UGC NET Questions (Paper – 1)

The series can be described by the formula (a_n = n^3 + 2) for n starting from 2. For n=2,3,4,5 and 6 the values are 8+2=10, 27+2=29, 64+2=66, 125+2=127 and 216+2=218, matching the given terms. The next index is n=7, giving (7^3+2 = 343+2 = 345). Thus 345 is the natural continuation of this cubic-based sequence.

Option A:
Option A, 337, does not arise from the expression (n^3+2) for any integer n that continues the index sequence. Using 337 would require abandoning the clear functional rule that explains all earlier terms. Therefore 337 is not a valid next term.

Option B:
Option B equals 345, which is exactly (7^3+2), following the same formula used to generate 10, 29, 66, 127 and 218. Because the rule holds for every term including the new one, 345 is the correct answer.

Option C:
Option C, 349, is larger than the value predicted by the cubic formula and cannot be justified without changing the rule. It does not match (n^3+2) for any integer n that logically follows. Hence 349 is not consistent with the pattern.

Option D:
Option D, 353, similarly fails to satisfy the relation (a_n = n^3+2) for the next integer in sequence. Adopting 353 would make the series lose its simple and elegant algebraic structure. Thus 353 is not the right continuation.

This sequence can be described recursively by a₁ = 3 and aₙ₊₁ = 3aₙ + 2n² for n ≥ 1. Using this rule we get a₂ = 3·3+2·1² = 11, a₃ = 3·11+2·2² = 41, a₄ = 3·41+2·3² = 141 and a₅ = 3·141+2·4² = 455, which matches the terms given. For n = 5 the next term is a₆ = 3·455+2·5² = 1365+50 = 1415. Thus 1415 is the only value that satisfies the same recurrence pattern for the next step.

Option A:
Option A, 1387, does not equal 3·455+2·5² and therefore fails to satisfy the recurrence relation. It would imply subtracting 28 from the required value without any change in the rule. Since the earlier terms fit the recurrence exactly, option A is inconsistent with the established pattern.

Option B:
Option B, 1415, is exactly the result of applying aₙ₊₁ = 3aₙ+2n² when n = 5. It respects the same combination of a multiple of the previous term and an index-dependent correction used in all earlier steps. Because the rule remains unchanged and still produces the next term, 1415 is the correct continuation of the series.

Option C:
Option C, 1431, overshoots the recurrence output by 16 and cannot be written as 3·455+2·5². It would require altering the coefficient or adding an extra constant at the last step, which is not supported by the data. Therefore option C is not a valid continuation.

Option D:
Option D, 1459, deviates even more from the computed value and likewise does not satisfy the stated recurrence. Using 1459 would break the close correspondence between the rule and the generated sequence, so option D is incorrect.

Statements C and E are the wrong statements here. A is correct because probabilities range from 0 to 1. B is true since probabilities of all mutually exclusive and exhaustive outcomes sum to 1. D correctly states that a certain event has probability 1. C is false because 0.7 indicates a more likely event than 0.4. E is false because P(A ∪ B) equals P(A) + P(B) only when A and B are mutually exclusive; otherwise the intersection must be subtracted. Therefore, the wrong statements are C and E only.

Option A:
Option A is incomplete because it selects only C as wrong and ignores E, which misapplies the addition rule for probabilities of unions.

Option B:
Option B is correct as it identifies both C and E as incorrect, thereby recognising that both comparative likelihood and union probabilities are mishandled in those statements.

Option C:
Option C is wrong since it mislabels A as wrong, even though A accurately describes the probability range, and still keeps C and E, mixing correct and incorrect judgments.

Option D:
Option D is incorrect because it classifies B and D as wrong despite them stating fundamental probability principles, and it omits E, which actually contains an error about unions.

Each term is a 5-letter consecutive block. Starting letters A(1), F(6), K(11), P(16) increase by +5. Next start is U(21), giving UVWXY.

Option A:
Option A starts at S(19), which is +3 from P, not +5.

Option B:
Option B is correct because U is +5 from P and V,W,X,Y follow consecutively.

