UGC NET Questions (Paper – 1)

The letters in each group are equally spaced by five positions. AFK is (1,6,11), BGL is (2,7,12), CHM is (3,8,13), DIN is (4,9,14). Next is (5,10,15) = EJO.

Option A:
Option A is correct because E, J and O preserve the constant five-letter spacing and the start-letter progression.

Option B:
Option B ends with Q (17) instead of O (15), breaking the spacing.

Option C:
Option C does not maintain equal gaps of five.

Option D:
Option D starts at F instead of E, breaking the start-letter sequence.

Variance is defined as the square of the standard deviation. Since standard deviation is the square root of variance, squaring the standard deviation gives the variance (a measure of dispersion).

Option A:
Variance is correct because it equals (standard deviation)² and measures how spread out data are around the mean using squared deviations.

Option B:
Mean is a measure of central tendency (average), not the square of standard deviation, so it does not match the definition.

Option C:
Range is max − min, another dispersion measure, but it is not equal to (standard deviation)².

Option D:
Median is the middle value of ordered data and is unrelated to squaring the standard deviation.

Each term after the first equals three times the previous term plus 2. We see 1×3+2 = 5, 5×3+2 = 17 and 17×3+2 = 53. Applying this rule again gives 53×3+2 = 159+2 = 161. Therefore, 161 is the correct next term in the sequence.

Option A:
Option A is 150, which would correspond to 53×3−9 and does not follow the "+2" pattern. No earlier step subtracts anything, so this is inconsistent. Hence, 150 cannot be correct.

Option B:
Option B is 155, implying 53×3−4 or some other ad hoc adjustment. This does not match the fixed addition of 2 visible in all previous transitions. Therefore, 155 is not suitable.

Option C:
Option C is 157, which does not arise from 53×3 plus a constant consistent with earlier terms. Using 157 would produce an irregular change. Thus, it breaks the simple rule.

Option D:
Option D equals 161 and is exactly 53×3+2, preserving the recursive relationship. The series 1, 5, 17, 53, 161 clearly shows the same operation being applied repeatedly. This makes 161 the correct answer.

The sequence is defined recursively by a₁ = 2 and aₙ₊₁ = 5aₙ + n for n ≥ 1. Using this rule we get a₂ = 5·2+1 = 11, a₃ = 5·11+2 = 57, a₄ = 5·57+3 = 288 and a₅ = 5·288+4 = 1444, which reproduces the given series. For n = 5 the next term is a₆ = 5·1444+5 = 7220+5 = 7225. Thus 7225 is the only value that fits the same recurrence structure for the sixth term.

Option A:
Option A, 7085, is significantly smaller than the recurrence output and does not equal 5·1444+5. It would require changing the multiplier or subtracting a large amount at the last step, neither of which is indicated by the previous terms. Hence option A is not concordant with the established pattern.

Option B:
Option B, 7155, is closer but still falls short of the correct value 7225, and it is not produced by the rule aₙ₊₁ = 5aₙ+n with n = 5. Choosing 7155 would introduce an unexplained discrepancy between the rule and the sequence. Therefore option B is incorrect.

Option C:
Option C, 7185, again does not satisfy 5·1444+5 and reflects an arbitrary reduction of 40 compared to the formula result. Since the recurrence has been obeyed exactly by all earlier terms, such a modification is unjustified. Option C is therefore incorrect.

Option D:
Option D, 7225, equals the exact value given by the recurrence for the sixth term. It continues the pattern of multiplying the previous term by 5 and adding the index, which explains all of the earlier numbers. Because it fits perfectly with the recursive rule, 7225 is the correct next term in the series.

The differences in the series are 3, 5, 7 and 9, which form an increasing sequence of odd numbers. To continue this pattern, the next difference should be 11. Adding 11 to the last term 26 gives 26 + 11 = 37. This maintains the smooth growth of the differences and fits well with the structure of the given series. Therefore, 37 is the only option that preserves the observed pattern.

