UGC NET Questions (Paper – 1)

The first letters are K, L, M and N, moving forward by one each time. The second letters are M, N, O and P, again moving forward by one. The third letters are O, P, Q and R, also advancing by one. The next term should therefore have letters O, Q and S, giving OQS.

Option A:
Option A is correct because it increases each letter of NPR by exactly one step: N→O, P→Q and R→S. This preserves the parallel +1 movement in all three positions. As it matches the transformation applied between each pair of successive terms, OQS is the proper continuation.

Option B:
Option B, OPR, keeps the first letter O but returns to P and R in the second and third positions. That fails to advance these positions as the pattern requires. Because not all three letters move forward, OPR does not fit the series.

Option C:
Option C, OQT, advances the third letter too far from R to T, a jump of two steps. The series shows only one-step shifts. This extra movement breaks the uniform transformation pattern, so OQT is not acceptable.

Option D:
Option D, ORU, changes the second and third letters unevenly and does not continue the simple +1 progression in all positions. It does not arise from applying the same shift to NPR. Therefore ORU cannot be the correct answer.

The series is described by the quadratic formula aₙ = 3n²+2n+1 for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 17, 34, 57 and 86, which match the given terms exactly. For n = 6 we obtain 3·6²+2·6+1 = 108+12+1 = 121. Hence the next term must be 121 to preserve the same quadratic relationship.

Option A:
Option A, 111, does not equal 3·6²+2·6+1 and therefore fails to extend the polynomial rule consistently. It would require lowering the value obtained from the formula by 10, which has no justification in the structure of the sequence. Thus 111 cannot be the correct continuation.

Option B:
Option B, 121, is exactly the value produced by the formula aₙ = 3n²+2n+1 when n = 6. It maintains the same dependence on n that explains all earlier terms. Because the pattern holds perfectly when this value is used, 121 is the correct next term.

Option C:
Option C, 115, lies between 111 and 121 but does not arise from substituting n = 6 into the quadratic expression. Selecting 115 would break the neat quadratic pattern and make the rule inconsistent. Therefore it is not the right answer.

Option D:
Option D, 129, is larger than the value predicted by the formula and again cannot be written as 3n²+2n+1 for n = 6. Using 129 would lose the tight algebraic link between n and aₙ, so it is not a valid continuation.

Each group is a block of four consecutive letters written in reverse order. The starting letters Z, X, V and T occupy positions 26, 24, 22 and 20, decreasing by two each time. The next starting position is 18, which is R, and the reversed block from R through O is RQPO. Thus RQPO continues both the backward step and the internal reverse structure.

Option A:
Option A, RSTU, is in forward order and does not follow the reversal rule.

Option B:
Option B, RQOP, is not a clean reverse of four consecutive letters (the last two letters are out of order).

Option C:
Option C, SRQP, starts at S instead of R, breaking the −2 start-step rule and also distorts the block.

Option D:
Option D is correct because it starts at R (two positions before T) and lists R, Q, P, O in strict descending consecutive order.

The differences between successive terms are 3, 9, 27 and 81. Each difference is obtained by multiplying the previous difference by 3, forming a geometric progression at the second level. The next difference should therefore be 81 × 3 = 243. Adding 243 to the last term 122 gives 122 + 243 = 365, which continues the pattern exactly.

Option A:
Option A gives 362, corresponding to a difference of 240 from 122, which does not match the expected value of 243. This breaks the geometric pattern in the differences. Therefore, 362 is not consistent with the rule.

Option B:
Option B offers 364, yielding a difference of 242 from 122, one less than the required 243, and again failing to preserve the exact growth in differences. Hence, 364 cannot be accepted as correct.

Option C:
Option C yields 365, which is 243 more than 122 and matches the predicted next difference in the geometric sequence 3, 9, 27, 81, 243. This keeps both the main series and the second-level pattern in harmony. Thus, 365 is the correct next term.

Option D:
Option D suggests 368, giving a difference of 246 from 122, which overshoots the expected difference and disrupts the clean factor-of-three rule. Therefore, 368 is not a valid continuation.

Observe that each term’s letters move one step forward from the previous term: ACE (1,3,5), BDF (2,4,6), CEG (3,5,7) and DFH (4,6,8). The next set of positions should be (5,7,9), which correspond to E, G and I. Thus EGI is the only group that continues the pattern of simultaneously advancing all three positions by one.

