UGC NET Questions (Paper – 1)

Statements A, C and D correctly summarise key properties of HCF and LCM. A gives the fundamental relationship a × b = HCF(a, b) × LCM(a, b) for positive integers. C is true because the LCM is defined as the least common multiple of the numbers. D is correct since when one number divides the other, the smaller is their HCF and the larger is their LCM. B is false because the HCF cannot exceed the smaller number, and E is false because distinct primes have HCF equal to 1, not one of them, so A, C and D only are correct.

Option A:
Option A is incomplete because it omits D, failing to mention the simple special case where one number divides the other, which is frequently used in problems. Even though it includes two true statements, it does not list all of them.

Option B:
Option B is correct as it gathers all the true properties given, including the product relationship, the multiple property of LCM and the special divisor case, while excluding the incorrect descriptions in B and E.

Option C:
Option C is wrong because it includes E, which misstates the HCF of distinct prime numbers. By adding this false statement, it makes the combination invalid.

Option D:
Option D is incomplete since it leaves out A, the central product formula linking HCF and LCM, so it does not provide the full set of correct statements among those listed.

Statements A and B are standard properties: a set includes itself as a subset and ∅ is a subset of every set. C is correct since a proper subset of a finite set must omit at least one element, so its cardinality is smaller. E is also correct because mutual inclusion implies equality of sets. D is false; no set is a proper subset of itself by definition. Thus A, B, C and E only are correct.

Option A:
Option A is incomplete because it omits E and therefore fails to mention the important characterisation of equality in terms of mutual subset relations, a common exam point.

Option B:
Option B is wrong since it adds D, which incorrectly states that a set can be a proper subset of itself, contradicting the strictness in the definition of “proper”.

Option C:
Option C is incorrect because it includes D and drops C, losing the crucial cardinality property of proper subsets and introducing the false claim that a set is a proper subset of itself.

Option D:
Option D is correct as it collects all true subset properties listed and excludes D, which conflicts with the standard definition of proper subset in set theory.

This sequence is described by the rule aₙ = n⁵ + n² + n for n starting from 1. For n = 1, 2, 3, 4 and 5 the expression produces 3, 38, 255, 1044 and 3155, which reproduces the given series exactly. Applying the same rule for n = 6 yields 7818. Therefore 7818 is the unique next term that keeps this quintic pattern intact.

Option A:
Option A, 7766, is smaller than the value from aₙ = n⁵ + n² + n at n = 6. To obtain 7766 we would need to reduce the correct result by 52 only for the sixth term. Such a sudden change is not suggested by earlier terms, so 7766 is inconsistent with the established rule.

Option B:
Option B, 7790, is closer but still not equal to the expression’s value for n = 6. It would require subtracting 28 from the polynomial outcome, which again alters the pattern at only one point. Because the formula matches all previous terms exactly, 7790 cannot correctly continue the series.

Option C:
Option C, 7806, is 12 less than the required value and likewise fails to satisfy aₙ = n⁵ + n² + n when n = 6. Accepting 7806 would mean slightly lowering the correct quintic result at the last step without justification. Hence option C is not compatible with the underlying pattern.

Option D:
Option D, 7818, coincides exactly with the value obtained from the rule for n = 6. It preserves the combination of fifth-power and square growth plus the linear term that explains the earlier numbers. Because it extends the same algebraic structure without any adjustment, 7818 is the correct answer.

The sum of the first n natural numbers 1 + 2 + 3 + ... + n is known to be n(n+1)/2. This result can be derived by pairing terms from the beginning and end of the series, each pair summing to n+1, and noting that there are n/2 such pairs. It is a standard formula widely used in mathematical reasoning and aptitude questions. Hence, n(n+1)/2 is the correct expression.

Option A:
The expression n² would only match the sum for specific values of n and does not hold in general. For example, if n = 3, the sum is 6, but n² is 9, which is incorrect. Therefore, n² cannot be the correct general formula for the sum of the first n natural numbers.

Option B:
The formula n(n+1)/2 is correct because it consistently gives the right sum for all positive integers n. For instance, for n = 4, it yields 4×5/2 = 10, which equals 1+2+3+4. This formula is an essential tool for solving problems involving series and counting in UGC NET Paper 1.