Option A:
Option A provides 37, which is obtained by adding the next odd difference 11 to 26. The difference sequence 3, 5, 7, 9, 11 remains perfectly consistent and shows a clear second level arithmetic pattern. This makes 37 the logically correct continuation of the number series.

Option B:
Option B gives 35, which corresponds to adding 9 to 26 and simply repeating the last difference instead of increasing it. This breaks the rule that differences themselves should grow by 2 each time. Hence, 35 does not correctly follow the pattern governing the series.

Option C:
Option C gives 38, which would require a difference of 12 from 26, overshooting the next expected odd difference of 11. This disrupts the systematic progression of differences and introduces an unjustified jump. Therefore, 38 is not a valid extension of the series.

Option D:
Option D suggests 40, which would create a difference of 14 from 26 and move even further away from the orderly sequence of odd differences. Such a large change is inconsistent with the established pattern of gradually increasing differences. Thus, 40 cannot be accepted as the next term.

In mathematical induction, there are two main parts: the base case and the inductive step. The inductive step assumes that the statement holds for n = k (the induction hypothesis) and then proves that it must also hold for n = k + 1. This step is called the inductive step because it extends the truth from one case to the next. Hence, the description in the question refers to the inductive step.

Option A:
The basis or base step is where we verify that the statement holds for the initial value, usually n = 1 or n = 0. It does not involve the assumption "if true for k then true for k + 1." Therefore, basis does not match the part described in the question.

Option B:
Verification is a general term that could refer to checking any part of the proof. It is not the standard technical name for the specific step that connects n = k to n = k + 1. Hence, it is too vague and not suitable as an answer.

Option C:
The inductive step is correct because it formalises the logical link needed to propagate the truth of the statement from one natural number to the next. Once both the base case and inductive step are established, the statement is proved for all natural numbers in the domain. Recognising this terminology is important for understanding proofs in discrete mathematics and reasoning.

Option D:
Concluding step might describe the final remarks of a proof, but it is not a standard name for the process of moving from k to k + 1. The main logical engine in induction is the inductive step, not a generic conclusion. Thus, this option is not accurate.

The terms follow the rule aₙ = n⁴ + 5n with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 26, 96, 276 and 650, exactly the given sequence. Applying the same expression for n = 6 produces 1326. Therefore 1326 is the unique value that continues this polynomial pattern without any change in the rule.

Option A:
Option A gives 1326, which is precisely the value obtained from aₙ = n⁴ + 5n when n = 6. It preserves the same functional relationship between n and aₙ that generates all earlier terms. Hence this option correctly represents the next term of the series.

Option B:
Option B proposes 1314, which is close but does not equal the value from n⁴ + 5n at n = 6. To obtain 1314 we would need to subtract 12 from the polynomial result only at this step. Since no such irregular adjustment appears in the previous terms, 1314 cannot be the correct continuation.

Option C:
Option C suggests 1338 as the next term. This number is 12 greater than the value predicted by the rule for n = 6. Introducing such an extra increment only at the sixth term would break the consistent algebraic pattern, so 1338 is not acceptable.

Option D:
Option D gives 1350, which deviates even further from the calculated value 1326. No single formula of the form n⁴ + 5n can generate the first five terms and then jump to 1350 next. Because it would require altering the rule at the last term, this option is incorrect.

B is correct: permutations count ordered arrangements. C is correct because nPr = nCr × r! and r! ≥ 1. D is correct since choosing 0 or all items yields one way. E is correct since choosing 1 item from n gives n ways. A is false because combinations ignore order. Hence B, C, D and E only are correct.

Option A:
Option A is incomplete because it omits D and E, which are standard special cases of combinations.

Option B:
Option B is incorrect because it omits E (true).

Option C:
Option C is incorrect because it omits C (true) even though B, D and E are true; it does not list all correct statements.

Option D:
Option D is correct because it includes all true statements and excludes A, which reverses the definition of combinations.

