UGC NET Questions (Paper – 1)

Each term after the first is obtained by doubling the previous term and subtracting 1. We have 3×2−1 = 5, 5×2−1 = 9, 9×2−1 = 17 and 17×2−1 = 33, confirming the rule. Applying it again gives 33×2−1 = 66−1 = 65. Hence, 65 is the correct next term.

Option A:
Option A is 63, which would require subtracting 3 from 33×2, not 1. The series consistently subtracts 1, so altering that constant is unjustified. Therefore, 63 is incorrect.

Option B:
Option B is 64, implying that we simply double 33 and subtract 2. Such a modification has no support from the earlier steps. Thus, 64 cannot match the pattern.

Option C:
Option C equals 65 and follows the rule 33×2−1 exactly. The extended sequence 3, 5, 9, 17, 33, 65 is perfectly consistent with the defined operation. This makes 65 the valid continuation.

Option D:
Option D is 67, which would result from 33×2+1 and introduces addition instead of subtraction. Since all previous transitions subtracted 1, this change breaks the rule. Therefore, 67 is not appropriate.

The series follows the rule (a_n = n^3 + 2) for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3, 10, 29, 66 and 127, which match the given terms. For n = 6 the same formula yields (6^3 + 2 = 216 + 2 = 218). Therefore 218 is the unique value that continues the same cubic pattern without any change.

Option A:
Option A equals 218, which is exactly the value of (n^3 + 2) when n = 6. It preserves the single algebraic rule that generates all previous terms. Because no adjustment or exception is needed to obtain 218, this option correctly extends the series.

Option B:
Option B, 214, does not equal (6^3 + 2) and would require subtracting an additional 4 from the formula’s output. This breaks the clean relationship between each term and its index. Hence 214 cannot be accepted as the correct next term.

Option C:
Option C, 222, is larger than the formula’s value and does not arise from (n^3 + 2) for the next integer. Choosing 222 would arbitrarily increase the term and destroy the exact cubic correspondence. Therefore 222 is not a valid continuation of the series.

Option D:
Option D, 226, deviates even further from 218 and again fails to satisfy the expression (n^3 + 2) for n = 6. It would imply altering the rule only at the last step, which is mathematically inconsistent. Thus 226 is not correct.

A proportion is an equation that states that two ratios are equal, such as a:b = c:d. It indicates that the relative relationship between the first pair of quantities is the same as that between the second pair. In mathematical aptitude, many word problems use proportions to relate parts and wholes or to scale quantities. Therefore, equality of two ratios is correctly described by the term proportion.

Option A:
Variation refers to how one quantity changes in relation to another, such as direct or inverse variation, but it is not specifically about equality of two numerical ratios. While variation also deals with relationships, it does not necessarily state that two ratios are equal. Hence, this term does not match the precise condition mentioned in the question.

Option B:
Proportion is correct because it captures the idea "two ratios are equal" in a single term. For instance, if 2:3 = 4:6, we say 2, 3, 4 and 6 are in proportion. This concept is widely used in problems involving scale diagrams, recipes and map reading. Thus, option B aligns perfectly with the wording of the question.

Option C:
Inverse usually refers to inverse proportion or inverse relationship where one quantity increases while the other decreases, maintaining a constant product. It does not refer to an equality of two ratios directly. Therefore, it is not an accurate response to the specific description in the stem.

Option D:
Percentage is a way of expressing a ratio out of 100, but it is not the term used solely for equality of two ratios. You can express each ratio as a percentage, yet the fact that percentages match does not change the underlying terminology: the equality is still called a proportion. Thus, this option is not the best answer.

The given series consists of perfect squares of natural numbers. Specifically, 1² = 1, 2² = 4, 3² = 9, 4² = 16 and 5² = 25. The next term should therefore be 6², which is 36. This continues the clear pattern of successive squares.

Option A:
Option A, 30, does not correspond to any perfect square and breaks the pattern of square numbers. It would require a different rule that is not evident in the series.

Option B:
Option B, 32, is also not a perfect square and falls between 25 and 36 without a clear basis from squaring integers.

