.

Option C:
Option C, 23398, is 36 greater than the required value and cannot be written as 3n⁵ + n² − 2 for n = 6. Adopting 23398 would arbitrarily increase the term at the final position, breaking the algebraic consistency. Thus option C is not a valid continuation of the sequence.

Option D:
Option D, 23434, deviates even more from the computed value and likewise fails to satisfy the rule at n = 6. Using 23434 would destroy the precise functional link between index and term. Hence option D is incorrect.

The series follows the rule (a_n = n^3 + 2) for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3, 10, 29, 66 and 127, which match the given terms. For n = 6 the same formula yields (6^3 + 2 = 216 + 2 = 218). Therefore 218 is the unique value that continues the same cubic pattern without any change.

Option A:
Option A equals 218, which is exactly the value of (n^3 + 2) when n = 6. It preserves the single algebraic rule that generates all previous terms. Because no adjustment or exception is needed to obtain 218, this option correctly extends the series.

Option B:
Option B, 214, does not equal (6^3 + 2) and would require subtracting an additional 4 from the formula’s output. This breaks the clean relationship between each term and its index. Hence 214 cannot be accepted as the correct next term.

Option C:
Option C, 222, is larger than the formula’s value and does not arise from (n^3 + 2) for the next integer. Choosing 222 would arbitrarily increase the term and destroy the exact cubic correspondence. Therefore 222 is not a valid continuation of the series.

Option D:
Option D, 226, deviates even further from 218 and again fails to satisfy the expression (n^3 + 2) for n = 6. It would imply altering the rule only at the last step, which is mathematically inconsistent. Thus 226 is not correct.

A correctly uses ∃! to express unique existence of an additive inverse. D correctly states that “exactly one” implies both existence and uniqueness. B is false (∃! means exactly one). C is false (uniqueness needs a comparison argument). E is false (∀x∃y is not equivalent to ∃y∀x in general). Thus A and D only are correct.

Option A:
Option A is correct as it includes the precise symbolic formulation and the correct meaning of uniqueness, excluding B, C and E which are incorrect.

Option B:
Option B is incorrect because it adds B, which misinterprets ∃! as “infinitely many”.

Option C:
Option C is wrong because it includes E, which is not generally equivalent to ∀x∃y P(x, y).

Option D:
Option D is incorrect because it includes B and E, both of which are false.

The starting letters W, T, Q and N occupy positions 23, 20, 17 and 14, decreasing by 3 each time. Each term after the starting letter gives the next three consecutive letters. Continuing this rule, the next starting letter is position 11, which is K, and the block from K is KLMN. Thus KLMN preserves both the minus three jump and the four letter consecutive structure.

Option A:
Option A is correct because K is exactly three positions before N, and the letters L, M and N follow consecutively. It continues the steady backward movement of the starting letters and keeps each group a simple sequence of four adjacent letters. This is fully consistent with the pattern in the given series.

Option B:
Option B, LMNO, starts at L, which is only two positions before N, and therefore changes the step size from minus three to minus two. This inconsistency in the backward jump breaks the arithmetic pattern underpinning the sequence. As a result, LMNO cannot be the correct next term.

Option C:
Option C, PQRS, goes forward from N instead of backward, reversing the direction of the starting letters. The series does not exhibit any forward jumps in its current segment, so PQRS does not align with the logic of the pattern.

Option D:
Option D, UVWX, also represents a forward jump, returning to letters that appeared earlier in the extended alphabetic ordering. It fails to continue the recorded progression of starting letters moving leftward by three positions. Hence UVWX is not a valid continuation.

The positions of the letters are B(2), F(6), J(10), N(14) and R(18). Each term increases by 4 positions (2→6, 6→10, 10→14, 14→18). Continuing this rule, the next position is 18 + 4 = 22, which corresponds to the letter V. Thus V is the only option that maintains the constant step size of 4 in the sequence.

Option A:
Option A, T, corresponds to position 20, which is only 2 steps ahead of R at 18. This contradicts the observed increment of 4 positions between consecutive terms. Because it does not respect the uniform difference, T cannot be the correct continuation.

Option B:
Option B, U, stands at position 21, giving a difference of 3 from R. The sequence does not display variable increments; it keeps a consistent jump of 4. Since U does not satisfy this constant difference property, it fails to fit the pattern.

