UGC NET Questions (Paper – 1)

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.

The word LEVEL has five letters in total, with L appearing twice and E appearing twice, while V appears once. The formula for permutations of n objects with repetitions is n factorial divided by the product of factorials of repeated counts. Thus, the number of distinct permutations is 5! ÷ (2! × 2!). Calculating this gives 120 ÷ 4 = 30. Hence, there are 30 different arrangements of the letters in LEVEL.

Option A:
Option A, 20, underestimates the actual number and cannot be obtained by the correct factorial expression 5! ÷ (2! × 2!). It likely comes from an incorrect guess or misapplied formula.

Option B:
Option B accurately uses the repetition-adjusted permutation formula, recognizing two repeated L’s and two repeated E’s. By evaluating 120 ÷ 4, it yields 30, which counts all unique arrangements without overcounting identical ones.

Option C:
Option C, 40, exceeds the true count and would require a different numerator or denominator in the formula. It suggests that the effect of repeated letters has not been properly accounted for.

Option D:
Option D, 60, is exactly half of 120 and may arise from dividing by only one 2! instead of both 2!. This partially correct adjustment still leads to overcounting because both L and E are repeated.

Statement A correctly specifies that the contrapositive of “if p then q” is “if not q then not p”. Statement C is true because a statement and its contrapositive always have the same truth table and are logically equivalent. Statement D is also correct, as the inverse of “if p then q” is “if not p then not q”. Statement B is false because it actually describes the inverse, not the converse; the converse should be “if q then p”. Therefore, the correct set of statements is A, C and D only.

Option A:
Option A is incomplete since it omits D and therefore fails to state the form of the inverse, even though A and C are true. Without D, the picture of how all four related forms are constructed remains partial. This makes the option unsuitable as the correct answer.

Option B:
Option B is incorrect because it excludes C and includes D alone, leaving out the explicit mention of logical equivalence between contrapositives. It does not gather all the correct statements and so cannot represent the full answer.

Option C:
Option C is wrong as it puts together C and D but omits A, so it never explicitly defines the contrapositive itself, while still not distinguishing it from other forms. The missing definition makes this combination insufficient.

Option D:
Option D is correct because it lists the definition of contrapositive, the equivalence of contrapositives and the definition of inverse, while excluding the incorrect wording in B. It therefore gives a coherent and accurate summary of conditional variants.

Solving 2x + 3 ≤ 11, we subtract 3 from both sides to get 2x ≤ 8. Dividing both sides by 2 yields x ≤ 4. On the real number line, this corresponds to all numbers less than or equal to 4. In interval notation, that set is written as (−∞, 4], with a square bracket at 4 indicating that 4 is included.

Option A:
Option A, (−∞, 4), excludes 4 itself and would represent x < 4, which is a strict inequality, not the given ≤ relation. Option B: Option B correctly captures x ≤ 4 by including all values up to and including 4, thus satisfying the inequality exactly. Option C: Option C, (−∞, 7], includes many values greater than 4, violating the upper bound imposed by 2x ≤ 8. Option D: Option D, [4, ∞), represents x ≥ 4, the opposite direction of the inequality, and only includes 4 and greater values.

This option is correct because a leap year adds one extra day to February, making it 29 days instead of 28. As a result, the total number of days in the year becomes 366. This adjustment keeps the calendar year synchronised with the astronomical year. Therefore, a leap year has 366 days.

Option A:
366 days equal 365 regular days plus 1 extra day. This extra day compensates for the fact that the Earth's orbit around the Sun is slightly more than 365 days. Thus, this option accurately describes the length of a leap year.

Option B:
365 days correspond to a common, non-leap year. Although this is the usual number of days in a year, it does not apply when February has 29 days. Hence, this option is not correct for a leap year.

Option C:
364 days would give a year composed of exactly 52 weeks, but standard calendars do not use this length. It neither matches common nor leap years. Therefore, this option is wrong.

Option D:
360 days is sometimes used in financial calculations as a simplification, but it is not the actual number of days in either a common or leap year. Thus, it does not answer the question correctly.

This series alternates between adding 3 and multiplying by 2. Starting with 5, we have 5 + 3 = 8, 8 × 2 = 16, 16 + 3 = 19, 19 × 2 = 38, 38 + 3 = 41 and 41 × 2 = 82. The next operation should be addition by 3. Thus, the next term is 82 + 3 = 85, which continues the alternating pattern exactly.

Option A:
Option A gives 83, which would require adding only 1 to 82 and introduces a new increment not used previously. This breaks the uniform "+3, then ×2" rule. Therefore, 83 is not consistent with the pattern.

Option B:
Option B offers 84, whose difference from 82 is 2, again incompatible with the fixed addition of 3 in the pattern. This makes the series irregular. Hence, 84 cannot be accepted.

Option C:
Option C suggests 87, which requires adding 5 to 82 and does not conform to the standard "+3" step seen throughout. This disrupts the structured alternation. Thus, 87 is not appropriate.

