UGC NET Questions (Paper – 1)

The sequence is generated by aₙ = n⁴ + n² − n for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 1+1−1 = 1, 16+4−2 = 18, 81+9−3 = 87, 256+16−4 = 268 and 625+25−5 = 645, which matches the given series exactly. For n = 6 we compute 6⁴+6²−6 = 1296+36−6 = 1326. Thus 1326 is the only value that preserves this quartic-plus-quadratic-minus-linear pattern.

Option A:
Option A, 1315, is close but not equal to 6⁴+6²−6. It would require subtracting 11 from the value dictated by the formula only for the sixth term. Such an adjustment is not supported by the earlier behaviour of the series and therefore makes option A incorrect.

Option B:
Option B, 1320, differs by 6 from the formula’s result and also does not satisfy n⁴+n²−n when n = 6. Choosing 1320 would arbitrarily reduce the correct term and break the functional rule connecting all terms. Hence this option cannot represent the correct continuation.

Option C:
Option C, 1323, is still not equal to 1296+36−6 and has no direct expression in terms of the established rule. It lies between some candidate values but does not come from the defined algebraic pattern. Therefore option C is not logically consistent with the observed sequence.

Option D:
Option D, 1326, matches exactly the output of the formula for n = 6. It continues the same combination of quartic, quadratic and linear components that produced all previous terms. Because this value arises naturally from the rule without modification, 1326 is the correct next term in the series.

Statements A and B correctly describe the length of non-leap and leap years and the divisibility rule used to identify leap years. Statement C is true because the weekly cycle repeats every 7 days, and E is also correct as calendar questions usually count odd days from a known reference day. Statement D is false because odd days are calculated modulo 7, not modulo 5. Therefore, the combination A, B, C and E only is correct.

Option A:
Option A is incomplete because it omits E and therefore misses the practical step of accumulating odd days from a reference, which is crucial for solving day-of-week problems. Without E, the method is not fully described.

Option B:
Option B is wrong as it includes D, which uses modulo 5 instead of modulo 7, and so miscalculates odd days. This incorrect rule makes the entire option unacceptable.

Option C:
Option C is incomplete since it leaves out A and B, thereby not explicitly stating the year-length facts and leap-year rule, even though it contains two true statements.

Option D:
Option D is correct because it gathers all the true statements and excludes D, the incorrect modulus rule, giving a complete and accurate picture of odd-day calculations in calendar questions.

A is correct because the unit covers both numerical aptitude and logical reasoning aspects. C is true as data interpretation in NET uses various graphical and tabular forms. E is also correct since exposure to diverse question types improves speed and accuracy. B is wrong because series, analogies, classification and coding–decoding are explicitly part of mathematical reasoning and aptitude. D is wrong because purely logical items fall within the same unit and are not excluded. Hence B and D only are the wrong statements.

Option A:
Option A selects only B as wrong and ignores D, even though D incorrectly narrows the scope of mathematical aptitude to numeric questions. It therefore does not capture all incorrect statements.

Option B:
Option B chooses only D as wrong and overlooks B, which wrongly claims that several reasoning types are unrelated to Unit 5. So it also fails to identify the full set of wrong statements.

Option C:
Option C is correct because it identifies both B and D as misrepresenting the syllabus while leaving A, C and E as accurate descriptions of the breadth and preparation strategy for the unit.

Option D:
Option D incorrectly includes C among the wrong statements, even though data interpretation indeed uses tables and graphs, and thus misclassifies a true statement along with B and D.

Statements A, B and C describe the standard logical notions of validity and soundness in deductive reasoning, so they are all true together. In a valid deductive argument, true premises guarantee a true conclusion, and validity is a matter of form rather than factual correctness of premises. A sound argument must be both valid and have all true premises, which is exactly what C states. Statements D and E are false, because deductive conclusions do not always go beyond the information in the premises, and an argument can be valid yet unsound if its premises are false. Therefore the combination containing A, B and C only is correct.

Option A:
Option A includes A and B, which are both true, but it omits C, which correctly defines a sound argument as valid with true premises. Because C is also true and is left out, this option does not capture all the correct statements and is therefore incomplete.

