CSAT (GS Paper II) 2024

Answer: (b) केवल 2 और 3

The passage ties food waste to land degradation and to the carbon dioxide released by the energy spent on wasted food, so statements 2 and 3 follow directly. Statement 1 is too extreme because of ‘solely’, and statement 4 is about technology the passage never mentions.

• ✗ 1. ‘Solely’ is too strong. The passage says food is lost ‘throughout the supply chain, from initial agricultural production to final household consumption’, so distribution is only one of several stages.
• ✓ 2. The passage says wastage ‘results in land degradation by about 45%’. Degraded land is less productive, so this is a direct inference.
• ✓ 3. The energy spent over wasted food produces about 3·5 billion tonnes of carbon dioxide a year, and decay releases other harmful gases. Less waste therefore means a smaller carbon footprint.
• ✗ 4. Post-harvest technology is not mentioned at all, let alone said to be unavailable. This goes outside the passage.

Remember · Strike out statements with ‘solely’, ‘only’ or ‘never’ unless the passage says so; every inference must rest on a line in the text.

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Answer: (c) 1 और 2 दोनों

The passage says food is lost at every stage from the farm to the household, and that tackling loss and wastage ‘in all forms’ is critical because of its heavy cost to land and climate. That presumes both a better distribution system and every citizen’s own effort.

• ✓ 1. Food is lost ‘throughout the supply chain’, and distribution is part of that chain. For loss ‘in all forms’ to be addressed, the distribution mechanism has to work better; the author takes this for granted.
• ✓ 2. Waste also happens at ‘final household consumption’, and its costs (land degradation, carbon dioxide, other harmful gases) fall on society as a whole. Asking that waste ‘in all forms’ be addressed presumes that every citizen shares this duty.

Remember · An assumption is what the author must take for granted. Check each stage the passage names: farm, supply chain, household.

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Answer: (c) 1 और 2 दोनों

The passage says that ‘without the fiscal backup, monetary policy eventually loses traction’, and that higher interest rates can even become inflationary if governments simply borrow more. So central banks need budgetary support, and what monetary policy achieves depends on fiscal choices.

• ✓ 1. This restates ‘Without the fiscal backup, monetary policy eventually loses traction’: higher interest rates alone do not bring inflation down if governments keep borrowing.
• ✓ 2. Whether higher rates are disinflationary or inflationary depends on whether governments cut spending and raise taxes or just borrow more to pay interest. That is, the effect of monetary policy depends on fiscal policy.

Remember · When a passage states a condition (‘without X, Y fails’), both ‘Y needs X’ and ‘Y depends on X’ are valid inferences.

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Answer: (d) न तो 1 और न ही 2

Neither statement is taken for granted by the passage. It describes how fiscal and monetary actions interact, so fiscal policy is not the ‘sole’ cause of higher prices, and it says nothing about long-term government bonds.

• ✗ 1. ‘Solely’ is extreme. The passage opens with inflation already rising and then explains how interest rates and government borrowing interact; it never makes fiscal policy the only cause of higher prices.
• ✗ 2. Government bonds are not discussed. If anything, the passage’s point about rising debt-service costs suggests that inflation and higher rates do affect government borrowing.

Remember · An assumption about something the passage never discusses is invalid; statements with ‘solely’ rarely survive.

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Answer: (b) 9

64 identical pieces means a 4 × 4 × 4 arrangement. Each direction needs 3 cuts to make 4 slices, so the least number is 3 + 3 + 3 = 9.

1. With a, b and c straight cuts in the three directions, the number of pieces is (a + 1)(b + 1)(c + 1).
2. We need (a + 1)(b + 1)(c + 1) = 64 with a + b + c as small as possible; the total is least when the three factors are equal: 4 × 4 × 4.
3. So a = b = c = 3, and the number of cuts = 3 + 3 + 3 = 9.
4. Check: other splits cost more, e.g. 8 × 4 × 2 needs 7 + 3 + 1 = 11 cuts.

Remember · Pieces = (a + 1)(b + 1)(c + 1) for a, b, c cuts in three directions; cuts are fewest when the factors are equal.

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Answer: (d) 0

Zero itself can be reached: 5 − 4 − 3 + 2 × 1 = 0, using minus twice and plus and times once each. No non-negative value can be smaller than 0.

1. The smallest possible non-negative value is 0, so first try to make 0.
2. 5 − 4 − 3 = −2, so the last part must give +2; and 2 × 1 = 2.
3. 5 − 4 − 3 + 2 × 1 = 5 − 4 − 3 + 2 = 0 (multiplication is done first).
4. Check the rule: − is used twice, + once and × once, so no symbol is used more than two times.

Remember · For a ‘smallest value’ question, test the smallest option first; one valid construction settles it.

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Answer: (c) 20%

For a fixed job, the number of men is inversely proportional to the number of days. Cutting the days from 6k to 5k needs 6/5 times the men, which is a 20% increase.

1. Work = men × days, so M × 6k = M′ × 5k.
2. M′ = M × 6k/5k = (6/5)M; the k cancels.
3. Increase = (6/5 − 1) × 100% = (1/5) × 100% = 20%.
4. Check: with k = 1, 5 men for 6 days = 30 man-days = 6 men for 5 days, and 6 is 20% more than 5.

Remember · For fixed work, new men = old men × old days ÷ new days. Common factors like k cancel out.

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Answer: (c) 6 घंटे 45 मिनट

X is the fastest, so X should take the first hour and every alternate hour after it. In six hours 21 of the 24 units are done, and X finishes the last 3 units in 45 minutes of the seventh hour.

1. Take the work as 24 units (LCM of 6 and 8): X does 4 units per hour, Y 3 and Z 3.
2. Nobody may work two hours in a row, so X can work at most every other hour; start with X so that he gets as many hours as possible.
3. Hours 1 to 6: X, Y, X, Z, X, Y gives 3 × 4 + 3 × 3 = 21 units, and all three have worked.
4. Hour 7 is X’s again: the remaining 3 units at 4 units per hour take 3/4 hour = 45 minutes.
5. Total = 6 hours 45 minutes.
6. Check: if Y or Z starts, X still gets only three of the first six hours (21 units), but hour 7 then goes to Y or Z at 3 units per hour, finishing at exactly 7 hours.

