UGC NET Questions (Paper – 1)

This series is governed by the rule aₙ = n⁴ + 2n³ + 4 with n starting at 1. For n = 1, 2, 3, 4 and 5 the expression gives 7, 36, 139, 388 and 879, matching the provided terms. Evaluating the same formula at n = 6 yields 1732. Therefore 1732 is the unique next term that preserves this quartic-plus-cubic pattern.

Option A:
Option A, 1708, is 24 less than the value obtained from aₙ = n⁴ + 2n³ + 4 when n = 6. Using 1708 would require lowering the exact polynomial result only at this index. Since no such adjustment occurs earlier, this option does not align with the pattern and is incorrect.

Option B:
Option B, 1720, is still below the required value and does not equal the formula’s output for n = 6. It partially approaches the right number but still breaks the precise functional relationship. Consequently, 1720 cannot be the correct continuation of the sequence.

Option C:
Option C, 1726, lies closer but is still 6 less than the correct value 1732. Accepting 1726 would again mean tampering with the polynomial outcome at only one term. Because the sequence is exactly modelled by aₙ = n⁴ + 2n³ + 4, any deviation at n = 6 makes this option wrong.

Option D:
Option D, 1732, is exactly the result of substituting n = 6 into the expression n⁴ + 2n³ + 4. It keeps the same structure of a dominant fourth-power term plus a cubic and constant term. As it fits perfectly into the established rule, 1732 is the correct next term.

Each term in the series can be written as n³ − 1 for successive integers n. For n = 2, 3, 4 and 5 we get 2³ − 1 = 7, 3³ − 1 = 26, 4³ − 1 = 63 and 5³ − 1 = 124. The next integer is 6, so the next term should be 6³ − 1. This equals 216 − 1 = 215, which maintains the same cubic minus one relationship.

Option A:
Option A gives 215, which is precisely 6³ − 1 and continues the formula applied in generating all previous terms. The series 7, 26, 63, 124, 215 clearly follows the rule n³ − 1 for n = 2 to 6. Hence, 215 is the logically correct next term.

Option B:
Option B offers 216, which is 6³ but does not follow the "minus one" adjustment that characterises every earlier term. Choosing 216 would change the defining rule of the sequence. Therefore, 216 is not consistent with the pattern.

Option C:
Option C presents 220, which has no direct connection with the values of n³ − 1 for consecutive integers. It appears as a random large number without algebraic justification in this context. Thus, 220 is not a valid continuation.

Option D:
Option D gives 230, which similarly does not correspond to any n³ − 1 in the progression and does not respect the underlying structure. Including 230 would make the series irregular and conceptually weak. Hence, it cannot be considered correct.

A valid argument is one in which, if the premises are true, the conclusion must be true because of the logical structure. Validity concerns the form of the argument rather than the actual truth of the premises. In mathematical reasoning and proofs, ensuring validity guarantees that no incorrect conclusion is drawn from correct assumptions. Therefore, the phrase "conclusion necessarily follows from the premises" directly describes a valid argument.

Option A:
A sound argument is valid and also has all true premises, which is a stronger condition than mere validity. The question, however, focuses only on the logical relation between premises and conclusion, not on whether the premises are true. Because soundness includes an extra requirement not mentioned, this term does not exactly match the description. Hence, it is not the best answer.

Option B:
A strong argument is a term more often used in inductive reasoning to indicate that the premises provide substantial support for the conclusion, but not certainty. Strength suggests high probability, not logical necessity. Since the question emphasises that the conclusion necessarily follows, "strong" is not the correct technical term.

Option C:
A weak argument gives little support to its conclusion and often suffers from logical gaps or insufficient evidence. It certainly does not guarantee that the conclusion follows from the premises. Thus, it is almost the opposite of what the question describes. Consequently, this option is clearly incorrect.

Option D:
Valid is correct because validity captures exactly the idea that the logical form forces the conclusion to follow from the premises. It does not depend on content but on structure, which is crucial in mathematical proofs and reasoning. Recognising validity helps in evaluating arguments on exams and in solving theoretical problems.

Let the original numbers of boys and girls be 7k and 5k. After 6 boys leave and 14 girls join, the new numbers are 7k − 6 and 5k + 14, and the ratio becomes 1:1. Hence 7k − 6 = 5k + 14, which gives 2k = 20 and k = 10. Therefore the original number of boys is 7k = 70. This matches the condition that the class becomes balanced after the given changes.