Option C:
Option C starts at V(22), which is +6 from P, not +5.

Option D:
Option D repeats an earlier term.

Each term shifts every letter by +2: ACEG→CEGI→EGIK→GIKM. Next is GIKM shifted by +2 to IKMO.

Option A:
Option A is correct because it applies the same +2 shift to each letter and keeps the +2 internal spacing.

Option B:
Option B does not come from adding +2 to each letter of GIKM.

Option C:
Option C breaks the internal spacing because M→N is +1.

Option D:
Option D starts with J instead of I, so it is not the +2 shift continuation.

The positions are Z(26), V(22), R(18), N(14) and J(10). Each move subtracts 4 from the previous position (26−4=22, 22−4=18, 18−4=14, 14−4=10). Continuing this pattern, the next position is 10 − 4 = 6, which is the letter F. Thus F uniquely maintains the constant backward step of 4.

Option A:
Option A, G, is at position 7, which is 3 steps after J instead of 4 steps before. This reverses both direction and step size, violating the observed pattern. Therefore G cannot be the correct continuation.

Option B:
Option B is correct because F is exactly 4 positions before J, matching the consistent −4 difference. It keeps both the direction (backward) and the magnitude of the jump unchanged. This strict alignment with the rule makes F the proper next term.

Option C:
Option C, E, lies at position 5, which is 5 positions before J. This would enlarge the backward step from 4 to 5 without any justification from previous terms. Hence E does not fit the constant difference structure of the series.

Option D:
Option D, H, is at position 8, which is 2 positions before J. That shrinks the backward step to 2, contradicting the established gap of 4. As a result, H cannot logically follow J in this pattern.

Statements A, B and C all describe essential features of seating puzzles. A is correct because in a row, left and right depend on the facing direction. B is correct as neighbours in a circle facing the centre are those immediately adjacent along the circumference. C is also true; when people face outward, the sense of left and right relative to the circle reverses compared to facing inward. D is false because assuming a common facing direction without explicit information can lead to wrong conclusions. Thus, A, B and C only are correct.

Option A:
Option A is incomplete because it lists only A and B and omits C, which explains the reversal of left and right when people face outward. Without C, understanding of circular arrangements is incomplete.

Option B:
Option B is correct as it collects all three true statements about orientation and neighbouring positions and excludes D, which would lead to improper assumptions in exam questions. It matches the conventions of NET seating arrangement problems.

Option C:
Option C is incorrect since it omits A and includes only B and C, losing the basic rule about left and right in linear arrangements, which are also commonly tested. This makes the option partial.

Option D:
Option D is wrong because it contains D, which incorrectly permits assumptions about facing directions, and omits B, so it both introduces error and drops a central property of circular seating.

Statements A, C and D summarise key properties of divisibility and remainders. A encodes the division algorithm representation. C is true because 6 has prime factors 2 and 3, so divisibility by 6 implies divisibility by both. D is correct since numbers with the same remainder differ by a multiple of the divisor. B is false as divisibility by 3 and 9 together does not force divisibility by 27, and E is false because a multiple of 4 need not be a multiple of 8. Therefore, A, C and D only are correct.

Option A:
Option A is correct because it collects the basic representation of numbers with remainders and two important consequences for divisibility, while excluding B and E, which overstate divisibility conditions.

Option B:
Option B is incorrect as it adds B, which wrongly claims that common divisibility by 3 and 9 forces divisibility by 27, contradicting simple counterexamples like 9 itself.

Option C:
Option C is wrong since it also incorporates E, which incorrectly asserts that multiples of 4 are automatically multiples of 8, and thus mixes false statements with true ones.

Option D:
Option D is incomplete because it drops A and only keeps C and D, leaving out the fundamental division algorithm expression that underpins the remainder concept in many NET questions.

Let the numbers be 6x and 7x. Their sum is given as 6x + 7x = 13x = 325. Solving this, we get x = 325/13 = 25. Therefore the two numbers are 6 × 25 = 150 and 7 × 25 = 175. The smaller number is 150.