Remember · In alternate-hour work problems, give the fastest worker the first slot and every second slot after it.

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Answer: (d) 200

The product is n raised to the power 2n for n = 1 to 25. Twos are plentiful, so count the fives: only 5, 10, 15, 20 and 25 contribute, giving 10 + 20 + 30 + 40 + 100 = 200.

1. Each zero at the end needs one 2 and one 5; there are far more 2s than 5s here, so count the 5s.
2. Only multiples of 5 carry a 5: 5¹⁰ gives 10 fives, 10²⁰ gives 20, 15³⁰ gives 30, 20⁴⁰ gives 40.
3. 25 = 5², so 25⁵⁰ = 5¹⁰⁰ gives 100 fives.
4. Total fives = 10 + 20 + 30 + 40 + 100 = 200, so there are 200 zeros at the end.
5. Check: forgetting that 25 holds two 5s gives 150, which is not even an option.

Remember · Trailing zeros = number of 5s (2s are always more). Remember 25 = 5², so its power counts double.

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Answer: (b) 8 जनवरी, 2023

He saves 1, 3, 5, 7, … rupees, so the total after n days is n². A number that is both a square and a cube is a sixth power, so n must be a perfect cube; n = 8 gives 64 = 8² = 4³.

1. Daily savings are the odd numbers 1, 3, 5, …; the sum of the first n odd numbers is n².
2. n² is also a perfect cube only when it is a sixth power, i.e. when n itself is a perfect cube.
3. The first cube after 1 is 8: total after 8 days = 64 = 8² = 4³.
4. So the date is 8th January, 2023.
5. Check: 7 days give 49 and 9 days give 81, neither a cube. (The ₹1 at the end of 1st January is trivially 1² = 1³, but that date is not among the options.)

Remember · Sum of the first n odd numbers = n². A number that is both a square and a cube is a sixth power.

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Answer: (a) नगर आयोजना पाठ्यक्रमों की पाठ्यचर्या में विविध और अंतर-विषयक उपागम (अप्रोच) शामिल होना चाहिए।

The passage credits the West’s iconic cities to planners trained in a curriculum that joined computing, statistics and optimization with the social sciences, geography and architecture. The logical inference is that planning education should be diverse and interdisciplinary.

• ✓ (a) The West produced ‘ingenious planners’ because mathematical subjects ‘were absorbed into the planning curriculum that had its roots in the social sciences, geography and architecture’, which is an interdisciplinary mix.
• ✗ (b) Bureaucracy and the training of administrators are never mentioned. The passage only says that Indian planning and its education differ from the West.
• ✗ (c) Local management of cities and a shortage of funds lie outside the passage.
• ✗ (d) Population density and poverty are not discussed; the passage is about how planners are trained, not about why planning is hard.

Remember · The right inference extends the passage’s own cause and effect; options that bring in new problems (funds, bureaucracy, poverty) go outside it.

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Answer: (a) केवल 1

The author holds up the systems-trained planners of the West and then notes that planning in India and its education differ. This presumes that India needs such modern urban professionals. Saying Indian universities have ‘no capacity or potential’ is far too extreme.

• ✓ 1. By praising the West’s planners trained in modern systems methods and contrasting Indian planning education with them, the author takes for granted that India needs professionals with this modern knowledge.
• ✗ 2. The passage only says Indian planning education differs from the West. ‘No capacity or potential’ is an absolute claim it never makes; a difference in approach is not an inability.

Remember · ‘Different’ does not mean ‘incapable’. Reject assumptions that turn a contrast into a total absence.

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Answer: (b) सोशल मीडिया आदर्श या नैतिक संस्थाएँ नहीं हैं, बल्कि कंपनियों द्वारा लाभ कमाने के लिए निर्मित उत्पाद हैं।

The passage says money buys a louder voice and that social media promote whatever drives user engagement, however inflammatory. Its central idea is that these platforms work as profit-driven products, not as neutral or moral institutions.

• ✗ (a) The passage denies a level marketplace (‘Not every voice on the internet commands the same kind of audience’) and says nothing about instant access to information.
• ✓ (b) Entities with high capital buy ‘a louder voice’, and the platforms focus on content that ‘generates more user engagement, regardless of how inflammatory’. This is the logic of a product built for profit, not of a moral institution.
• ✗ (c) The passage says the quality of politics suffers; it never claims that social media were created to strengthen democracy.
• ✗ (d) The passage does not argue that social media are inevitable or needed for a well-informed life; its tone is critical.

Remember · The central idea must cover the whole passage and match its tone; a critical passage rules out options that praise.

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Answer: (c) 1 और 2 दोनों

The opening line says not every voice gets the same audience and that money buys a louder voice, so the internet is not equally open to all. The passage also says the quality of politics suffers when digital political discourse becomes a contest to outshine rivals.

• ✓ 1. ‘Not every voice on the internet commands the same kind of audience’, and entities ‘with high capital’ can buy a louder voice. Voices are not equally heard, so the internet is not inclusive enough.
• ✓ 2. The passage says that if political discourse in the digital sphere is about outshining one’s opponent till the election is won, ‘then the quality of politics suffers’.

Remember · When a statement restates a line of the passage in general words, it is valid. Read the first and last sentences closely.

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Answer: (b) 3 और 37 से किन्तु 2 से नहीं

222 = 2 × 3 × 37 and 333 = 3² × 37, so both terms, and hence their sum, are divisible by 3 and 37. But 222³³³ is even and 333²²² is odd, so the sum is odd and not divisible by 2.

1. Factorise the bases: 222 = 2 × 3 × 37 and 333 = 3 × 3 × 37.
2. Both 222³³³ and 333²²² contain the factors 3 and 37, so their sum is divisible by 3 and by 37.
3. 222³³³ is even; 333²²² is odd (a power of an odd number stays odd). Even + odd = odd.
4. So the sum is divisible by 3 and 37 but not by 2.

Remember · Factorise the bases first: a sum is divisible by any factor common to both terms. Even + odd is always odd.