Each term can be written as n² + 3 for consecutive integers n. For n = 1, 2, 3, 4, 5 and 6, we obtain 1² + 3 = 4, 2² + 3 = 7, 3² + 3 = 12, 4² + 3 = 19, 5² + 3 = 28 and 6² + 3 = 39. The next integer is 7, so the next term should be 7² + 3 = 49 + 3 = 52. This maintains the same square-based functional rule.

Option A:
Option A gives 50, which does not correspond to the expression n² + 3 for the next integer in sequence. Using 50 would break the direct link between n and the term value. Therefore, 50 is not consistent with the pattern.

Option B:
Option B yields 52, which matches 7² + 3 and fits the established formula perfectly. The extended series 4, 7, 12, 19, 28, 39, 52 follows a single algebraic rule from n = 1 to n = 7. Hence, 52 is the correct next term.

Option C:
Option C suggests 54, which has no basis in the n² + 3 pattern for any consecutive n after 6. It introduces an unjustified deviation from the rule. Thus, 54 is not an appropriate continuation.

Option D:
Option D offers 56, which similarly fails to fit the n² + 3 relationship and appears arbitrary in the context of the given sequence. Therefore, 56 cannot be accepted as correct.

Each term after the first is obtained by multiplying the previous term by 2 and then adding 1. We have 6×2+1 = 13, 13×2+1 = 27, 27×2+1 = 55 and 55×2+1 = 111. Applying the same rule again gives 111×2+1 = 223. Thus 223 is the only option that maintains the consistent recursive transformation.

Option A:
Option A gives 221, which would require subtracting 1 after doubling, i.e., 111×2−1, and introduces a new operation not used earlier. This contradicts the strictly “×2+1” structure observed in the series. Therefore 221 is not the correct continuation.

Option B:
Option B gives 225, which can be written as 111×2+3. This breaks the constant “+1” adjustment and suggests a different, unsupported rule. Hence 225 is not aligned with the series pattern.

Option C:
Option C gives 227, corresponding to 111×2+5, again using a different constant at the end of the multiplication. This has no basis in the previously observed behaviour, so 227 cannot be accepted.

Option D:
Option D gives 223, exactly equal to 111×2+1. It extends the same recursive rule that generates every earlier term, keeping the pattern simple and coherent. For this reason, 223 is the correct next term.

Statements A and B correctly describe basic facts about directions and displacement, and D is true because right-triangle paths allow the use of Pythagoras theorem to compute straight-line distance. Statement C is false since turning right from north leads to east, not west, and E is false because direction sense can be interpreted through coordinate geometry and vector concepts, making it quite related. Therefore, the set A, B and D only is correct.

Option A:
Option A is incomplete as it omits D and thus misses the important connection between direction sense paths and right-triangle geometry, which is central in many distance questions.

Option B:
Option B is correct because it includes all three true statements and excludes C and E, both of which misstate directional relationships or conceptual links. It accurately reflects how NET-level direction questions combine basic orientation with simple geometry.

Option C:
Option C is wrong since it includes C, the incorrect turning rule, and leaves out A, failing to mention the fundamental opposition of cardinal directions. This combination is therefore not acceptable.

Option D:
Option D is incorrect because it includes C and excludes B, thereby mixing an incorrect turning direction with a partial description of displacement and Pythagorean use.

The differences between terms are 5, 10, 15, 20 and 25. These are multiples of 5 increasing by 5 each time. The next difference should therefore be 30. Adding 30 to the last term 81 gives 81 + 30 = 111, which maintains the clear pattern in the differences.

Option A:
Option A gives 107, corresponding to a difference of 26 from 81, which is not a multiple of 5 and breaks the multiples pattern. Thus, 107 is not consistent with the second-level structure.

Option B:
Option B offers 109, which would produce a difference of 28 from 81, again not a multiple of 5 and unrelated to the previous increments. Therefore, 109 cannot be accepted as correct.

Option C:
Option C yields 111, which is 30 more than 81 and completes the sequence of differences 5, 10, 15, 20, 25, 30. The pattern stays simple and mathematically sound. Hence, 111 is the correct next term.