Option A:
Option A is correct because it emerges directly from solving the equation based on the ratio and the sum. Substituting 150 and 175 back into the problem, their sum is 325 and their ratio is 150:175, which reduces to 6:7 on dividing by 25. This confirms the consistency of the solution.

Option B:
Option B, 175, is the larger of the two numbers. The question explicitly asks for the smaller number, so choosing 175 would ignore this requirement. Although 175 is part of the correct pair, it does not answer the query.

Option C:
Option C, 200, would imply that the other number is 125 to maintain the sum 325. The ratio 125:200 simplifies to 5:8, not 6:7, so this pair does not match the stated ratio. Therefore 200 cannot be the smaller number satisfying both conditions.

Option D:
Option D, 225, paired with 100, would give a sum of 325 but a ratio of 100:225, which simplifies to 4:9. This is different from the required 6:7, so 225 is not a valid candidate for the smaller number under the given constraints.

This series is given by the rule (a_n = 2n^3 + 3n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 6, 29, 82, 177 and 326, which match the listed terms. For n = 6 we compute (2×6^3 + 3×6^2 + 1 = 432 + 108 + 1 = 541). Thus 541 is the unique next term that maintains this cubic-plus-square pattern.

Option A:
Option A, 529, is 12 less than the formula’s result and does not equal (2×6^3 + 3×6^2 + 1). It would imply altering the rule at the last term, which is not supported by the sequence. Therefore 529 is not correct.

Option B:
Option B, 541, coincides exactly with the value produced by the expression for n = 6. It preserves all the coefficients and the constant term used for earlier values. For this reason, 541 is the correct continuation.

Option C:
Option C, 547, overshoots the computed value and cannot be written as (2n^3 + 3n^2 + 1) for the next index. Choosing 547 would break the algebraic consistency, so it is not valid.

Option D:
Option D, 553, deviates even more from the formula and again fails to satisfy the same relationship for n = 6. Adopting 553 would distort the pattern, so it is not the right answer.

To find the compound ratio a:d, we multiply the fractional forms of each given ratio: (3/4) × (5/6) × (2/3). This product equals 30/72, which simplifies to 5/12 on reducing by the common factor 6. So the overall ratio of a to d is 5:12. Hence 5:12 is the only option that correctly represents the compound ratio of a to d.

Option A:
Option A is correct because it follows directly from the multiplication of the three given ratios and proper simplification of the resulting fraction. When we reduce 30/72, we divide numerator and denominator by 6 to obtain 5/12. This simplified form represents the true relationship between a and d after chaining all three ratios.

Option B:
Option B, 5:18, would correspond to a fraction 5/18 which is smaller than 5/12 and does not emerge from multiplying 3/4, 5/6 and 2/3. If we tried to reconstruct the intermediate ratios from 5:18, they would not match any of the given values. Therefore, 5:18 cannot be the correct compound ratio.

Option C:
Option C, 10:27, appears similar in form but is not equivalent to 5:12 when written as a fraction. 10/27 is approximately 0.37, whereas 5/12 is approximately 0.4167, so they are clearly different values. It does not result from multiplying the given three ratios in sequence.

Option D:
Option D, 5:9, would give a fraction 5/9 which is larger than 5/12 and does not match the product 30/72 after simplification. If 5:9 were correct, reversing the calculation would not recover the original ratios 3:4, 5:6 and 2:3. Hence, 5:9 is an incorrect representation of a:d.

Initially, acid = (3/8) × 40 = 15 litres and water = (5/8) × 40 = 25 litres. Let x litres of acid be added. Then acid becomes 15 + x while water remains 25, and the new ratio is (15 + x):25 = 7:5. Cross-multiplying gives 5(15 + x) = 7 × 25, so 75 + 5x = 175 and 5x = 100, which gives x = 20. Thus, 20 litres of acid must be added.

Option A:
Option A, 12 litres, would make the acid quantity 27 litres while water remains 25 litres, giving a ratio 27:25, which does not simplify to 7:5. It therefore fails to meet the required target ratio and underestimates the amount of acid needed.