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Answer: (a) 20%

Selling at the honey’s cost price and still gaining 20% means 120 litres of mixture must be sold for what 100 litres of honey cost. So 20 litres of water go into every 100 litres of honey, i.e. 20% of the honey.

1. Take 100 litres of honey at ₹1 per litre: cost = ₹100.
2. A 20% gain means sales must bring ₹120; at ₹1 per litre that is 120 litres of mixture.
3. Water added = 120 − 100 = 20 litres, which is 20% of the honey.
4. Check: water is 20/120 = 16.67% of the mixture, which is not an option, so the question means water as a percentage of honey.

Remember · When free water is sold at the price of the pure item, the gain percent equals water as a percent of the pure item.

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Answer: (d) 9

30³⁰ = 3³⁰ × 10³⁰, so the digit just before the zeros is the unit digit of 3³⁰. Powers of 3 end in 3, 9, 7, 1 in a cycle of 4, and 30 leaves remainder 2 on division by 4, giving 9.

1. 30³⁰ = 3³⁰ × 10³⁰; the factor 10³⁰ supplies the 30 zeros at the end.
2. The unit digits of 3¹, 3², 3³, 3⁴ are 3, 9, 7, 1, and then they repeat every 4 powers.
3. 30 = 4 × 7 + 2, so 3³⁰ ends like 3², i.e. in 9.
4. So the rightmost digit before the zeros is 9.

Remember · Split off the powers of 10 first; then use the four-step unit-digit cycle (remainder 0 means the fourth digit).

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Answer: (c) 3

If both numbers leave remainder 1, the divisor divides 420 and 426, and hence their HCF, 6. The divisors of 6 that are greater than 1 are 2, 3 and 6, so three numbers work (1 cannot leave a remainder of 1).

1. Remainder 1 means the divisor divides 421 − 1 = 420 and 427 − 1 = 426.
2. HCF(420, 426) = HCF(420, 6) = 6.
3. Divisors of 6 are 1, 2, 3 and 6. A divisor must be larger than the remainder, so 1 is ruled out.
4. Possible divisors: 2, 3 and 6, i.e. 3 numbers.
5. Check: 421 = 6 × 70 + 1 and 427 = 6 × 71 + 1.

Remember · Same remainder r: the divisor divides each number minus r. Count the HCF’s divisors that are larger than r.

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Answer: (b) 4

A bottle size must divide both 399 and 532 exactly, i.e. it must divide their HCF, 133 = 7 × 19. The number 133 has four divisors: 1, 7, 19 and 133.

1. 399 = 3 × 7 × 19 and 532 = 2² × 7 × 19.
2. HCF(399, 532) = 7 × 19 = 133.
3. Every common divisor divides 133: 1, 7, 19 and 133.
4. So 4 different bottle sizes are possible.
5. Check: 532 − 399 = 133, and 399 = 3 × 133, 532 = 4 × 133.

Remember · ‘Equal bottles for both’ means common divisors; their number equals the number of divisors of the HCF.

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Answer: (c) केवल 1 और 3

The primes below 10 are 2, 3, 5 and 7, and three distinct ones can be chosen in only four ways, giving sums 10, 12, 14 and 15. Unit digits 0 and 5 occur; 9 does not.

1. Primes less than 10: 2, 3, 5, 7. Choosing 3 distinct ones gives just 4 cases.
2. 2 + 3 + 5 = 10; 2 + 3 + 7 = 12; 2 + 5 + 7 = 14; 3 + 5 + 7 = 15.
3. Possible unit digits: 0, 2, 4, 5.
4. So statements 1 (0) and 3 (5) are correct; statement 2 (9) is not.

• ✓ 1. 2 + 3 + 5 = 10 ends in 0.
• ✗ 2. None of the four possible sums (10, 12, 14, 15) ends in 9.
• ✓ 3. 3 + 5 + 7 = 15 ends in 5.

Remember · When the set is tiny, list every case (four sums here) instead of reasoning in general.

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Answer: (d) न तो 1 और न ही 2

The passage describes a gap between what class 8 children can actually do and what they and their parents expect. It never compares the role of parents with that of governments, and it does not suggest a new, uniform national curriculum as the remedy.

• ✗ 1. Parents appear only as people who hold expectations for their children. The passage never weighs their role against the government’s.
• ✗ 2. Competencies fall short ‘even based on curricular expectations of class 8’, so the problem is weak learning against the existing curriculum, not the absence of a new or uniform one. A nationwide uniform curriculum is the reader’s idea, not the author’s premise.

Remember · An assumption must be needed by the author’s argument; a remedy the author never hints at is not an assumption.

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Answer: (c) सार्वजनिक नीति से यह सुनिश्चित होना चाहिए कि युवाजनों के सामर्थ्य और उपलब्धियाँ उनकी अपेक्षाओं के अनुरूप हों।

The passage sets high expectations (high school, college, no return to farming) against low actual competencies. Its central concern is closing that gap, which is what option (c) says.

• ✗ (a) Poverty is not discussed, and ‘total eradication … will resolve’ is an extreme claim the passage never makes.
• ✗ (b) Monetary incentives to parents or teachers are not mentioned anywhere.
• ✓ (c) Parents and children expect the ‘graduates’ to go on to high school and college, yet ‘abilities in terms of academic competencies are far lower than they should be’. Aligning competencies with expectations is the point the passage builds towards.
• ✗ (d) The passage notes that hardly anyone wants to go back to agriculture, but it never says pass-outs should do so, and the demographic dividend is not mentioned.

Remember · The central idea usually joins the passage’s two halves; here, high expectations versus low competencies.

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Answer: (d) विज्ञान को सत्य का अनुसरण अवश्य करना चाहिए किन्तु सामाजिक कल्याण के लिए उत्तरदायी भी होना चाहिए।

The author contrasts the older view, in which the only duty of science was to tell the truth, with today’s accepted view that science has a social responsibility. Science should therefore pursue truth and also answer for social welfare.