Option C:
Option C is correct because V is at position 22, exactly 4 places after 18. This preserves the arithmetic progression of positions and ensures that the series continues smoothly. Any deviation from this step size would change the core structure of the series, so V is the only valid next term.

Option D:
Option D, W, lies at position 23, which is 5 steps beyond R. This would increase the gap toward the end of the series without any indication of such change earlier. The absence of a growing difference in previous steps means W cannot be justified by the pattern.

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.

Statements A, B and C correctly describe mixture setups, the alligation shortcut and the fact that equal quantities lead to an average percentage. Statement E is true because both algebraic and alligation approaches are valid for such problems. Statement D is false since in an alligation diagram the differences from the mean are written on the opposite side to the corresponding values, forming the cross pattern. Therefore, A, B, C and E only are correct.

Option A:
Option A is incomplete because it omits E and hence does not reflect the flexibility of solving mixture problems either by equations or by alligation.

Option B:
Option B is correct as it gathers all the true statements and excludes D, which misdescribes the alligation diagram placement rule.

Option C:
Option C is wrong because it includes D (false), even though it contains some true statements.

Option D:
Option D is wrong because it includes D (false) and omits A, losing the general description of mixture problems.

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.

Statement B is the only wrong statement in this group. A is correct because data sufficiency focuses on whether information is enough, not on computing the final number. C is true since some questions are solvable with one statement but not with the other alone. D is also correct because in some cases, even combining statements fails to give a unique answer. E is correct since we generally presume the given data are consistent unless told otherwise. B is false because two insufficient statements may remain insufficient when combined.

Option A:
Option A is incorrect as it treats both A and B as wrong, even though A accurately states the objective in data sufficiency problems. Misclassifying A makes the option invalid.

Option B:
Option B is correct because it isolates B as the sole incorrect statement, capturing the fact that insufficiency of each statement alone does not guarantee sufficiency when combined.

Option C:
Option C is wrong because it wrongly marks C as wrong as well, even though C describes a common situation where one statement alone can be sufficient.

Option D:
Option D is incorrect since it ignores B altogether and claims only C and D are wrong, even though D and C correctly describe typical sufficiency patterns and B is the problematic rule.

The series consists of overlapping three letter blocks in forward order, while the starting letter moves backward by one each time. Starting letters S, R, Q and P occupy positions 19, 18, 17 and 16, decreasing by 1. The next starting letter must therefore be O at position 15, leading to the block OPQ. This group keeps the internal order and the backward movement of the starting position.

Option A:
Option A, ONP, uses the correct starting letter O but arranges the remaining letters out of consecutive order. The letters N and P do not appear in correct sequence following O. Because each group in the series is strictly built from consecutive letters in increasing order, ONP is not acceptable.

Option B:
Option B, PQR, repeats the last given group instead of moving to the next starting letter. The pattern does not involve repetition; it consistently shifts the starting point backward along the alphabet. For that reason, PQR cannot be the next term.

Option C:
Option C, NOP, starts at N, which is two positions before P rather than one. Although NOP is a consecutive block, it does not correctly mirror the one step backward movement seen in the starting letters. Therefore NOP breaks the established positional rule.

Option D:
Option D is correct because OPQ begins with O, exactly one step before P, and continues with P and Q in order. It maintains the overlapping structure and the uniform backward shift of the starting letter. This agreement with both the structure and movement of the series makes OPQ the logical continuation.

The differences between consecutive terms are 2, 4, 8 and 16. These are powers of two doubling at each step. To continue this pattern, the next difference should be 32. Adding 32 to the last term 31 gives 31 + 32 = 63, which extends the series in a perfectly consistent way.

Option A:
Option A gives 61, which corresponds to a difference of 30 from 31 and fails to match the next power of two. Since the previous differences are 2, 4, 8 and 16, the logical continuation is 32, not 30. Hence, 61 is not correct.

Option B:
Option B yields 63, which is 32 more than 31 and completes the power-of-two difference sequence 2, 4, 8, 16, 32. The pattern remains clean and conceptually strong. Therefore, 63 is the correct next term.

Option C:
Option C suggests 65, giving a difference of 34 from 31, which does not fit into the powers-of-two framework. Such an increment is arbitrary with respect to the earlier differences. Thus, 65 cannot be accepted.

Option D:
Option D offers 67, which is 36 greater than 31 and again unrelated to the powers-of-two idea. Using 67 would disrupt the elegant second-level pattern that defines the series. Therefore, 67 is not the right answer.