Option A:
Option A, EFI, uses E and F correctly as increasing letters but breaks the pattern at the middle position. The second letter should move from 6 to 7, which is G, not F. Since the triple shift must occur uniformly across all letters, EFI does not fully align with the rule.

Option B:
Option B is correct because its letters E, G and I map exactly to positions 5, 7 and 9, matching the expected progression. It continues the uniform +1 shift from DFH in every letter. This consistency across all three positions makes EGI the precise continuation of the series.

Option C:
Option C, EHI, places H instead of G in the middle, jumping two positions forward from F instead of one. This disrupts the uniform one-step movement in the second letter. As the pattern requires equal increments, EHI cannot be accepted as the next term.

Option D:
Option D, FGJ, alters all three letters differently and no longer represents a simple one-step shift from DFH. It especially distorts the first letter, which should move from D to E, not to F. Because it does not respect the common transformation applied to every letter, FGJ fails the pattern test.

The volume of a cube is proportional to the cube of its edge, so if V₁:V₂ = 27:64, then (a₁³):(a₂³) = 27:64. Taking cube roots gives a₁:a₂ = 3:4. Surface area is proportional to the square of the edge, so the surface area ratio is a₁²:a₂² = 3²:4² = 9:16. Thus, the required ratio of surface areas is 9:16.

Option A:
Option A, 3:4, is the ratio of the edges, not of the surface areas. While it correctly comes from the cube root of the volume ratio, it does not account for the fact that surface area scales with the square of the linear dimension rather than the first power.

Option B:
Option B, 4:3, reverses the correct edge ratio and therefore also reverses the implied surface area ratio. It would suggest that the cube with smaller volume has larger surface area, which contradicts the geometric relationships given by the volume ratio.

Option C:
Option C, 27:64, simply repeats the volume ratio and ignores the need to convert from a three-dimensional measure (volume) to a two-dimensional measure (surface area). Since these scale differently, the same ratio cannot apply to both.

Option D:
Option D is correct because it follows the sequence of taking the cube root to find the edge ratio and then squaring that to find the surface area ratio. This two-step transformation, from volume to side to area, is exactly what the geometry of similar cubes requires.

Inside each group the jumps between letters follow a fixed but asymmetric pattern, and this pattern is repeated from term to term. Comparing BILO, CJMP, DKNQ and ELOR shows that each letter in a given position moves forward by one place from one group to the next. If we shift every letter of ELOR forward by one step, E becomes F, L becomes M, O becomes P and R becomes S, giving FMPS. Since this option alone keeps both the internal spacing pattern and the uniform column wise shift, FMPS is the correct next term.

Option A:
Option A, FMPT, matches the first three letters of the true next group but changes the last letter so that the spacing from P to T differs from the spacing found in the earlier sequences. This breaks the repeated structure of the last position. Because the final jump is inconsistent with the established pattern, FMPT cannot be the correct answer.

Option B:
Option B, GMQT, shifts some letters too far ahead and does not arise from a simple one step advance from ELOR. Its second and third letters no longer align with the smooth vertical progressions in the previous groups. As it fails to respect both the column wise movement and the internal gaps, GMQT does not fit the series.

Option C:
Option C is correct because FMPS is exactly one alphabetical step ahead of ELOR in every position, while still preserving the asymmetric internal distance pattern. The starting letters E and F line up with the earlier chain from B through E, and the other columns continue similarly. This strict mechanical derivation confirms FMPS as the proper continuation.

Option D:
Option D, ENQS, keeps E and N from the last term and changes only the remaining letters, which contradicts the rule that all letters move together. The internal pattern among its letters also differs from that in the given groups. Therefore ENQS is not a valid extension of the series.

The pattern here arises from the formula aₙ = 2n⁴−1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 2·1⁴−1 = 1, 2·2⁴−1 = 31, 2·3⁴−1 = 161, 2·4⁴−1 = 511 and 2·5⁴−1 = 1249. For n = 6 we compute 2·6⁴−1 = 2·1296−1 = 2592−1 = 2591. Therefore 2591 is the only term that correctly extends the quartic-based pattern.