• ✗ (a) The commitment of scientists is not the author’s theme.
• ✗ (b) The passage does not call science a product of civilized society, and it says nothing about promoting scientific awareness.
• ✗ (c) The passage says the idea of science as a social enterprise began at the industrial revolution; it does not say that science made the revolution possible.
• ✓ (d) Newton and Galileo acknowledged only the duty ‘to tell the truth’, while ‘We take it for granted now that science has a social responsibility.’ Truth together with social responsibility is the author’s view.

Remember · When a passage contrasts an older view with the present one, the answer often keeps the old duty and adds the new one.

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Answer: (c) A, A, C, D

Split the sequence into blocks of five letters. Each block is ABCD with one letter doubled, and the doubled letter moves along: AABCD, ABBCD, ABCCD, ABCDD. The blanks are A, A, C, D.

1. Write it in blocks of five: A_BCD | _BBCD | ABC_D | ABC_D.
2. The second block has B doubled in ABCD, so each block is ABCD with one letter doubled.
3. The doubled letter moves A → B → C → D: AABCD, ABBCD, ABCCD, ABCDD.
4. So the blanks are A (block 1), A (block 2), C (block 3) and D (block 4).
5. Check: the completed series is AABCD ABBCD ABCCD ABCDD, a clean pattern.

Remember · In letter series with blanks, split the string into equal blocks (try the length of the repeating unit plus one).

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Answer: (a) ₹ 30,000

Out of ₹2,000, P gets ₹1,200 and Q ₹800, a ratio of 3 : 2. Profit is shared in the ratio of capital × time, so 8(Q + 14,000) : 10Q = 3 : 2, which gives Q = ₹16,000 and P = ₹30,000.

1. Shares add to 2,000 and differ by 400, so P gets ₹1,200 and Q ₹800; ratio 3 : 2.
2. Let Q’s capital be x; then P’s capital is x + 14,000.
3. 8(x + 14,000) : 10x = 3 : 2 → 16(x + 14,000) = 30x → 14x = 2,24,000 → x = 16,000.
4. P’s capital = 16,000 + 14,000 = ₹30,000.
5. Check: 30,000 × 8 = 2,40,000 and 16,000 × 10 = 1,60,000, a ratio of 3 : 2.

Remember · Profit share is proportional to capital × time. Turn the profit clue into a ratio first, then solve one equation.

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Answer: (b) 56·25%

Take R’s salary as 100: Q gets 80 and P gets 64. R’s salary is 36 more than P’s, and 36 out of 64 is 56.25%.

1. Let R’s salary be 100. Q’s is 20% lower: 80. P’s is 20% lower than Q’s: 64.
2. R − P = 100 − 64 = 36.
3. R is more than P by 36/64 × 100 = 56.25%.
4. Check: 64 × 1.5625 = 100.

Remember · Take 100 as the base. ‘More than P’ means divide the difference by P’s value, not by 100.

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Answer: (d) 93·75%

The correct result is 4x and the mistaken one x/4. The result falls by 4x − x/4 = 15x/4, which is 15/16 of the correct result, i.e. 93.75%.

1. Take the number as 16. Correct result = 16 × 4 = 64; mistaken result = 16 ÷ 4 = 4.
2. Change = 64 − 4 = 60.
3. Percentage change = 60/64 × 100 = 93.75% (a fall).
4. Check in general: (4 − 1/4)/4 = 15/16 = 0.9375.

Remember · Pick a convenient number and measure the change against the correct result, which is the base.

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Answer: (b) 20%

20% failed in English and 30% in Hindi, with 15% failing both. So 5% failed only English and 15% failed only Hindi: 20% failed in exactly one subject.

1. Failed in English = 100 − 80 = 20%; failed in Hindi = 100 − 70 = 30%.
2. Failed in both = 15%.
3. Failed only in English = 20 − 15 = 5%; failed only in Hindi = 30 − 15 = 15%.
4. Failed in only one subject = 5 + 15 = 20%.
5. Check: passed in both = 100 − (20 + 30 − 15) = 65%, and 65 + 20 + 15 = 100.

Remember · Work with the failure sets: ‘only one’ = (A − both) + (B − both).

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Answer: (a) 30 वर्ष

n years ago the father was the son’s present age, so n is the gap between their ages. Using ‘father now = 4 × son n years ago’ gives father : son = 8 : 5, so with a total of 130 they are 80 and 50, a difference of 30 years.

1. Let the father’s age be F and the son’s S. ‘n years back I was as old as you are now’: F − n = S, so n = F − S.
2. Son’s age n years back = S − n = S − (F − S) = 2S − F.
3. F = 4(2S − F) → 5F = 8S → F : S = 8 : 5.
4. F + S = 130 → 13 parts = 130 → 1 part = 10, so F = 80 and S = 50.
5. Difference = 80 − 50 = 30 years.
6. Check: n = 30; the son was 20 thirty years ago, and 4 × 20 = 80.

Remember · ‘I was your present age n years ago’ means n equals the age gap; turn the rest into a ratio.

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Answer: (d) 1, 2 और 3

All three are standard metric equivalences: a cubic metre holds 1000 litres, a metric ton (tonne) is 1000 kg, and a hectare is a 100 m × 100 m square, i.e. 10000 m².

1. 1 m³ = 100 cm × 100 cm × 100 cm = 10,00,000 cm³, and 1 litre = 1000 cm³, so 1 m³ = 1000 litres.
2. 1 metric ton (tonne) = 1000 kg by definition.
3. 1 hectare = 100 m × 100 m = 10000 m².
4. All three statements are correct.

• ✓ 1. 1 m³ = 10,00,000 cm³ and 1 litre = 1000 cm³, so 1 m³ = 1000 litres.
• ✓ 2. A metric ton (tonne) is defined as 1000 kg.
• ✓ 3. A hectare is the area of a square of side 100 m: 100 × 100 = 10000 m².

Remember · Know the core metric links: 1 litre = 1000 cm³, 1 m³ = 1000 litres, 1 hectare = 10000 m², 1 tonne = 1000 kg.

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Answer: (d) न तो 1 और न ही 2

The author stresses that science and mankind cannot be separated, and asks whether war is the inevitable result of science or whether the fault lies elsewhere. The passage never says mankind survives because of science, nor that science will decide mankind’s destiny.

