UGC NET Questions (Paper – 1)

Each term in the series can be represented as n² − 1 for successive integers n starting from 2. For n = 2, 3, 4, 5 and 6 we get 2² − 1 = 3, 3² − 1 = 8, 4² − 1 = 15, 5² − 1 = 24 and 6² − 1 = 35. The next integer is 7, so the next term should be 7² − 1, which equals 49 − 1 = 48. This keeps the expression governing the series intact.

Option A:
Option A gives 46, which is not of the form n² − 1 for any integer that naturally follows 6 in the progression. Using 46 would break the simple relationship between term position and value. Hence, 46 is not consistent with the pattern.

Option B:
Option B yields 48, matching 7² − 1 and continuing the formula-based structure. The extended sequence 3, 8, 15, 24, 35, 48 clearly corresponds to 2² − 1 through 7² − 1. Therefore, 48 is the correct next term.

Option C:
Option C suggests 50, which has no direct explanation in terms of the n² − 1 rule for the next integer. It introduces a deviation from the well-defined polynomial pattern. Thus, 50 cannot be accepted as correct.

Option D:
Option D presents 52, which again does not correspond to 7² − 1 and appears arbitrary relative to the rest of the series. Choosing 52 would disrupt the simple square-based relationship. Therefore, 52 is not appropriate.

Statements A and B correctly define sufficient and necessary conditions by linking the holding of one property to the presence of another, and C is true because some conditions are both necessary and sufficient, forming a biconditional. Statement E is also true as many reasoning errors arise from confusing what is merely necessary with what is sufficient. Statement D is false because from “p is sufficient for q” it does not follow that q is sufficient for p; the implication generally does not reverse. Therefore, the correct combination is the one that collects A, B, C and E and excludes D.

Option A:
Option A is incomplete because it omits E and therefore does not acknowledge the common reasoning mistake of confusing necessary and sufficient conditions in practical thinking. While A, B and C are true, the set does not cover the full conceptual picture.

Option B:
Option B is also incomplete since it leaves out B, so it does not explicitly state how a necessary condition restricts the possibility of Q without P. Although A, C and E are true, missing B means the definition of necessity is not fully represented.

Option C:
Option C is correct because it gathers all true statements, including the definitional ones and the observation about everyday confusion, and it rightly rejects D, which incorrectly reverses the implication between sufficient conditions. This option best reflects the logic of conditionals.

Option D:
Option D is wrong because it includes D, which describes an invalid inference pattern, and it omits A, thereby failing to give the correct account of sufficiency. Mixing one false statement with an incomplete set of true ones makes this option unacceptable.

The differences between the terms are 5, 10, 15, 20 and 25. These differences form a sequence of multiples of 5, increasing by 5 each time. The next difference should therefore be 30. Adding 30 to the last term 79 gives 79 + 30 = 109, which respects the established pattern in the differences.

Option A:
Option A gives 104, which is 25 more than 79 and simply repeats the last difference instead of increasing it. This breaks the rule that the increments themselves must grow by 5. Thus, 104 is not consistent with the pattern.

Option B:
Option B offers 107, corresponding to a difference of 28 from 79, which is not a multiple of 5 and does not fit the multiples-of-five structure. Therefore, 107 cannot be accepted as correct.

Option C:
Option C yields 109, which is 30 greater than 79 and continues the sequence of differences 5, 10, 15, 20, 25, 30. This makes the series mathematically coherent. Hence, 109 is the correct next term.

Option D:
Option D suggests 111, giving a difference of 32 from 79, which does not align with the multiples-of-5 pattern. Including 111 would disrupt the clear second-level arithmetic progression. Therefore, 111 is not appropriate.

Statements A, B and C provide the definition, nth term and sum formula for APs. Statement D is also correct because terms equidistant from the ends have the same average (equal to the average of first and last). Statement E is false since d = 0 gives a constant sequence, which is still an AP. Thus A, B, C and D only are correct.

Option A:
Option A is incomplete as it omits D (symmetry property).

Option B:
Option B is incomplete because it omits B (nth term formula), so it misses a core true statement.

Option C:
Option C is incomplete since it omits A (definition), so it does not include the full true set.

Option D:
Option D is correct because it unites all true statements and excludes E (false).

