UGCNET Questions

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.

Let the original price be 100 units for ease of calculation. A 20% increase raises the price to 120 units. A subsequent 20% decrease is applied to this new price, so the reduction is 20% of 120, which is 24. Subtracting 24 from 120 gives 96. The final price is therefore 96, which is 4 units less than the original 100, indicating a net decrease of 4%.

Option A:
Option A, 0% change, would be correct only if an increase and an equal decrease cancelled each other exactly, but this happens only when both percentages are applied to the same base value. Here the decrease is applied to an already increased price.

Option B:
Option B, 2% decrease, underestimates the net effect and would arise from a miscalculation or an incorrect averaging of percentages without considering the changed base.

Option C:
Option C accurately tracks the price through both changes and compares the final value with the initial value. The computation demonstrates that the successive operations do not cancel out but instead yield a 4% net decrease.

Option D:
Option D, 6% decrease, overstates the net reduction and cannot be derived from the multiplicative factors 1.2 and 0.8 that represent the two changes.

The rule is that a number n maps to n² + 4. For n = 2, we get 2² + 4 = 4 + 4 = 8, and for n = 3, we get 3² + 4 = 9 + 4 = 13, matching the given examples. Applying the same rule to n = 5 yields 5² + 4 = 25 + 4 = 29. Hence, 5 maps to 29 under this square-plus-four rule.

Option A:
Option A, 21, could arise from a different linear rule like 4n + 13, but it does not agree with the n² + 4 pattern shown by the examples.

Option B:
Option B, 29, is exactly 5² + 4 and therefore fits the defined function rule for this analogy.

Option C:
Option C, 31, would correspond to 5² + 6, which changes the constant term and breaks the consistency of the mapping.

Option D:
Option D, 35, is even further from 29 and does not follow from squaring and then adding a fixed constant.

The pattern in the series is that each term is obtained by doubling the previous term and adding 1. Specifically, 3 × 2 + 1 = 7, 7 × 2 + 1 = 15, 15 × 2 + 1 = 31 and 31 × 2 + 1 = 63. Continuing this rule, the next term is 63 × 2 + 1 = 127. Hence, 127 is the only number that preserves the established doubling plus one pattern.

Option A:
Option A, 95, could come from adding 32 to 63, but there is no basis for adding a fixed difference when the earlier transitions are multiplicative with an extra 1. It breaks the observed rule of doubling.

Option B:
Option B applies the same operation used at every step of the series, doubling 63 and then adding 1. This keeps the structure consistent across all terms and therefore correctly predicts the next number.

Option C:
Option C, 125, is close numerically but would require adding 62, again inconsistent with the doubling plus one pattern seen earlier.

Option D:
Option D, 119, might reflect adding 56, but such a jump is not supported by any of the prior steps in the sequence.

To solve the inequality 2x − 5 > 7, we first add 5 to both sides, giving 2x > 12. Then we divide both sides by 2, which is positive, so the inequality direction remains the same, yielding x > 6. Thus, the solution set consists of all real numbers greater than 6, and this is the equivalent inequality in terms of x.

Option A:
Option A captures the full result of correctly isolating x. It incorporates both the addition and division steps and expresses the final constraint directly as x > 6.

Option B:
Option B, x > 5, would follow only if the inequality were 2x − 5 > 5, not 2x − 5 > 7. It represents a weaker condition than required.

Option C:
Option C, x > 4, ignores more of the numerical shift and significantly broadens the solution set beyond what the original inequality allows.

Option D:
Option D, x > 3, is even looser and does not align with the operations needed to isolate x in the given inequality.