UPSC CSE Prelims 2026: CSAT Official Paper (Previous Year Paper)

Shortcut Trick

For I: The function f(t) = t + 1/t has its minimum at t = 1. Since (1/3, 2) includes 1, the function decreases then increases, so it is not always increasing.

For II: Let g(t) = √(1 + t²) / t = √(1/t² + 1). As t increases, 1/t² strictly decreases.

Since x < y, it follows that 1/x² > 1/y² ⇒ √(1/x² + 1) > √(1/y² + 1).

Thus, g(x) > g(y) is always true for the given range.

∴ The correct answer is II only.

Alternate Method

Given: 1/3 < x < y < 2

For Statement I: x + 1/x < y + 1/y

⇒ Let x = 0.5 and y = 1.0 (both in range 1/3 < x < y < 2)

⇒ x + 1/x = 0.5 + 2 = 2.5

⇒ y + 1/y = 1 + 1 = 2

⇒ 2.5 is not less than 2, so Statement I is False.

For Statement II: √(1 + y²)/y < √(1 + x²)/x

⇒ Let g(t) = √(1 + t²)/t = √((1 + t²)/t²) = √(1/t² + 1)

⇒ Since x < y ⇒ x² < y² ⇒ 1/x² > 1/y²

⇒ 1/x² + 1 > 1/y² + 1

⇒ √(1/x² + 1) > √(1/y² + 1)

⇒ Therefore, Statement II is True.

∴ The correct answer is II only.

Additional Information

Monotonic Functions

A function is monotonic if its derivative does not change sign. f(t) = t + 1/t has f'(t) = 1 − 1/t², which is negative for t < 1 and positive for t > 1.

Inequality Property

For positive values, if a > b, then 1/a < 1/b and √a > √b. These properties are essential for comparing fractional expressions.

AM-GM Inequality

For any positive number x, x + 1/x ≥ 2. The minimum value 2 occurs exactly at x = 1.

Given measuring cylinders:

1 litre, 6 litres, 25 litres and 100 litres

We need to measure 298 litres.

To minimize the number of measurements, use the largest cylinder as much as possible.

298 = 100 + 100 + 100 − 1 − 1

So, measure:

100 litre cylinder → 3 times

1 litre cylinder → 2 times

Total measurements = 3 + 2 = 5

Hence, the minimum number of measurements required is 5.

Hence, the correct answer is "Option 2".

Shortcut Trick

Total weight required = 20 kg. Available weights: 1 kg, 2 kg, 5 kg, 10 kg.

Count of 1 kg weights must be between 8 and 11:

→ Case 1: 8 × 1 kg | Remaining 12 kg → {10,2}, {5,5,2}, {2×6} (3 ways)

→ Case 2: 9 × 1 kg | Remaining 11 kg → {5,2,2,2} (1 way)

→ Case 3: 10 × 1 kg | Remaining 10 kg → {10}, {5,5}, {2×5} (3 ways)

→ Case 4: 11 × 1 kg | Remaining 9 kg → {5,2,2} (1 way)

Total ways = 3 + 1 + 3 + 1 = 8.

∴ The correct answer is 8 ways.

Alternate Method

Given:

Weights available = {1, 2, 5, 10} kg

Target weight = 20 kg

Condition = 8 ≤ (1 kg weights) ≤ 11

Calculations:

Let n₁, n₂, n₅, n₁₀ be the number of weights of 1 kg, 2 kg, 5 kg, and 10 kg respectively.

⇒ Total Weight: 1 × n₁ + 2 × n₂ + 5 × n₅ + 10 × n₁₀ = 20

⇒ Where 8 ≤ n₁ ≤ 11

⇒ If n₁ = 8: 2n₂ + 5n₅ + 10n₁₀ = 12. Solutions: (n₁₀=1, n₅=0, n₂=1); (n₁₀=0, n₅=2, n₂=1); (n₁₀=0, n₅=0, n₂=6). (3 ways)

⇒ If n₁ = 9: 2n₂ + 5n₅ + 10n₁₀ = 11. Solutions: (n₁₀=0, n₅=1, n₂=3). (1 way)

⇒ If n₁ = 10: 2n₂ + 5n₅ + 10n₁₀ = 10. Solutions: (n₁₀=1, n₅=0, n₂=0); (n₁₀=0, n₅=2, n₂=0); (n₁₀=0, n₅=0, n₂=5). (3 ways)

⇒ If n₁ = 11: 2n₂ + 5n₅ + 10n₁₀ = 9. Solutions: (n₁₀=0, n₅=1, n₂=2). (1 way)

⇒ Total ways = 3 + 1 + 3 + 1 = 8 ways.

∴ The correct answer is 8 ways.

Additional Information

Partition of a Number

The number of ways to write a positive integer n as a sum of positive integers from a given set is a classic combinatorial problem often solved using generating functions or systematic counting.

Coin Change Logic

When denominations (weights) are given, the number of ways to make a total sum is restricted by the parity (even/odd) of the target. For example, if all remaining weights are even, the remainder must be even to have any solutions.

Constraint Handling

In problems with constraints (like 8 to 11 weights), always isolate the variable with the constraint first and then calculate independent cases for each possible value of that variable.

The logic for determining the number of pieces produced by cutting a solid cube with n cuts, where pieces remain unmoved, is based on the distribution of cuts along the three axes (x, y, and z).

The total number of pieces x_n is calculated using the formula:

x_n = (c_x + 1) × (c_y + 1) × (c_z + 1)

where c_x + c_y + c_z = n (total cuts).

Condition for the third cut:

After 3 cuts (n = 3), the pieces are identical. This can happen in three possible configurations:

 

Analysis for the fourth cut (n = 4):

We add one additional cut to the configurations identified above to find the possible values for x₄:

1. From configuration (3, 0, 0):

• Add cut to same axis: (4, 0, 0) → x₄ = (4+1) × 1 × 1 = 5
• Add cut to new axis: (3, 1, 0) → x₄ = (3+1) × (1+1) × 1 = 8

2. From configuration (2, 1, 0):

• Add cut to first axis: (3, 1, 0) → x₄ = (3+1) × (1+1) × 1 = 8
• Add cut to second axis: (2, 2, 0) → x₄ = (2+1) × (2+1) × 1 = 9
• Add cut to third axis: (2, 1, 1) → x₄ = (2+1) × (1+1) × (1+1) = 12

3. From configuration (1, 1, 1):

• Add cut to any axis: (2, 1, 1) → x₄ = (2+1) × (1+1) × (1+1) = 12

The possible values for x₄ are 5, 8, 9, and 12. Any value higher than the maximum achievable with 4 cuts (which is 12 for orthogonal cuts) is impossible.

Therefore, 16 is not a possible value for x₄.

Hence, the correct answer is "Option 1".

Shortcut Trick

Net increase in total marks for the first correction = 100 − 0 = 100.

Since the average increase is 4, Total students (n) = 100 ÷ 4 = 25.

Second correction: 81 is replaced by 56, so total marks decrease by 81 − 56 = 25.

Net change in average for second correction = −25 ÷ 25 = −1.

Overall change in average (y − x) = 4 − 1 = 3.

∴ The correct answer is 3.

Alternate Method

Given: Initial average = x, Corrected average = y. Correction 1: 0 → 100. Correction 2: 81 → 56.

Formula Used: Average = Sum of observations ÷ Number of observations (n)

⇒ Let Total Students = n and Initial Sum = S

⇒ Initial Average x = S ÷ n

⇒ After 1st correction: (S + 100) ÷ n = x + 4

⇒ S/n + 100/n = x + 4 ⇒ x + 100/n = x + 4 ⇒ 100/n = 4 ⇒ n = 25

⇒ Final Correction 2: New Sum = S + 100 + (56 − 81) = S + 100 − 25 = S + 75

⇒ Final Average y = (S + 75) ÷ 25 = S/25 + 75/25

⇒ y = x + 3

⇒ y − x = 3

∴ The correct answer is 3.

Additional Information

Net Change in Average

Net Change in Average = (Net Change in Sum) ÷ (Total Number of Items). If the total sum increases, the average increases, and vice versa.

Correction of Errors

When data is misread, the Correct Sum = (Wrong Sum − Wrong Value + Correct Value). This adjustment helps find the true average without recalculating every entry.

Average Property

If every observation in a set is increased or decreased by a constant value 'k', the average also increases or decreases by that same value 'k'.

Logic followed here is:

The validity of the conclusions is determined by analyzing the specific differences mentioned between HMPV (Human Metapneumovirus) and SARS-CoV-2 in terms of their severity, mortality rates, and long-term health effects.

Analysis of Statement 1:

The passage focuses on the physiological and clinical differences between the two viruses, such as the severity of illness and the presence of long-term symptoms. While it mentions SARS-CoV-2 and HMPV have different epidemiology, it does not explicitly state that the primary reason SARS-CoV-2 became a pandemic was specifically due to its novelty or lack of prior exposure. Without direct evidence in the text linking pandemic status solely to novelty, this conclusion cannot be definitively drawn from the given information. Therefore, Statement 1 is invalid.

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Analysis of Statement 2:

The passage provides a clear comparison of the impacts of the two viruses:

Because HMPV causes milder illness and lacks post-viral symptoms, its impact on the individual and the healthcare system is fundamentally different from SARS-CoV-2. This disparity implies that public health responses, treatment protocols, and management strategies should be tailored to the specific threat level of each virus. Thus, they should not be dealt with in a similar manner. Therefore, Statement 2 is valid.

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The pattern observed is that only the second statement logically follows from the differences in clinical outcomes described in the text.

Therefore, the correct conclusion is 2 only.

The correct answer is None of the above

Key Points

• The passage highlights the distinction between HMPV (Human Metapneumovirus) and SARS-CoV-2, emphasizing that HMPV causes milder illness, has a longer history of documentation, and lacks the long-term post-viral symptoms associated with COVID-19.
• Statement 1 is incorrect because the text focuses on describing the characteristics and historical context of the viruses rather than proposing or evolving new methodologies or frameworks for scientific analysis.
• Statement 2 is incorrect because the primary intent of the author is to contrast the severity and social impact of the two viruses to show why they should not be treated in the same manner. Merely 'establishing similarities and differences' is too broad and does not capture the specific intent of highlighting the fundamental differences in their impact on human populations.
• Statement 3 is incorrect as the passage discusses epidemiology, pathology, and symptomatology, but it does not provide information regarding remedies, medical treatments, or cures for HMPV or SARS-CoV-2.
• Since none of the provided statements accurately capture the author's intent of differentiating the severity and historical context of the two viruses, the correct choice is None of the above.

Additional Information

• Human Metapneumovirus (HMPV) was first identified in 2001 and belongs to the Pneumoviridae family (formerly Paramyxoviridae). It is a common cause of upper and lower respiratory tract infections.
• Unlike the SARS-CoV-2 pandemic, which was driven by a novel virus to which the global population had no pre-existing immunity, HMPV has been circulating for decades, leading to a level of community exposure that prevents widespread pandemic-level mortality in most healthy individuals.
• The distinction in long-term outcomes is a major differentiator; while SARS-CoV-2 can lead to Long COVID (post-acute sequelae), HMPV is generally associated with acute recovery without chronic post-viral syndromes.

Based on the clinical characteristics and the nature of the viruses described in the passage, let us analyze the statements to draw logical inferences:

Analysis of Statement 1:

The passage and context suggest that SARS-CoV-2 was a novel virus when it emerged, meaning it was previously unknown to the human immune system and medical researchers. In contrast, Human Metapneumovirus (HMPV), while less discussed in popular media, has been identified and studied for a significantly longer period in clinical history. Thus, it has a longer documentation of studies. Therefore, Statement 1 is a valid inference.

Analysis of Statement 2:

The text explicitly states that HMPV causes milder illness and that deaths are very rare. This is a direct contrast to the impact of SARS-CoV-2, which resulted in a global pandemic with high rates of severity and mortality. Because their epidemiological impacts and clinical outcomes are fundamentally different, they result in different outcomes for those affected. Therefore, Statement 2 is a valid inference.

Analysis of Statement 3:

The passage highlights that HMPV is characterized by no long-term post-viral symptoms. This is one of the most significant differentiators from SARS-CoV-2, which is widely known for causing long-term impacts (often termed as Long COVID). The presence of chronic post-viral issues in one and their absence in the other makes this a rational differentiator. Therefore, Statement 3 is a valid inference.

Analysis of Statement 4:

The provided text does not mention or provide data regarding seasonal specificity for both viruses. While many respiratory viruses are seasonal, SARS-CoV-2 has demonstrated the ability to cause waves of infection across various seasons and climates, making it difficult to infer seasonal specificity as a common factor based strictly on the logical flow of this comparison. Therefore, Statement 4 is not a supported inference.

Conclusion:

Statements 1, 2, and 3 reflect the logical and rational conclusions that can be drawn regarding the differences in novelty, clinical outcomes, and long-term impacts of the two viruses.

Therefore, the correct answer is 1, 2 and 3.

Given expression:

(x × y) + z

x, y and z take distinct values from 2, 3, 4, 5.

To find the largest value (M):

Choose the largest values for multiplication:

x = 5, y = 4, z = 3

M = (5 × 4) + 3

M = 20 + 3 = 23

To find the smallest value (N):

Choose the smallest distinct values:

x = 2, y = 3, z = 4

N = (2 × 3) + 4

N = 6 + 4 = 10

Therefore,

M − N = 23 − 10 = 13

Hence, the correct answer is "Option 3".

Shortcut Trick

Combine proportions: Since y ∝ x² and x ∝ 1/z, then y ∝ 1/z².

