UGC NET Questions (Paper – 1)

This sequence is generated by the formula aₙ = n³+5 for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 yields 1³+5 = 6, 2³+5 = 13, 3³+5 = 32, 4³+5 = 69 and 5³+5 = 130. For n = 6 the same expression gives 6³+5 = 216+5 = 221. Therefore 221 is the correct continuation that keeps the series consistent.

Option A:
Option A, 221, comes directly from the rule aₙ = n³+5 when n = 6. It preserves the cubic-plus-constant pattern that perfectly explains all previous terms. Because no modification of the rule is needed, this option fits seamlessly as the next term.

Option B:
Option B, 211, does not equal 6³+5 and is obtained by subtracting an unrelated amount from the correct value. This breaks the neat pattern connecting the terms to their indices. Thus 211 cannot be accepted as the correct next term.

Option C:
Option C, 215, is closer to 221 but still does not agree with the generating formula. It would require changing the constant 5 at the end, which is consistently applied to earlier terms. Hence 215 is not a valid continuation.

Option D:
Option D, 229, overshoots the correct cubic value and again fails to match the rule aₙ = n³+5. Selecting 229 would destroy the simple uniform structure of the sequence, so it is not the right answer.

A’s one day work is 1/12 of the job and B’s one day work is 1/18 of the job. When they work together, their combined daily work is 1/12 + 1/18. The least common multiple of 12 and 18 is 36, so their combined rate becomes 3/36 + 2/36 = 5/36 of the work per day. The total time to complete one whole work is therefore 1 ÷ (5/36) = 36/5 = 7.2 days.

Option A:
Option A, 7 days, would require a combined rate of 1/7 per day, which is larger than the actual 5/36 and therefore inconsistent with the calculated work rates. It underestimates the time needed.

Option B:
Option B, 7.2 days, corresponds exactly to 36/5 days, which arises from adding the reciprocals of 12 and 18 correctly. This option respects the standard reciprocal method and therefore matches the true combined time.

Option C:
Option C, 8 days, implies a daily work rate of 1/8, which is smaller than 5/36. It overestimates the time required and is not supported by the given individual efficiencies.

Option D:
Option D, 9 days, further slows the work rate to 1/9 per day and contradicts the fact that together A and B can accomplish more work per day than either alone.

To compare the fractions, we can either convert them to decimal form or cross multiply. The fraction 3/5 equals 0.6, 4/7 is approximately 0.571, 5/9 is approximately 0.556 and 6/13 is approximately 0.462. Among these decimal values, 0.6 is the largest. Therefore, 3/5 represents the greatest fraction in the given list.

Option A:
Option A, 3/5, clearly exceeds the others when converted into decimals or compared via cross multiplication with a common denominator. It corresponds to 0.6, which is larger than the approximate decimal values of the remaining fractions.

Option B:
Option B, 4/7, is slightly less than 0.6 and therefore smaller than 3/5, even though it is greater than the other two fractions listed. It cannot be the greatest overall.

Option C:
Option C, 5/9, falls between 0.55 and 0.56, making it smaller than both 3/5 and 4/7, so it is not the maximum value.

Option D:
Option D, 6/13, is noticeably less than 0.5, making it the smallest among the four fractions rather than the largest.

Statements A, B, C and D give standard first-order logical translations of everyday sentences. A correctly uses a universal conditional for “all teachers are researchers”. B uses an existential statement with a conjunction for “some students are hardworking”. C appropriately translates “no books are boring” as a universal statement with a negated predicate. D is correct for “some teachers are not researchers” using an existential with a conjunction of teacher and not researcher. E is false because a universal statement does not, by itself, guarantee existence in formal logic, so A, B, C and D only are correct.

Option A:
Option A is incomplete since it omits D and therefore fails to include the important pattern for expressing “some S are not P” in quantifier form. Without D, the range of translation patterns is narrower than NET questions usually expect.

Option B:
Option B is correct because it gathers all four accurate translations and excludes E, which misinterprets what can be inferred from a universal statement alone. It reflects the standard mapping from natural language to quantified logic.