Option D:
Option D yields 85, which is exactly 82 + 3 and matches the next stage of the "+3, then ×2" cycle. The series 5, 8, 16, 19, 38, 41, 82, 85 remains coherent. Therefore, 85 is the correct next term.

Lengths increase 2,3,4,5 so next length is 6. Starting letters A(1), C(3), F(6), J(10) have gaps +2,+3,+4 so next gap is +5 → O(15). Six consecutive letters from O are OPQRST.

Option A:
Option A starts at L, not O, so it fails the starting-letter rule.

Option B:
Option B starts at M, not O, so it fails the starting-letter rule.

Option C:
Option C is correct because it starts at O and has exactly 6 consecutive letters.

Option D:
Option D starts at Q, not O, so it skips the required start.

The differences between the terms are 3, 6, 9 and 12. These differences form a simple arithmetic progression where each difference increases by 3. Following this logic, the next difference should be 15. Adding 15 to the last term 40 gives 55, which continues the pattern in a consistent manner.

Option A:
Option A yields 55, which is 15 greater than 40 and extends the difference sequence to 3, 6, 9, 12, 15. This smooth increase by 3 at the second level is exactly what the pattern suggests. Therefore, 55 is the correct next term in the series.

Option B:
Option B gives 53, which corresponds to a difference of 13 and does not match the expected multiple of 3. This breaks the uniform pattern of the differences. Hence, 53 cannot be considered correct.

Option C:
Option C suggests 57, which would produce a difference of 17 from 40, again not a multiple of 3 and out of step with the existing sequence of differences. Thus, 57 is not consistent with the pattern.

Option D:
Option D offers 60, which is 20 more than 40 and introduces a much larger difference than the expected 15. This disrupts the clear second-level arithmetic structure and makes the series irregular. Therefore, 60 is not an appropriate continuation.

Statements A, C and D express standard partnership principles. A is correct because profit shares are proportional to capital multiplied by duration of investment. C correctly defines a sleeping partner’s role. D is true as reducing capital during the year reduces the effective capital–time product. B is false since equal capital and time lead to a 1:1 ratio, and E is false because time is crucial when investments are made for different periods. Hence, A, C and D only are correct.

Option A:
Option A is correct because it combines the basic rule of capital–time proportionality with the concept of sleeping partners and the effect of withdrawing capital, while excluding B and E, which misstate profit ratios and the role of time.

Option B:
Option B is incorrect because it includes B, which incorrectly fixes the profit share at 1:2 even when capital and time are equal, thus contradicting the proportional rule.

Option C:
Option C is wrong as it also includes E, wrongly claiming that time is irrelevant in partnership problems, and therefore mixes incorrect statements with correct ones.

Option D:
Option D is incomplete since it omits A and includes only C and D, failing to explicitly state the capital–time proportionality on which partnership questions are based.

The terms of this sequence are generated by aₙ = n⁵ + n³ + 2 with n beginning at 1. Substituting n = 1, 2, 3, 4 and 5 gives 4, 42, 272, 1090 and 3252, which agrees perfectly with the series. For n = 6 we obtain a₆ = 6⁵ + 6³ + 2 = 7776 + 216 + 2 = 7994. Hence 7994 is the unique value that preserves this quintic-plus-cubic pattern.

Option A:
Option A, 7970, is 24 less than the formula’s output at n = 6. To accept 7970 we would have to subtract 24 from the correct result only at the sixth term, even though earlier values match the formula exactly. This breaks the algebraic consistency of the series, so option A is incorrect.

Option B:
Option B, 7982, is 12 smaller than the correct value 7994 and cannot be produced by n⁵ + n³ + 2 for n = 6. It represents a near miss rather than a true continuation of the rule. Since the pattern is purely functional in n, any such deviation must be rejected.

Option C:
Option C, 7994, is exactly the result of substituting n = 6 into the expression n⁵ + n³ + 2. It maintains the same structure in which the fifth-power term dominates and the cubic and constant terms supply additional increments. Because it fits all given data and extends the rule without modification, this option is correct.

Option D:
Option D, 8006, is 12 greater than the computed value and again cannot arise from the same polynomial at n = 6. Accepting 8006 would introduce an unjustified extra amount at the final step. Therefore this option does not correctly extend the number series.

This option is correct because 3³ means 3 multiplied by itself three times. Calculating 3 × 3 × 3 gives 27. This follows directly from the definition of exponentiation. Therefore, 3³ equals 27.

Option A:
Squaring 3 gives 3² = 9, which is too small. The exponent 3 requires one more multiplication by 3. Hence, 9 is not the correct value for 3³.

Option B:
The result 27 comes from multiplying 3 by itself three times in succession. The sequence 3, 9 and then 27 confirms that 3³ = 27. This directly matches the required calculation, so this option is correct.