Using the ratio: y₁ ÷ y₂ = (z₂ ÷ z₁)²

50 ÷ 98 = (z₂ ÷ (7/25))² ⇒ 25/49 = (25z₂ ÷ 7)²

Taking square root on both sides: 5/7 = 25z₂ ÷ 7 ⇒ 25z₂ = 5 ⇒ z₂ = 1/5.

∴ The correct answer is 1/5.

Alternate Method

Given: y ∝ x², x ∝ 1/z, z = 7/25, x = 5, y = 50. Find z when y = 98.

Formula Used: y = k₁x² and x = k₂/z

Step 1: Find constants from initial values.

⇒ y = k₁x² ⇒ 50 = k₁ × 5² ⇒ 50 = 25k₁ ⇒ k₁ = 2

⇒ x = k₂/z ⇒ 5 = k₂ / (7/25) ⇒ 5 = k₂ × (25/7) ⇒ k₂ = 35/25 = 7/5

Step 2: Find new x when y = 98.

⇒ y = 2x² ⇒ 98 = 2x² ⇒ x² = 49 ⇒ x = 7 (as x > 0)

Step 3: Find new z using x = 7 and k₂ = 7/5.

⇒ x = (7/5) / z ⇒ 7 = 7 / (5z) ⇒ 35z = 7

⇒ z = 7/35 = 1/5

∴ The correct answer is 1/5.

Additional Information

Direct Variation

When two variables change in the same ratio, y ∝ x or y = kx. Here, y/x remains constant.

Inverse Variation

When one variable increases and the other decreases proportionally, x ∝ 1/z or xz = k. Their product is constant.

Joint Variation

Describes a situation where a variable depends on two or more other variables, e.g., y ∝ xz² ⇒ y = kxz².

Chain Rule of Variation

If A ∝ B and B ∝ C, then A ∝ C. If A ∝ B and B ∝ 1/C, then A ∝ 1/C.

Based on the passage provided, here is the step-by-step analysis of the given assumptions:

Assumption 1: Agricultural productivity has marched ahead in the West because of the economies of scale facilitated by the adoption of AI tools.

The passage mentions that in the US and Europe, generative AI tools have started offering precision farming at scale. However, it does not explicitly state that agricultural productivity has 'marched ahead' or that 'economies of scale' were the specific cause of such progress. Since the answers must be based solely on the passage, this assumption is not valid.

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Assumption 2: Affordable AI tools rendered available in local languages can help AI-based solutions reach more and more small farmers.

The text suggests that for at-scale integration and accessibility of AI tools in India, it would be helpful to develop Indian languages-based AI tools for smallholder farmers and partner with AgTechs to create affordable AI solutions. This implies that language support and affordability are critical factors for reaching a larger number of farmers. Therefore, this assumption is valid.

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Assumption 3: Though penetration is as yet low, critical application areas deploying AI tools are already in use in India.

The passage starts by stating that India is starting to deploy AI for critical use cases such as weather forecasting, pest detection and control, and crop yield optimisation. It immediately follows this by noting that penetration is limited to a small subset of tech-savvy farmers. This directly confirms that critical applications are in use despite low overall penetration. Therefore, this assumption is valid.

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The valid assumptions are 2 and 3 only.

Therefore, the correct answer is 2 and 3 only.

The correct answer is Option 1.

 

Key Points

• Statement 1 is not correct: The passage mentions that AI penetration is currently limited to a small subset of tech-savvy farmers in India. It does not state or imply that these farmers will drive the AgTech companies of the future.
• Statement 3 is not correct: While the passage states that AI tools in the US and Europe offer precision farming at scale, it does not claim that these tools have replaced traditional agricultural practices entirely. Replacement is an overstatement not supported by the text.
• Statement 2 is correct: The text explicitly suggests that for accessibility, it would be helpful to disseminate AI-based advisory services through government programmes.
• Statement 4 is correct: The passage confirms that in the US and Europe, AI tools are already integrating large datasets to provide real-time agronomic insights.

Therefore, the incorrect statements are 1 and 3.

 

Additional Information

• The passage highlights that India is already deploying AI for critical use cases like weather forecasting and pest detection.
• To reach smallholder farmers, the passage emphasizes the development of Indian languages-based AI tools and affordable solutions through partnerships with AgTechs.

Given:

P = 4

R = 1

Q ≠ 5

S ≠ 2

T ≠ 3

Remaining ranks to assign are: 2, 3, 5

Checking Statement I:

If S = 3,

Then Q and T must take 2 and 5.

Since Q ≠ 5,

Q = 2 and T = 5

But the statement says T = 2, which is false.

So, Statement I does not follow.

Checking Statement II:

If Q = 3,

Then S and T must take 2 and 5.

Since S ≠ 2,

S = 5 and T = 2

But the statement says T = 5, which is false.

So, Statement II also does not follow.

Therefore, neither I nor II is correct.

Hence, the correct answer is "Option 4".

Given:

1. P1 matches Q3
2. P4 matches Q1 or Q2
3. Q4 matches P3 or P5
4. Q3 and Q5 are adjacent

Possible Arrangement:

Here:

1. P1 matches Q3
2. P4 matches Q1
3. Q4 matches P3
4. Q3 and Q5 are adjacent

 

Checking Statement I:

Q3 is placed at the second position, but it can also be placed at the middle position in another valid arrangement.

So, Statement I is possible.

Checking Statement II:

In the above arrangement, P2 is placed at an extreme end.

So, Statement II is also possible.

Therefore, both I and II are possible.

Hence, the correct answer is "Option 3".

Given:

1. A and two others travel by X → X has 3 persons.
2. Only E travels with G.
3. C travels by Z.
4. B does not travel by Y.
5. A and B do not travel by the same car.

 

Arrangement:

This arrangement satisfies all the conditions.

Checking Statement I:

No one travels alone.

X has 3 persons, Y has 2 persons, Z has 2 persons.

So, Statement I is correct.

Checking Statement II:

“Only D travels with F” means only two persons are in that car.

But A also travels with D and F in car X.

So, Statement II is false.

Checking Statement III:

Only C travels with B.

Car Z contains only B and C.

So, Statement III is correct.

Therefore, Statements I and III only are correct.

Hence, the correct answer is "Option 2".

The relations given in the statements can be represented as logical implications:

1. X is incorrect → Z is incorrect
2. Y is incorrect → W is correct
3. W is correct → X is incorrect

 

Step 1: Combining the logical relations

By combining statements 2 and 3: If Y is incorrect, then W is correct, and if W is correct, then X is incorrect.

This gives the sequence: Y is incorrect → W is correct → X is incorrect.

Now, incorporating statement 1: If X is incorrect, then Z is incorrect.

This forms the complete logical chain: Y is incorrect → W is correct → X is incorrect → Z is incorrect.

 

Step 2: Evaluating Conclusion I

Conclusion I: If X is correct, then so is Y.

From our chain, we have the relation: Y is incorrect → X is incorrect.

The contrapositive of this statement is: If X is correct, then Y is correct.

Therefore, Conclusion I is correct.

 

Step 3: Evaluating Conclusion II

Conclusion II: If Z is correct, then it is not necessary that Y is correct.

From our chain, we have the relation: Y is incorrect → Z is incorrect (derived from Y is incorrect → W → X is incorrect → Z is incorrect).

The contrapositive of this statement is: If Z is correct, then Y is correct.

This means if Z is correct, Y must be correct. The statement that it is 'not necessary' contradicts this logical derivation.

Therefore, Conclusion II is incorrect.

 

Based on the analysis, only Conclusion I follows from the given statements.

Correct Answer: I only

Hence, the correct answer is "Option 1".

Shortcut Trick

Phase 1 (Same direction): X covers 7 rounds, Y covers 5 rounds. Meetings = |7 − 5| = 2.

Phase 2 (Opposite direction): X needs 21 − 7 = 14 rounds. Time is doubled, so Y speed × 2 and time × 2 ⇒ Y rounds = 5 × 4 = 20.

Meetings in Phase 2 = 14 + 20 = 34.

Total meeting points = 2 + 34 = 36. Excluding the final stopping point: 36 − 1 = 35.

∴ The correct answer is 35.

Alternate Method

Given: Initial rounds X = 7, Y = 5. Final rounds X = 21. Speed of Y doubles and direction reverses after 5 rounds.

Formula: Meetings (Same dir) = |Rounds₁ − Rounds₂|; Meetings (Opposite dir) = Rounds₁ + Rounds₂.

Phase 1: Same Direction (0 to T₁)

⇒ Rounds by X = 7; Rounds by Y = 5.

⇒ Relative rounds = 7 − 5 = 2. Meetings occur at relative distance 1L and 2L.

⇒ Meeting times: t = 0.5T₁ and t = T₁ (at the starting point).

Phase 2: Opposite Direction (T₁ to T_final)

⇒ Speed of X (V_x) = 7/T₁; Speed of Y (V_y1) = 5/T₁.

⇒ New Speed of Y (V_y2) = 2 × (5/T₁) = 10/T₁.

⇒ Remaining rounds for X = 21 − 7 = 14 rounds.

⇒ Time for Phase 2 (T₂) = 14 / (7/T₁) = 2T₁.

⇒ Rounds by Y in Phase 2 = V_y2 × T₂ = (10/T₁) × 2T₁ = 20 rounds.

⇒ Relative rounds in opposite direction = 14 + 20 = 34.

⇒ Meetings in Phase 2 occur at relative distances 1L, 2L, ..., 34L.

⇒ The 34^th meeting is at the final stop (14 + 20 = 34).

Total Meetings:

⇒ After start and before stop = (Meetings in Phase 1) + (Meetings in Phase 2 excluding last one).

⇒ Total = 2 + (34 − 1) = 2 + 33 = 35.

∴ The correct answer is 35.

Additional Information

Circular Track Meetings (Same Direction)

When two objects move in the same direction, the number of times they meet is given by the difference in the number of rounds covered: |n − m|.

Circular Track Meetings (Opposite Direction)

When two objects move in opposite directions, the number of times they meet is given by the sum of the number of rounds covered: n + m.

Relative Speed on Circular Track

Relative speed in same direction is V₁ − V₂ and in opposite direction is V₁ + V₂. Time between consecutive meetings is L / V_relative.

Shortcut Trick

Let the number of Correct answers be C and Wrong answers be W.

Case 1: 5C − 2W = 69

Case 2: 4C − 1W = 84 ⇒ 8C − 2W = 168 (Multiplying by 2 to equate W)

Subtracting Eq 1 from Eq 2: (8C − 5C) = 168 − 69 ⇒ 3C = 99 ⇒ C = 33

Putting C = 33 in Case 2: 4(33) − W = 84 ⇒ 132 − W = 84 ⇒ W = 48

Total questions = C + W = 33 + 48 = 81

∴ The correct answer is 81.

Alternate Method

Given:

Correct answer marks (Case 1) = +5 | Wrong answer marks (Case 1) = −2 | Score = 69

Correct answer marks (Case 2) = +4 | Wrong answer marks (Case 2) = −1 | Score = 84

Formula Used:

Total Marks = (Number of Correct × Marks per Correct) − (Number of Wrong × Negative marks per Wrong)

Total Questions = Correct + Wrong (Since all questions were attempted)

Let Correct answers be x and Wrong answers be y.

⇒ 5x − 2y = 69 --- (i)

⇒ 4x − y = 84 --- (ii)

⇒ From (ii), y = 4x − 84

⇒ Substitute y in (i): 5x − 2(4x − 84) = 69

⇒ 5x − 8x + 168 = 69

⇒ −3x = 69 − 168

⇒ −3x = −99

⇒ x = 33

⇒ y = 4(33) − 84 = 132 − 84 = 48

⇒ Total questions = x + y = 33 + 48 = 81

∴ The correct answer is 81.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by + c = 0 can be solved using Substitution, Elimination, or Cross-multiplication methods.

Negative Marking Calculation

Total marks = (Correct × Marks/Correct) − (Wrong × Penalty/Wrong). If unattempted questions exist, they are excluded from the sum of Correct and Wrong.

Consistency of Equations

A system has a unique solution if a₁/a₂ ≠ b₁/b₂. In this problem, 5/4 ≠ (−2)/(−1), ensuring a unique count of questions.

The correct answer is Option 2.

 

Key Points

• Conclusion 1 is invalid. According to the Juvenile Justice (Care and Protection of Children) Act, 2015, it is the heinous offences (offences with a minimum punishment of seven years) that allow for the possibility of trying adolescents above 16 as adults. The 2021 amendment introduced serious offences, but the passage mentions that in the opinion of activists, these do not merit the transfer of a case to the adult criminal justice system.
• Conclusion 2 is valid. The passage explicitly states that the JJ Act, amended in 2021, categorises an offence that has no minimum sentence but has a maximum sentence of seven years or more as a serious offence. This confirms that the classification of a 'serious offence' is based on the maximum sentence rather than the minimum.
• Since only the second conclusion logically follows from the text provided, the correct code is 2 only.

Therefore, the correct answer is 2 only.

 

Additional Information

• Heinous Offences: Defined by the JJ Act 2015 as crimes carrying a minimum punishment of seven years. Examples like culpable homicide or negligent death often fall outside this because they may lack a prescribed minimum sentence.
• Serious Offences: A category created by the 2021 amendment to include crimes with a maximum sentence of 7+ years but no minimum sentence, addressing a legal gap highlighted by various court cases.
• Age Threshold: The provision for trying juveniles as adults applies specifically to adolescents above 16 years of age involved in specific categories of crime.

