UGC NET Questions (Paper – 1)

Statement A is correct as series questions almost always apply some rule linking neighbouring terms. Statement B is true since letter series may involve transformations like skipping or reversing. Statement D is also correct because the intended pattern must be consistent with all terms and is typically the simplest one that works. Statement C is false (first two figures are not always enough), and E is false (blind guessing is risky). Thus, A, B and D only are correct.

Option A:
Option A is correct because it includes all true statements and leaves out C and E.

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

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

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

To find x and y, we solve the system of linear equations using elimination or substitution. Solving 3x + 2y = 26 and 2x + 3y = 19 gives x = 8 and y = 1. Their difference x − y is therefore 8 − 1 = 7, a clean integer result consistent with both equations.

Option A:
Option A, 5, would be the difference if the values of x and y were closer together, but solving the system shows a larger gap between them. It does not match the algebraic relationships specified.

Option B:
Option B, 6, again does not arise from any correct elimination or substitution procedure. When checked against the solved values of x and y, it fails to match their actual difference.

Option C:
Option C reflects the correct outcome after systematically eliminating one variable and solving for the other. Substituting back gives consistent values of x and y whose difference is exactly 7, agreeing with all the given equations.

Option D:
Option D, 8, overshoots the true difference and would imply values of x and y that do not satisfy both equations simultaneously.

AIQ, BJR, CKS and DLT each consist of three letters that are widely separated by the same fixed distance in the alphabet. At the same time, the first letters move A, B, C, D, the second letters I, J, K, L and the third letters Q, R, S, T, all in simple one step sequences. Extending these sequences gives E as the next first letter, M as the next second letter and U as the next third letter. Together they form EMU, which preserves both the wide equal spacing inside the group and the column wise progression, so EMU is the correct continuation.

Option A:
Option A, EMT, uses E and M correctly but leaves the last letter at T, failing to move the third column forward by one. This breaks the clear sequence Q, R, S, T, U in the last position. Because it does not update every column consistently, EMT cannot be the right answer.

Option B:
Option B is correct because EMU keeps the same distance between its letters as the earlier groups and advances each column exactly one step. The triple A→B→C→D→E, I→J→K→L→M and Q→R→S→T→U continues without interruption. This perfect alignment with the observed pattern makes EMU the only valid next term.

Option C:
Option C, FNV, moves the starting letter two positions ahead of D instead of one and also shifts the others in a way that does not maintain the fixed internal gap. The vertical sequences would be broken in every column. Therefore FNV does not fit the series.

Option D:
Option D, ENU, begins with E but skips one step in the second column by jumping from L to N. This distorts the regular one step middle column progression. As it fails to keep the sequences intact, ENU is not an acceptable continuation.

Average speed is defined as total distance divided by total time, not the average of the two speeds. The car travels 120 km each way, so the total distance is 240 km. The time taken in the first leg is 120 ÷ 40 = 3 hours and in the return leg is 120 ÷ 60 = 2 hours, giving a total of 5 hours. Dividing 240 by 5 yields 48 km/h as the average speed for the entire journey.

Option A:
Option A, 45 km/h, might be guessed by taking a simple average of the two speeds or by an approximate calculation, but it does not correspond to the correct total distance over total time.

Option B:
Option B uses the exact definition of average speed and calculates both distances and times explicitly. It also matches the harmonic mean formula for a round trip with equal distances, which simplifies to 2ab/(a + b).

Option C:
Option C, 50 km/h, exceeds the true average because more time is spent at the lower speed, pulling the overall average below the midpoint between 40 and 60.

Option D:
Option D, 52 km/h, is even higher and clearly inconsistent with spending a substantial part of the journey at just 40 km/h.

The first-level differences are 19, 35, 55 and 79. The second-level differences are 16, 20 and 24, which themselves increase by 4. Continuing this pattern, the next second-level difference should be 28, so the next first-level difference becomes 79+28 = 107. Adding 107 to 199 yields 306, which keeps both levels of the pattern coherent.

Option A:
Option A gives 300, corresponding to a difference of 101 from 199. This would make the second-level differences 16, 20, 24 and 22, breaking the rule that these grow by 4 each time. Thus 300 is not consistent with the established structure.

Option B:
Option B gives 306, implying a difference of 107 from 199. The differences become 19, 35, 55, 79, 107 and the second-level differences are 16, 20, 24, 28. This preserves the steady +4 growth at the second level, so 306 is the correct continuation.

Option C:
Option C gives 310, which corresponds to a difference of 111 from 199. That would produce a second-level jump from 79 to 111 equal to 32, not 28, making the pattern irregular. Hence 310 cannot be accepted.

Option D:
Option D gives 314, yielding a difference of 115 from 199. The last second-level step would become 36, which further disrupts the neat arithmetic progression. Therefore 314 is not a valid next term.

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.

Each term in the series can be expressed as n³ + 1 for successive integers n. When n = 1, 2, 3, 4 and 5, we obtain 1³ + 1 = 2, 2³ + 1 = 9, 3³ + 1 = 28, 4³ + 1 = 65 and 5³ + 1 = 126. The next integer is 6, so the next term should be 6³ + 1. This equals 216 + 1 = 217, which perfectly matches the observed rule.