Each term can be written as n² + 2 for successive natural numbers n. When n = 1, 2, 3, 4 and 5, we get 1² + 2 = 3, 2² + 2 = 6, 3² + 2 = 11, 4² + 2 = 18 and 5² + 2 = 27. The next integer is 6, so the next term should be 6² + 2. This equals 36 + 2 = 38, which maintains the exact functional rule of the sequence.

Option A:
Option A gives 38, which matches 6² + 2 and extends the pattern generated by the expression n² + 2. The sequence becomes 3, 6, 11, 18, 27, 38, corresponding to n = 1 to 6. This clear algebraic rule supports 38 as the correct next term.

Option B:
Option B offers 36, which is a perfect square but does not fit the formula n² + 2 because 36 would correspond to 6², missing the constant addition of 2. Using 36 would break the specific relationship seen in all earlier terms. Therefore, 36 is not consistent with the governing pattern.

Option C:
Option C gives 40, which is larger than 6² + 2 by 2 and does not correspond to any n² + 2 within the sequence. It appears arbitrary and fails to match the functional rule that defines the series. Hence, 40 cannot be accepted as the correct answer.

Option D:
Option D suggests 42, which also does not align with the expression n² + 2 for the next integer. It introduces a new, unsupported relationship between consecutive terms. Thus, 42 breaks the pattern and is not the correct continuation.

Statements A, B, C and E are correct as they capture the essence of mixture problems: combining components, summing component quantities, using the rule of alligation as a shortcut and representing scenarios algebraically or graphically. Statement D is false because mixture questions are naturally modelled by weighted averages where proportions act as weights. Thus, the combination A, B, C and E only is correct.

Option A:
Option A is incomplete as it leaves out E, not mentioning that visual or diagrammatic approaches are often used to make mixture relationships clearer in exams. Without E, the modelling variety is understated.

Option B:
Option B is also incomplete because it omits A and thereby fails to state directly that mixtures can involve various types of components like solutions or grains, which is basic contextual information.

Option C:
Option C is correct because it collects all the true statements about mixture structures, alligation shortcuts and representation methods, while excluding D, which incorrectly denies the connection with weighted averages. This matches the treatment in UGC NET aptitude.

Option D:
Option D is wrong as it includes D, which wrongly claims that weighted averages cannot be used for mixtures, and omits B, which accurately describes how component quantities combine. The presence of D invalidates the set.

This series is described by the rule aₙ = n⁴ + 5n² for n starting from 1. Evaluating for n = 1, 2, 3, 4 and 5 yields 1+5 = 6, 16+20 = 36, 81+45 = 126, 256+80 = 336 and 625+125 = 750. For n = 6 we obtain 6⁴+5·6² = 1296+180 = 1476. Therefore 1476 is the only term that keeps this quartic-plus-quadratic pattern intact for the next position.

Option A:
Option A, 1440, is lower than the formula’s output and does not equal n⁴+5n² for n = 6. It would require subtracting 36 from the correctly computed term only at the sixth step, which is not implied by the earlier values. This inconsistency makes option A incorrect.

Option B:
Option B, 1452, still fails to match 1296+180 and thus does not follow the same expression. Although numerically close, it cannot be produced by the rule for the sixth term. Hence option B would break the algebraic structure of the series and is not acceptable.

Option C:
Option C, 1464, lies between some candidate values but again does not satisfy aₙ = n⁴+5n² when n = 6. Selecting 1464 would mean slightly reducing the correct result without any justification from the preceding pattern. Therefore option C is not logically consistent with the sequence.

Option D:
Option D, 1476, coincides exactly with the value computed from the rule for n = 6. It preserves both the powers and coefficients that define the behaviour of the earlier terms. Because this option emerges naturally from the same formula, 1476 is the correct continuation of the number series.

Statement A correctly describes the overall concentration as a weighted average of component concentrations. Statement B is a special case for equal quantities, where the weights are equal and the average is arithmetic. Statement C states the classical alligation rule connecting mixing ratio to the differences from the mean. Statement D is false because adding pure water does not change the solute amount, and E is false since diluting with water lowers concentration but does not decrease the amount of solute. Hence A, B and C only are correct, which corresponds to option A.