Option B:
Option B selects B and C but leaves out A, so it fails to mention the fundamental guarantee that in a valid deductive argument true premises ensure a true conclusion. Since A is a correct statement about validity and is missing, this combination cannot be considered fully correct.

Option C:
Option C includes A, B and C but adds D, which is false because deductive reasoning need not extend beyond the information in the premises; often the conclusion is logically contained in them. Including D alongside the true statements makes the overall set incorrect.

Option D:
Option D is correct because it lists exactly those statements that are true: A about the truth-preserving nature of validity, B about structure, and C about soundness. It excludes D and E, which misrepresent deductive reasoning by suggesting that every valid argument is automatically sound and that conclusions must always go beyond the premises.

Proof by contradiction is an indirect method of proof where we assume that the statement to be proved is false and then logically deduce a contradiction. This contradiction shows that the assumption of falsity cannot hold, so the original statement must be true. It is often used in proving results that are difficult to handle directly. Hence, the method described in the question is called proof by contradiction.

Option A:
Proof by induction is a different technique that involves a base step and an inductive step to show that a statement holds for all natural numbers. It does not rely on assuming the negation of the statement and reaching a contradiction. Therefore, induction does not match the description in the question.

Option B:
Proof by contradiction is correct because it fits exactly the process of assuming the negation and deriving an impossibility or inconsistency. For example, classic proofs of the irrationality of √2 use this method. Understanding this technique helps students tackle theoretical questions in mathematical reasoning.

Option C:
Proof by construction involves explicitly constructing an example or object that satisfies the required properties. It does not necessarily involve any assumption that the statement is false. Therefore, construction is conceptually different from the method described in the stem.

Option D:
Proof by example is typically used to demonstrate the existence of at least one object with a given property, but it cannot prove statements that are meant to hold universally. It also does not depend on assuming the negation and producing a contradiction. Thus, this option is not correct for the method mentioned.

Equal real roots occur when the discriminant of a quadratic equation is zero. For the equation 2x² − kx + 8 = 0, the discriminant is b² − 4ac, which becomes k² − 4(2)(8). Setting this equal to zero gives k² − 64 = 0. Solving, we get k² = 64 and hence k = ±8, but the given options only include 8 as a valid choice. Therefore, k = 8 satisfies the equal roots condition using the standard discriminant rule.

Option A:
Option A, 4√2, when squared gives 32, and substituting into the discriminant k² − 64 yields 32 − 64 = −32, which is negative. A negative discriminant would produce complex, not equal real, roots.

Option B:
Option B leads to k² = 64, so the discriminant becomes 64 − 64 = 0. This matches the precise condition for equal real roots in a quadratic and is consistent with the formula b² − 4ac = 0.

Option C:
Option C, 16, gives k² = 256 and a discriminant of 256 − 64 = 192, which is positive and corresponds to two distinct real roots rather than equal ones.

Option D:
Option D, 16√2, squares to 512, making the discriminant 512 − 64 = 448. This again yields unequal real roots, not the required equal roots.

Statements A and B correctly describe linear inequalities in two variables as half-planes bounded by straight lines, and D accurately notes the convention of using dashed lines for strict inequalities. Statement E is also true because many optimisation and allocation problems, such as resource constraints, are modelled with systems of inequalities. Statement C is false because solution sets are typically infinite regions rather than single points. Therefore, the combination A, B, D and E only is correct.

Option A:
Option A is correct as it includes all true statements and excludes C, which wrongly restricts solutions to a single point. It reflects the graphical interpretation used for inequality questions at the NET level.

Option B:
Option B is incomplete since it omits E and thus does not emphasise that such inequalities can model practical constraints in optimisation problems like linear programming. Without E, the application aspect is missing.

Option C:
Option C is wrong because it contains C, the claim about a single point, and also omits A, so it both admits a false statement and leaves out a key concept about half-planes. This combination does not represent the topic accurately.

Option D:
Option D is incorrect because it includes C and therefore endorses a misunderstanding of the size of solution sets, even though some of the other statements in the option are correct. The presence of C makes the option unacceptable.

The pattern begins with a multiplication by 2 and is then followed by repeated multiplication by 3. Specifically, 2 × 2 = 4, 4 × 3 = 12, 12 × 3 = 36, 36 × 3 = 108 and 108 × 3 = 324. To continue the series, we again multiply by 3, giving 324 × 3 = 972. This keeps the multiplicative behaviour consistent.