• ✗ 1. Mankind’s survival is not discussed. The passage is about the relation between science and mankind, not about mankind depending on science to exist.
• ✗ 2. The author leans the other way: science should not be treated as ‘some abstract third party’, and the closing question suggests the fault may ‘lie elsewhere’, that is, with men. Science is not presented as the force that decides destiny.

Remember · ‘Emphatically conveyed’ needs a point the author clearly makes; reject statements giving science a decisive role the author questions.

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Answer: (b) केवल 2

The aspect ‘easily overlooked’ is the relation between science and mankind: science is not an abstract third party but is shaped by men. The author questions, rather than assumes, that the horrors of war are the inevitable result of science.

• ✗ 1. The author asks ‘Is this the inevitable result of the progress of science or does the fault lie elsewhere?’ and says science is ‘conveniently blamed’. That is the opposite of assuming inevitability.
• ✓ 2. The overlooked aspect is ‘the relations between science and mankind’. Since the two ‘cannot be divorced’ and science is not an abstract third party, science is what man has made of it.

Remember · A question the author raises is not an assumption. Words like ‘conveniently blamed’ show where the author really stands.

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Answer: (c) स्थानीय मूल्य और सार्वभौमिक मूल्य

Through travel the merchant learns ‘what is parochial and what is universal’, and the passage calls learning ‘fundamental common values’ the greatest gain of travel. Parochial means local, so the contrast is between local and universal values.

• ✗ (a) Imagination and understanding are not contrasted anywhere in the passage.
• ✗ (b) Communities and nationalities are not the subject; the passage speaks of values that are ‘valid everywhere’ but moulded differently in different places.
• ✓ (c) The merchant comes to know ‘what is parochial and what is universal in his choice’; parochial means local.
• ✗ (d) Friends and foes are not mentioned; the passage speaks of living ‘at ease among their fellows’.

Remember · Find the passage’s own contrasting pair (parochial/universal) and pick the option that restates it.

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Answer: (a) केवल 1 और 2

Travel, the passage says, shows how the same values are shaped differently among people, and learning fundamental common values is its greatest gain for those who seek it. Nowhere does it say that experienced travellers can resolve differences between people.

• ✓ 1. Meeting people abroad shows how shared values like justice, love and honour are ‘variously moulded and often differently handled’; this is an understanding of humans.
• ✓ 2. ‘the art of learning fundamental common values is perhaps the greatest gain of travel to those who wish to live at ease among their fellows.’
• ✗ 3. The traveller learns to recognise common values so that he can live at ease among others. Resolving differences amongst other people is a much bigger claim that the passage does not make.

Remember · Separate ‘understands’ from ‘solves’: a passage about learning rarely supports claims about settling others’ differences.

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Answer: (d) 8

X + Y = 11 × (sum of the digits), and it must equal 99, so the digits add up to 9. Both numbers must be two-digit, so neither digit can be 0: X can be 18, 27, 36, 45, 54, 63, 72 or 81, which is 8 values.

1. Let X = 10a + b and Y = 10b + a; then X + Y = 11(a + b).
2. The greatest two-digit number is 99, so 11(a + b) = 99 and a + b = 9.
3. Y must also be a two-digit number, so b cannot be 0; this rules out X = 90 (its reverse is 09).
4. a = 1 to 8 gives X = 18, 27, 36, 45, 54, 63, 72, 81, i.e. 8 values.

Remember · A number plus its reverse is 11 × digit sum. Watch for a 0 digit that makes the reverse a one-digit number.

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Answer: (b) 42 kg

Four women weigh 4 × 63 = 252 kg, and six boys weigh the same. So the average boy weighs 252 ÷ 6 = 42 kg.

1. Weight of 4 women = 4 × 63 = 252 kg.
2. Weight of 6 boys = weight of 4 women = 252 kg.
3. Average weight of a boy = 252 ÷ 6 = 42 kg.
4. Check: 6 × 42 = 252 = 4 × 63.

Remember · In a chain of equal totals, use only the two links the question needs and ignore the rest.

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Answer: (a) 3 बार

The hands coincide once every 12/11 hours, and in this window that happens at about 10:54:33, 12:00 and 1:05:27. There is no separate meeting in the 11 o’clock hour, because the meeting after 11 falls exactly at 12:00.

1. The hands coincide every 12/11 hours = 65 5/11 minutes, counting from 12:00.
2. Working back and forward from 12:00: 10:54 6/11, 12:00 and 1:05 5/11; the next one, 2:10 10/11, is after 2:00 p.m.
3. So the hands coincide 3 times between 10:00 a.m. and 2:00 p.m.
4. Check: the hands meet only 11 times in 12 hours, so one hour-slot (11 to 12) has no meeting of its own.

Remember · Hands coincide 11 times in 12 hours; the meetings of the 11 o’clock and 12 o’clock hours are the same one, at 12:00.

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Answer: (c) 2031

2025 is an ordinary year. Adding odd days year by year (2025: 1, 2026: 1, 2027: 1, 2028: 2, 2029: 1, 2030: 1) reaches 7 at the end of 2030, so 2031 begins on the same weekday. It is also an ordinary year, so its calendar matches 2025.

1. An ordinary year has 1 odd day; a leap year has 2.
2. Odd days from the start of 2025: 1 + 1 + 1 + 2 (2028 is a leap year) + 1 + 1 = 7, a whole number of weeks, by the end of 2030.
3. So 1st January 2031 falls on the same weekday as 1st January 2025 (a Wednesday).
4. 2031, like 2025, is not a leap year, so the whole calendar repeats.
5. Check: by the start of 2029 only 5 odd days have gathered, and by the start of 2033 there are 10, which leaves 3; neither gives a match.

Remember · For an ordinary year, add odd days until the total is a multiple of 7, then confirm the target year is also ordinary.

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Answer: (d) 1, 2 और 3

With p, q odd and r, s even, statement 1 contains the even factor qs and statement 2 the even factor s, so both are even. In statement 3, q + r and p + s are both odd, so the product is odd. All three are correct.