The correct answer is Option 3.

 

Key Points

• According to the JJ Act, 2015, a heinous offence is specifically defined as one that carries a minimum punishment of seven years.
• The passage explicitly states that offences lacking a prescribed minimum punishment (like culpable homicide or death by negligence) cannot be classified as heinous offences under the 2015 Act. Therefore, Statement 1 is correct.
• The JJ Act, 2021 amendment introduced a new classification called a serious offence.
• This category includes crimes that have no minimum sentence but carry a maximum sentence of seven years or more. This aligns perfectly with the description in the passage. Therefore, Statement 2 is correct.
• Since both statements accurately reflect the provisions mentioned in the passage, the correct code is Both 1 and 2.

Therefore, the correct answer is Option 3.

 

Additional Information

• JJ Act, 2015: Focuses on the minimum sentence (7 years) to classify an offence as heinous, which allows for trying minors aged 16-18 as adults.
• JJ Act, 2021: Addressed a gap in the law by categorising offences with high maximum sentences but no minimum sentences as serious offences.
• Activist Perspective: The passage notes that activists believe serious offences should not merit the transfer of a juvenile's case to the adult criminal justice system.

Shortcut Trick

Time taken by Drone 2 to travel from Camp to Spot = 5:05 PM − 5:02 PM = 3 minutes.

Since speed is constant, Drone 1 also took 3 minutes; thus, Sound was received at 5:02 PM − 3 min = 4:59 PM.

Distance = Speed × Time = 30 m/s × 180 s = 5400 m.

Time for Sound to reach camp = 5400 m ÷ 300 m/s = 18 seconds.

Explosion Time = Sound Arrival Time − Sound Travel Time = 4:59:00 PM − 18 s = 4:58:42 PM.

∴ The correct answer is 4:58:42 PM.

Alternate Method

Given:

Speed of Drone (v_d) = 30 m/s

Speed of Sound (v_s) = 300 m/s

Arrival of Picture 1 = 5:02 PM

Arrival of Picture 2 = 5:05 PM

Formula Used:

Distance = Speed × Time | Time = Distance ÷ Speed

⇒ Let the Explosion occur at time T₀ and distance D from the camp.

⇒ Time when the Sensor receives sound (T_s) = T₀ + D/300

⇒ Drone 1 starts at T_s and reaches the spot at 5:02 PM.

⇒ T_s + D/30 = 5:02 PM — (1)

⇒ Drone 2 starts at 5:02 PM and reaches the spot at 5:05 PM.

⇒ 5:02 PM + D/30 = 5:05 PM

⇒ D/30 = 3 minutes = 180 seconds

⇒ D = 180 × 30 = 5400 m

⇒ From equation (1): T_s = 5:02 PM − 3 min = 4:59:00 PM

⇒ Time of explosion T₀ = T_s − D/300

⇒ T₀ = 4:59:00 PM − (5400 / 300) s

⇒ T₀ = 4:59:00 PM − 18 seconds = 4:58:42 PM

∴ The correct answer is 4:58:42 PM.

Additional Information

Speed-Distance Relationship

For a fixed distance, the time taken is inversely proportional to the speed (t ∝ 1/v). Here, the ratio of speeds of sound and drone is 10:1, so the ratio of times taken is 1:10.

Sound Propagation

Sound travels at a constant speed in a uniform medium. The speed of sound in air is approximately 330-344 m/s at standard temperature, though 300 m/s is used here for simplification.

Time Conversion

Always convert minutes to seconds (1 min = 60 s) when dealing with speeds in m/s to maintain unit consistency in calculations.

Shortcut Trick

The unit digit of 6 raised to any positive integer power is always 6.

The cyclicity of 7 is 4 (7, 9, 3, 1).

Divide the power 307 by 4: 307 ÷ 4 → Remainder = 3.

Unit digit of 7³⁰⁷ is the same as 7³, which is 3 (since 343 ends in 3).

Required unit digit = (6 × 3) = 18 → 8.

∴ The correct answer is 8.

Alternate Method

Given Expression: 6¹²⁹ × 7³⁰⁷

Step 1: Finding unit digit of 6¹²⁹

⇒ Powers of 6: 6¹=6, 6²=36, 6³=216...

⇒ Unit digit of 6¹²⁹ = 6.

Step 2: Finding unit digit of 7³⁰⁷

⇒ Cyclicity of 7: 7¹=7, 7²=49, 7³=343, 7⁴=2401.

⇒ Divide the exponent 307 by 4.

⇒ 307 = (4 × 76) + 3.

⇒ Unit digit of 7³⁰⁷ = Unit digit of 7³ = 3.

Step 3: Calculating final unit digit

⇒ Unit digit of (6¹²⁹ × 7³⁰⁷) = (Unit digit of 6¹²⁹) × (Unit digit of 7³⁰⁷)

⇒ Unit digit = 6 × 3 = 18.

⇒ Unit digit = 8.

∴ The correct answer is 8.

Additional Information

Cyclicity of Digits

Digits 0, 1, 5, and 6 have a cyclicity of 1. Their unit digit remains unchanged regardless of the power.

Cyclicity of 4 and 9

These digits have a cyclicity of 2. For 4: 4^odd ends in 4, 4^even ends in 6. For 9: 9^odd ends in 9, 9^even ends in 1.

Cyclicity of 2, 3, 7, and 8

These digits have a cyclicity of 4. To find the unit digit, divide the power by 4 and use the remainder as the new power. If remainder is 0, use 4 as the power.

Shortcut Trick

Let initial Salary = 100, then Saving = 10 and Expenditure = 90.

New Salary = 112 (12% increase) and New Expenditure = 90 + 9 = 99 (10% increase).

New Saving = 112 − 99 = 13.

Percentage change in Saving = [(13 − 10) ÷ 10] × 100 = 30%.

∴ The correct answer is 30% increase.

Alternate Method

Given: Initial savings = 10%, Salary increase = 12%, Expenditure increase = 10%.

Formula Used: Salary = Expenditure + Savings; % Change = [(New − Old) ÷ Old] × 100.

⇒ Let Initial Salary = 100

⇒ Initial Savings = 10% of 100 = 10

⇒ Initial Expenditure = 100 − 10 = 90

⇒ New Salary = 100 + 12% of 100 = 112

⇒ New Expenditure = 90 + 10% of 90 = 90 + 9 = 99

⇒ New Savings = 112 − 99 = 13

⇒ Percentage Change in Savings = [(13 − 10) ÷ 10] × 100

⇒ Percentage Change = (3 ÷ 10) × 100 = 30%

∴ The correct answer is 30% increase.

Additional Information

Income-Expenditure Relationship

The basic identity used in these problems is Income = Expenditure + Savings. Any percentage change in two components allows for the calculation of the third.

Percentage Change Formula

Percentage change is calculated as [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive result indicates an increase, while a negative result indicates a decrease.

Base Value Selection

In percentage problems involving comparisons, choosing 100 as the Base Value for the original quantity (like Salary) simplifies calculation of subsequent changes.

Given:

1. P has one son and one daughter.
2. S is the mother of T.
3. S is R's spouse.
4. Q and R are children of P.

Since R is the spouse of S and S is female, R must be male.

Therefore:

R = son of P

Q = daughter of P

So, Q is the sister of R.

And R is the husband of S.

Therefore, Q is the sister of the husband of S.

Hence, the correct answer is "Option 4".

Shortcut Trick

List the powers of 2 starting from the first three-digit number:

2⁶ = 64 (Two-digit)

2⁷ = 128 (First three-digit)

2⁸ = 256 (Three-digit)

2⁹ = 512 (Three-digit)

2¹⁰ = 1024 (First four-digit)

∴ The correct answer is 3.

Alternate Method

Given: Three-digit numbers range from 100 to 999.

Expression: The numbers must be of the form 2ⁿ where n is an integer.

⇒ For a number to be a three-digit number, 100 ≤ 2ⁿ < 1000.

⇒ If n = 6, 2⁶ = 64 (Not in range).

⇒ If n = 7, 2⁷ = 128 (In range).

⇒ If n = 8, 2⁸ = 256 (In range).

⇒ If n = 9, 2⁹ = 512 (In range).

⇒ If n = 10, 2¹⁰ = 1024 (Not in range).

⇒ The numbers are 128, 256, and 512.

∴ The correct answer is 3.

Additional Information

Powers of 2 and Digit Boundaries

2¹⁰ is 1024, which is the smallest power of 2 that exceeds the three-digit limit (999). It is often used as a benchmark (1 KB = 1024 bytes).

Logarithmic Relation

To find the number of digits in 2ⁿ, use the formula: Number of digits = floor(n × log₁₀2) + 1. For 3 digits, the result must be 3.

Range of n-digit Numbers

An n-digit number N satisfies 10ⁿ⁻¹ ≤ N < 10ⁿ. For n = 3, the range is 100 to 999.

The logic followed here is based on the evaluation of the provided statements in the context of environmental policy and economic intervention as described in the passage.

For Statement 1:

The passage discusses the transition from dirty fuels (like coal) to cleaner alternatives (like natural gas) as a strategy for air quality improvement. This implies that the energy source used is the fundamental factor in the battle for cleaner air. Thus, this statement is a valid inference.

<br>

For Statement 2:

The passage focuses on the economic impact, industrial competitiveness, and pricing mechanisms. It does not mention the public will or societal conviction regarding the bans. As there is no supporting evidence in the text for these social aspects, this statement cannot be inferred.

<br>

For Statement 3:

The text highlights that there is judicial approval for a policy that intervenes fiscally (by managing price differences) to ensure that cleaner fuel remains cheaper and more accessible than dirty fuel. This facilitate benevolent pricing for eco-friendly energy. Thus, this statement is a valid inference.

<br>

Based on the analysis, only Statement 1 and Statement 3 are correct inferences from the text.

Therefore, the correct answer is 1 and 3.

The correct answer is 2 only.

Key Points

• Statement 1 refers to the cessation of operations at power plants within the capital. While facilities like the Badarpur Thermal Power Station and Rajghat Power House were indeed shut down to mitigate air pollution, these closures were phased based on environmental standards and age rather than a summary, instantaneous shutdown of all units. Hence, statement 1 is incorrect.
• Statement 2 pertains to the logistical requirements of the Supreme Court mandate in the M.C. Mehta case, which ordered the conversion of public transport to Compressed Natural Gas (CNG). To ensure the viability of this ban on diesel, the government had to guarantee a robust clean fuel supply infrastructure to maintain urban mobility. Hence, statement 2 is correct.
• Statement 3 discusses the prohibition of solid fuels. While the Commission for Air Quality Management (CAQM) recently moved to restrict coal in industrial applications, historically, the legal focus was on banning specific high-sulfur 'dirty' fuels like Pet Coke and Furnace Oil, rather than an absolute blanket ban on all coal across every sector in the National Capital Region (NCR) during that period. Hence, statement 3 is incorrect.

 Additional Information

• Clean Fuel Policy and Economics:
• The transition to Natural Gas is sensitive to pricing; if Natural Gas remains significantly more expensive than coal or diesel due to taxation, industries lose their competitive edge.
• CNG is primarily composed of Methane (CH₄) and is favored in the NCR because it emits significantly lower levels of Particulate Matter (PM 2.5) and Nitrogen Oxides (NOx) compared to traditional fuels.
• The Supreme Court has consistently used Article 21 of the Indian Constitution to intervene in environmental policy, advocating for fiscal measures that make cleaner energy sources more accessible to the public.

Solution

Given Statements:

Every red is blue.

Every blue is green.

Every green is yellow.

The least possible Venn diagram for the given statement is:

Conclusions:

P. Every blue is yellow - True (Since all blue are green and all green are yellow, the entire blue circle lies within the yellow circle.)

Q. Every red is green - True (Since all red are blue and all blue are green, the entire red circle lies within the green circle.)

R. Every red is yellow - True (Since red is inside blue, blue is inside green, and green is inside yellow, it follows that all red are yellow.)

So, P, Q and R follow.

∴ The correct answer is P, Q and R.

We need to count how many times digit 5 appears in all two-digit positive integers (10 to 99).

Case 1: 5 in the tens place

Numbers are:

50, 51, 52, 53, 54, 55, 56, 57, 58, 59

Here, 5 appears 10 times.

Case 2: 5 in the units place

Numbers are:

15, 25, 35, 45, 55, 65, 75, 85, 95

Here, 5 appears 9 times.

Note: In 55, the digit 5 appears twice, and both occurrences are counted separately.

Total occurrences of digit 5:

10 + 9 = 19

Hence, the correct answer is "Option 2".

Shortcut Trick

Ignore the middle segment (2 km at 10 kmph) as it is identical for both X and Y.

Total time for X (remaining parts) = (1/5 + 3/4) × 60 = 12 + 45 = 57 minutes.

Total time for Y (remaining parts) = (1/4 + 3/5) × 60 = 15 + 36 = 51 minutes.

Time difference = 57 − 51 = 6 minutes.

∴ The correct answer is 6 minutes.

Alternate Method

Given: Total journey = 6 km. X and Y travel in three segments of 1 km, 2 km, and 3 km.

Formula: Time = Distance ÷ Speed | 1 hour = 60 minutes.

Total Time taken by X (T_X):

⇒ T_X = (1/5 + 2/10 + 3/4) hours

⇒ T_X = (0.2 + 0.2 + 0.75) hours = 1.15 hours

⇒ T_X in minutes = 1.15 × 60 = 69 minutes

Total Time taken by Y (T_Y):

⇒ T_Y = (1/4 + 2/10 + 3/5) hours

⇒ T_Y = (0.25 + 0.2 + 0.6) hours = 1.05 hours

⇒ T_Y in minutes = 1.05 × 60 = 63 minutes

Difference in arrival time:

⇒ 69 minutes − 63 minutes = 6 minutes

∴ The correct answer is 6 minutes.