Option C:
The expression n(n−1)/2 actually represents the number of ways to choose 2 items from n, that is, the combination nC2, not the sum of the first n natural numbers. While related to combinatorics, it does not correspond to the series 1+2+...+n. Thus, this option is not appropriate.

Option D:
n²/2 would give half the square of n, which again does not match the series sum in general. For example, if n = 4, n²/2 = 8, but the sum 1+2+3+4 is 10. Therefore, this expression is not valid for the sum described.

A is correct because one-hour work is the reciprocal of the total time needed. B is correct as an emptying pipe can be assigned a negative work rate. C is also correct since combined effect is the sum of filling and emptying rates with appropriate signs, and E is true because these problems are a direct application of time and work concepts. D is wrong because two filling pipes working together have a combined time less than either individual time; for 6 and 8 hours, the joint time is 24/7 hours, not 14 hours. Hence D only is the wrong statement.

Option A:
Option A correctly isolates D as the single incorrect statement, leaving the accurate descriptions of one-hour work, negative rates and time–work analogy intact. It reflects the usual modelling of pipes and cisterns problems in aptitude tests.

Option B:
Option B incorrectly treats both C and D as wrong, even though C correctly describes adding work rates with signs. By misclassifying C, this option fails the requirement to pick only wrong statements.

Option C:
Option C wrongly includes B as wrong along with D, despite B providing a standard way of modelling emptying actions with negative work. This misidentification makes the option invalid.

Option D:
Option D is incorrect because it also labels A as wrong, even though A properly defines one-hour work as the fraction 1/5. Thus it rejects a true statement while trying to catch the false one.

For a quadratic equation ax² + bx + c = 0, the sum of the roots is given by −b/a. In the given equation, a = 1 and b = −5, so the sum of the roots is −(−5)/1 = 5. Thus, the correct sum of the roots is 5.

Option A:
A value of 6 is actually the product of the roots for this quadratic, given by c/a = 6/1 = 6. This represents a different relationship between coefficients and roots. Since the question specifically asks for the sum of the roots, 6 is not the correct answer.

Option B:
4 does not correspond to either the sum or product derived from the standard formulas for this equation. It might result from a miscalculation when solving the quadratic. Therefore, 4 cannot be justified as the sum of the roots.

Option C:
5 is correct because it directly results from applying the formula −b/a with b = −5 and a = 1. If we factor the equation as (x−2)(x−3) = 0, the roots are 2 and 3, whose sum is also 5, confirming the calculation. This dual reasoning strengthens the validity of option C.

Option D:
2 is one of the roots of the equation but not the sum of both roots. Confusing a single root with the sum of all roots leads to this incorrect choice. Thus, 2 cannot satisfy the requirement of adding both roots together.

Statement A gives the correct general definition of average speed. Statement C is the standard formula for average speed when equal distances are covered at two speeds, which equals the harmonic mean of the speeds. Statement D is also true because when equal times are spent at each speed, the average speed simplifies to the arithmetic mean. Statement B is false because the arithmetic mean does not apply for equal-distance segments, and E is false since confusing equal-distance with equal-time cases leads to wrong answers. Therefore, A, C and D only are correct, corresponding to option C.

Option A:
Option A is incomplete as it lists only A and C and omits D, failing to mention the equal-time case where the arithmetic mean does apply. This makes the description of average speed patterns incomplete.

Option B:
Option B is incorrect because it includes E, which says there is no need to distinguish between equal-distance and equal-time cases, an assertion that contradicts typical exam traps. It therefore contains a false statement.

Option C:
Option C is correct since it distinguishes the definitions for general, equal-distance and equal-time situations and explicitly excludes B and E, which misapply or trivialise the conditions.

Option D:
Option D is wrong as it contains B, the false claim about arithmetic mean for equal-distance travel, and it omits A, so it both adds an error and leaves out the basic definition of average speed.

This sequence is given by the formula (a_n = n^4 + n^3 + n) for n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1+1+1 = 3, 16+8+2 = 26, 81+27+3 = 111, 256+64+4 = 324 and 625+125+5 = 755. For n = 6 we obtain (6^4 + 6^3 + 6 = 1296 + 216 + 6 = 1518). Therefore 1518 is the correct next term.