1. Parity rules: odd ± even = odd; any product with an even factor is even; odd × odd = odd.
2. Statement 1: (p − r)² is odd, but qs = odd × even is even, so the product is even. Correct.
3. Statement 2: s is even, so (q − s)q²s is even. Correct.
4. Statement 3: q + r is odd, so (q + r)² is odd; p + s is odd; odd × odd = odd. Correct.
5. Check with p = 1, q = 3, r = 2, s = 4: (−1)² × 12 = 12 (even); (−1) × 9 × 4 = −36 (even); 5² × 5 = 125 (odd).

• ✓ 1. qs is odd × even = even, so the whole product is even whatever (p − r)² is.
• ✓ 2. The factor s is even, so the product is even.
• ✓ 3. q + r and p + s are each odd + even = odd, and a product of odd numbers is odd.

Remember · For parity questions, plug in small numbers of the right parity and test each statement; it is faster than algebra.

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Answer: (c) 17·5°

At 4:25 the minute hand is at 25 × 6° = 150° from 12, and the hour hand at 4 × 30° + 25 × 0.5° = 132.5°. The angle between them is 17.5°.

1. Minute hand moves 6° per minute: 25 × 6 = 150° from the 12.
2. Hour hand moves 30° per hour plus 0.5° per minute: 4 × 30 + 25 × 0.5 = 132.5°.
3. Angle = 150 − 132.5 = 17.5°.
4. Check with the formula |30H − 5.5M| = |120 − 137.5| = 17.5°.

Remember · Angle between the hands = |30H − 5.5M|; if the result is more than 180°, subtract it from 360°.

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Answer: (c) 1 और 2 दोनों

The passage says conventional classrooms emphasise ‘fixed duration over learning effectiveness’ and put every learner through the same lectures, in the same way, for the same time. The two statements restate these two ideas.

• ✓ 1. Classrooms work ‘by emphasizing fixed duration over learning effectiveness’, and after the fixed duration the model ‘moves on’ whatever has been learnt. Time spent, not competency reached, is the goal.
• ✓ 2. ‘every learner is subjected to the same set of lectures in the same way for the same duration’ is one-size-fits-all. The author calls it ‘the tyranny of the classroom’ because it makes no allowance for different learners.

Remember · A statement that closely paraphrases a sentence of the passage is a valid inference, even if its wording is a little sharper.

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Answer: (d) न तो 1 और न ही 2

The passage blames the conventional classroom for ‘10 percent employability’ among graduates and warns against repeating it in skill education. Both statements defend the present system, so neither can be an assumption of this passage.

• ✗ 1. The unorganized sector is not mentioned, and the author clearly wants the classroom model changed: ‘Repeating the same ineffectual script … will not produce different results.’
• ✗ 2. The passage cites ‘10 percent employability in our graduates’, which contradicts the claim that the present system yields enough skilled workers. The demographic dividend is not mentioned.

Remember · An assumption must support the author’s argument; statements that contradict the author’s stand are invalid.

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Answer: (a) केवल 1

The passage says parental concern is often ‘a disguise for love of power’ and that parents’ views are so dogmatic that the young seldom confide in them. So the adolescent’s discomfort comes from domineering parents; a lack of respect among young people of ‘modern times’ is not suggested.

• ✓ 1. Parental solicitude is ‘often a disguise for love of power’, and parents express views ‘so dogmatic that the young seldom confide in them’. Dominating, assertive parents make the adolescent uncomfortable.
• ✗ 2. The passage describes a conflict that arises at adolescence in general, not a feature of ‘modern times’. The young keep things secret because parents are dogmatic, not because they lack respect.

Remember · Watch for added qualifiers like ‘of modern times’; if the passage speaks generally, a time-bound assumption is invalid.

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Answer: (a) जब बच्चे वयस्क होने के अपनी राह पर होते हैं, माता-पिता सामान्यतः अधिक सहायता नहीं कर सकते।

Because parents tend to be possessive and dogmatic, adolescents ‘seldom confide in them’ and go their own way in secret. The central idea is that parents often cannot help much as children move towards adulthood.

• ✓ (a) The young ‘seldom confide in them, and usually go their own way in secret’, so parents, as they usually behave, are of little help at this stage. ‘In general’ and ‘may not’ keep the claim suitably soft.
• ✗ (b) The passage criticises how parents get involved, not whether they should be; calling their involvement ‘unnecessary’ goes too far.
• ✗ (c) Nuclear families are not mentioned.
• ✗ (d) The passage blames parental dogmatism; it does not label adolescents stubborn, disobedient or careless.

Remember · Prefer the option with soft words (‘may not’, ‘in general’) over absolutes like ‘unnecessary’ or ‘not capable’.

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Answer: (b) 56

Count the digit 5 place by place in 1 to 260. It appears 26 times in the units place and 30 times in the tens place (50–59, 150–159, 250–259), a total of 56.

1. Units place: 5, 15, 25, …, 255, which is 26 numbers.
2. Tens place: 50–59, 150–159 and 250–259, 10 each, 30 in all.
3. Hundreds place: there is no page from 500 upwards, so 0.
4. Total fives = 26 + 30 = 56.
5. Check: 55, 155 and 255 each use two fives; they are counted once in the units count and once in the tens count, as they should be.

Remember · Count a digit place by place (units, tens, hundreds); numbers like 55 are then automatically counted twice.

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Answer: (b) 84

After the first 1, the sequence runs in blocks 1 | 2, 1 | 3, 2, 1 | 4, 3, 2, 1 | …. The first 28 terms are the lone 1, the full blocks up to 6 (21 terms, sum 56) and the first 6 terms of the block 7, 6, …, 1 (sum 27): 1 + 56 + 27 = 84.

1. Group the terms: 1 | 1 | 2, 1 | 3, 2, 1 | 4, 3, 2, 1 | 5, 4, 3, 2, 1 | …; after the first 1, block k is k, k − 1, …, 1.
2. Blocks 1 to 6 have 1 + 2 + 3 + 4 + 5 + 6 = 21 terms; with the first 1 that makes 22 terms.
3. Their sum = 1 + (1 + 3 + 6 + 10 + 15 + 21) = 1 + 56 = 57.
4. Terms 23 to 28 are the first 6 terms of block 7: 7 + 6 + 5 + 4 + 3 + 2 = 27.
5. Sum of the first 28 terms = 57 + 27 = 84.