Additional Information

Average Speed Formula

Average Speed = Total Distance ÷ Total Time. For multiple segments, it is ∑d_i ÷ ∑(d_i/s_i).

Unit Conversion

To convert speed from kmph to m/s, multiply by 5/18. To convert time from hours to minutes, multiply by 60.

Time, Speed, and Distance Relationship

When distance is constant, time is inversely proportional to speed (T ∝ 1/S). When time is constant, distance is directly proportional to speed (D ∝ S).

Given:

There are 7 cubes. 6 cubes have equal weight and 1 cube is lighter.

We need to find the lightest cube using a balance.

First Attempt:

Divide the 7 cubes into:

1. 3 cubes on left side
2. 3 cubes on right side
3. 1 cube aside

Case 1: If both sides balance, then the remaining cube is the lightest.

Case 2: If one side is lighter, then the lightest cube is among those 3 cubes.

Second Attempt:

From the suspected 3 cubes, compare:

1 cube vs 1 cube

If one side is lighter, that cube is the answer. If both balance, then the third cube is the lightest.

Therefore, the minimum number of attempts required is 2.

Hence, the correct answer is "Option 1".

The correct answer is Option 3.

 

Key Points

• The passage discusses the evolution of customer-service technology, comparing current AI advancements with previous waves like email and voice menus.
• It emphasizes that while AI might indeed change the job landscape, its effects could be salutary (beneficial) for the workforce.
• The core argument is that human agents will be freed up from repetitive or stressful duties (like handling irate customers) to focus on more creative and rewarding tasks, such as using feedback to improve products.
• Therefore, the statement that reflects the redirection of human intervention toward fulfilling tasks most accurately captures the point of view of the passage.

Therefore, the correct answer is Option 3.

 

Additional Information

• Option 1 is incorrect because the passage suggests that human agents will have different, more rewarding work to do, rather than no work at all.
• Option 2 focuses on the degree of human redundancy, whereas the passage emphasizes the qualitative shift in the nature of human work.
• Option 4 is inaccurate because the passage explicitly mentions that previous waves of technology also stoked concerns about job losses; AI is not the first to raise such alarms.

The correct answer is Option 4.

 

Key Points

• Statement 1 is correct: The passage explicitly states that previous waves of customer-service technology, such as email and voice menus, consistently 'stoked concerns of job losses.'
• Statement 2 is correct: The author notes that instead of jobs simply disappearing, AI might allow human agents to be 'freed up' to focus on creative and rewarding tasks, such as improving products through feedback. This implies the discovery of alternative channels for engagement.
• Statement 3 is incorrect: The passage suggests that AI could actually lead to beneficial (salutary) outcomes by reducing the time employees spend dealing with irate customers, thereby making the work environment less stressful, not more.
• Therefore, both conclusions 1 and 2 are supported by the text.

Therefore, the correct answer is 1 and 2.

 

Additional Information

• The term salutary in the passage means producing good effects or being beneficial, which directly contradicts the idea that technology inevitably leads to stressful outcomes.
• The passage emphasizes a qualitative shift in labor, where automation handles repetitive tasks while humans pivot toward high-value, creative roles.
• The phrase fail to materialise regarding job losses from email and voice menus supports the idea that historical fears have often been misplaced.

The correct answer is Option 3.

 

Key Points

• Statement 3 is the only valid conclusion as the passage explicitly states that Rai Sahib held to old values and believed that a landlord who didn't look after his tenants was less than human. This directly connects traditional values with concern for tenants.
• Statement 1 is incorrect because the passage contrasts Rai Sahib with selfish Zamindars who would ignore the welfare of cattle to make money, implying that not all landlords possess such goodness.
• Statement 2 is not a generalized conclusion provided by the text; while the cattle from nearby villages used the ground, the focus of the passage is on Rai Sahib's personal decision to preserve it rather than the universal purpose of all such grounds.
• Statement 4 is incorrect because the passage mentions there was a cool freshness in the morning, but Hori anticipated scorching 'loo' winds for the rest of the day, indicating it would get hotter, not cooler.
• Therefore, the only correct conclusion based strictly on the text is 3.

Therefore, the correct answer is 3 only.

 

Correct Conclusion: "Landlords who believe in tradition (old values) tend to be more concerned about their tenants."

 

Additional Information

• The passage highlights the character of Rai Sahib as an exception to the typical selfish Zamindars of that era.
• The term 'loo' refers to the strong, dusty, gusty, hot, and dry summer afternoon wind from the west which blows over the western Indo-Gangetic Plain region of North India and Pakistan.
• The phrase 'less than human' emphasizes the moral weight Rai Sahib placed on the duty of a landlord toward his dependents.

The correct answer is Option 4.

 

Key Points

• Statement 1 is incorrect because the passage only refers to a specific bit of land reserved for grazing. It does not suggest that all of Rai Sahib’s landholdings were excluded from farming; in fact, as a Zamindar with tenants, his other lands were naturally used for cultivation.
• Statement 2 is incorrect because the text highlights that Rai Sahib held old values and cared for his tenants, explicitly distinguishing him from selfish or greedy landlords.
• Statement 3 is incorrect because the passage clearly states that farmers had offered a good price for the land. Therefore, it is easily ascertainable that he could have earned money by leasing it.
• Statement 4 is correct based on the passage, as it mentions Rai Sahib felt any landlord who did not look after his tenants was less than human, showing he valued their livelihood.
• Since statements 1, 2, and 3 are false according to the text, they are the not correct statements.

Therefore, the correct answer is 1, 2 and 3.

 

Additional Information

• The passage emphasizes the moral character of Rai Sahib, who prioritizes the common good (village grazing) over personal profit.
• The mention of the loo wind and the cool freshness of the morning provides the environmental context in which Hori is reflecting on the landlord's kindness.
• The term Zamindar refers to a landowner, and the passage explores the ethical responsibilities expected of that position according to traditional values.

Given:

Starts facing East

Travels 4 km straight

Turns right and travels 3 km

Turns left and travels 2 km

Turns left and travels 3 km

 

Step-by-step movement:

Minimum distance between initial and final point = 6 km

Final direction faced = North

Hence, the correct answer is "Option 4".

Logic followed here is.

Logic: Each number in the sequence, other than the first two, is the sum of the two immediately preceding numbers.

Given:

First number = 4

Second number = 7

<br>

Calculating the terms of the sequence:

3rd number = 1st number + 2nd number = 4 + 7 = 11

4th number = 2nd number + 3rd number = 7 + 11 = 18

5th number = 3rd number + 4th number = 11 + 18 = 29

6th number = 4th number + 5th number = 18 + 29 = 47

<br>

The sequence is summarized below:

<br>

Therefore, the sixth number in the sequence is 47.

Shortcut Trick

Company A ratio (M : F) = 13 : 10 (Total = 23 units)

Company B ratio (M : F) = 7 : 5

To make female workers equal, multiply Company B ratio by 2 → 14 : 10 (Total = 24 units)

Required Ratio (Total A : Total B) = 23 : 24

∴ The correct answer is 23 : 24.

Alternate Method

Given:

Ratio of Male to Female in Company A = 13 : 10

Ratio of Male to Female in Company B = 7 : 5

Formula Used:

Total workers = Male workers + Female workers

⇒ Let the number of female workers in both companies be 10x.

⇒ For Company A: Male = (13/10) × 10x = 13x

⇒ Total workers in Company A = 13x + 10x = 23x

⇒ For Company B: Male = (7/5) × 10x = 14x

⇒ Total workers in Company B = 14x + 10x = 24x

⇒ Ratio of Total A to Total B = 23x : 24x = 23 : 24

∴ The correct answer is 23 : 24.

Additional Information

Equating Ratios

When one component is common or equal in two ratios, multiply the entire ratio by a constant to make that component's numerical value the same in both sets.

Combined Ratio

If A : B = x : y and B : C = p : q, the combined ratio A : B : C is found by equating the value of B in both ratios.

Proportion Rule

If four quantities a, b, c, d are in proportion, then a/b = c/d, which implies a × d = b × c.

Shortcut Trick

Let the two distinct numbers be a and b.

We know: HCF × LCM = a × b.

Given: HCF × LCM = a³ ⇒ a × b = a³ ⇒ b = a².

Statement I: Difference = a² − a = a(a − 1). Product of two consecutive integers is always even.

Statement II: Since b = a², one number is clearly a perfect square.

∴ The correct answer is Both I and II.

Alternate Method

Given:

Two distinct numbers: n₁ and n₂.

Product of HCF and LCM = (n₁)³.

Formula Used:

HCF × LCM = Product of the numbers

Step-by-step Calculation:

⇒ Let the numbers be x and y.

⇒ x × y = x³

⇒ y = x³ ÷ x

⇒ y = x²

⇒ Since the numbers are distinct, x ≠ 1 (as 1² = 1).

Evaluating Statement I:

⇒ Difference = y − x = x² − x

⇒ Difference = x(x − 1)

⇒ For any integer x, one of x or (x − 1) must be even.

⇒ Even × Odd = Even. Thus, Statement I is correct.

Evaluating Statement II:

⇒ Since y = x², y is a perfect square.

⇒ Thus, Statement II is correct.

∴ The correct answer is Both I and II.

Additional Information

Fundamental Property of HCF and LCM

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This property holds only for two numbers.

Consecutive Integers Property

The product of n consecutive integers is always divisible by n! (n factorial). For 2 consecutive integers, the product is always divisible by 2! = 2 (i.e., it is even).

Perfect Squares

A number is a perfect square if it can be expressed as k² where k is an integer. Examples include 4, 9, 16, 25, etc.

Shortcut Trick

Divisibility rule of 11: Sum of digits at odd positions − Sum of digits at even positions = 0, 11, 22...

For the number 4x5y790: (4 + 5 + 7 + 0) − (x + y + 9) = 16 − (x + y + 9) = 7 − (x + y).

For divisibility by 11, 7 − (x + y) must be 0 or −11.

⇒ x + y = 7 or x + y = 18.

In both cases, (7 ÷ 11) or (18 ÷ 11) leaves a remainder of 7.

∴ The correct answer is 7.

Alternate Method

Given: The number is 4x5y790 and it is divisible by 11.

⇒ Sum of digits at odd positions = 4 + 5 + 7 + 0 = 16

⇒ Sum of digits at even positions = x + y + 9

⇒ Difference = |16 − (x + y + 9)| = |7 − (x + y)|

⇒ Since the number is divisible by 11, the difference must be a multiple of 11.

⇒ Case 1: 7 − (x + y) = 0 ⇒ x + y = 7

⇒ Case 2: 7 − (x + y) = −11 ⇒ x + y = 18 (since max sum of two digits is 9 + 9 = 18)

⇒ When x + y = 7 is divided by 11: 7 ÷ 11 → Remainder = 7

⇒ When x + y = 18 is divided by 11: 18 ÷ 11 → Remainder = 7

∴ The correct answer is 7.

Additional Information

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is 0 or a multiple of 11 (e.g., 11, 22, 33...).

Digit Sum Constraints

For any two single digits x and y, the minimum sum (x + y) is 0 and the maximum sum is 9 + 9 = 18.

Remainder Theorem Basics

If a number N is smaller than the divisor D, the remainder when N is divided by D is N itself (e.g., 7 ÷ 11 leaves remainder 7).

The correct answer is Option 3.

 

Key Points

• Statement 1 is correct: The passage describes Hugh Grant's decision to nap as 'quietly radical' because it represents an act of mild surrender to the body's needs in a public setting, effectively acting against societal performance metrics and expectations of constant engagement.
• Statement 2 is correct: The author explicitly clarifies that this act of sleeping was a surrender 'not to laziness, but to limits.' It highlights that acknowledging the physical limitations caused by factors like age (noting he is a 64-year-old man) should not be mistaken for a lack of industry or laziness.
• Statement 3 is correct: The term 'leading man' typically denotes the protagonist or lead actor in a film; however, the author uses it metaphorically to suggest that by choosing to rest, Grant has become a figurehead or 'leading man' for the sleep-deprived, portraying a 'resting hero' rather than the typical 'action hero'.
• Since all three statements accurately reflect the themes and specific wording of the passage, the correct code is 1, 2 and 3.

Therefore, the correct answer is 1, 2 and 3.

 

Additional Information

• Context: The event took place at Wimbledon during the men's singles quarterfinals match between Flavio Cobolli and Novak Djokovic.
• Royal Box: This is a prestigious seating section at the Centre Court of Wimbledon, making the actor's nap even more noticeable and socially significant.
• Forty winks: A common English idiom meaning to take a short sleep or nap during the day.
• Unexpected ace: The author uses this tennis metaphor to describe the nap as a surprising yet successful move by Hugh Grant, subverting the high-intensity environment of the sports arena.

The correct answer is Option 2.

 

Key Points

• Statement 1 is incorrect: According to the passage, Hugh Grant was watching the men's singles quarterfinals, not the semifinal match.
• Statement 2 is incorrect: While the phrase 'unexpected ace' is a tennis term used to describe the nap, the passage does not view the act with opprobrium (harsh criticism). Instead, the author uses phrases like 'praise be to Grant' and 'elegant case for surrender,' showing a positive and empathetic perspective.
• Statement 3 is correct: The passage explicitly states that Grant made a case for surrender to the body's quiet wisdom over society's relentless performance metrics. This indicates he prioritized his physical needs over societal expectations.
• Since only the third statement aligns with the facts and tone of the passage, the correct code is 3 only.

