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.

Statements A and B explain how combined graphs display related quantities with different graphical forms in the same figure. Statement C is correct because separate vertical scales mean each series must be read against its own axis. Statement E is also true; exam questions often focus on relative trends and comparisons rather than exact calculations. Statement D is false since assuming a common scale when there are two axes can lead to misinterpretation. Therefore, A, B, C and E only are correct, as given in option D.

Option A:
Option A is incomplete because it includes only A, B and C and omits E, which stresses the interpretative focus on trends that is common in NET data questions.

Option B:
Option B is incorrect because it omits B (a correct example of bar–line encoding), so it does not include all true statements even though A, C and E are true.

Option C:
Option C is incorrect because it excludes A (the general definition of combined graphs) and keeps only B, C and E, making it incomplete for “select correct statements”.

Option D:
Option D is correct as it gathers the correct structural and interpretative points about combined graphs and explicitly excludes D, which wrongly assumes a single scale for all 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.

Here the pattern is aₙ = n⁴ + 2n³ + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1+2+1 = 4, 16+16+1 = 33, 81+54+1 = 136, 256+128+1 = 385 and 625+250+1 = 876, which matches the sequence exactly. For n = 6 we compute 6⁴+2·6³+1 = 1296+432+1 = 1729. Therefore 1729 continues the exact same quartic-plus-cubic pattern and is the correct next term.

Option A:
Option A, 1713, is smaller than the formula’s output and does not equal n⁴+2n³+1 for n = 6. It would imply subtracting 16 from the correct value without any justification in the earlier terms. Hence option A violates the established algebraic rule and is incorrect.

Option B:
Option B, 1729, matches precisely the result from the expression n⁴+2n³+1 when n = 6. It maintains the same combination of powers of n that generated all previous numbers in the series. Because no change to the functional relationship is needed, this option correctly extends the sequence.

Option C:
Option C, 1735, slightly overshoots the correct value and again fails to satisfy the expression for n = 6. Choosing 1735 would mean adding an arbitrary 6 only at the last step, which is inconsistent with the pattern. Therefore option C cannot be the correct answer.

Option D:
Option D, 1745, is even further from the predicted value and has no basis in the rule aₙ = n⁴ + 2n³ + 1. Using 1745 would destroy the exact mapping between the index and the term values. Thus option D is not a valid continuation of the given series.

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.

This option is correct because the sum of the interior angles of a triangle in Euclidean geometry is always 180 degrees. This can be shown using parallel lines and alternate interior angles. Every triangle can be thought of as half of a straight line angle sum. Therefore, the correct total is 180 degrees.

Option A:
90 degrees is the measure of a right angle, not the sum of all angles in a triangle. Only one angle in a right triangle is 90 degrees. Thus, this value is too small for the total.

Option B:
270 degrees exceeds the known sum of angles in a triangle and does not correspond to any standard triangle property. It is closer to the sum for different configurations, not a simple triangle. Hence, this option is not valid.

Option C:
360 degrees is the angle measure around a point, not inside a triangle. It equals the sum of interior angles of a quadrilateral, not a triangle. Therefore, it cannot be the correct answer.

Option D:
180 degrees is the established sum for any triangle in a plane. No matter the shape of the triangle, the three interior angles always add up to 180 degrees. This property is widely used in geometry problems, confirming this option as correct.

Statements A, B, C and E correctly characterise the biconditional connective. A and C together state that p ↔ q is true when p and q match in truth value and false otherwise. B is correct since p ↔ q decomposes into (p → q) and (q → p). E is true because swapping p and q in a biconditional does not change its meaning. D is false; with two propositional variables, the truth table has four rows, and the biconditional is true in exactly two of them (both true or both false), not three. Therefore, A, B, C and E only form the correct set.

Option A:
Option A is correct because it collects all four accurate properties and excludes D, which miscounts the true rows of the biconditional. It matches the logical behaviour of “if and only if” as tested in NET reasoning.

Option B:
Option B is incomplete as it leaves out E, so it does not mention the commutative nature of the connective, even though that is an important structural property. Without E, the description is not fully comprehensive.

Option C:
Option C is incorrect because, while B and C and E are true, it omits A, which directly states the key condition about matching truth values. The option therefore does not explicitly identify the central truth condition of the biconditional.

Option D:
Option D is wrong because it retains D, the statement that there are three true rows, which is factually incorrect, and it omits B. Once a wrong statement is included, the option cannot represent the exact set of correct statements.

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.

Statements A and B correctly describe the multiplication and addition rules of counting, and D is true because many questions can be answered directly by reasoning about choices without naming permutations or combinations explicitly. Statement C is false since the multiplication rule in its simplest form assumes independence or appropriately defined conditional paths, and it cannot be applied blindly when choices are dependent. Statement E is false because counting principles are explicitly part of NET Unit 5. Thus, the combination A, B and D only is correct.

Option A:
Option A is correct as it includes the key rules and the note that problems can sometimes be solved without formal notation, while excluding C and E, which misrepresent the applicability of the multiplication rule and the syllabus.

Option B:
Option B is incomplete because it omits D, thereby failing to acknowledge that intuitive counting can sometimes bypass formal permutation or combination formulas in exam settings. Without D, the practical approach is not highlighted.

Option C:
Option C is wrong because it includes C, which incorrectly asserts that the multiplication rule applies regardless of dependence, and omits A, leaving out the fundamental statement of the rule itself.

Option D:
Option D is incorrect since it accepts C and thus endorses an overgeneralisation of the multiplication rule, even though A and D are true. The presence of C makes the overall combination logically unsound.