Remember · In block sequences, count terms block by block (1, 2, 3, … terms) and handle the last, partial block separately.

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Answer: (d) ₹ 1,350

Take Q’s cost as 100: R costs 120 and P costs 150, a total of 370 units. As 370 units = ₹3,330, one unit = ₹9, so P costs 150 × 9 = ₹1,350.

1. Let Q = 100. R is 20% more: 120. P is 25% more than R: 150.
2. Total = 100 + 120 + 150 = 370 units = ₹3,330, so 1 unit = ₹9.
3. P = 150 × 9 = ₹1,350.
4. Check: Q = ₹900, R = ₹1,080, P = ₹1,350, and 900 + 1,080 + 1,350 = 3,330.

Remember · Chain the percentages from the cheapest item taken as 100, then scale to the given total.

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Answer: (a) 9

Two two-digit numbers add up to at most 197, so the leading 1 of 1CE is a carry. In the tens column A + C + carry = 10 + C, which forces A = 9 with a carry of 1 from the units column.

1. AB + CD ≤ 99 + 98 = 197, so the sum 1CE lies between 100 and 197.
2. Tens column: A + C + c = C + 10, where c (0 or 1) is the carry from the units column.
3. So A + c = 10. A is a single digit, so c = 1 and A = 9.
4. Check: 97 + 35 = 132 fits, with A = 9, B = 7, C = 3, D = 5, E = 2 all different.

Remember · In digit-sum puzzles, write each column as an equation with its carry; the leftmost column usually fixes a digit at once.

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Answer: (d) चार त्रिक

Since x ≤ 7, 2y ≤ 6 and so 3z < 6, which forces z = 1 and y = 2 or 3. That gives (5, 2, 1), (6, 2, 1), (7, 2, 1) and (7, 3, 1): four triplets.

1. x ≤ 7 and x > 2y, so 2y ≤ 6, i.e. y ≤ 3.
2. 2y > 3z with 2y ≤ 6 means 3z < 6, so z = 1; then 2y > 3 gives y = 2 or 3.
3. y = 2: x > 4, so x = 5, 6 or 7, giving three triplets.
4. y = 3: x > 6, so x = 7, giving one triplet.
5. Total = 4 triplets: (5, 2, 1), (6, 2, 1), (7, 2, 1), (7, 3, 1).

Remember · In chained inequalities, start from the tightest bound (here x ≤ 7) and work down to the smallest variable.

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Answer: (c) 1 और 2 दोनों

4O + 6M + 8A = 2(O + 2M + 5A) simplifies to O + M = A, which is statement 2. Statement 1 then follows, because 3O + 5M + 9A differs from 4O + 6M + 8A by A − O − M = 0.

1. Let the prices be O, M and A. Then 4O + 6M + 8A = 2O + 4M + 10A.
2. So 2O + 2M = 2A, i.e. O + M = A. Statement 2 is correct.
3. (3O + 5M + 9A) − (4O + 6M + 8A) = A − O − M = 0. Statement 1 is correct.
4. Check with O = 1, M = 2, A = 3: 4 + 12 + 24 = 40 = 2 × (1 + 4 + 15), and 3 + 10 + 27 = 40 as well.

• ✓ 1. The two totals differ by A − O − M, which is 0 because O + M = A.
• ✓ 2. Simplifying the given condition gives exactly O + M = A.

Remember · Simplify the given condition first; then test each statement by subtracting one side from the other.

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Answer: (b) इसका उपयोग कार्बन प्रच्छादन (सिक्वेस्ट्रेशन) की एक विधि के रूप में किया जा सकता है।

Carbon dioxide pumped into basalt ends up locked in stone (limestone). The practical suggestion is to use this as a way of storing carbon dioxide permanently, that is, carbon sequestration.

• ✗ (a) The passage says nothing about cost or about commercial limestone for building; its point is where the carbon dioxide goes.
• ✓ (b) ‘Basically carbon dioxide is converted into stone’: locking carbon dioxide away in solid rock underground is carbon sequestration.
• ✗ (c) The rock’s calcium and magnesium are used up in forming limestone; the method does not make basalt a source of these minerals.
• ✗ (d) The acidic mixture is only a step in the process; making acid is not what the finding is useful for.

Remember · Ask what problem the finding solves. The key word here is carbon dioxide, so the answer concerns climate, not construction.

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Answer: (c) 1 और 2 दोनों

Iceberg meltwater brings iron and nutrients that boost phytoplankton, the base of ocean food chains, and giant icebergs account for up to 20 percent of the carbon stored in the Southern Ocean. So icebergs affect productivity and food chains, and their melting bears on climate and on fisheries.

• ✓ 1. Phytoplankton are the ocean’s primary producers. Meltwater from giant icebergs ‘supports hitherto unexpectedly high levels of phytoplankton growth’, so icebergs bear on primary productivity and the food chains built on it.
• ✓ 2. Giant icebergs are ‘responsible for storing up to 20 percent of carbon in the Southern Ocean’, so their melting affects carbon storage and hence climate; and the phytoplankton growth they drive is the base on which fish stocks depend.

Remember · Link each fact to its direct consequence: phytoplankton lead to food chains and fisheries; carbon storage leads to climate.

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Answer: (c) यह खोज जैविक, पारिस्थितिक रूप से संधारणीय पीड़कनाशी दवाइयों (पेस्टिसाइड) के विकास में सहायता कर सकती है।

The passage shows that caterpillar frass switches off a plant’s defences, and says the find could throw light on compounds linked to how plants respond to pathogens. The practical message is that understanding these natural compounds can help develop organic, ecologically sound ways to protect crops.

• ✗ (a) The caterpillars feed on corn, a crop; nothing suggests using them against weeds.
• ✗ (b) The compounds concern how plants respond to pathogens; ‘clinically useful’ medicines for people go well beyond the passage.
• ✓ (c) Knowing the natural compounds that turn plant defences off (and on) can guide the development of organic, ecologically sustainable products to protect crops from pests and pathogens.
• ✗ (d) Genetic modification of caterpillars is not suggested anywhere.