Option A:
Option A gives 960, which is not equal to 324 multiplied by any simple factor following the established rule. It does not preserve the sequence of multiplications by 3. Hence, 960 is not appropriate.

Option B:
Option B offers 966, which is 6 less than 324 × 3 and alters the multiplication pattern by subtracting an extra number. This is not supported by earlier terms. Therefore, 966 cannot be accepted.

Option C:
Option C suggests 968, which does not correspond to 324 multiplied by a constant ratio used earlier and appears arbitrary. This disrupts the regular multiplicative structure. Thus, 968 is not correct.

Option D:
Option D yields 972, which equals 324 × 3 and continues the run of factors from 3 onward. The series 2, 4, 12, 36, 108, 324, 972 stays well-organised under a clear rule. Therefore, 972 is the correct next term.

The nth term of this series is given by the expression (a_n = n^3 + n) with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 1³+1 = 2, 2³+2 = 10, 3³+3 = 30, 4³+4 = 68 and 5³+5 = 130, matching the given terms. For n = 6, the expression yields 6³+6 = 216+6 = 222. Hence 222 is the term that keeps the sequence consistent with a single cubic formula.

Option A:
Option A, 214, does not arise from (n^3+n) for any integer n that logically follows 1 to 5 in this context. Choosing 214 would force a change in the algebraic rule that explains all earlier terms. Therefore 214 is not compatible with the series.

Option B:
Option B, 218, is slightly below the value given by 6³+6 and cannot be expressed as (n^3+n) with an integer n continuing the order. This inconsistency shows that 218 does not respect the functional pattern behind the sequence.

Option C:
Option C, 220, lies between 218 and 222 but still does not equal 6³+6. Using 220 would abandon the clean polynomial structure and introduce an arbitrary number. Hence 220 cannot be treated as the correct continuation.

Option D:
Option D, 222, matches exactly the value produced by substituting n = 6 into (a_n = n^3+n). Because this single expression generates all the given terms and 222 as well, this option preserves the mathematical relationship perfectly and is therefore correct.

Each term is a 5-letter block; starting letters move by +2: A, C, E, G → next I. So the next block is IJKLM.

Option A:
Option A starts at H (+1 from G), not +2.

Option B:
Option B is correct because it starts at I (+2 from G) and continues consecutively.

Option C:
Option C starts at J (+3 from G), not +2.

Option D:
Option D repeats an earlier term and does not continue the slide.

Each term in the series is obtained by multiplying the previous term by 2, giving a constant ratio of 2. We have 3×2 = 6, 6×2 = 12 and 12×2 = 24, which confirms the rule. Applying the same operation to the last term gives 24×2 = 48. Thus, 48 is the only term that continues the geometric pattern correctly.

Option A:
Option A suggests 30, which would correspond to adding 6 rather than multiplying by 2. This change in operation does not match any of the earlier transitions in the series. Therefore, 30 does not fit the rule of constant multiplication.

Option B:
Option B maintains the ratio of 2 because 48 ÷ 24 = 2, just as 24 ÷ 12 and 12 ÷ 6 equal 2. The extended sequence 3, 6, 12, 24, 48 clearly follows a doubling rule. This makes 48 the unique correct choice under standard number-series logic.

Option C:
Option C gives 36, corresponding to a factor of 1.5 from 24, which has not appeared anywhere in the series. Introducing this new ratio would destroy the simplicity and consistency of the pattern. Hence, 36 cannot be considered correct.

Option D:
Option D offers 54, which would require multiplying 24 by 2.25. Since none of the previous steps uses such a factor and all use an exact doubling, this option contradicts the established rule. Therefore, it is not the right answer.

A is correct because conditional probability measures A given B. B is the definition formula P(A|B) = P(A ∩ B)/P(B) (when P(B) > 0). C is true because Bayes’ theorem is used to update probabilities of hypotheses/causes when evidence is known. D is true for independent events since B does not change the probability of A. E is false because P(A|B) and P(not A|B) are complementary conditional probabilities and must sum to 1, not exceed 1. Therefore, A, B, C and D only are correct.