Option D:
Option D suggests 113, giving a difference of 32 from 81, which neither follows the multiple-of-5 rule nor the linear growth seen earlier. This choice would disrupt the logic of the series. Therefore, 113 is not appropriate.

The given relation “x varies directly as y and inversely as z” can be written as x = k·y/z for some constant k. Using x = 16 when y = 8 and z = 3, we have 16 = k·8/3, so k = 16 × 3 / 8 = 6. For y = 9 and z = 2, x = 6·9/2 = 54/2 = 27. Hence, the new value of x is 27.

Option A:
Option A, 18, would correspond to x = 6·y/z = 6·9/? with some other z value, but it is not obtained when z = 2. If we plug x = 18 into x = k·y/z with k = 6, we get 18 = 6·9/z, implying z = 3, which contradicts the specified z = 2. So 18 does not satisfy the given condition.

Option B:
Option B, 24, arises if we mistakenly treat the change in y as proportional but ignore the change in z. With k = 6, x would be 24 only when y/z = 4, which is not the case for y = 9 and z = 2 where y/z = 4.5. Thus, 24 is not consistent with the correct variation formula.

Option C:
Option C is correct because it uses the same constant of proportionality k = 6 found from the first condition and correctly substitutes y = 9 and z = 2. The resulting value 27 is obtained by straightforward application of x = k·y/z and maintains both the direct dependence on y and inverse dependence on z.

Option D:
Option D, 30, would require y/z to be 5 when k = 6, since 30 = 6·5. But the actual ratio y/z is 9/2 = 4.5 in the new situation. Therefore, 30 does not follow from the prescribed relationship between x, y and z and is not an acceptable solution.

The terms follow the rule (a_n = 3n^3 - n^2 + 1) for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3, 21, 73, 177 and 351, which match the series exactly. For n = 6 we compute (3×6^3 - 6^2 + 1 = 648 - 36 + 1 = 613). Therefore 613 is the value that continues the same cubic pattern.

Option A:
Option A, 601, is smaller than the value obtained from the formula and does not equal (3×6^3 - 6^2 + 1). It would imply subtracting an extra 12 without any justification in the rule. Hence 601 is not consistent with the sequence.

Option B:
Option B, 607, is still below 613 and again cannot be derived from the expression for n = 6. Choosing 607 would break the precise connection between index and term. Thus 607 is not a valid continuation.

Option C:
Option C, 613, exactly equals the result of applying (a_n = 3n^3 - n^2 + 1) with n = 6. It preserves both the cubic and quadratic contributions that explain all previous terms. For this reason, 613 is the correct next term.

Option D:
Option D, 625, overshoots the formula’s prediction and fails to satisfy the same expression for n = 6. Adopting 625 would change the rule at the last step and disrupt the pattern, so it is not correct.

Statement A is correct since with equal maxima the candidate corresponding to the larger ratio part has the higher score. Statement C is true because when group sizes differ, the overall average must weight each group average by its size. D is also correct since percentages place different exams on a common scale and make comparisons meaningful. E is correct because ratios and percentages adjust for different totals and show relative performance better than raw marks alone. B is false because simply averaging group means without considering sizes can give wrong overall values, so A, C, D and E only are correct.

Option A:
Option A is incomplete; although it lists A and C, it omits D and E and thus fails to acknowledge the role of percentages and relative measures in comparing performance. As a result, it does not give the full conceptual picture required.

Option B:
Option B is also incomplete because it leaves out E and includes only A, C and D, so it does not stress the comparative advantage of ratios and percentages over raw scores in unequal situations.

Option C:
Option C is wrong because it includes only C, D and E and omits A, losing the simple but important fact that the greater ratio component corresponds to higher marks under equal maxima. The combination is therefore not fully correct.

Option D:
Option D is correct because it brings together the ideas of ratio comparison, weighted averaging and the usefulness of percentages and relative measures, while excluding B, which misapplies the averaging rule. It mirrors the reasoning needed for NET data interpretation problems on exam marks.