Option C:
Option C is wrong because it leaves out B, ignoring the existential construction for “some students are hardworking”, and still does not address E’s incorrect existential inference. As a result, it is not fully correct.

Option D:
Option D is incorrect since it includes E, which claims existence from a universal statement, an inference that is not valid in general first-order semantics, and also omits A, missing the core universal conditional pattern.

The starting letters L, O, R and U are at positions 12, 15, 18 and 21 in the alphabet, increasing by 3 each time. Each group is made of three consecutive letters in forward order. Continuing the same pattern, the next starting position is 24, which corresponds to X, and the following letters are Y and Z, giving XYZ. Thus XYZ maintains both the constant plus three gap and the consecutive structure.

Option A:
Option A, WXY, starts at W, which is position 23 and only two steps ahead of U, so it does not satisfy the required three position jump. While WXY is a set of consecutive letters, it does not arise from the same arithmetic rule for starting letters. Therefore it cannot represent the logical continuation of the series.

Option B:
Option B, VWX, begins at V, which is position 22 and only one step beyond U. This reduces the gap rather than maintaining the plus three pattern. Since all previous transitions show a fixed increase of three, VWX is inconsistent with the established trend and thus incorrect.

Option C:
Option C is correct because X is exactly three positions after U and XYZ forms a clean block of three consecutive letters. It extends the same numerical pattern in the first letter and keeps the internal arrangement of letters unchanged. This dual consistency makes XYZ the only suitable next term.

Option D:
Option D, YZA, uses Y as the starting letter, which is four positions ahead of U, overshooting the required step. It also wraps into the next cycle of the alphabet before the pattern naturally demands it. For these reasons, YZA does not accurately continue the given series.

An equivalence relation is defined as a relation on a set that is reflexive, symmetric and transitive. Reflexive means every element is related to itself, symmetric means the relation is two-way and transitive means relations chain consistently. When all three properties hold, the relation partitions the set into equivalence classes. Therefore, a relation with these three properties is called an equivalence relation.

Option A:
A partial order is a relation that is reflexive, antisymmetric and transitive, not symmetric. It is used to model ordered structures rather than equivalence. Since the question explicitly includes symmetry, partial order cannot be the correct term.

Option B:
A symmetric relation only satisfies the symmetry property and may or may not be reflexive or transitive. The question, however, demands all three properties simultaneously. Thus, describing the relation as merely symmetric is incomplete and incorrect.

Option C:
A transitive relation satisfies just the transitivity property and may fail to be reflexive or symmetric. Again, this does not capture the full set of conditions mentioned in the question. Therefore, transitive alone is not the correct description.

Option D:
Equivalence is correct because it encapsulates the combined requirements of reflexivity, symmetry and transitivity. Such relations are crucial in classifying objects into groups where elements are "equivalent" under some criterion. Recognising this definition is important in topics like modular arithmetic and partitioning sets.

Starting letters Z(26), V(22), R(18), N(14) decrease by −4. Next start is J(10), giving JKL.

Option A:
Option A starts at K (11), which is −3 from N, not −4.

Option B:
Option B is correct because J is −4 from N and K,L follow consecutively.

Option C:
Option C starts at H (8), which is −6 from N, not −4.

Option D:
Option D starts at G (7), which is −7 from N, not −4.

Statements A, B, C and E correctly describe fundamental ideas of ratio and proportion. Ratios compare quantities by division, proportions equate two ratios and ad = bc is the standard cross-multiplication rule. It is also true that meaningful ratios compare quantities in the same units. Statement D is false because there is no requirement that a + c equals b + d in a valid proportion. Hence the combination including A, B, C and E but excluding D is the only fully correct set.

Option A:
Option A is correct because it includes all true statements and excludes D, which imposes a wrong condition on proportions. It provides a complete and accurate summary of the key properties tested in NET-level questions on ratio and proportion.

Option B:
Option B is incomplete since it omits E and therefore ignores the practical requirement about same units, which is important for interpreting real-life ratio problems. This omission makes the set of statements only partially correct.