Option A:
Option A gives 217, which fits the formula n³ + 1 when n = 6. Extending the sequence to 2, 9, 28, 65, 126, 217 keeps the entire series under one simple cubic rule. This confirms that 217 is the correct next term.

Option B:
Option B suggests 218, which is one more than 217 and does not correspond to n³ + 1 for any integer in the natural continuation. It breaks the tight relationship between n and the terms. Therefore, 218 cannot be accepted as correct.

Option C:
Option C offers 225, which is a square number (15²) but not of the form n³ + 1 in this context. It introduces a different kind of pattern unrelated to the existing terms. Hence, 225 does not match the logic of the series.

Option D:
Option D provides 230, which likewise fails to satisfy the n³ + 1 relation for any integer following 5 in sequence. Choosing 230 would abandon the clearly established cubic structure. Thus, 230 is not the right answer.

Statements A and B correctly describe arithmetic and geometric series as having constant difference and constant ratio respectively. Statement C is also true because many exam series mix patterns, for example by forming two interleaved sub-sequences. Statement E is correct since checking whether values rise, fall or oscillate helps to guess the right pattern. Statement D is false as some questions deliberately use interleaving or alternating rules. Therefore, the only combination that gathers all true statements and excludes the false one is A, B, C and E only.

Option A:
Option A is incorrect because although A, B and C are all true, the option omits E, which is also a correct and useful principle for analysing series. Since it does not list all the true statements, it cannot be the right answer.

Option B:
Option B is correct because it includes each true statement (A, B, C and E) and excludes D, which wrongly denies the possibility of interleaved or alternating sub-series in exam questions. It therefore matches the complete set of correct statements.

Option C:
Option C is wrong because it includes D, which incorrectly rules out interleaving of sub-series, and at the same time leaves out B, which correctly characterises geometric series. The presence of a false statement and the omission of a true one make this combination invalid.

Option D:
Option D is incorrect because it retains D along with B and E and omits A and C, so it both includes a false restriction about patterns and drops two true statements about arithmetic and mixed series.

Each term of the series can be written as n⁴+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1⁴+1 = 2, 2⁴+1 = 17, 3⁴+1 = 82, 4⁴+1 = 257 and 5⁴+1 = 626. The next term corresponds to n = 6, which gives 6⁴+1 = 1296+1 = 1297. Therefore 1297 is the only value that continues the same quartic pattern.

Option A:
Option A matches the value obtained from the rule aₙ = n⁴+1 when n = 6. It extends the exact same relationship that holds for all the earlier terms of the sequence. Because no change in the functional rule is needed to reach 1297, this option is fully consistent with the observed pattern.

Option B:
Option B, 1257, is close in magnitude but does not equal 6⁴+1 and therefore fails to satisfy the quartic formula for n = 6. Choosing it would require altering the rule that clearly generates the previous terms. Hence it cannot represent the correct continuation of the series.

Option C:
Option C, 1273, similarly does not arise from the expression n⁴+1 for any integer n that follows the existing index order. It breaks the tight link between term position and value that is evident in the given numbers. Thus 1273 is not a valid next term.

Option D:
Option D, 1327, overshoots the correct quartic value and cannot be expressed as 6⁴+1. Adopting 1327 would mean abandoning the neat polynomial structure and introducing an arbitrary adjustment. Therefore it is not the correct answer.

Starting letters W(23), T(20), Q(17), N(14) decrease by −3 each time. Next start is K(11), giving KLM.

Option A:
Option A starts at J(10), which is −4 from N, not −3.

Option B:
Option B is correct because K is −3 from N and L,M follow consecutively.

Option C:
Option C starts at L(12), which is −2 from N, not −3.

Option D:
Option D moves forward from N (breaks the backward pattern).

This sequence is described by the rule aₙ = n⁴ + 4n³ + 2 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 into the expression gives 7, 50, 191, 514 and 1127, which are the terms listed in the question. Evaluating the same formula for n = 6 produces 2162. Therefore 2162 is the unique next term that maintains this quartic-plus-cubic structure.

Option A:
Option A, 2146, is 16 less than the value obtained from aₙ = n⁴ + 4n³ + 2 when n = 6. Accepting 2146 would require subtracting 16 from the polynomial outcome only at the sixth term. Since earlier terms agree perfectly with the rule, this option is not compatible with the pattern.

Option B:
Option B, 2150, is 12 less than the correct value and again does not equal the expression’s output for n = 6. It provides a near miss but still disturbs the exact functional relationship linking index and term. Consequently, option B cannot correctly continue the sequence.

Option C:
Option C, 2156, is 6 less than 2162 and likewise fails to match the formula’s value. Choosing 2156 would introduce a small but unjustified deviation from the pattern at the final step. Because the series is exactly captured by aₙ = n⁴ + 4n³ + 2, option C must be rejected.

Option D:
Option D, 2162, coincides exactly with the number generated by the rule for n = 6. It preserves both the fourth-power and cubic components and keeps the constant term unchanged. As it fits seamlessly within the existing algebraic structure, 2162 is the correct next term in the number series.