Option C:
Option C, 35, is again not a perfect square and therefore cannot follow the pattern of 1², 2², 3², 4² and 5².

Option D:
Option D, 36, is 6² and naturally follows 5² = 25 in the sequence of square numbers. It maintains the structure of the series and is therefore the correct next term.

This option is correct because one of the fundamental axioms of probability states that the total probability of all outcomes in a sample space is 1. When outcomes are mutually exclusive and exhaustive, exactly one of them must occur. Therefore, adding their probabilities must give 1. This ensures that some outcome always happens.

Option A:
A total probability of 0 would mean that no outcome can occur, which contradicts the idea of running an experiment. Probabilities must be non-negative and at least one outcome must have positive probability. Thus, the sum cannot be 0 for an actual experiment.

Option B:
The value 1 represents certainty that some outcome from the sample space will occur. Since mutually exclusive and exhaustive outcomes cover all possibilities without overlap, their probabilities must add up to 1. This matches the basic probability axiom, making this option correct.

Option C:
A sum of 1/2 would imply that there is only a 50% chance that any outcome in the declared space occurs. That would mean half the probability lies outside the listed outcomes, so the set would not be exhaustive. Hence, this value cannot be correct here.

Option D:
The value 100 without the percent sign does not fit the standard probability scale from 0 to 1. Even if interpreted as a percentage, it would correspond to 1 on the probability scale, but the question expects the probability value, not the percentage number. Therefore, 100 is not the right answer.

Analogical reasoning involves identifying structural or functional similarity between two situations and transferring knowledge from the known situation to the new one. In mathematical aptitude, we often use analogies in number and letter series when we infer the next term by comparing to a familiar pattern. The strength of an analogy depends on the relevance and depth of the similarities, not merely superficial resemblance. This fits the description of drawing conclusions by recognising similarity between situations.

Option A:
Deductive reasoning operates by applying a general rule to a specific case rather than by comparing two similar situations. It focuses on logical implication from premises to conclusion. Since the question emphasises similarity-based reasoning, deductive reasoning does not match the required description. Therefore, this option is not correct.

Option B:
Inductive reasoning generalises from many specific instances to formulate a general rule or law. It does not primarily rely on comparing two different situations for similarity. Even though inductive reasoning might sometimes use analogies, the technical term for similarity-based inference between cases is analogical reasoning, not induction. Hence, this option is not the most appropriate.

Option C:
Analogical reasoning is correct because it is defined as reasoning that maps similarities between two domains or problems to draw new conclusions. In aptitude tests, analogies like "2:4::3:6" directly test this type of reasoning. Recognising this concept helps students approach analogy-based questions in UGC NET Paper 1 more effectively. Therefore, this option accurately represents the reasoning described in the question.

Option D:
Causal reasoning focuses on cause–effect relationships rather than on similarity between different situations. It tries to understand how one event brings about another, such as how increased study time causes better performance. The question does not mention cause and effect, but similarity, so causal reasoning is not the correct answer.

The nth term of this series can be expressed as (a_n = 2n^2 + 1) for n starting from 2. For n = 2, 3, 4, 5 and 6, the values are 2·4+1 = 9, 2·9+1 = 19, 2·16+1 = 33, 2·25+1 = 51 and 2·36+1 = 73, which exactly match the given sequence. For n = 7, the expression yields 2·49+1 = 99. Hence 99 is the correct continuation of the series.

Option A:
Option A, 95, does not equal 2n²+1 for any integer n that naturally follows 2 to 6 here. Using 95 would force a change in the generating rule and so is inconsistent with the polynomial pattern.

Option B:
Option B, 97, also fails to satisfy the formula 2n²+1 for the next integer n. It appears close numerically but does not arise from the same expression, so it breaks the functional relationship.

Option C:
Option C, 98, lies between 97 and 99 but again does not equal 2·49+1. Adopting 98 would mean abandoning the clear square-based rule that explains all earlier terms.

Option D:
Option D, 99, matches precisely the value predicted by the formula for n = 7. Since the same expression generates each term, 99 preserves the structure and therefore is the correct next term.