Option A:
Option A is incomplete because it omits D, which is a key property of independent events (conditioning does not change probability).

Option B:
Option B is correct because it includes all true statements (A, B, C, D) and excludes E, which wrongly claims the sum of complementary conditional probabilities is greater than 1.

Option C:
Option C is incorrect because it includes E (false) and also omits A and D, both of which are true statements about conditional probability and independence.

Option D:
Option D is incorrect because it includes E (false) and excludes B and C, which are fundamental definitions/uses of conditional probability and Bayes’ theorem.

A, B and D correctly explain fundamental ideas about simple and compound interest. Simple interest remains based on the original principal, compound interest periodically adds interest to principal to generate interest-on-interest, and increasing compounding frequency (for the same nominal rate and duration) increases the final amount. Statement C is wrong because compound interest for the same principal, rate and time is greater than or equal to simple interest, not less, and E is wrong because rate and time are central factors in every interest formula. Hence the set A, B and D only is the correct combination of true statements.

Option A:
Option A includes A and B but omits D, thereby ignoring the important effect of compounding frequency on the final amount. Although it contains two true statements, the absence of D means it does not cover all correct information listed.

Option B:
Option B selects B and D but leaves out A, failing to mention the defining feature of simple interest, namely that it is calculated only on the original principal. Without A, understanding of the contrast between simple and compound interest is incomplete.

Option C:
Option C is incorrect because it includes E, which wrongly claims that rate and time are irrelevant, and excludes B, one of the key statements about compound interest. The presence of a clearly false statement makes this option invalid.

Option D:
Option D is correct since it groups the three true statements that cover simple interest behaviour, the nature of compounding and the effect of compounding frequency, while excluding C and E, both of which misrepresent interest calculations.

This series alternates between adding 3 and multiplying by 2. Starting from 2, we have 2 + 3 = 5, 5 × 2 = 10, 10 + 3 = 13, 13 × 2 = 26 and 26 + 3 = 29. Following the same alternation, the next operation should be multiplication by 2. Therefore, the next term is 29 × 2 = 58. This continues the alternating rule without any break.

Option A:
Option A yields 58, which honours the exact up–down pattern of "+3, then ×2" seen throughout the series. The extended sequence 2, 5, 10, 13, 26, 29, 58 shows the rule being applied consistently. Hence, 58 is the correct next term.

Option B:
Option B gives 56, which would require adding 27 to 29 or applying a different operation unrelated to the earlier steps. It does not arise from either "+3" or "×2" as used in the pattern. Therefore, 56 is not consistent with the series.

Option C:
Option C suggests 60, requiring an addition of 31 or some new rule not indicated in previous terms. This breaks the elegant alternation between addition and multiplication. Thus, 60 cannot be considered the correct continuation.

Option D:
Option D offers 62, which again does not result from multiplying 29 by 2 or adding 3 to it. It forces a new operation into the series and disrupts the established structure. Hence, 62 is not the right answer.

This sequence follows the rule (a_n = n^4 + n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1+1+1 = 3, 16+4+1 = 21, 81+9+1 = 91, 256+16+1 = 273 and 625+25+1 = 651. For n = 6 the formula gives (6^4 + 6^2 + 1 = 1296 + 36 + 1 = 1333). Therefore 1333 is the unique next term that maintains this quartic-plus-square pattern.

Option A:
Option A, 1317, is 16 less than the formula’s result and does not equal (6^4 + 6^2 + 1). It would introduce an unexplained shortfall at the last term. Hence 1317 is not consistent with the series.

Option B:
Option B, 1325, is still lower than 1333 and similarly fails to arise from the expression for n = 6. Choosing 1325 would alter the rule only for the final term, which is logically unsound. Thus 1325 is not a valid continuation.

Option C:
Option C, 1341, overshoots the correct value and cannot be written as (6^4 + 6^2 + 1). It breaks the precise algebraic fit that exists between the terms and their indices. Therefore 1341 is not correct.

Option D:
Option D, 1333, matches exactly the output of the rule (a_n = n^4 + n^2 + 1) when n = 6. It preserves the same quartic and quadratic contributions that generate all earlier terms, so 1333 is the correct next term.