Therefore, the correct answer is 3 only.

 

Additional Information

• Forty winks: This is an idiomatic expression meaning a short sleep or a nap during the day.
• Royal Box: An exclusive seating area at Wimbledon, typically reserved for the Royal Family and their guests.
• Radical Action: The author describes the nap as 'quietly radical' because it goes against the societal norm of constant engagement and performance, especially in a high-profile public setting.

The correct answer is Option 1.

 

Key Points

• According to the passage, in the fine-tuning phase, participants (both leaders and followers) attempt to extract remaining benefits from a technology that is increasingly exhausted. This results in both categories having similar per capita incomes. Therefore, statement 3 is correct.
• The passage outlines a chronological sequence of technological developments: scavenging (3 million years ago), followed by hunting (1 million years ago), followed by farming (12 thousand years ago), and finally industrial technology (over 200 years ago).
• Statement 4 accurately reflects this timeline: industrial technology appeared after scavenging; scavenging occurred before hunting; and hunting was followed by farming. Thus, statement 4 is correct.
• The convergence theory uses these phases to explain how the labour-productivity gap is narrowed over time.

Therefore, the correct answer is 3 and 4 only.

 

Correct Conclusion: The fine-tuning phase involves extracting leftover benefits from exhausted tech, and the historical progression moved from scavenging to hunting, farming, and then industrial technology.

 

Additional Information

• Statement 1: This is incorrect because the passage explicitly states that during the breakaway phase, the per capita income of technology followers is unchanged, rather than describing the progress as merely slow.
• Statement 2: This is incorrect because the passage says the catch-up phase involves followers adopting new technology to close the gap; it does not mention that technology leaders stagnate during this period.

The correct answer is Option 2.

 

Key Points

• The convergence theory is defined by the process through which countries work to close the labour-productivity gap existing between themselves and the technology leader. Statement 2 is considered correct as it identifies this gap as the central focus of the theory.
• The passage describes technology leaders as the entities that initiate a new technology (like industrial technology or farming), meaning they logically arrive at and define different economic eras earlier than the followers. Thus, statement 3 is correct.
• The model segments the process into three distinct phases: the breakaway phase, the catch-up phase, and the fine-tuning phase.
• Based on the passage analysis, only statements 2 and 3 accurately reflect the provided information.

Therefore, the correct answer is 2 and 3.

 

Additional Information

• Statement 1 is incorrect: The passage explicitly states that the convergence model divides economic eras (not nations) into three phases: breakaway, catch-up, and fine-tuning.
• Statement 4 is incorrect: The timeline mentioned begins 3 million years ago (scavenging). Statement 4 appears to be a random summation of the numbers mentioned in the text (3,000,000 + 1,000,000 + 12,000 + 200), which does not represent the actual time period covered by the theory.
• During the breakaway phase, the per capita income of leaders rises while that of followers remains unchanged.
• During the fine-tuning phase, technology becomes exhausted, and both leaders and followers reach similar income levels.

Shortcut Trick

Equal amounts of copper and zinc in Alloy R means copper is 50%.

Apply Alligation on Copper concentration:

Alloy P (20%) and Alloy Q (60%) mix to give Alloy R (50%).

Ratio P : Q = (60 − 50) : (50 − 20) = 10 : 30 = 1 : 3.

∴ The correct answer is 1 : 3.

Alternate Method

Given:

Alloy P: Copper = 20%, Zinc = 80%

Alloy Q: Copper = 60%, Zinc = 40%

Alloy R: Copper = 50%, Zinc = 50% (since they are in equal amounts)

Calculations:

⇒ Let the quantity of Alloy P be x and Alloy Q be y.

⇒ Quantity of Copper in R = (20% of x) + (60% of y)

⇒ Total quantity of R = x + y

⇒ Concentration of Copper in R = 50% = 1/2

⇒ [0.2x + 0.6y] / [x + y] = 0.5

⇒ 0.2x + 0.6y = 0.5x + 0.5y

⇒ 0.6y − 0.5y = 0.5x − 0.2x

⇒ 0.1y = 0.3x

⇒ x/y = 0.1 / 0.3 = 1/3

∴ The correct answer is 1 : 3.

Additional Information

Alligation Rule

It is used to find the ratio in which two ingredients at given prices/concentrations must be mixed to produce a mixture at a target price/concentration. Ratio = (Higher Value − Mean Value) : (Mean Value − Lower Value).

Concentration in Mixtures

When components are mixed, the weight of a specific element (like Copper) in the final mixture is the sum of its weights in the individual alloys.

Percentage to Ratio Conversion

Equal amounts of two components in a mixture always represent a 1 : 1 ratio or 50% concentration for each component.

Shortcut Trick The difference between a 2-digit number and its reverse is always 9 × (Difference of its digits). Let the digits be x and y. Then, A − B = 9(x − y). Given A − B is a multiple of 27 ⇒ 9(x − y) = 27, 54, 81... ⇒ (x − y) must be 3, 6, or 9. For (x − y) = 3: Pairs are (4,1), (5,2), (6,3), (7,4), (8,5), (9,6) → 6 numbers. For (x − y) = 6: Pairs are (7,1), (8,2), (9,3) → 3 numbers. (x − y = 9 ⇒ 9,0 is excluded as B must be 2-digit). ∴ The correct answer is 9.

Alternate Method Given: A is a 2-digit number (10x + y), B is its reverse (10y + x). A > B and B is also a 2-digit number (y ≠ 0). Formula Used: A − B = (10x + y) − (10y + x) = 9(x − y).

⇒ x and y are digits where 1 ≤ x, y ≤ 9 and x ≠ y. ⇒ Since A − B is a multiple of 27, 9(x − y) = 27 × k. ⇒ (x − y) = (27 × k) ÷ 9 = 3k. ⇒ If k = 1: x − y = 3. Possible (x,y) pairs: (4,1), (5,2), (6,3), (7,4), (8,5), (9,6). Total = 6. ⇒ If k = 2: x − y = 6. Possible (x,y) pairs: (7,1), (8,2), (9,3). Total = 3. ⇒ If k = 3: x − y = 9. Possible (x,y) pairs: (9,0). But y ≠ 0 because B is a 2-digit number. ⇒ Total possible values of A = 6 + 3 = 9. ∴ The correct answer is 9.

Additional Information Difference of Reversed Numbers The difference between any 2-digit number and its reverse is always divisible by 9. The quotient is the difference of the digits. Sum of Reversed Numbers The sum of any 2-digit number and its reverse is always divisible by 11. (10x + y) + (10y + x) = 11(x + y). 2-Digit Number Constraints For a number 10x + y to be a 2-digit number, the leading digit x must be from {1, 2, ..., 9}. If its reverse is also a 2-digit number, then y must also be from {1, 2, ..., 9}.

The logic followed here is**:** The letters of the word are shifted forward and backward alternatively by 1 position (+1, −1, +1, −1).

ZERO is coded as ADSN →

The pattern observed is a sequential shift of +1 and −1 applied to each letter.

 

Similarly, let's find the code for STOP:

Therefore, the code for STOP is TSPO.

Hence, the correct answer is "Option 2".

Shortcut Trick

Area width is common (3'), so solve for length 100' using tile lengths 3', 7', and 11'.

To maximize tiles (x), use the smallest tile: (31 × 3') + (1 × 7') = 100' ⇒ x = 31 + 1 = 32 tiles.

To minimize tiles (y), use the largest tile: (8 × 11') + (4 × 3') = 100' ⇒ y = 8 + 4 = 12 tiles.

Difference x − y = 32 − 12 = 20.

∴ The correct answer is 20.

Alternate Method

Given: Dimensions of rectangular area = 3' × 100'. Tile sizes = 3' × 3', 3' × 7', 3' × 11'.

Since one dimension (3') is common to the area and all tiles, we simplify the problem to a one-dimensional length of 100' to be covered by segments of 3', 7', and 11'.

Let a, b, and c be the number of tiles of length 3', 7', and 11' respectively.

Equation: 3a + 7b + 11c = 100

Step 1: Finding the maximum number of tiles (x)

To maximize the count, we use the smallest tile (3') as much as possible.

⇒ If we use only 3' tiles: 100 ÷ 3 = 33.33 (Not possible exactly).

⇒ If a = 31: 3 × 31 = 93. Remainder = 100 − 93 = 7.

⇒ We can use one 7' tile (b = 1).

⇒ Total tiles x = 31 (of 3') + 1 (of 7') = 32 tiles.

Step 2: Finding the minimum number of tiles (y)

To minimize the count, we use the largest tile (11') as much as possible.

⇒ If we use only 11' tiles: 100 ÷ 11 = 9.09 (Not possible exactly).

⇒ If c = 9: 11 × 9 = 99. Remainder = 1. (No tile fits).

⇒ If c = 8: 11 × 8 = 88. Remainder = 100 − 88 = 12.

⇒ We can cover 12' using four 3' tiles (a = 4).

⇒ Total tiles y = 8 (of 11') + 4 (of 3') = 12 tiles.

Step 3: Calculating x − y

⇒ x − y = 32 − 12 = 20.

∴ The correct answer is 20.

Additional Information

Linear Diophantine Equations

Equations like 3a + 7b + 11c = 100 require integer solutions. The minimum sum (a+b+c) occurs when the variable with the largest coefficient is maximized.

Optimization Strategy

To maximize the total number of items, prioritize the smallest unit; to minimize, prioritize the largest unit while ensuring the total sum is met exactly.

Constraints in Tiling

When a tile dimension matches the area dimension exactly, the problem effectively becomes 1D, simplifying 2D area coverage to a length-partitioning problem.

Shortcut Trick

Since the distance segments (0-200 km and 600-800 km) and their respective final speeds (V/8) are the same in both scenarios, the time difference depends only on the 400 km middle segment.

In Scenario 1, middle segment speed = V/2. In Scenario 2, middle segment speed = V/4.

Time difference ⇒ Distance ÷ (Speed₂) − Distance ÷ (Speed₁) = 4 hours.

400 ÷ (V/4) − 400 ÷ (V/2) = 4 ⇒ 1600/V − 800/V = 4.

800/V = 4 ⇒ V = 200 km/h.

∴ The correct answer is 200 kmph.

Alternate Method

Given:

Total Journey = 800 km. Let original Speed = V km/h.

Minor Accident: Speed becomes 1/2 of existing. Mechanical Defect: Speed becomes 1/4 of existing.

Calculations:

Total time for Scenario 1 (T₁):

⇒ T₁ = 200/V + 400/(V/2) + 200/(V/8)

⇒ T₁ = 200/V + 800/V + 1600/V = 2600/V

Total time for Scenario 2 (T₂):

⇒ T₂ = 200/V + 400/(V/4) + 200/(V/8)

⇒ T₂ = 200/V + 1600/V + 1600/V = 3400/V

According to the question, Scenario 2 takes 4 hours more:

⇒ T₂ − T₁ = 4

⇒ 3400/V − 2600/V = 4

⇒ 800/V = 4

⇒ V = 800/4 = 200 km/h.

∴ The correct answer is 200 kmph.

Additional Information

Relative Speed

When two objects move in the same direction, relative speed = V₁ − V₂. When they move in opposite directions, relative speed = V₁ + V₂.

Average Speed

Average Speed = Total Distance ÷ Total Time. If distances are equal for two speeds a and b, Average Speed = 2ab ÷ (a + b).

Time and Distance Relationship

For a constant distance, speed is inversely proportional to time (V ∝ 1/T). For a constant time, distance is directly proportional to speed (D ∝ V).

Shortcut Trick

Calculate weighted scores: (Score₁ × 0.2) + (Score₂ × 0.3) + (Score₃ × 0.5).

C1 = 5.2 | C2 = 5.1 | C3 = 5.2 | C4 = 4.5 | C5 = 5.0.

Ranking the scores in descending order: 5.2, 5.2, 5.1, 5.0, 4.5.

For exactly 3 selections, the 3rd highest score is the cutoff.

∴ The correct answer is 5.1.

Alternate Method

Given: weightage w₁ = 0.2, w₂ = 0.3, w₃ = 0.5.

Formula Used: Total Weighted Score = Σ(Score_i × weight_i)

⇒ Candidate 1 Score: (5 × 0.2) + (4 × 0.3) + (6 × 0.5) = 1.0 + 1.2 + 3.0 = 5.2

⇒ Candidate 2 Score: (4 × 0.2) + (6 × 0.3) + (5 × 0.5) = 0.8 + 1.8 + 2.5 = 5.1

⇒ Candidate 3 Score: (3 × 0.2) + (2 × 0.3) + (8 × 0.5) = 0.6 + 0.6 + 4.0 = 5.2

⇒ Candidate 4 Score: (9 × 0.2) + (4 × 0.3) + (3 × 0.5) = 1.8 + 1.2 + 1.5 = 4.5

⇒ Candidate 5 Score: (8 × 0.2) + (8 × 0.3) + (2 × 0.5) = 1.6 + 2.4 + 1.0 = 5.0

⇒ Sorted Scores: 5.2 (C1), 5.2 (C3), 5.1 (C2), 5.0 (C5), 4.5 (C4)

⇒ For exactly 3 candidates, the cutoff is the 3rd highest value.

∴ The correct answer is 5.1.

Additional Information

Weighted Average

It is calculated as Σ(x_i × w_i) / Σw_i. When the sum of weights equals 1, the formula simplifies to Σ(x_i × w_i).