Option A:
Option A, 1494, is smaller than the formula’s output and does not equal (6^4 + 6^3 + 6). It would impose an unjustified reduction at the last step, so 1494 is not a valid continuation.

Option B:
Option B, 1506, is still below 1518 and fails to satisfy the expression for n = 6. Choosing 1506 would break the quartic-plus-cubic structure of the sequence. Thus 1506 is not correct.

Option C:
Option C, 1518, matches exactly the value produced by the rule when n = 6. It preserves the same combination of quartic and cubic terms that generated all the earlier numbers and hence is the correct next term.

Option D:
Option D, 1530, overshoots the computed value and cannot be written as (n^4 + n^3 + n) for the next index. Using 1530 would distort the carefully balanced pattern, so it is not appropriate.

Each group here has three letters that are equally spaced in the alphabet by the same large distance. In BKT, CLU, DMV and ENW the jump from the first to the second letter and from the second to the third letter is the same in every term, and the starting letters move forward step by step from B to C to D to E. To extend the pattern, the next starting letter must be F and the same fixed distance must be maintained to reach the second and third letters. The only option that begins with F and preserves the equal spacing is FOX, so it is the correct next term.

Option A:
Option A, EOX, has a first letter that does not move forward from ENW and changes the spacing unevenly between the three letters. The second letter is pushed too far ahead, and the differences between letters are no longer identical. Because it violates both the progression of starting letters and the constant gap condition, EOX cannot be the right continuation.

Option B:
Option B is correct because FOX has the same fixed distance between F, O and X as the earlier groups have between their letters. It also respects the simple forward movement of starting letters from B through E to F. By satisfying both the internal and external regularities of the series, FOX uniquely continues the pattern.

Option C:
Option C, GPY, keeps equal spacing between its letters but jumps the starting letter two steps ahead of E instead of one. This disrupts the smooth sequence of starting letters and does not match the rate of change seen across the given groups. Hence GPY is not consistent with the overall pattern of the series.

Option D:
Option D, FNX, starts with the correct letter F but fails to keep the same distance to the second and third letters that the earlier groups use. The internal jumps differ, so the structure of the group no longer mirrors that of BKT, CLU, DMV and ENW. Therefore FNX is not a valid next term.

The differences are 6, 9, 12 and 15, which form an arithmetic pattern increasing by 3 each time. The next difference should therefore be 18. Adding 18 to the last term 48 gives 66. Hence, 66 is the correct next term in the series.

Option A:
Option A is 60, which gives a difference of 12 from 48, repeating an earlier step instead of moving to 18. This disrupts the neat progression of 6, 9, 12, 15, 18. Therefore, 60 is not correct.

Option B:
Option B equals 66 and produces the expected new difference of 18. With this choice, the difference sequence becomes 6, 9, 12, 15, 18, which maintains a clear rule of adding 3 each time. Thus, 66 is the valid continuation.

Option C:
Option C is 72, implying a difference of 24, which jumps beyond the required 18 and overshoots the pattern. No prior step supports such a large increase. Hence, 72 cannot be chosen.

Option D:
Option D is 78, giving a difference of 30 from 48, which is even further away from the expected 18. This would completely break the second-level arithmetic structure. Therefore, 78 is not appropriate.

Statement C is wrong because the correct negation of “All X are Y” is “Some X are not Y”, not “No X are Y”; denying a universal requires asserting the existence of at least one counterexample, not universal negation. Statements A and B correctly identify universal and existential quantifications, D correctly notes that the negation of “Some X are Y” is “No X are Y”, and E accurately explains the meaning of “Some X are not Y”. Therefore, the only incorrect statement is C, so the option that lists C only is correct.

Option A:
Option A is incorrect because it treats A and C as wrong together, even though A correctly states that “All teachers are researchers” is universally quantified. By marking A as wrong, this option misrepresents basic quantifier classification.

Option B:
Option B is wrong since it pairs C with D as wrong, despite D correctly expressing the negation of an existential statement. Including D among wrong statements contradicts standard quantifier negation rules.

Option C:
Option C is incorrect as it groups B, C and E as wrong, even though B and E correctly explain existential quantification and the meaning of “Some X are not Y”. This option therefore rejects true statements along with the single false one.