Option C:
The number 18 does not arise from any standard power of 3. It would require 3 × 6, not 3 × 3 × 3. Therefore, 18 is not correct here.

Option D:
The number 21 also does not match the cube of 3. It lies between 16 and 25 but has no connection to 3³. Thus, this option is incorrect.

The pattern is captured by the polynomial aₙ = n⁵ − 2n² + 4 with n starting from 1. For n = 1, 2, 3, 4 and 5 this formula yields 3, 28, 229, 996 and 3079, exactly the given sequence. Evaluating the same expression at n = 6 gives 7708. Thus 7708 is the only value that continues this quintic pattern consistently.

Option A:
Option A, 7708, is precisely the value obtained from aₙ = n⁵ − 2n² + 4 when n = 6. It maintains the strong fifth-power growth combined with the quadratic subtraction and constant term that shape the earlier numbers. Because this rule explains all five terms and leads directly to 7708 next, this option is correct.

Option B:
Option B, 7684, is 24 less than the required value and does not equal the formula’s output for n = 6. To accept 7684 we would need to diminish the polynomial result only at this index. Such a modification is not supported by the rest of the sequence, so option B is incorrect.

Option C:
Option C, 7724, is 16 greater than the pattern value and again cannot be produced by n⁵ − 2n² + 4 at n = 6. Using 7724 would inflate the term beyond the rule’s prediction, breaking the algebraic structure. Therefore option C is not a valid continuation.

Option D:
Option D, 7740, deviates even more from 7708 and likewise fails to satisfy the expression at n = 6. Adopting 7740 would ignore the exact polynomial relationship linking the terms. Hence option D is not acceptable.

A is correct because with downstream speed (b + s) and upstream speed (b − s), their average is b, the still-water speed. B is correct as the stream’s speed is half the difference of downstream and upstream speeds. D is true because if stream speed is zero, upstream and downstream speeds coincide, and E applies the standard time = distance ÷ speed relation used in these problems. C is false because upstream travel is slower and hence takes more time than downstream for the same distance. Therefore, A, B, D and E only are correct.

Option A:
Option A lists A, B and D but omits E, failing to mention the basic time formula that underlies most calculations in these problems. Because it does not include all true statements, it is incomplete.

Option B:
Option B is correct as it groups all true statements and excludes C, which reverses the comparison of upstream and downstream times. It reflects how relative speed, still-water speed and stream speed are handled in aptitude questions.

Option C:
Option C wrongly includes C, which claims upstream time is always less, and omits A, failing to describe the average-speed relationship. Including C makes this option logically incorrect.

Option D:
Option D includes C as well, and therefore it too inherits the error that contradicts basic time–speed relationships, despite including some correct statements. Hence it cannot be the right answer.

The highest common factor (HCF) is the largest number that divides both given numbers exactly. The factors of 24 include 1, 2, 3, 4, 6, 8, 12 and 24, while the factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18 and 36. The common factors are 1, 2, 3, 4, 6 and 12, of which 12 is the largest. Therefore, the HCF of 24 and 36 is 12.

Option A:
4 is a common factor of both numbers, but it is not the largest. Since there exists a greater common factor, namely 12, choosing 4 would underestimate the HCF. Thus, this option is incorrect.

Option B:
6 is also a common factor but not the highest one. Although 6 divides both 24 and 36, 12 is even larger and still divides both exactly. Therefore, 6 cannot represent the highest common factor.

Option C:
8 is a factor of 24 but not of 36, because 36 ÷ 8 is not an integer. Since it does not divide both numbers, it cannot be a common factor at all, let alone the highest. Hence, this option must be rejected.

Option D:
12 is correct because it is the greatest integer that divides both 24 and 36 without remainder. Recognising HCF and LCM is essential in simplifying fractions and solving many number-based aptitude problems. Therefore, this option correctly answers the question.

Each term is obtained by doubling the previous term and adding 1. We see 2 × 2 + 1 = 5, 5 × 2 + 1 = 11, 11 × 2 + 1 = 23, 23 × 2 + 1 = 47 and 47 × 2 + 1 = 95. Applying this rule again gives 95 × 2 + 1 = 190 + 1 = 191. This preserves the recursive structure of the series.

Option A:
Option A gives 187, which does not equal 95 × 2 + 1 and would require a different constant or operation. This breaks the "times two plus one" rule seen earlier. Therefore, 187 is not consistent with the pattern.

Option B:
Option B offers 189, which corresponds to adding −1 after doubling 95 or some other ad hoc change. This is not supported by earlier transitions. Hence, 189 cannot be accepted.

Option C:
Option C suggests 190, which is simply 95 × 2 and omits the addition of 1 that is present in every previous step. This makes the rule incomplete and inconsistent. Thus, 190 is not appropriate.

Option D:
Option D yields 191, which exactly matches 95 × 2 + 1 and continues the recursive pattern in a straightforward way. The series remains internally coherent. Therefore, 191 is the correct next term.