Selection Cutoff

The cutoff score is the minimum score required to remain in the selection pool based on the number of vacancies available.

Weightage Impact

Components with higher weightages (like Component 3 with 0.5) have a greater impact on the final score than those with lower weightages.

The logical analysis of the statements and their relationships is as follows:

Statement Analysis:

• S1: Focuses on the existential necessity of protecting the environment due to the impact of climate change.
• S2: Highlights that a scientific consensus has not been reached regarding the extent of human intervention in climate change.
• S3: Describes how environmental activism includes climate alarmism or extremist views, which are then utilized by climate change deniers.

Evaluating the Relationships:

1. S3 is a counterpoint to S1:

This is incorrect. S1 argues for the necessity of environmental protection. S3 describes the nature of certain types of activism and how they are perceived. Describing the flaws or extremes of activism (alarmism) does not logically negate or act as a counterpoint to the fundamental need for environmental protection mentioned in S1.

2. S3 is unconnected to S1 and S2:

This is incorrect. All three statements are closely thematically linked to the global discourse on climate change. S1 sets the stakes, S2 explains the scientific uncertainty, and S3 explains the socio-political reactions to that uncertainty. They are inherently connected.

3. S2 could be the reason for S3:

This is correct. The lack of scientific consensus (S2) on the exact extent of human contribution creates a vacuum of certainty. This uncertainty often leads different groups to adopt extremist points of view, such as climate alarmism (to spur action despite the uncertainty) or denial (using the lack of consensus as an excuse to reject the phenomenon). Thus, S2 provides the context or reason for the behaviors described in S3.

Conclusion:

Based on the logical deduction, only the third relationship is valid.

Therefore, the correct code is 3 only.

Hence, the correct answer is "Option 4".

The logic for matching the relationship categories with their communication types is based on the nature of hierarchy and the setting of the interaction.

Analysis of the Relationships:

A. Between cricket captain and team members:

In a sports team, there is a strong bond and camaraderie among players, making the interaction Informal. However, to execute strategies and maintain discipline on the field, the captain must be Firm in their decision-making. Thus, it matches with 1. Informal and firm.

B. Between judge and lawyers in court:

The courtroom is a highly regulated and official environment. The communication follows strict legal protocols, making it Formal. The judge’s orders and rulings must be Firm to uphold the authority of the court. Thus, it matches with 4. Formal and firm.

C. Between Vice Chancellor and Deputy Registrar:

This is a standard administrative relationship within a university hierarchy. The communication is Formal due to the official designations, but it is often Open-ended to allow for administrative discussions, reporting, and problem-solving. Thus, it matches with 3. Formal and open-ended.

D. Between peers and coworkers:

Since peers are at the same hierarchical level, their interaction is usually Informal and relaxed. Their communication is Open-ended as it involves brainstorming, collaboration, and social dialogue without rigid constraints. Thus, it matches with 2. Informal and open-ended.

 

The Final Matching Table:

 

Therefore, the correct match is a-1, b-4, c-3, d-2.

Hence, the correct answer is "Option 2".

The correct answer is a-3, b-4, c-2, d-1

Key Points

• Memorandum (Memo): A Memorandum is a tool of Internal Communication used within an organization to convey information, policies, or instructions. Its primary purpose is to intimate a directive or a specific action required by the staff or departments.
• Flyer: A Flyer is a form of paper advertisement or notice intended for wide distribution. It is typically handed out in public places or posted on boards to disseminate non-targeted information to a large, general audience rather than a specific individual.
• Bcc (Blind Carbon Copy): In email communication, Bcc is used to send a copy of the message to a recipient whose identity is hidden from the other recipients. The main purpose is to inform confidentially, ensuring that the primary recipients are unaware that the Bcc recipient has also received the information.
• Minutes: Minutes, or Minutes of a Meeting, are the official written record of the proceedings of a meeting. Their essential function is to record decisions, track progress on action items, and provide a legal or formal summary of what was discussed and agreed upon.

Additional Information

• Circulars: These are used to communicate information to a large number of people within an organization simultaneously, often regarding general updates or changes in rules.
• Cc (Carbon Copy): Unlike Bcc, the Cc field in an email is used to keep stakeholders in the loop where all recipients can see who else has received the message.
• Notice: A Notice is a formal means of communication used to announce events, meetings, or instructions to a specific group of people, often displayed on a Notice Board.
• Reports: Formal documents that present information in an organized format for a specific audience and purpose, often involving the analysis of data or findings.

Logic followed here is based on the principles of Organizational Behavior and Conflict Management within a culturally diverse environment. Effective leadership requires proactive strategies that bridge gaps in understanding rather than superficial measures or passive avoidance.

Analysis of the Directives:

1. Direct personnel to practise activities that are the cultural markers of diverse groups:

This directive is generally considered ineffective for conflict resolution. Forcing employees to adopt or practice specific cultural markers (like rituals or religious practices) can be seen as intrusive, insincere, or even a violation of personal boundaries. It does not address the root cause of interpersonal friction.

2. Allow conflicts to resolve naturally over time:

This approach reflects laissez-faire leadership or an avoidance strategy. In a professional office setting, leaving conflicts unaddressed often leads to an escalation of tension, reduced team cohesion, and a decline in productivity. It sets a poor precedent that the leadership is unwilling to maintain a professional standard of conduct.

3. Encourage personnel to seek each other's perspectives:

This is a fundamental technique in active listening and empathy building. In a culturally diverse personnel group, conflicts often arise from miscommunication or differing cultural norms. Encouraging employees to understand the viewpoint and context of others helps in de-escalating tension and finding a mutually acceptable resolution.

The most appropriate directive for resolving conflict is to promote cross-cultural communication and perspective-taking.

Therefore, among the given directives, only 3 constitutes an appropriate action for a leader.

Hence, the correct answer is 3 only.

The correct answer is a-2, b-4, c-1, d-3

Key Points

• Semantic Barriers (A-2): These barriers are concerned with the encoding and decoding of messages. They arise when the same words or symbols carry different meanings for the sender and the receiver. Misunderstanding the meaning of a word or using technical jargon that the receiver does not understand are classic examples of semantic barriers.
• Cognitive Barriers (B-4): These relate to the mental processes of the individual. Information overload occurs when a person is presented with more information than their brain can effectively process and categorize at one time, leading to a breakdown in communication because the receiver cannot absorb all the details.
• Organisational Barriers (C-1): These barriers arise from the structure, rules, and environment within an organization. A lack of feedback is often an organizational issue where the hierarchy or communication flow does not provide a formal or informal channel for the receiver to respond, making the communication one-way and ineffective.
• Affective Barriers (D-3): These are emotional or psychological barriers. Affective refers to feelings and attitudes. Fear of social stigma is a psychological state where an individual avoids communicating certain ideas or feelings due to the emotional dread of being judged or rejected by society.

Additional Information

• Physical Barriers: These include environmental factors such as noise, distance, or technical glitches in the communication medium (e.g., poor telephone connection).
• Cultural Barriers: These occur when people from different backgrounds use different customs, gestures, or social norms, which can lead to misinterpretation of non-verbal cues.
• Psychological Barriers: Closely related to affective barriers, these include premature evaluation, lack of attention, and poor retention, which prevent the receiver from hearing the message objectively.
• Filtering: This is a specific organisational barrier where information is manipulated so that it will be seen more favorably by the receiver, common in upward communication within a hierarchy.

The correct answer is 1 and 2 only

Key Points

• To determine the feasibility of a family-oriented vacation resort, the most critical data points are those that directly influence the demand and economic viability of the project.
• Size of family: This is a primary factor for a family-oriented resort. It helps designers and investors determine the room configuration (e.g., suites, interconnected rooms), the capacity of common areas, and the type of recreational facilities required for different age groups.
• Budget: This is the cornerstone of any feasibility study. Knowing the price sensitivity and the average amount a family is willing to spend on accommodation and services allows the developer to position the resort (e.g., luxury, mid-range, or budget) and calculate the Return on Investment (ROI).
• Number of earners in the family: While this provides a hint at socio-economic status, it is often considered intrusive and indirect. The total budget (Option 2) is a much more reliable and direct metric for feasibility than the internal financial structure of the family.
• Food allergies and dietary restrictions: This is an operational detail. While important for guest satisfaction once the resort is running, it does not significantly impact the initial decision of whether or not setting up the resort is feasible or profitable.

Additional Information

• A Feasibility Study in tourism involves market research, site analysis, and financial forecasting.
• Primary Data Collection via questionnaires on-location helps in understanding the preferences of the target demographic (current visitors to the lake destination).
• Other important factors for such a study would include seasonality, proximity to transport hubs, availability of utilities, and local environmental regulations regarding the mountain ecosystem.
• The Questionnaire Design must focus on quantitative data that can be translated into statistical trends for the final feasibility report.

The correct answer is 1 only

Key Points

• A circular letter is a form of mass communication used to transmit the same information to a large number of people at the same time. This confirms that Statement 1 is correct, as they are addressed to a group of people rather than a single individual.
• The content of a circular letter is standardized and formal. Since the same letter is sent to many people, it does not contain customized or non-standard content specific to one person. Therefore, Statement 2 is incorrect.
• Appraisals, salary increments, and promotions are highly confidential and personal matters. These are always communicated via individualized letters or private memos to maintain privacy, never through a public or group circular. Thus, Statement 3 is incorrect.
• Circular letters are actually more cost-effective than personalized letters. Because one common draft is prepared and printed/emailed in bulk, it significantly reduces the administrative cost and time per recipient. This makes Statement 4 incorrect.

Additional Information

• Internal Circulars are used within an organization to inform employees about policy changes, office holidays, or internal meetings.
• External Circulars are used to reach out to customers, suppliers, or shareholders regarding new product launches, changes in business address, or changes in the partnership structure of a firm.
• The primary advantage of a circular is that it ensures consistency in the information received by everyone in the target audience, preventing miscommunication.
• A well-drafted circular letter should be concise, clear, and follow a professional tone suitable for the intended group.

Shortcut Trick

Ratio of Profit Share = (Investment × Time) for each partner.

Ratio A : B : C = (1/3 × 1/3) : (1/4 × 1/4) : [(1 − 1/3 − 1/4) × 1]

Ratio = 1/9 : 1/16 : 5/12 ⇒ Multiply by LCM(9, 16, 12) = 144.

Ratio = 16 : 9 : 60 | Total parts = 16 + 9 + 60 = 85 units.

85 units = ₹ 17,000 ⇒ 1 unit = ₹ 200.

C's share = 60 × 200 = ₹ 12,000.

∴ The correct answer is ₹ 12,000.

Alternate Method

Given:

Total Profit = ₹ 17,000

Let Total Capital = 12 units (LCM of 3 and 4) and Total Duration = 12 months.

Formula Used:

Profit Share ∝ (Capital × Time)

Calculations:

⇒ A's capital = 1/3 × 12 = 4 units; A's time = 1/3 × 12 = 4 months.

⇒ B's capital = 1/4 × 12 = 3 units; B's time = 1/4 × 12 = 3 months.

⇒ C's capital = 12 − (4 + 3) = 5 units; C's time = 12 months.

⇒ Ratio of Profit (A : B : C) = (4 × 4) : (3 × 3) : (5 × 12)

⇒ Ratio = 16 : 9 : 60

⇒ Sum of ratios = 16 + 9 + 60 = 85

⇒ C's share = (60 / 85) × 17,000

⇒ C's share = 60 × (17,000 / 85)

⇒ C's share = 60 × 200 = ₹ 12,000

∴ The correct answer is ₹ 12,000.

Additional Information

Basic Partnership Rule

Profit is always distributed in the ratio of the product of Capital (C) and Time (T): P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Active vs. Sleeping Partner

An Active partner manages the business and may receive a salary or percentage of profit before the remaining profit is split. A Sleeping partner only invests capital.

Compound Partnership

When partners invest different amounts for different time periods, it is called a Compound partnership. The profit sharing ratio is the ratio of their effective capitals.

Shortcut Trick

Remaining amount of Chemical A = 10 × (1 − 1/10)³ = 10 × 0.729 = 7.29 litres.

Amount of Chemical B = Total Volume − Final A = 10 − 7.29 = 2.71 litres.

Approximate percentage of Chemical B = (2.71 ÷ 10) × 100 = 27.1% ≈ 27%.

∴ The correct answer is 27%.

Alternate Method

Given:

Initial quantity of Chemical A (V) = 10 litres

Quantity replaced each time (x) = 1 litre

Number of operations (n) = 3

Formula Used:

Final quantity of original liquid = V × (1 − x ÷ V)ⁿ

Calculations:

⇒ Final volume of Chemical A = 10 × (1 − 1 ÷ 10)³

⇒ Final volume of Chemical A = 10 × (9 ÷ 10)³

⇒ Final volume of Chemical A = 10 × (729 ÷ 1000) = 7.29 litres

⇒ Final volume of Chemical B = Total Volume − Final A = 10 − 7.29 = 2.71 litres

⇒ Percentage of Chemical B = (2.71 ÷ 10) × 100

⇒ Percentage of Chemical B = 27.1% ≈ 27%

∴ The correct answer is 27%.

Additional Information

General Replacement Formula

For a container with volume V, if x litres are replaced with another liquid n times, the remaining quantity of the original liquid is V(1 − x/V)ⁿ.

Concentration in Mixtures

The concentration of the added liquid (B) is always [1 − (Final A / Total Volume)] × 100.