Option A:
Option A is correct because it is directly obtained from solving the linear equation formed by the new equal numbers of boys and girls. With 70 boys and 50 girls initially, removing 6 boys and adding 14 girls yields 64 boys and 64 girls. The ratio 64:64 is indeed 1:1, confirming the validity of 70 as the original boy strength.

Option B:
Option B, 64, represents the number of boys after 6 leave in the correct solution, not the original count. Treating 64 as the original figure would lead to a different number of boys after departure and thus fail to produce equality with the adjusted number of girls. Therefore 64 does not satisfy the original ratio 7:5 along with the transformation.

Option C:
Option C, 80, would give girls as (80 × 5/7) ≈ 57.14 to maintain the original ratio, which is not an integer and hence not viable for a headcount. Even if we forced integers near these values, the changed numbers would not become equal after subtracting and adding the specified students.

Option D:
Option D, 84, similarly leads to a non-integer girls count when used with ratio 7:5, and the subsequent changes would not result in a 1:1 ratio. This contradicts both the initial and final conditions, so 84 cannot be the original number of boys.

This sequence is generated by the rule aₙ = 3n⁴ + 2n³ + 1 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 into the expression gives 6, 65, 298, 897 and 2126, matching the given terms. For n = 6 we compute a₆ = 3·6⁴ + 2·6³ + 1 = 3·1296 + 2·216 + 1 = 3888 + 432 + 1 = 4321. Therefore 4321 is the only value consistent with this polynomial pattern.

Option A:
Option A, 4297, is 24 less than the formula’s output for n = 6 and cannot be produced by 3n⁴ + 2n³ + 1. Accepting 4297 would require lowering the correct value only at the last step, even though earlier terms follow the rule exactly. Hence option A is not a valid continuation.

Option B:
Option B, 4321, matches exactly the value obtained when n = 6 is substituted into the rule. It preserves the dominance of the quartic term with an added cubic component and a constant, just as in previous terms. Because it extends the sequence in a way that fully respects the algebraic structure, this option is correct.

Option C:
Option C, 4309, is 12 less than the correct term 4321 and again does not equal 3·6⁴ + 2·6³ + 1. It is numerically close but fails to reflect the exact functional relationship, so it cannot be accepted.

Option D:
Option D, 4333, is 12 greater than the proper value and cannot be generated by the same expression at n = 6. Choosing 4333 would create an artificial jump in the series that is not implied by the pattern. Therefore this option is incorrect.

D and E are wrong because they reverse the meanings of tautology and contingency. A tautology is true in all possible situations, not always false, and a contingent proposition is true in some situations and false in others, not true in all. Statements A and B correctly describe what counts as a statement in logic, and C accurately expresses the idea that contradictory statements cannot both be true simultaneously. Thus the pair D and E only is the correct set of wrong statements.

Option A:
Option A is incorrect because it selects D alone, but E is also wrong; contingent propositions are not true in all possible situations. By ignoring E, Option A fails to identify all the incorrect statements.

Option B:
Option B is also incorrect because it focuses only on E and ignores D, which misdefines tautology as always false. Since both D and E are wrong, singling out only one of them does not satisfy the requirement of the question.

Option C:
Option C is correct because it groups together D and E, the two statements that misrepresent the key logical categories of tautology and contingency. It leaves A, B and C untouched, acknowledging that they are true characterisations of statements and contradictions.

Option D:
Option D is wrong because it includes C, which is actually true: contradictory statements cannot both be true at the same time. By treating C as if it were wrong along with D, this option misidentifies the logical property of contradiction.

Statements A, B and C express the basic axioms about range of probabilities and values for sure and impossible events. Statement D gives the general addition rule for the probability of a union of two events. Statement E is false because probabilities are never negative. Therefore, A, B, C and D only form the correct set.

Option A:
Option A is incomplete as it omits D (union/addition rule).

Option B:
Option B is incomplete because it omits B (sure event) and does not include all core axioms together.

Option C:
Option C is incomplete since it omits A (range restriction 0 ≤ P(A) ≤ 1).

Option D:
Option D is correct because it includes all true statements and excludes E, which suggests an impossible negative probability.

The pattern here is given by aₙ = 3n⁴ + 2n² − n, with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3·1⁴+2·1²−1 = 4, 3·2⁴+2·2²−2 = 54, 3·3⁴+2·3²−3 = 258, 3·4⁴+2·4²−4 = 796 and 3·5⁴+2·5²−5 = 1920, which matches the given series exactly. For n = 6 the same expression gives 3·6⁴+2·6²−6 = 3888+72−6 = 3954. Hence 3954 is the next term that preserves this quartic-plus-quadratic-minus-linear relationship.