Option A:
Option A, 2511, is far below the calculated value and does not equal 2n⁴−1 for any n that continues the sequence. It ignores the rapid growth implied by the quartic expression. Thus 2511 is not a valid next term.

Option B:
Option B, 2541, is still different from 2591 and cannot be obtained from 2·6⁴−1. Choosing it would break the exact algebraic rule that explains all the previous terms. Hence 2541 cannot be the correct answer.

Option C:
Option C, 2591, matches the result of evaluating 2n⁴−1 when n = 6. It preserves the quartic dependence on n and continues the rapid increase seen in the sequence. For these reasons, 2591 is the correct continuation.

Option D:
Option D, 2651, overshoots the correct value and is not equal to 2n⁴−1 for any appropriate n in order. Adopting 2651 would distort the pattern and introduce an unjustified jump. Therefore it is not a valid option.

BCFI, CDGJ, DEHK and EFIL all share the same pattern of unequal gaps between consecutive letters, and each new group is obtained by moving every letter one place forward in the alphabet. Looking at the columns, the first letters go B, C, D, E, the second C, D, E, F, the third F, G, H, I and the fourth I, J, K, L. Continuing these sequences, the next group should have first letter F, second G, third J and fourth M. That forms FGJM, which keeps both the internal spacing pattern and the column wise advancement, making it the correct answer.

Option A:
Option A, FGKM, uses F and G as the first two letters but changes the last two so that the third and fourth columns do not follow their observed sequences. The internal distance between K and M also differs from that in the earlier groups. Because it fails to preserve both aspects of the pattern, FGKM is not the correct continuation.

Option B:
Option B is correct because FGJM comes from shifting every letter in EFIL forward by one step and it maintains the same uneven yet consistent gaps inside the group. The series of first letters B, C, D, E extends to F and the other columns behave similarly. This faithful extension of the structure confirms FGJM as the next term.

Option C:
Option C, GHJM, starts with G instead of F, breaking the simple one step increase for the first letters. Its other letters also do not match the vertical progressions exactly. Therefore GHJM does not fit the rule governing the series.

Option D:
Option D, EGJM, retains some letters from EFIL but does not move all positions forward simultaneously. That contradicts the clear pattern that every letter in a term is advanced together to create the next term. Hence EGJM cannot be the correct answer.

Let the two numbers be 4x and 9x. Their geometric mean is √(4x × 9x) = √(36x²) = 6x, which is given as 12. Thus, 6x = 12, so x = 2. The numbers are 8 and 18, and the larger of the two is 18. This directly uses the definition of geometric mean as the square root of the product.

Option A:
Option A, 8, is the smaller number arising from the correct solution and does not answer the question asking for the larger number. Although 8 does participate in the correct ratio and geometric mean, it is not the required value.

Option B:
Option B, 12, is the geometric mean itself, not one of the original numbers. The geometric mean lies between the two numbers in magnitude and is not equal to either when the ratio is not 1:1. Therefore, 12 cannot be the larger number.

Option C:
Option C, 16, would produce a partner number of 64/9 ≈ 7.11 if we tried to keep the ratio 4:9, which is not an integer and also would not give a geometric mean of 12. So 16 does not fit the given constraints on the ratio and mean.

Option D:
Option D is correct because with x = 2, the numbers 8 and 18 satisfy both the ratio 4:9 and the geometric mean 12. Checking, √(8 × 18) = √144 = 12, which confirms that 18 is the correct larger number.

The positions of these letters are Z(26), W(23), T(20), Q(17) and N(14). The difference between consecutive terms is −3 each time (26→23, 23→20, 20→17, 17→14). Continuing the same rule, the next position is 14 − 3 = 11, which is the letter K. Therefore K is the unique letter that preserves the constant backward step of 3.

Option A:
Option A, L, is at position 12, which is only 2 steps behind N at 14. This contradicts the strict pattern of subtracting 3 from the position each time. Because L does not maintain the consistent decrement, it cannot be part of the series.

Option B:
Option B, M, corresponds to position 13, giving a difference of 1 from N. There is no indication that the size of the backward step is reducing as the series progresses. Since the pattern clearly shows a constant −3, M fails to fit the rule.

Option C:
Option C, J, is at position 10, which is 4 positions behind N. This would increase the backward jump to −4, altering the original rule of the sequence. As there is no evidence of changing step sizes, J does not correctly continue the series.