Option D:
Option D is correct because it isolates C as the only statement that misstates a quantifier negation, while implicitly affirming the accuracy of A, B, D and E. It preserves the canonical equivalences used in UGC NET logic questions.

Simple interest is computed using the formula I = (P × R × T) ÷ 100. Substituting the given values P = 5,000, R = 8% and T = 3 years gives I = (5,000 × 8 × 3) ÷ 100. The product 5,000 × 8 × 3 is 120,000, and dividing by 100 yields 1,200. Therefore, the simple interest over 3 years is ₹1,200.

Option A:
Option A, ₹1,000, corresponds to a lower effective rate or shorter time period and does not match the product in the interest formula. It underestimates the interest earned.

Option B:
Option B, ₹1,100, is not an exact multiple of P × R × T ÷ 100 and therefore cannot result from correctly applying the formula.

Option C:
Option C aligns perfectly with the given parameters. It reflects three years of interest at 8% per annum on ₹5,000 and matches both the multiplication and division steps.

Option D:
Option D, ₹1,300, overstates the interest and could only be obtained by using a higher rate or an additional year, neither of which is specified.

The sequence is generated by the rule aₙ = n⁴−n²+2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1−1+2 = 2, 16−4+2 = 14, 81−9+2 = 74, 256−16+2 = 242 and 625−25+2 = 602. For n = 6 the expression gives 1296−36+2 = 1262. Hence 1262 is the term that preserves this quartic-minus-square pattern.

Option A:
Option A, 1222, is 40 less than the value predicted by n⁴−n²+2 for n = 6. It does not satisfy the formula and would require adjusting the result in a way that is not supported by the earlier data. Therefore 1222 is not the correct continuation.

Option B:
Option B, 1238, also fails to match 1296−36+2 and breaks the tight link between index and term. It represents a partial correction but still does not arise from the generating rule. Thus 1238 cannot be accepted.

Option C:
Option C, 1292, overshoots the correct value and cannot be written as n⁴−n²+2 for n = 6. Adopting 1292 would abandon the exact quartic pattern that fits the sequence so well. Hence it is not valid.

Option D:
Option D, 1262, exactly equals 6⁴−6²+2 and therefore continues the same polynomial relationship. It keeps the structure of the series fully consistent from the first term to the next one. For this reason, 1262 is the correct next term.

Statements A, B and C correctly give the basic angular speeds for clock hands: 360 degrees per full rotation, 6 degrees per minute for the minute hand and 0.5 degrees per minute for the hour hand. Statement E is also true because many questions involve finding when the hands coincide or form a given angle by using relative speed. Statement D is false since at 3:00 the angle between the hands is 90 degrees, not 60. Thus A, B, C and E together form the correct set.

Option A:
Option A is incomplete as it omits E and therefore does not mention the key technique of using relative speed, which is central for higher-level clock questions in NET. This makes the option only partially accurate.

Option B:
Option B is incomplete because it omits B (minute hand speed), so it does not include all true statements even though A, C and E are correct.

Option C:
Option C is correct because it includes all true statements and excludes D, which misstates the 3:00 angle. It represents the full conceptual toolkit for handling typical clock-angle questions.

Option D:
Option D is incorrect since it includes D, the wrong value for the angle at 3:00, and it omits A, ignoring the basic 360-degree fact. This mixture of an error with an omission makes the option invalid.

Statements A and B are correct because adding the same number or multiplying by a positive number does not change the inequality direction. Statement C is also correct as multiplying by a negative number reverses the inequality sign, a key exam point. Statement E is true since solutions of such inequalities are commonly expressed as intervals or rays on the real line. D is false because if a > b and b > c, then a > c, not a < c. Therefore A, B, C and E only form the correct set. Option A: Option A is incomplete since it leaves out C and E. Without mentioning the effect of multiplying by a negative and the interval representation, it does not give a full picture of inequality properties. Option B: Option B is also incomplete because it omits E and thereby ignores the standard graphical representation of solution sets on the number line. Option C: Option C is correct as it combines the basic algebraic rules for manipulating inequalities with the interval depiction of solutions, while excluding D, which contradicts the transitivity of the “greater than” relation. Option D: Option D is wrong since it includes D, which reverses the transitive relation, and thus mixes a clearly false statement with true ones.