Successive Percentage Decrease

Replacing 1L from 10L is equivalent to a successive 10% decrease in the quantity of the original liquid for each operation.

Shortcut Trick

Total daily earnings = (Deliveries × ₹5) + (Distance in km × ₹2).

Conditions: Earnings > ₹500 and Distance ≤ 100 km.

For 90 deliveries: ₹450 + 2 × Distance. This only exceeds ₹500 if Distance > 25 km.

Since the minimum distance per delivery is not given, the target cannot be guaranteed.

∴ The correct answer is The question cannot be answered due to insufficient data.

Alternate Method

Given:

Rate per delivery = ₹5

Rate per km = ₹2

Target Earnings > ₹500

Maximum Distance = 100 km

Formula Used:

Total Earnings (E) = 5n + 2D

Where n = number of deliveries and D = total distance in km.

Calculations:

⇒ Let us test the highest option, n = 90.

⇒ E = 5(90) + 2D = 450 + 2D

⇒ To satisfy E > 500: 450 + 2D > 500

⇒ 2D > 50 ⇒ D > 25 km

⇒ Although D ≤ 100 km, we are not told that 90 deliveries must cover more than 25 km.

⇒ If the boy makes 90 deliveries to very close locations (e.g., total 10 km), he earns 450 + 20 = ₹470, which is less than the target.

⇒ Without a minimum distance constraint per delivery, no number in the options (80, 85, 90) definitely meets the target.

∴ The correct answer is The question cannot be answered due to insufficient data.

Additional Information

Linear Inequalities

When solving for a target value with multiple variables, all independent variables must be constrained or linked to find a definite solution.

Insufficient Data in Aptitude

A solution is deemed indeterminable if the outcome depends on a variable whose range or relationship with other factors is not fully defined.

Definite vs Possible Outcomes

A definite outcome must hold true for all possible values of the variables within their given limits (e.g., for D between 0 and 100).

Shortcut Trick Possible sums m without 5-coin: 1+2+10=13, 1+2+20=23, 1+10+20=31, 2+10+20=32. Check Statement 2: Sum of digits for these is {4, 5, 4, 5}. None are > 5. All sums with digit sum > 5 must contain a 5-coin. (Sufficient) Check Statement 1: m=32 is not prime and has no 5-coin, while m=8 is not prime and has a 5-coin. (Insufficient) ∴ The correct answer is statement 2 alone is sufficient. Alternate Method Given: Coins = {1, 2, 5, 10, 20}. Number of coins = 3. Total amount = m.

Step 1: Analyze Statement 1 (m is not a prime number) ⇒ Composite values of m: {8, 16, 26, 27, 32, 35} ⇒ If m = 32 (2 + 10 + 20), X does not have a 5-coin. ⇒ If m = 8 (1 + 2 + 5), X has a 5-coin. ⇒ Statement 1 alone is not sufficient. Step 2: Analyze Statement 2 (Sum of digits of m > 5) ⇒ m values with digit sum > 5: {8, 16, 17, 26, 27, 35} ⇒ Case 1: m = 8 → {1, 2, 5} ⇒ Case 2: m = 16 → {1, 5, 10} ⇒ Case 3: m = 17 → {2, 5, 10} ⇒ Case 4: m = 26 → {1, 5, 20} ⇒ Case 5: m = 27 → {2, 5, 20} ⇒ Case 6: m = 35 → {5, 10, 20} ⇒ Every possible case contains a 5-denomination coin. ⇒ Statement 2 alone is sufficient. ∴ The correct answer is statement 2 alone is sufficient. Additional Information Data Sufficiency Logic If one statement leads to a unique 'Yes' or 'No' answer, it is sufficient. If it leads to both possibilities, it is insufficient. Prime and Composite Numbers A prime number has exactly two factors (1 and itself). A composite number (not prime) has more than two factors. Sum of Digits Property The sum of digits is often used in divisibility rules (e.g., divisibility by 3 or 9) and can help uniquely identify sets of numbers in constrained problems.

Shortcut Trick

Statement 2 implies √x is real and less than 1, so 0 ≤ x < 1.

For any value in the range 0 < x < 1, the mathematical property x² < x < √x always holds.

Combining both statements results in the range x² < y < √x.

Since the variable x lies between the lower bound (x²) and upper bound (√x) of y, y can be either smaller or larger than x.

Because multiple possibilities exist (y < x or y > x), the question cannot be determined uniquely.

∴ The correct answer is cannot be answered even using both statements.

Alternate Method

Given: x and y are distinct real numbers.

From Statement 2: y < √x < 1

⇒ For √x to be a real number < 1, x must be in the range 0 ≤ x < 1.

From Statement 1: x² < y < 1

Combining both statements for the range 0 < x < 1:

⇒ x² < y < √x

Testing with values to verify indeterminacy:

⇒ Let x = 0.25 (then x² = 0.0625 and √x = 0.5).

⇒ The range for y is 0.0625 < y < 0.5.

⇒ Case 1: If y = 0.1, then y < x (0.1 < 0.25).

⇒ Case 2: If y = 0.4, then y > x (0.4 > 0.25).

⇒ Both cases satisfy all given conditions, meaning no unique comparison is possible.

∴ The correct answer is cannot be answered even using both statements.

Additional Information

Inequalities for Proper Fractions

For any real number x such that 0 < x < 1, the order of powers is xⁿ < x < x^1/n for all n > 1.

Data Sufficiency Logic

A question is considered 'answerable' only if the statements lead to a unique and consistent conclusion (always x > y or always y > x).

Domain of Square Root

The expression √x is defined for all x ≥ 0. If √x < 1, it directly restricts the base variable to 0 ≤ x < 1.

Logic followed here is based on the parity rules (even or odd properties) for the addition and multiplication of integers.

To determine if x is an even integer:

For Statement 1: x² y² is even →

If the product of two squared integers is even, then at least one of the base integers must be even. This leads to the following cases:

Since x can be either even or odd while satisfying the condition, Statement 1 alone is not sufficient.

<br>

For Statement 2: 1 + x² + y² is odd →

If 1 + x² + y² is odd, then the term (x² + y²) must be even (because odd − 1 = even). For the sum of two squares to be even, both x and y must share the same parity:

Since x can still be either even or odd, Statement 2 alone is not sufficient.

<br>

Combining both Statement 1 and Statement 2:

From Statement 2, we know that x and y must be either both even or both odd.

Now we apply Statement 1, which states that x² y² is even. We check the two possibilities from Statement 2:

• If x and y are both odd: x² × y² = Odd × Odd = Odd. This contradicts Statement 1.
• If x and y are both even: x² × y² = Even × Even = Even. This satisfies Statement 1.

The only case that satisfies both statements is when both x and y are even. Therefore, x is even.

Therefore, the question can be answered using both statements together, but cannot be answered using either statement alone.

Shortcut Trick

Statement 1: Sum of any two numbers is even ⇒ Numbers must be either all odd or all even.

Statement 2: (p − 1)q is even ⇒ If any q is odd, then (p − 1) must be even, so p is odd (making T all odd); or all q are even (making T all even).

Both statements confirm that T is a homogeneous set (contains only one type), but neither statement specifies whether that type is X (odd) or Y (even).

Since T could be {1, 3} (from X) or {2, 4} (from Y), we cannot determine if every number is from Y.

∴ The correct answer is the question cannot be answered even using any of the statements.

Alternate Method

Given: X = {certain odd numbers}, Y = {certain even numbers}, T ⊆ X ∪ Y.

Case Analysis:

⇒ Let Statement 1 be true: p + q = Even. This happens if (Odd + Odd) or (Even + Even). Thus, T ⊆ X OR T ⊆ Y. (Insufficient)

⇒ Let Statement 2 be true: (p − 1)q = Even.

⇒ If T ⊆ X (Odd), then (p − 1) is (Odd − 1) = Even. (Even × Odd = Even). Statement holds.

⇒ If T ⊆ Y (Even), then q is Even. (p − 1 × Even = Even). Statement holds.

⇒ If T contains both 2 (even) and 3 (odd): Let p = 2, q = 3. (2 − 1) × 3 = 1 × 3 = 3 (Odd). This violates the statement. So T must be homogeneous.

⇒ Combining both: T is still either all Odd or all Even. We cannot uniquely identify if it is Y.

∴ The correct answer is the question cannot be answered even using any of the statements.

Additional Information

Sum and Difference Properties

The sum or difference of two odd numbers is always even, and the sum or difference of two even numbers is always even.

Product Properties

The product of any number and an even number is even. The product of two odd numbers is always odd.

Data Sufficiency Logic

If two different sets (one subset of X and one subset of Y) both satisfy the conditions but lead to different answers to the question, the statements are insufficient.

Shortcut Trick

Factorize 116: 116 = 2² × 29. Since y > 1, y² must be 4, so x = 29 (Prime). Statement 1 is Sufficient.

Factorize 261: 261 = 3² × 29 = 9 × 29 or 3 × 87. Since x can be 9 (composite) or 3 (prime), Statement 2 is Insufficient.

∴ The correct answer is Statement 1 alone is sufficient.

Alternate Method

Given: x, y, z are integers > 1.

Analysis of Statement 1:

⇒ xy² = 116

⇒ Prime factorization: 116 = 2 × 2 × 29 = 2² × 29

⇒ Since y > 1, y² must be a perfect square factor of 116. The only such factor > 1 is 4 (where y = 2).

⇒ Therefore, x = 116 ÷ 4 = 29.

⇒ 29 is a prime number. Statement 1 alone is sufficient.

Analysis of Statement 2:

⇒ xz = 261

⇒ Factor pairs of 261: (3, 87), (9, 29), (29, 9), (87, 3).

⇒ If x = 3, it is prime. If x = 9, it is not prime (composite). If x = 87, it is composite.

⇒ Multiple possibilities exist for x. Statement 2 alone is not sufficient.

∴ The correct answer is Statement 1 alone is sufficient.

Additional Information

Prime Number Definition

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11, 29, etc.).

Perfect Square Factors

When an equation involves y² as a factor of a constant, identifying the perfect square factors (1, 4, 9, 16...) of that constant helps isolate possible values for y and x.

Data Sufficiency Logic

A statement is sufficient only if it leads to a unique 'Yes' or 'No' answer. If a statement allows for both 'Yes' and 'No' cases, it is insufficient.

The correct answer is Option 1.

 

Key Points

• Statement 1 is valid: The passage explicitly states that Kalagram unfolded as a mosaic of India's diverse regions, with each represented by seven meticulously crafted 'Sanskriti Angans'.
• Statement 2 is valid: The text mentions that these thematic zones were inspired by iconic temples like the Dakshineshwar Kali Temple and the Brahma Mandir, and they served as treasure troves of regional artistry.
• Statement 3 is invalid: While the passage mentions 230 master artisans shaping India's ancient history, it emphasizes that they used age-old techniques. The statement's use of the term contemporary craftsmanship contradicts the passage's focus on traditional methods.
• Statement 4 is invalid: The passage describes the village as a mosaic of diverse regions, but it does not claim that art forms from all regions of India were showcased. Using the word all makes the conclusion too absolute.

Therefore, the correct answer is 1 and 2 only.

 

Additional Information

• Kalagram was a cultural village set up at the Maha Kumbh Mela to showcase India's heritage.
• Specific art forms mentioned include Bengal's Pattachitra paintings, Assam's bamboo crafts, Tamil Nadu's Thanjavur paintings, and Madhya Pradesh's tribal sculpture.
• The Sanskriti Angans were described as living galleries, indicating an immersive and active display of traditional arts.
• The craftsmanship involved 230 master artisans who acted as the bridge between ancient history and visual creations to behold.

The correct answer is Option 1.

 

Key Points

• The passage mentions four specific states and their respective art forms: Bengal (Pattachitra paintings), Assam (bamboo crafts), Tamil Nadu (Thanjavur paintings), and Madhya Pradesh (tribal sculpture).
• Upon closer inspection, only two of these states are associated with paintings: Bengal and Tamil Nadu.
• The other two states are associated with bamboo crafts (Assam) and tribal sculpture (Madhya Pradesh), which are not categorized as paintings.
• Therefore, the statement that paintings from four states have been mentioned is incorrect.

Therefore, the correct answer is Option 1.

 

Additional Information

• Option 2: Mentioning that Kalagram is likened to stepping into another world is a common contextual description for such immersive cultural exhibitions (though it may appear in the full passage text).
• Option 3: The passage explicitly states that these art forms were showcased in living galleries, referring to the Angans where 230 master artisans worked.
• Option 4: The text mentions the exhibition was inspired by iconic temples like the Dakshineshwar Kali Temple and the Brahma Mandir, confirming that the zones/Angans were themed after well-known Indian temples.
• The passage highlights the use of age-old techniques by 230 master artisans to shape India's ancient history into contemporary creations.

Given:

A + B + C + D = 17

A + C = 6

P + Q + S + D = 15

P + Q + R + B = 17

P = R and Q = S

From:

A + B + C + D = 17

A + C = 6

So,

B + D = 17 − 6 = 11

Now use:

P + Q + S + D = 15

Since Q = S,

P + 2Q + D = 15 .......... (1)

Also,

P + Q + R + B = 17

Since P = R,

2P + Q + B = 17 .......... (2)

Adding (1) and (2):

3P + 3Q + B + D = 32

We know B + D = 11

So,

3P + 3Q + 11 = 32

3(P + Q) = 21

P + Q = 7

And,

A + C = 6

Therefore,

P + Q = 7 > 6 = A + C

Hence, P and Q together weigh more than the total weight of A and C.

Hence, the correct answer is "Option 2".