Option A:
Option A is correct because it includes the general weighted-average idea, its equal-quantity special case and the alligation rule, while excluding D and E, which misunderstand how solute amount changes under dilution.

Option B:
Option B is incorrect since it includes D, which wrongly asserts that adding pure water increases solute amount, mixing an error with accurate statements.

Option C:
Option C is wrong because it includes E (false: adding water alone does not reduce solute amount) and omits B (true equal-quantity averaging), so it does not match the true set.

Option D:
Option D is incorrect as it contains D (false) and omits A (the key weighted-average principle), making it both incorrect and incomplete.

This option is correct because the set {0, 1, 2, 3, ...} includes zero and all positive integers. This collection is commonly called the set of whole numbers. It extends the natural numbers by adding zero. Therefore, the notation given describes the whole numbers.

Option A:
Whole numbers consist of zero and all positive counting numbers. The set in the question starts at 0 and continues with 1, 2, 3 and so on, which matches this definition. Hence, whole numbers is the correct term.

Option B:
Integers include negative numbers as well as zero and positive numbers. The set given in the question does not include any negative integers. Thus, the term integers is too broad and does not precisely describe the set.

Option C:
Natural numbers usually refer to positive counting numbers starting from 1, and sometimes definitions vary on including 0. Since the set explicitly lists 0, calling it only natural numbers can be ambiguous. Therefore, this option is not the best fit.

Option D:
Rational numbers include all numbers that can be written as a fraction of integers, including many non-integer values. The set listed contains only special integer values and is much more restricted. Hence, rational numbers is not the correct classification.

The first and last letters P and T remain fixed in each word, while the middle vowel changes in alphabetical order: A, E, I, O. The next vowel after O in this sequence is U. Therefore the next word formed with the same consonants is PUT.

Option A:
Option A, PET, repeats the second term rather than moving to a new vowel. The pattern clearly shows a progression of different vowels in order. Since PET does not introduce the next vowel, it cannot be the correct continuation.

Option B:
Option B, PAT, is the first word of the series and would move the pattern backward. There is no indication that the sequence cycles back to earlier words. Hence PAT does not fit as the next term.

Option C:
Option C is correct because it uses U as the next vowel after O while keeping P and T unchanged. This preserves both the consonant framework and the alphabetical order of the vowels. As a result, PUT is the only word that logically follows POT.

Option D:
Option D, PUTT, adds an extra T at the end, changing the word length and structure. The series maintains three-letter words, so this alteration in length disrupts the pattern. Therefore PUTT is not an appropriate continuation.

The pattern can be represented by a₁ = 7 and aₙ₊₁ = 2aₙ + 3ⁿ + n for n ≥ 1. Applying this rule, we have a₂ = 2·7 + 3¹ + 1 = 18, a₃ = 2·18 + 3² + 2 = 47, a₄ = 2·47 + 3³ + 3 = 124 and a₅ = 2·124 + 3⁴ + 4 = 333, which coincides with the given terms. For n = 5 the next term is a₆ = 2·333 + 3⁵ + 5 = 666 + 243 + 5 = 914. Therefore 914 is the unique value that satisfies this recurrence and extends the sequence.

Option A:
Option A, 890, is 24 less than the recurrence value at n = 5 and cannot be obtained from 2a₅ + 3⁵ + 5. Choosing 890 would require subtracting 24 from the correctly computed term only at this point. This change breaks the exact rule defining the series, so option A is not correct.

Option B:
Option B, 914, exactly matches the number produced by the recurrence when a₅ = 333 and n = 5. It preserves the doubling of the previous term together with the exponential component and linear adjustment. Because this description explains each step in the sequence, 914 is the correct next term.

Option C:
Option C, 902, is 12 less than the required value and does not equal 2·333 + 243 + 5. It approximates the correct term but fails to follow from the specified recurrence relation. Hence option C cannot be accepted as the continuation.

Option D:
Option D, 926, is 12 greater than the correct value and again fails to satisfy the rule at n = 5. To get 926 we would need to inflate the additive part, altering the recurrence at the final step. Thus option D is not a valid extension of the number series.