Option A:
Option A, 3924, is 30 less than the value dictated by the formula and does not satisfy aₙ = 3n⁴+2n²−n for n = 6. It would require a sudden downward shift that is not reflected in the behaviour of previous terms. Therefore option A is not consistent with the underlying rule.

Option B:
Option B, 3954, matches exactly the result of applying 3n⁴+2n²−n when n = 6. It continues the same combination of polynomial components that generated each earlier term in the series. Because no change to the algebraic structure is necessary, 3954 is the correct continuation.

Option C:
Option C, 3984, overshoots the correct value by 30 and again does not come from the formula for the sixth term. Choosing 3984 would arbitrarily increase the term and disturb the precise mapping between n and aₙ. Thus option C is not logically acceptable.

Option D:
Option D, 3996, deviates even more from the correct result and has no basis in the expression 3n⁴+2n²−n. Using 3996 would destroy the internal consistency of the series and cannot be considered correct.

The terms are factorials: 1! = 1, 2! = 2, 3! = 6, 4! = 24 and 5! = 120. Following this rule, the next term should be 6!. Calculating 6! gives 1×2×3×4×5×6 = 720. Therefore, 720 is the correct continuation of the factorial sequence.

Option A:
Option A matches the pattern exactly because it equals 6!, the next factorial in order. The series 1, 2, 6, 24, 120, 720 clearly shows factorial growth of 1, 2, 3, 4, 5 and 6. This consistency confirms 720 as the right answer.

Option B:
Option B is 600, which has no direct factorial interpretation in this context. It is neither 5! nor 6!, so it does not follow the factorial rule. Hence, 600 cannot be correct.

Option C:
Option C is 480, which also does not match any factorial value subsequent to 120. Choosing 480 would break the clearly defined multiplicative pattern. Therefore, it is not suitable.

Option D:
Option D is 240, which is only double 120 and would correspond to multiplying by 2 rather than by increasing integers. That is a different pattern from the factorial one, so 240 is not the correct next term.

For a fixed principal P, simple interest is proportional to the product of rate and time, SI ∝ R × T. In the first case, R × T = 9 × 4 = 36. In the second case, R × T = 6 × 7 = 42. Thus, the ratio of the first interest to the second is 36:42, which simplifies to 6:7 on dividing both terms by 6.

Option A:
Option A, 3:4, might be a rough guess comparing 9% and 12% or other partial values, but it does not arise from the exact products 36 and 42. Simplifying 36:42 gives 6:7, not 3:4, so 3:4 misrepresents the proportion of interest amounts.

Option B:
Option B is correct because it strictly follows the rule that simple interest is directly proportional to R × T when principal is constant. Using 36 and 42 as the proportional measures and simplifying leads uniquely to 6:7. This ratio precisely captures the slight advantage of the second scheme’s longer duration and different rate.

Option C:
Option C, 4:5, would require the ratio 36:45, but the second product is 42, not 45. Therefore, this option does not align with the actual calculation of rate–time products.

Option D:
Option D, 5:6, simplifies to 30:36, which again does not match 36:42. Using 5:6 would imply different underlying R × T values than those given, so it cannot represent the true ratio between the two simple interests.

This sequence is generated by the rule aₙ = n⁵ + 2n³ + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 the expression yields 4, 49, 298, 1153 and 3376, matching the given terms exactly. Evaluating the same polynomial for n = 6 gives 8209. Therefore 8209 is the unique next term that preserves this quintic-plus-cubic structure.

Option A:
Option A, 8185, is 24 less than the value produced by aₙ = n⁵ + 2n³ + 1 at n = 6. Adopting 8185 would require lowering the correct polynomial result only at this term. Such an arbitrary correction is not reflected in earlier values, so option A cannot be correct.

Option B:
Option B, 8209, equals exactly the number obtained when n = 6 is substituted into the rule. It maintains the same higher-degree dependence on n that explains the rapid growth of the series. Because no change in coefficients or constants is needed, 8209 correctly continues the number series.

Option C:
Option C, 8225, is slightly greater than the pattern value and does not satisfy the formula for n = 6. Choosing 8225 would add an unjustified extra amount to the correct term, breaking the clean algebraic rule. Thus this option is not a valid continuation.