Option B:
Option B, 16 litres, leads to acid = 31 litres and water = 25 litres, so the ratio becomes 31:25. This cannot be simplified to 7:5, indicating that 16 litres is still insufficient to reach the desired composition.

Option C:
Option C is correct because it yields acid = 35 litres and water = 25 litres, giving a ratio 35:25, which reduces to 7:5 on dividing both numbers by 5. This matches the requirement exactly and confirms that 20 litres is the correct quantity to add.

Option D:
Option D, 24 litres, would result in acid = 39 litres and water = 25 litres, giving a ratio 39:25. This ratio is more acid-heavy than 7:5 and does not simplify to it, showing that 24 litres of acid overshoots the intended mixture ratio.

C and E are wrong because they contradict basic properties and uses of ratios. Ratios can indeed be simplified by dividing both terms by the same non-zero number, in the same way that fractions are simplified, so C is false. E is also false because ratio and proportion are central to many aptitude problems, especially those involving sharing, mixtures and scaling. A, B and D all correctly describe comparison, equality in proportion and sharing applications, so the wrong statements are exactly C and E.

Option A:
Option A is incorrect because it identifies only C as wrong and overlooks E, which dismisses the importance of ratios in aptitude, a claim that clearly conflicts with the UGC NET syllabus. Leaving out E means not all wrong statements are captured.

Option B:
Option B is also incorrect since it treats only E as wrong and ignores C, which incorrectly states that ratios cannot be simplified. A proper understanding of ratio includes the idea of simplification.

Option C:
Option C is correct because it brings together both C and E, the two statements that misrepresent ratio properties and their examination relevance. It implicitly acknowledges that A, B and D provide accurate descriptions used in aptitude questions.

Option D:
Option D is wrong because it adds B, which is true about cross products in a proportion, to the set of supposed wrong statements. Including B makes the option inconsistent with the mathematical definition of proportion.

The pattern alternates between multiplying by 2 and adding 1, and multiplying by 2 and subtracting 1. Specifically, 3 × 2 + 1 = 7, 7 × 2 − 1 = 13, 13 × 2 + 1 = 27, 27 × 2 − 1 = 53 and 53 × 2 + 1 = 107. The next step should follow the "×2 − 1" rule. Thus, 107 × 2 − 1 = 214 − 1 = 213, which fits the alternating structure.

Option A:
Option A gives 211, which would correspond to subtracting 3 after doubling and does not follow the precise "minus one" part of the pattern. This breaks the carefully balanced alternation between adding and subtracting 1. Therefore, 211 is not consistent with the rule.

Option B:
Option B yields 213, which is obtained by 107 × 2 − 1 and continues the alternating pattern exactly. The series 3, 7, 13, 27, 53, 107, 213 remains governed by a single, well-defined recursive rule. Hence, 213 is the correct next term.

Option C:
Option C suggests 215, which would require adding 1 after doubling instead of subtracting 1 at this step. That would repeat the previous type of operation and disrupt the alternation. Thus, 215 is not the appropriate continuation.

Option D:
Option D offers 217, which is even further away and would demand a completely different adjustment after doubling. This option ignores the subtle alternation at the heart of the sequence. Therefore, 217 cannot be accepted as correct.

Statements A, B, C and E describe core properties of means. A gives the definition of arithmetic mean. B is true because the overall mean of two groups must fall between their group means, weighted by group sizes. C is correct in explaining that weighted averages give more influence to values with higher weights. E holds since adding a constant to all values shifts the mean by that constant. D is false because the mean need not be one of the data values, especially for continuous data. Hence A, B, C and E only are correct.

Option A:
Option A is incomplete because it leaves out E and therefore ignores the important transformation property showing how means shift under addition of a constant. It does not include all correct statements.

Option B:
Option B is wrong since it excludes C and includes only A, B and E, failing to mention the key feature of weighted averages, which are frequently tested in aptitude problems.

Option C:
Option C is correct because it collects all four true statements while excluding D, which wrongly asserts that the mean must be a data point. This combination matches the conceptual understanding required in NET statistics questions.

Option D:
Option D is incorrect as it omits A and does not include all four true statements together, so it cannot be the correct combination.