Option D:
Option D is correct because K is at position 11, exactly 3 positions before N. This maintains the uniform backward movement that characterises the sequence. Keeping the decrement constant is essential for logical consistency, so K is the only valid next term.

In this pattern the first and second letters gradually move forward through the alphabet while the third letter moves backward from the end. AMZ, BNY, COX and DPW show that each column increases by one step for the first two letters and decreases by one step for the last letter. Continuing this synchronized movement, A,B,C,D lead naturally to E as the next first letter, the middle letters M,N,O,P give Q as the next, and the last letters Z,Y,X,W give V. This produces EQV, which alone respects the exact motions of every position and is therefore the correct next term.

Option A:
Option A, EQW, uses the correct first two letters but keeps the last letter at W instead of moving it back to V. This breaks the clear backward march of the final letters from Z towards the middle of the alphabet. Since the last position is not updated consistently, EQW cannot be the right continuation.

Option B:
Option B, FPV, advances the first letter too far and also disturbs the coordinated step of the middle and last letters. The resulting group does not come from applying the same one step shift in each column that produced the earlier terms. Because of this mismatch, FPV does not fit the pattern.

Option C:
Option C is correct because EQV is obtained by exactly the same type of movement that takes AMZ to BNY, BNY to COX and COX to DPW. Each first letter moves one place forward, each middle letter moves one place forward and each last letter moves one place backward. This perfect alignment with the existing rule makes EQV the only consistent choice.

Option D:
Option D, ERU, moves the last letter too far back while pushing the middle letter ahead in a way that does not match the established sequences. The column wise progressions would be broken if this group were inserted. Therefore ERU cannot be considered the next term in the series.

The terms are squares of consecutive even numbers: 2² = 4, 4² = 16, 6² = 36 and 8² = 64. The next even number in sequence is 10. Squaring 10 gives 100. Therefore, 100 is the next term that preserves this structure.

Option A:
Option A is 90, which is not a perfect square and cannot be written as k² for an integer k. It breaks the pattern of exact squares. Hence, this option is not suitable.

Option B:
Option B is 96, again not a perfect square of an integer. There is no clear way to match it with the rule based on even squares. Therefore, 96 is not correct.

Option C:
Option C equals 100, which is 10² and follows 2², 4², 6² and 8² naturally. The extended series 4, 16, 36, 64, 100 completes the pattern of consecutive even squares. This makes 100 the valid continuation.

Option D:
Option D is 108, which lies between 10² and 11² but is not itself a square. Since the rule is precisely about squares, 108 cannot be accepted.

This series is generated by the expression (a_n = 2n^4 - n + 3) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2−1+3 = 4, 32−2+3 = 33, 162−3+3 = 162, 512−4+3 = 511 and 1250−5+3 = 1248. For n = 6 we compute (2×6^4 - 6 + 3 = 2592 - 3 = 2589). Therefore 2589 is the correct next term.

Option A:
Option A, 2589, coincides with the value of (2n^4 - n + 3) when n = 6. It maintains the same combination of quartic and linear terms that explain all previous numbers. Because it fits the rule exactly, 2589 is the correct continuation.

Option B:
Option B, 2557, is 32 less than the computed value and does not satisfy the expression for n = 6. It would require a sudden drop that has no basis in the pattern, so 2557 is not valid.

Option C:
Option C, 2613, overshoots the predicted value and cannot be obtained from the generating rule. Selecting 2613 would arbitrarily increase the last term and break the algebraic structure. Thus 2613 is not correct.

Option D:
Option D, 2645, deviates even further from 2589 and is not the output of (2n^4 - n + 3) for any integer n continuing the sequence. Adopting 2645 would disrupt the pattern, so it is not the right answer.

Each group uses a sliding window of consecutive prime positions: (2,3,5)=BCE, (3,5,7)=CEG, (5,7,11)=EGK, (7,11,13)=GKM. Next is (11,13,17)=KMQ.

Option A:
Option A uses 8 which is not prime.

Option B:
Option B uses 12 which is not prime (and breaks the prime-based rule).

Option C:
Option C is correct because K(11), M(13), Q(17) are consecutive prime positions after the prior window.

Option D:
Option D starts with L(12), which is not a prime position.