Let the two numbers be 4x and 5x. Their sum is 4x + 5x = 9x, which is given as 81. Solving 9x = 81 gives x = 9. Therefore, the numbers are 4 × 9 = 36 and 5 × 9 = 45, and the larger number is clearly 45. This uses the standard technique of expressing numbers with a common factor and solving a single linear equation.

Option A:
Option A, 36, is the smaller of the two numbers in the pair 36 and 45. While 36 does fit the ratio 4:5 when paired with 45, the question specifically asks for the larger number, so choosing 36 would ignore the direction of comparison. It therefore does not answer what is being asked.

Option B:
Option B is correct because it corresponds to 5x with x = 9, the larger term in the ratio 4:5. When we check, 36 + 45 = 81 and 36:45 simplifies to 4:5 on dividing both numbers by 9. This double verification confirms that 45 is the only number that fits both the ratio and sum conditions as the larger value.

Option C:
Option C, 40, might seem plausible as a mid-range value, but if 40 were the larger number, the smaller would have to be 40 × (4/5) = 32 to maintain the ratio 4:5. The pair 32 and 40 has a sum of 72, not 81, so this option fails the sum condition.

Option D:
Option D, 48, would give a smaller number of 48 × (4/5) = 38.4 if the ratio 4:5 were preserved, which is not even an integer. Moreover, 38.4 + 48 ≠ 81, so this choice violates both the integer requirement and the sum requirement.

The number of ways to choose r objects from n distinct objects is given by the combination formula nCr = n!/[r!(n−r)!]. Here n = 5 and r = 2, so 5C2 = 5!/[2!3!] = (5×4)/(2×1) = 10. Therefore, there are 10 different pairs of students that can be chosen from 5.

Option A:
A value of 5 would arise if we incorrectly assumed that each student could only form one pair or confused combinations with simple counting. However, with 5 students, there are more than 5 distinct pairs. So, this option is an underestimate and is not correct.

Option B:
A value of 8 does not match the formula 5C2 and suggests a miscalculation. No standard counting principle or correct formula would yield 8 for this problem. Therefore, this option is incorrect.

Option C:
A value of 9 is also inconsistent with the formula for combinations. It lies between nearby values but does not have any combinatorial justification here. Since 5C2 has been clearly computed as 10, 9 cannot be the right answer.

Option D:
10 is correct because it results directly from applying the combination formula and correctly accounts for all unordered pairs of students. It avoids double counting pairings like (A,B) and (B,A), which are considered the same pair in combinations. This understanding is essential in many selection problems in aptitude tests.

A correctly describes contradiction: assume the negation of what you want to prove and reach a contradiction. B is correct: proving the contrapositive is equivalent to proving the original conditional. D is correct: both are valid classical methods. C is false because contradiction proves the conclusion true (not both premises and conclusion false). E is false because contradiction proofs can be valid even when non-constructive. Therefore A, B and D only are correct.

Option A:
Option A is incomplete because it omits D, the statement confirming both methods are valid.

Option B:
Option B is incorrect because it includes C, which misstates the outcome of a contradiction proof.

Option C:
Option C is wrong because it omits A, leaving out the defining idea of contradiction.

Option D:
Option D is correct because it includes exactly the true statements and excludes C and E, which are false.

The positions of the letters are D(4), G(7), J(10), M(13) and P(16). Each step increases the position by 3 (4→7→10→13→16). Continuing this constant difference, the next position is 16 + 3 = 19, which corresponds to S. Therefore S is the only letter that preserves the fixed +3 pattern of the series.

Option A:
Option A is correct because S is exactly 3 positions after P in the alphabet. It keeps the same arithmetic progression of positions that has been followed throughout the series. By maintaining this constant increment, S fits perfectly as the next term.

Option B:
Option B, R, is only 2 positions after P(16→18), which breaks the consistent step size of 3. Choosing R would change the underlying rule from a fixed difference to a variable one. Hence R cannot logically continue the established sequence.

Option C:
Option C, T, is 4 positions after P(16→20), overshooting the required increment. This would increase the step size without any prior indication of such change. As the pattern clearly uses equal jumps, T is not a valid continuation.

Option D:
Option D, U, lies 5 positions after P(16→21), producing an even larger jump. This completely disrupts the constant difference structure of the series. Therefore U cannot be accepted as the next term.