This option is correct because combinations count the number of ways to select r objects from n distinct objects when order does not matter. The standard notation for this is nCr. It is also called the binomial coefficient and is given by n!/[r!(n−r)!]. Therefore, nCr represents the required count.

Option A:
nPr is used for permutations, where order of arrangement matters. It counts more cases than combinations because each selection can be arranged in many orders. Since the question specifies "without regard to order," nPr is not suitable.

Option B:
rCn is not the conventional way of writing combinations of r from n. It reverses the parameters and is not used in standard notation. This makes the option incorrect.

Option C:
rPn again refers to permutations but with reversed parameters compared to typical notation. It also considers ordering, which the question explicitly ignores. Thus, this symbol does not answer the question correctly.

Option D:
nCr correctly denotes the number of combinations of n objects taken r at a time. It ignores order and is widely used in probability and counting problems. This precisely matches the description given, so this option is correct.

This option is correct because the sum of the first n natural numbers is given by n(n+1)/2. For n = 10, we have 10 × 11 / 2 = 110 / 2 = 55. Therefore, the sum of the numbers from 1 to 10 is 55.

Option A:
The value 45 would be the sum only if n were 9, since 9 × 10 / 2 = 45. However, we are summing up to 10, so 45 is too small. Hence, this option is wrong.

Option B:
A sum of 50 would correspond to a different sequence or set of terms. It does not match the formula result for the first 10 natural numbers. Therefore, it is not correct.

Option C:
A total of 60 would exceed the correct sum. The formula and direct addition both show that 55 is the actual sum, so 60 is too large. Thus, this option is incorrect.

Option D:
The value 55 satisfies both the formula and direct addition 1 + 2 + ... + 10. This consistency confirms that 55 is the correct sum for the first 10 natural numbers.

This option is correct because when equal quantities are mixed, the concentration of the mixture is the simple average of the two concentrations. The average of 10% and 20% is (10 + 20)/2 = 15%. Since both quantities are equal, no other weighting is needed. Therefore, the resulting concentration is 15%.

Option A:
If the mixture concentration remained at 10%, the presence of the stronger 20% solution would have no effect. This contradicts the idea of mixing. Therefore, 10% cannot be the concentration after mixing equal amounts.

Option B:
When equal volumes are combined, each solution contributes equally to the final concentration. Averaging 10% and 20% gives 15%, which lies between the two original values. This matches our expectation for a mixture of equal parts, so 15% is correct.

Option C:
A concentration of 20% would ignore the dilution effect of the weaker 10% solution. The final concentration cannot be as high as the stronger solution when they are mixed equally. Thus, 20% is not correct.

Option D:
30% exceeds both of the original concentrations. Mixing cannot produce a concentration stronger than the strongest component without adding extra solute. Therefore, 30% is impossible in this context.

This sequence is generated by the formula (a_n = n^3 + n^2) with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1³+1² = 2, 2³+2² = 12, 3³+3² = 36, 4³+4² = 80 and 5³+5² = 150. For n = 6 the formula gives 6³+6² = 216+36 = 252. Thus 252 maintains a single algebraic rule that explains all the terms.

Option A:
Option A, 252, matches exactly the value obtained by substituting n = 6 into the rule (a_n = n^3+n^2). Since this rule already accounts for every earlier term, extending it to 252 keeps the series perfectly consistent. Hence 252 is the correct continuation.

Option B:
Option B, 254, cannot be expressed as 6³+6² and does not result from the same expression for any natural n in order. Using 254 would break the clean connection between term position and value. Therefore it is not the right answer.

Option C:
Option C, 256, is numerically close but again does not obey the formula (n^3+n^2). It has no direct link to the pattern that generates the given sequence and so is not acceptable.

Option D:
Option D, 258, deviates even further from the predicted value and similarly lacks a basis in the generating rule. Selecting 258 would force an ad hoc change in the pattern, making it incorrect as the next term.

This option is correct because the product of an even integer with any integer is always even. An even number can be written as 2k, and multiplying 2k by any integer m gives 2(km), which is still divisible by 2. Therefore, if at least one factor is even, the product is even.

Option A:
Even numbers are multiples of 2, and when such a number multiplies any integer, the factor of 2 remains in the result. This ensures the product is also a multiple of 2 and hence even. Thus, this option correctly states the property described.