Option D:
Option D, 8241, deviates even more from the expected value and likewise cannot be produced by n⁵ + 2n³ + 1 at n = 6. Using 8241 would destroy the precise mapping between index and term that characterises the series. Therefore option D is incorrect.

Let the three consecutive even numbers be n − 2, n and n + 2, where n is the middle one. Their sum is (n − 2) + n + (n + 2) = 3n. According to the given condition, 3n = 66, so n = 66 ÷ 3 = 22. Hence, the middle even number is 22.

Option A:
Option A, 20, would give the three numbers 18, 20 and 22, whose sum is 60, not 66. Therefore, it does not satisfy the condition set in the problem.

Option B:
Option B fits perfectly: with 22 as the middle number, the three numbers become 20, 22 and 24, and their sum is 20 + 22 + 24 = 66, matching the given total.

Option C:
Option C, 24, would generate the sequence 22, 24 and 26, whose sum is 72, exceeding the required sum.

Option D:
Option D, 26, would produce 24, 26 and 28, summing to 78, which is also inconsistent with the problem statement.

The first letters are D(4), H(8), L(12) and P(16), each increasing by 4. The second letters are G(7), K(11), O(15) and S(19), again increasing by 4. Continuing this pattern, the next first letter is 16 + 4 = 20 (T) and the next second letter is 19 + 4 = 23 (W). Thus TW is the only pair that keeps the +4 increment intact for both positions.

Option A:
Option A, TW, is correct because it aligns perfectly with the constant step size applied to both letters. It places T as the next in the first-letter progression and W as the next in the second-letter progression. This combination preserves the arithmetic structure of the series.

Option B:
Option B, UX, shifts both letters further than necessary and changes the relative spacing between them. It does not correspond to the next set of positions obtained by adding 4. Because it breaks the consistent arithmetic progression, UX cannot be accepted.

Option C:
Option C, SV, uses S as the first letter, which should actually be the second letter based on previous terms. It also introduces a different spacing between the letters, disturbing the fixed difference of four positions for each component across terms. Hence SV does not satisfy the precise pattern.

Option D:
Option D, TV, gives the correct first letter T but moves the second letter only to V, which is position 22 instead of 23. This violates the required +4 movement in the second component. Since both positions must follow identical step sizes, TV cannot be the correct continuation.

The differences between the terms are 3, 6, 12, 24 and 48. Each difference is double the previous one, forming a geometric progression at the second level. The next difference should therefore be 96. Adding 96 to the last term 97 gives 97 + 96 = 193, which follows this doubling pattern exactly.

Option A:
Option A gives 187, corresponding to a difference of 90 from 97, which does not match the required next difference of 96. This breaks the doubling rule that defines the series. Hence, 187 is not consistent with the pattern.

Option B:
Option B offers 189, giving a difference of 92 from 97 and again failing to match the expected 96. This disrupts the geometric progression in the differences. Therefore, 189 cannot be accepted.

Option C:
Option C suggests 191, whose difference from 97 is 94, still not the correct doubled value of 48. This makes the second-level pattern inconsistent. Thus, 191 is not appropriate.

Option D:
Option D yields 193, which is 96 more than 97 and continues the difference sequence 3, 6, 12, 24, 48, 96 in a perfectly structured way. This maintains the integrity of the series. Hence, 193 is the correct next term.

Statement C is wrong because coding–decoding questions are deliberately constructed so that the code is systematically related to the original word, often through shifts, reversals or other consistent transformations. Statements A, B, D and E are all correct: codes frequently use alphabet shifts, reversals or swaps, comparing several coded words is a key strategy and these questions clearly test pattern recognition and analysis. Since only C misrepresents the nature of coding–decoding, the set of wrong statements consists of C alone.

Option A:
Option A is incorrect because it labels both A and C as wrong, even though A correctly describes a very common rule where letters are replaced by those at fixed distances in the alphabet. Treating A as wrong would contradict standard practice in many aptitude questions.

Option B:
Option B is wrong because it includes E among the wrong statements, wrongly suggesting that coding–decoding questions do not assess pattern recognition. In reality, E is an accurate statement about the purpose of such questions.

Option C:
Option C is incorrect as it groups B, C and D together and thereby misclassifies B and D, which are true descriptions of common patterns and solution strategies. Including true statements in the “wrong” set shows that this option does not satisfy the requirement of the question.

Option D:
Option D is correct since it isolates C as the only statement that denies the logical structure underlying coding–decoding questions and leaves all accurate descriptions intact. It reinforces the idea that these items are structured and analysable, not random.