Option C:
Option C is wrong because it includes D, the incorrect condition about sums of antecedents and consequents, even though it also contains some true statements. The presence of D makes the overall combination incorrect.

Option D:
Option D is wrong because it omits A (the basic definition of ratio) and also includes E while missing other true statements; it does not match the complete set of correct statements.

The consecutive differences are 12, 18, 24 and 30. These differences form an arithmetic progression with common difference 6. Therefore the next difference should be 36. Adding 36 to the last term 93 gives 129, so 129 continues the series in line with the observed second-level pattern.

Option A:
Option A gives 123, which implies a difference of 30 from 93. That would repeat the last difference and make the sequence of gaps 12, 18, 24, 30, 30, breaking the steady increment by 6. Thus 123 is not appropriate.

Option B:
Option B gives 126, corresponding to a difference of 33 from 93. In that case the last increase among differences would be only 3, not 6, distorting the structure. Therefore 126 does not fit the rule.

Option C:
Option C gives 129, implying a difference of 36 from 93. The difference sequence becomes 12, 18, 24, 30, 36, a clean arithmetic progression with constant step 6. This strong internal consistency makes 129 the correct next term.

Option D:
Option D gives 132, leading to a difference of 39 from 93. That makes the last increment among differences equal to 9, which is incompatible with the constant step of 6. Hence 132 cannot be accepted.

The series follows the rule (a_n = 4n^3 + 2n^2 + n + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 8, 43, 130, 293 and 556, which match the terms given. For n = 6 we compute (4×6^3 + 2×6^2 + 6 + 1 = 864 + 72 + 6 + 1 = 943). Hence 943 is the correct next term.

Option A:
Option A, 931, is 12 less than the value provided by the formula and does not equal (4×6^3 + 2×6^2 + 6 + 1). It indicates an unsupported decrease at the final term, so 931 is not valid.

Option B:
Option B, 943, matches exactly the output of the expression for n = 6. It maintains the balance between the cubic, quadratic and linear components, keeping the sequence structurally sound. Therefore 943 is the correct continuation.

Option C:
Option C, 955, overshoots the computed value and cannot be obtained from the given rule. Selecting 955 would artificially inflate the term and disrupt the algebraic pattern. Hence 955 is not correct.

Option D:
Option D, 961, deviates even further from 943 and again fails to satisfy the formula. Using 961 would destroy the precise relationship between n and a_n, so it is not an appropriate choice.

Discount is the reduction given on the marked price, so here the discount amount is 2000 − 1800 = ₹200. The discount percentage is calculated as (Discount ÷ Marked price) × 100, giving (200 ÷ 2000) × 100 = 10%. Therefore, the discount offered on the article is 10%.

Option A:
An 8% discount on ₹2000 would be 0.08 × 2000 = ₹160, leading to a selling price of ₹1840, which does not match the given price of ₹1800. Since the actual discount is larger, 8% underestimates the reduction. Thus, this option is incorrect.

Option B:
A 10% discount exactly matches a reduction of ₹200 from ₹2000, which produces the stated selling price of ₹1800. The computation is straightforward, confirming that the discount percentage is 10%. Hence, option B is the correct choice.

Option C:
A 12% discount would be 0.12 × 2000 = ₹240, giving a selling price of ₹1760, which is lower than what is mentioned. This overestimates the discount and therefore cannot be accurate for the given data.

Option D:
A 15% discount implies 0.15 × 2000 = ₹300, resulting in a selling price of ₹1700. This is even further from the actual selling price. Therefore, 15% is clearly not the correct discount percentage in this case.

In the first hour, from 8:00 to 9:00 a.m., the first bus travels 40 km. After 9:00 a.m., the second bus starts and the relative speed between the two buses is 60 − 40 = 20 km/h. To close a gap of 40 km at a relative speed of 20 km/h, the second bus needs 40 ÷ 20 = 2 hours. Adding 2 hours to 9:00 a.m. gives 11:00 a.m. as the overtaking time.