The terms follow the pattern aₙ = 3n⁴ − 2n² + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 3·1⁴−2·1²+1 = 2, 3·2⁴−2·2²+1 = 41, 3·3⁴−2·3²+1 = 226, 3·4⁴−2·4²+1 = 737 and 3·5⁴−2·5²+1 = 1826, exactly matching the given sequence. For n = 6 this formula gives 3·6⁴−2·6²+1 = 3·1296−72+1 = 3888−72+1 = 3817. Hence 3817 is the unique next term that keeps the polynomial rule intact.

Option A:
Option A, 3817, is precisely the value obtained from 3n⁴−2n²+1 when n = 6. It continues the same quartic pattern that accurately generates all of the previous terms. Because no change in coefficients or constants is needed, this option is fully consistent with the structure of the sequence.

Option B:
Option B, 3805, is smaller than the formula’s output and does not satisfy 3·6⁴−2·6²+1. Choosing 3805 would require subtracting an extra 12 only at the final term. Such an arbitrary adjustment breaks the clean algebraic relationship and therefore cannot be correct.

Option C:
Option C, 3825, is larger than the correct value and again does not arise from substituting n = 6 into the rule. It would imply inflating the term by 8 compared to the polynomial prediction. Since the earlier terms fit the rule exactly, this deviation is unjustified and makes option C wrong.

Option D:
Option D, 3841, deviates even further from the formula’s result. It has no basis in the expression 3n⁴−2n²+1 for the next index and would destroy the precise mapping between n and aₙ. Therefore option D is not a valid continuation of the given series.

The groups show both a sliding start and an increasing length. AB has length 2, BCD length 3, CDEF length 4 and DEFGH length 5, with each new group starting one letter later than the previous one. The next starting letter should be E and the length should increase to 6, giving EFGHIJ. Therefore EFGHIJ is the only option that satisfies both the starting position and length progression.

Option A:
Option A, EFGH, begins at the correct starting letter E but has length 4 instead of 6. It fails to reflect the clear pattern of increasing the group length by one at each step. Since it does not incorporate all the required letters up to J, EFGH is not the correct continuation.

Option B:
Option B is correct because it starts at E and includes exactly six consecutive letters, E through J. This matches the observed rule that each new term begins one letter later and has one additional letter compared to the previous one. By capturing both dimensions of the pattern, EFGHIJ logically follows DEFGH.

Option C:
Option C, CDEFGH, uses an earlier starting point C and repeats a length that does not align with the series stage. It effectively moves backward in terms of the starting letter while extending the length, which is not consistent with the observed movement. Hence CDEFGH cannot be chosen as the next term.

Option D:
Option D, ABCDEF, resets the pattern to start again at A and does not respect the sliding behaviour. It also gives a length that does not follow directly from the preceding term. For these reasons, ABCDEF is not a valid continuation.

The terms satisfy the formula aₙ = n³+n+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1+1+1 = 3, 8+2+1 = 11, 27+3+1 = 31, 64+4+1 = 69 and 125+5+1 = 131. For n = 6 the same rule produces 216+6+1 = 223. Consequently, 223 is the only value that continues this cubic-plus-linear pattern.

Option A:
Option A, 223, comes directly from applying aₙ = n³+n+1 with n = 6. It preserves the structure where each term combines a cubic part, the index itself and a constant. Because this formula works for all terms including the next, 223 is the correct answer.

Option B:
Option B, 213, is 10 less than the computed value and does not satisfy the generating expression. It breaks the exact algebraic pattern that has been consistent so far. Hence 213 is not a valid continuation of the series.

Option C:
Option C, 217, is closer but still not equal to 6³+6+1. Adopting 217 would mean altering the constant part or the linear component arbitrarily, which is not justified by the data. Thus 217 is not correct.

Option D:
Option D, 231, exceeds the correct value and again fails to equal n³+n+1 for n = 6. Using 231 would destroy the clean matching between the formula and the sequence, so it is not the right option.

The differences alternate between +2 and +4: 7→9 (+2), 9→13 (+4), 13→15 (+2), 15→19 (+4) and 19→21 (+2). Following this pattern, the next difference should be +4. Adding 4 to 21 gives 25. Therefore, 25 is the next term that maintains the alternating pattern.

Option A:
Option A is 23, which would add only 2 to 21 and repeat the smaller jump twice in a row. That contradicts the +2, +4 alternation visible in the earlier terms. Hence, 23 is not correct.

Option B:
Option B equals 25, obtained by 21 + 4 and continuing the established +2, +4, +2, +4, +2, +4 sequence of differences. This keeps the pattern consistent and regular. Thus, 25 is the appropriate next term.

Option C:
Option C is 27, representing a jump of 6 from 21, which has not appeared anywhere in the difference sequence. Introducing +6 here would break the simple alternation between +2 and +4. Therefore, 27 cannot be chosen.

Option D:
Option D is 29, which corresponds to an increase of 8 from 21 and departs even further from the observed pattern. As there is no justification for such a large jump, this option is not acceptable.