Shortcut Trick

Bind Q and U together as a single block (QU) because Q must always be followed by U.

Remaining letters from QUEUE are E, U, E.

Total units to arrange = 4: (QU), E, U, E.

Since E is repeated 2 times, Total arrangements = 4! ÷ 2!.

Calculation: 24 ÷ 2 = 12.

∴ The correct answer is 12 words.

Alternate Method

Given word: QUEUE (Letters: Q, U, E, U, E)

Constraint: Q is always followed by U.

Method: String Method (Grouping)

⇒ Total letters in QUEUE = 5 (Q × 1, U × 2, E × 2).

⇒ Treat (QU) as 1 entity. The remaining letters are U, E, E.

⇒ Total entities (n) = 4.

⇒ Repetitions: E repeats 2 times (p = 2).

⇒ Total ways = n! ÷ p!

⇒ Total ways = 4! ÷ 2! = (4 × 3 × 2 × 1) ÷ (2 × 1)

⇒ Total ways = 24 ÷ 2 = 12.

∴ The correct answer is 12 words.

Additional Information

Permutation of Multiset

When 'n' objects have 'p' identical items of one kind, 'q' of another, etc., the total arrangements are given by n! ÷ (p! × q!).

The String Method (Grouping)

Used when specific items must always stay together. Treat the group as a single unit and multiply by internal arrangements if the order within the group is not fixed.

Factorial Notation

n! represents the product of all positive integers up to n. For example, 4! = 4 × 3 × 2 × 1 = 24.

Shortcut Trick

The common mark interval is the LCM of jump lengths 4, 6, and 5.

LCM(4, 6, 5) = 60 feet.

Common marks follow the form: 199 + 60k (where k is an integer).

Range condition: 195 ≤ 199 + 60k ≤ 1000 ⇒ −4 ≤ 60k ≤ 801.

Possible values for k are 0, 1, 2, ..., 13 (Total 14 values).

∴ The correct answer is 14 times.

Alternate Method

Given:

Jump length of X = 4'

Jump length of Y = 6'

Jump length of Z = 5'

Initial common mark = 199' at 8 AM

Target range = [195', 1000']

Calculations:

⇒ Common landing occurs every LCM(4, 6, 5) feet.

⇒ 4 = 2², 6 = 2 × 3, 5 = 5

⇒ LCM = 2² × 3 × 5 = 60'

⇒ Since they land on 199' at 8 AM, other marks are 199 ± 60k.

⇒ Checking marks below 199': 199 − 60 = 139' (Outside range [195, 1000]).

⇒ Checking marks above 199': 199 + 60k ≤ 1000

⇒ 60k ≤ 1000 − 199 ⇒ 60k ≤ 801

⇒ k ≤ 801 ÷ 60 ⇒ k ≤ 13.35

⇒ Integer values of k are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.

⇒ Total count = 14.

∴ The correct answer is 14 times.

Additional Information

LCM (Least Common Multiple)

The smallest number that is a multiple of two or more numbers. Used to find synchronized events or common points in periodic cycles.

Arithmetic Progression (AP)

The common landing marks form an AP: a, a+d, a+2d... where a = 199 and d = 60. The number of terms n is found using T_n = a + (n − 1)d.

Relative Distance and Time

The time mentioned (8 AM and 9 AM) ensures the physical possibility of the event but does not affect the spatial distribution of the marks.

The logic for determining the valid conclusions is based on a direct analysis of the statements provided in the passage.

Analysis of Conclusions:

<br>

Detailed Breakdown:

1. Conclusion 1: This is a direct paraphrase of the authors call to view sport as a way of life rather than just a competitive activity.
2. Conclusion 2: Seclusion refers to the state of being private and away from people. The passage addresses this by noting children rarely leave the house and suggesting sport as a solution to mitigate this behavior.
3. Conclusion 3: While the author argues sport is more than just medals and quotas, the inclusion of these terms confirms that securing a job via a sports quota is a recognized outcome of sporting achievement (laurels).
4. Conclusion 4: The passage lists several skills like time management and teamwork. These are essential corporate sector skills. Since the passage says they can be learned through sport, the statement that they cannot be learned is incorrect.

<br>

Therefore, the valid conclusions are 1, 2 and 3.

The correct answer is Option 2.

 

Key Points

• Statement 1 is incorrect. While the passage mentions that parents often say their children are inactive and rarely leave the house, it does not claim that the parents themselves are not encouraging. It only highlights their observations regarding their children's behavior.
• Statement 2 is correct. The passage explicitly mentions that from sport, one can learn time management, teamwork, and coordination. These are essential life skills that contribute to personal development.
• Statement 3 is correct. The text advocates for a culture where sport is seen as a way of life, describing it as a path to building a healthier, happier society. Improving the health and happiness of its members directly helps evolve the nature of society.
• Therefore, only statements 2 and 3 are supported by the passage.

Therefore, the correct answer is 2 and 3.

 

Additional Information

• Mental Health Benefits: The passage highlights that sport is a tool to tackle modern issues like depression and anxiety among the current generation.
• Beyond Competition: It emphasizes that the value of sport transcends traditional rewards like winning medals or securing jobs through a sports quota.
• Social Impact: By fostering coordination and teamwork, sport serves as a foundational element for a more integrated and functional community.

Let:

1. Number of forward jumps = F
2. Number of backward jumps = B

 

Given:

F + B = 31 .......... (1)

Each forward jump = +5 ft

Each backward jump = −2 ft

Net distance = 15 ft forward

So,

5F − 2B = 15 .......... (2)

From (1):

B = 31 − F

Substitute in (2):

5F − 2(31 − F) = 15

5F − 62 + 2F = 15

7F = 77

F = 11

Then,

B = 31 − 11 = 20

Difference between forward and backward jumps:

20 − 11 = 9

Hence, the correct answer is "Option 4".

Solution

⇒ P and R are adjacent to Q. This means Q is seated between P and R.

⇒ T and R are adjacent to S. This means S is seated between T and R.

⇒ Combining these clues, we get a partial sequence: P - Q - R - S - T.

⇒ S and W are on opposite chairs. In an 8-seat circular arrangement, opposite means there are 3 persons between them on either side.

⇒ U and W are adjacent to V. This means V is seated between U and W.

⇒ The condition states that while going clockwise from P, one meets R before T. This confirms the clockwise order of the first sequence: P → Q → R → S → T.

⇒ Based on the clues, the final clockwise arrangement starting from Q is: Q → R → S → T → U → V → W → P.

⇒ Clockwise from P, R (2nd seat) is met before T (4th seat). This satisfies all conditions.

⇒ Now, moving clockwise from Q to meet W, we cross the following persons: R, S, T, U, and V.

⇒ Counting these persons: R(1), S(2), T(3), U(4), V(5).

∴ The correct answer is 5.

Given:

Table top dimensions = 6' × 10'

Two blue rectangles of dimensions = 2.5' × 8'

Each blue rectangle shares exactly two sides with the table edges.

Since each blue rectangle must touch two edges of the table, it can only be placed at a corner.

Possible corner placements are:

1. Top-left
2. Top-right
3. Bottom-left
4. Bottom-right

Now we count distinct visible patterns from above.

The two blue rectangles can form:

1. Both on top side corners
2. Both on bottom side corners
3. Both on left side corners
4. Both on right side corners

These give 4 different observable patterns.

Hence, the correct answer is "Option 2".

pThe correct answer is Neither 1 nor 2

Key Points

• Statement 1 is incorrect because it uses the term Real value in a general sense, whereas the passage specifically discusses Real Value Added (RVA). In economic accounting, these are distinct terms; Real Value Added is the difference between the real value of output and the real value of inputs. Furthermore, the statement uses the term output cost, but the passage refers to output prices. Price (what a seller receives) and Cost (what a producer spends) are not interchangeable in this context.
• Statement 2 is incorrect because the passage states that double deflation is a method used by most countries, but it is not universally adopted. The passage explicitly highlights India as an example of an economy that does not use this practice, instead deflating inputs and outputs using the same deflator (single deflation). The use of the word universally makes the statement factually incorrect based on the text.
• The passage explains that when input prices (like oil) fall while output prices and quantities remain the same, the actual physical contribution (the Real Value Added) should remain constant. However, if an economy uses a single deflator, it erroneously suggests an increase in productivity or real growth that did not actually occur.

Additional Information

• Double Deflation is an accounting technique where Gross Output is deflated by an output price index and Intermediate Consumption is deflated by an input price index to arrive at Real Gross Value Added (GVA).
• GVA is defined as: GVA = Value of Output − Value of Intermediate Consumption.
• In India, the reliance on Single Deflation, often using the Wholesale Price Index (WPI) as a proxy, has been a subject of debate among economists, especially when global commodity prices for raw materials fluctuate significantly compared to finished goods.
• The System of National Accounts (SNA), the international standard for GDP calculation, generally recommends the use of double deflation to provide a more accurate picture of an economy's manufacturing sector.

The correct answer is 1 only

Key Points

• Deflation strategies involve the process of converting Nominal Values (current prices) into Real Values (constant prices) to remove the impact of inflation. Statement 1 is valid because if an inappropriate or lower Deflator is used while actual input costs are rising sharply, the Real Value Added of manufacturers can be artificially inflated, making their performance look better than the actual physical growth suggests.
• Double Deflation is a method used in National Income Accounting to estimate the Real Gross Value Added (GVA). It requires deflating the value of Gross Output by an Output Price Index and separately deflating the Intermediate Consumption (inputs) by an Input Price Index.
• Statement 2 is invalid because deflating both input and output against a single price index is known as Single Deflation. Double deflation specifically mandates the use of two distinct indices to account for the different price movements of what a factory sells versus what it buys.
• In the manufacturing sector, if input prices (like raw materials) rise faster than output prices, using a single deflator can lead to a significant overestimation of Real GDP growth.

Additional Information

• The Gross Value Added (GVA) is calculated as: GVA = Gross Value of Output − Value of Intermediate Consumption.
• The Wholesale Price Index (WPI) is often used as a proxy deflator for the manufacturing sector in many developing economies, though experts argue that the Producer Price Index (PPI) provides a more accurate measure for Double Deflation.
• Real GDP is a better indicator of economic health than Nominal GDP because it reflects the actual volume of goods and services produced, excluding the 'noise' of price changes.
• Many economists have raised concerns about India's GDP calculation methods, specifically the shift to GVA at basic prices and the reliance on single deflation during periods of volatile commodity prices.

Logic followed here is.

Logic: The string is formed by the repetition of a specific sequence of characters a and b. By analyzing the fixed characters provided in the string, we can deduce the base pattern.

Step 1: Map the given characters and their positions

The provided string has a total of 15 positions. The known characters are distributed as follows:

Step 2: Identify the repeating pattern →

Since the string length is 15, let us test a 5-character repeating pattern. We divide the string into three equal segments:

• Segment 1 (Positions 1-5): x b x a x
• Segment 2 (Positions 6-10): a x x a x
• Segment 3 (Positions 11-15): a x b a b

By comparing the characters at corresponding positions across the three segments, we can deduce the base pattern:

• Position 1 of the pattern: From index 6 and index 11 → a
• Position 2 of the pattern: From index 2 → b
• Position 3 of the pattern: From index 13 → b
• Position 4 of the pattern: From index 4, 9, and 14 → a
• Position 5 of the pattern: From index 15 → b

The identified pattern is a b b a b.

 

Similarly, let us apply the pattern to the string →

By mapping the 7th and 8th positions to the second segment of the repeating pattern, we find:

7th position = b

8th position = b

Therefore, the characters x x at the 7th and 8th positions are bb.

Hence, the correct answer is "Option 4".

Shortcut Trick

The total exponent of 10 in the expression is the minimum of the exponents of 2 and 5.

Exponent of 5 → (2 × 25) + (2 × 32) = 50 + 64 = 114.

Exponent of 2 → (5 × 75) = 375.

Max power of 10 available = 114 ∴ 10^m+3 = 10¹¹⁴ (since 1000 = 10³).

m + 3 = 114 ⇒ m = 111.

∴ The correct answer is 111.

Alternate Method

Given: 10^m × 1000 × n = 75²⁵ × 25³² × 32⁷⁵

Formula: (a^b)^c = a^{b × c} and 10 = 2 × 5.

⇒ 10^m × 10³ × n = (3 × 5²)²⁵ × (5²)³² × (2⁵)⁷⁵

⇒ 10^m+3 × n = 3²⁵ × 5⁵⁰ × 5⁶⁴ × 2³⁷⁵

⇒ 10^m+3 × n = 3²⁵ × 5¹¹⁴ × 2³⁷⁵

⇒ 10^m+3 × n = 3²⁵ × 2^{375 − 114} × (2¹¹⁴ × 5¹¹⁴)

⇒ 10^m+3 × n = (3²⁵ × 2²⁶¹) × 10¹¹⁴

⇒ Since n is not divisible by 10, compare the powers of 10:

⇒ m + 3 = 114

⇒ m = 114 − 3 = 111

∴ The correct answer is 111.

Additional Information

Power of 10 in a Product

To find the number of trailing zeros or the highest power of 10 in an expression, calculate the powers of 2 and 5 in its prime factorization. The answer is always min(power of 2, power of 5).

Exponent Laws

Key rules used: (x^a)^b = x^a×b and x^a × x^b = x^a+b and (x × y)^a = x^a × y^a.

Divisibility by 10

A number is not divisible by 10 if its prime factorization does not contain at least one pair of (2 × 5). If it contains only 2s or only 5s (but not both), it is not divisible by 10.