Option A:
Option A, 10:00 a.m., would correspond to only one hour of catch-up time, which would close a distance of 20 km at a relative speed of 20 km/h, leaving 20 km still between the buses.

Option B:
Option B, 10:30 a.m., gives 1.5 hours of catch-up and would close 30 km of the gap, not the full 40 km, so the faster bus would still be behind.

Option C:
Option C, 11:00 a.m., correctly accounts for the full 2 hours needed at a relative speed of 20 km/h to cover the initial 40 km head start, exactly matching the required condition for overtaking.

Option D:
Option D, 11:30 a.m., overshoots the required catch-up time; by then, the second bus would have been ahead of the first bus for half an hour already.

The differences between consecutive terms are 1, 2, 3, 4 and 5. These differences increase by 1 at each step, forming a simple arithmetic sequence. To continue this pattern, the next difference should be 6. Adding 6 to the last term 18 yields 18 + 6 = 24. This ensures that the growth at the level of differences remains smooth and systematic.

Option A:
Option A gives 22, which corresponds to a difference of 4 from 18 and repeats an earlier increment instead of moving to 6. This disrupts the steadily increasing sequence of differences. Therefore, 22 is not consistent with the pattern.

Option B:
Option B offers 23, giving a difference of 5, which again has already appeared in the series and fails to advance the difference pattern to 6. Thus, 23 cannot be accepted as correct.

Option C:
Option C yields 24, which is 6 more than 18 and extends the differences to 1, 2, 3, 4, 5, 6 in order. This maintains a clear and logical structure at the second level. Hence, 24 is the correct next term.

Option D:
Option D suggests 25, creating a difference of 7 from 18 and jumping beyond the expected increment. This makes the difference progression irregular. Therefore, 25 is not a valid continuation.

The first-level differences are 13, 21, 29 and 37. These differences form an arithmetic progression with common difference 8. To keep this structure, the next difference should be 45. Adding 45 to the last term 107 gives 152, so 152 continues both the original sequence and the pattern among the differences.

Option A:
Option A gives 148, which corresponds to a difference of 41 from 107. This would produce differences 13, 21, 29, 37, 41, making the last increment among gaps equal to 4 instead of 8. Thus 148 does not respect the observed second-level behaviour.

Option B:
Option B gives 150, implying a difference of 43 from 107. The final step among differences would be 6, which still fails to match the constant increment of 8. Hence 150 does not align with the pattern.

Option C:
Option C gives 152, which implies a difference of 45 from 107. The sequence of differences becomes 13, 21, 29, 37, 45 and the increments remain 8 throughout. This perfect consistency at the second level makes 152 the correct next term.

Option D:
Option D gives 154, yielding a difference of 47 from 107. That would make the last increment among differences equal to 10, breaking the simple arithmetic progression. Therefore 154 is not a valid continuation.

The number of students who like at least one of the two subjects is given by the union of the sets of students liking Mathematics and Science. Using the inclusion–exclusion principle, we have n(M ∪ S) = n(M) + n(S) − n(M ∩ S) = 35 + 30 − 15 = 50. This counts each student only once, even if they like both subjects. Therefore, 50 students like at least one of the two subjects.

Option A:
A value of 45 would incorrectly subtract too many students or miscount those in the intersection. If we tried to get 45 from the given numbers, we would be effectively assuming a different overlap than 15. Since the data provided are precise, 45 does not satisfy the formula for the union. Hence, this option is incorrect.

Option B:
The value 50 is correct because it properly applies the inclusion–exclusion formula for two sets. It ensures that the 15 students who like both subjects are not double counted. In many reasoning questions, this technique is crucial for handling overlapping groups accurately.

Option C:
A value of 55 would imply that n(M ∪ S) is larger than either individual group but fails the inclusion–exclusion calculation 35 + 30 − 15. This would effectively overcount the intersection instead of subtracting it once. Therefore, 55 cannot be the right answer.

Option D:
Taking 60 as the answer would incorrectly assume that every student in the class likes at least one of the two subjects. The union we calculated is only 50, which means 10 students like neither Mathematics nor Science. Thus, 60 is not consistent with the given data.