Option B:
An odd product occurs only when both factors are odd, because then neither factor contributes a factor of 2. Here, the condition explicitly says at least one integer is even, so the product cannot be odd. Therefore, this option is wrong.

Option C:
A prime number has exactly two positive divisors, 1 and itself. The product of two integers with one even factor can be composite and need not be prime. So, prime is not guaranteed and is not the correct description.

Option D:
A negative product depends on the signs of the integers, not on whether they are even or odd. An even factor does not force the product to be negative. Hence, negativity is unrelated to the parity condition in the question.

This option is correct because three consecutive integers can be written as n−1, n and n+1. Their sum is (n−1) + n + (n+1) = 3n. Dividing by 3 gives n, which is the middle integer. Therefore, the average equals the middle integer.

Option A:
The smallest integer in the set is n−1. The average of n−1, n and n+1 is n, which is larger than the smallest. Thus, the average cannot equal the smallest integer.

Option B:
The largest integer in the set is n+1. Since the average is n, which lies between n−1 and n+1, it is less than the largest value. Therefore, it is not equal to the largest integer.

Option C:
Saying the average equals the sum of the integers would ignore the division by the number of terms. The sum is 3n, while the average is 3n ÷ 3 = n. Thus, "sum of" does not describe the average correctly.

Option D:
The middle integer n is exactly the average of the three consecutive integers. It balances the smaller and larger values symmetrically. Hence, the average is always equal to the middle integer, making this option correct.

The phrase "for every real number x" indicates that the statement is being made about all real numbers without exception. In logic, this is expressed using the universal quantifier, typically denoted by ∀. It asserts that the property x² ≥ 0 holds for each element in the domain of discourse. Therefore, the statement employs the universal quantifier.

Option A:
Universal quantifier is correct because it captures the idea of generality over an entire set. In symbolic form, the statement would be written as ∀x ∈ ℝ, x² ≥ 0. This form is frequently used in mathematical proofs and theoretical reasoning in aptitude tests and higher mathematics.

Option B:
An existential quantifier is used when we assert that there exists at least one element in the domain with a given property, usually expressed as "there exists." The given statement, however, does not say "there exists some x" but "for every x," making the existential quantifier inappropriate.

Option C:
Negative quantifier is not a standard term in formal logic. Negation can apply to quantified statements, but the main quantifiers are universal and existential. Since the question refers to a well-defined quantifier symbol, this option does not match known terminology.

Option D:
Dual quantifier is not a commonly used name in basic logic courses or in the UGC NET syllabus. While universal and existential quantifiers can be viewed as duals in a theoretical sense, the specific statement here plainly uses the universal form. Thus, this is not the correct label.

A prime number is an integer greater than 1 with exactly two distinct positive divisors, namely 1 and the number itself. This definition excludes 1 because it has only one positive divisor and excludes composite numbers because they have more than two divisors. In mathematical reasoning, primes are considered the building blocks of the integers. Thus, the description in the question is the standard definition of a prime number.

Option A:
Composite numbers are positive integers greater than 1 that have more than two distinct positive divisors. For example, 6 is composite because it is divisible by 1, 2, 3 and 6. Since the question clearly states "exactly two distinct positive divisors," composite numbers do not fit this requirement. Therefore, this option is incorrect.

Option B:
A multiple is a number that can be obtained by multiplying a given number by an integer, such as 12 being a multiple of 3. This term describes a relationship between numbers and does not specify how many divisors a number has. Hence, it does not answer the definition provided in the stem.

Option C:
Prime is correct because a prime has no positive divisors other than 1 and itself, satisfying the condition of exactly two distinct positive divisors. Examples include 2, 3, 5, 7 and 11. Recognising prime numbers is important in tests involving factorisation, LCM and HCF. Thus, this option directly matches the given description.

Option D:
Even numbers are those divisible by 2, but not all integers greater than 1 divisible by 2 have exactly two divisors. For example, 4 is even but has three positive divisors: 1, 2 and 4. Therefore, "even" is not equivalent to the definition provided and cannot be the correct answer.