Remember · Choose the application that stays within the passage’s field (plants, pests, crop protection); avoid jumps to medicine or genetic engineering.

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Answer: (c) 10

32⁵ = (2⁵)⁵ = 2²⁵, so the sum is 2²⁵ + 2²⁷ = 2²⁵ × (1 + 4) = 5 × 2²⁵. It has the factors 2 and 5, so it is divisible by 10.

1. 32 = 2⁵, so 32⁵ = 2²⁵.
2. 32⁵ + 2²⁷ = 2²⁵ + 2²⁷ = 2²⁵ × (1 + 2²) = 5 × 2²⁵.
3. This contains 2 × 5 = 10, so it is divisible by 10.
4. Its only prime factors are 2 and 5, so it is not divisible by 3, 7 or 11.

Remember · Write every term with the same base, take out the common power, and read off the factors.

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Answer: (c) 5

For k = 3 and k = 4 there is only one pair with p < q: (1, 2) and (1, 3). For k = 5 there are two, (1, 4) and (2, 3), so 5 is the smallest k that does not fix p and q.

1. k = 3: only (1, 2). Unique.
2. k = 4: only (1, 3), since (2, 2) breaks p < q. Unique.
3. k = 5: (1, 4) and (2, 3). Not unique.
4. So the smallest such k is 5.

Remember · For a ‘smallest value’ question, test the options in increasing order and stop at the first that works.

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Answer: (c) दक्षिण

Assume he starts facing North and trace the walk: the office ends up 200 m West and 200 m South, i.e. South-West. For it to be North-East the whole path must be turned through 180°, so he actually started walking South.

1. Try starting North: 100 m North; left, so 100 m West; left again, so 300 m South; right, so 100 m West.
2. Net: 300 − 100 = 200 m South and 100 + 100 = 200 m West, so the office would be South-West.
3. The office is actually North-East, exactly opposite, so rotate everything by 180°: the first walk was South.
4. Check: 100 m South; left, 100 m East; left, 300 m North; right, 100 m East. Net 200 m North and 200 m East, i.e. North-East.

Remember · Assume North, solve, then rotate the whole figure to match the given final direction.

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Answer: (d) उत्तर-पूर्व

Walking West and turning left, he faces South. A 225° clockwise turn from South (bearing 180°) brings him to bearing 405° − 360° = 45°, which is North-East.

1. Walking West, a left turn makes him face South.
2. Clockwise from South: 90° brings West, 180° brings North, and 225° is 45° beyond North towards East.
3. So he now faces North-East.
4. Check with bearings: 180° + 225° = 405°, i.e. 45°, which is North-East.

Remember · Convert to bearings (N 0°, E 90°, S 180°, W 270°), add clockwise turns, and subtract 360° if needed.

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Answer: (d) न तो निष्कर्ष–I और न ही निष्कर्ष–II निकलता है

Being the largest producer says nothing about exports or imports: a country may consume most of what it produces, and may still import some. Neither conclusion follows from the statement.

• ✗ I Large production does not mean large exports; home consumption may take most of it. The statement gives no trade data.
• ✗ II Producing the most does not rule out importing, for example particular milk products or to meet demand in some years.

Remember · A conclusion must follow from the statement alone; production figures never settle questions of trade.

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Answer: (c) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर दिया जा सकता है, किन्तु अकेले किसी भी एक कथन का प्रयोग कर नहीं दिया जा सकता है

m + n > mn with m > n forces n = 1, but leaves m open; mn = 24 alone allows several pairs. Together, n = 1 and m = 24.

1. m + n > mn is the same as mn − m − n + 1 < 1, i.e. (m − 1)(n − 1) < 1.
2. For natural numbers this means (m − 1)(n − 1) = 0; since m > n ≥ 1, n = 1. Statement I fixes n = 1 but any m ≥ 2 works, so it is not sufficient.
3. Statement II alone: mn = 24 allows (24, 1), (12, 2), (8, 3), (6, 4) and their reversals, so it is not sufficient.
4. Together: n = 1 and mn = 24 give m = 24. Check: 24 + 1 = 25 > 24.

Remember · Rewrite m + n > mn as (m − 1)(n − 1) < 1; with natural numbers, one of them must be 1.

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Answer: (c) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर दिया जा सकता है, किन्तु अकेले किसी भी एक कथन का प्रयोग कर नहीं दिया जा सकता है

Download time = size ÷ transfer rate. Statement I gives only the size and Statement II only the rate; together they fix the time.

1. Time = size of the software ÷ transfer rate.
2. Statement I gives the size (12 megabytes) but no rate, so it is not sufficient.
3. Statement II gives the rate (2.4 kilobytes per second) but no size, so it is not sufficient.
4. Together: 12 megabytes = 12,000 kilobytes (taking 1 MB = 1000 KB), so time = 12,000 ÷ 2.4 = 5,000 seconds; with 1 MB = 1024 KB it is 5,120 seconds. Either way the time is fixed.

Remember · In data sufficiency, write the formula first and tick off which statement supplies each quantity.

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Answer: (a) किसी आदर्श लोकतंत्र में जिम्मेदार मीडिया द्वारा वास्तविकता को विकृत नहीं करना चाहिए।

The passage insists that in a robust democracy reality, however inconvenient, must be expressed transparently, and that a responsible media must ‘circulate the content of reality without distortion’. That is option (a).

• ✓ (a) ‘A normatively responsible media … has the responsibility to circulate the content of reality without distortion’ in a robust democracy.
• ✗ (b) The passage demands undistorted reality; it never treats fake news as a natural part of democracy.
• ✗ (c) The passage is about truthful expression of reality, not about removing every restriction on free speech; ‘any kind’ is extreme.
• ✗ (d) Controlling media or leaders is not discussed; the passage says what a responsible media ought to do.

Remember · The crux usually sits in the passage’s prescriptive sentence (‘has the responsibility to…’).

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Answer: (d) अंतर-संबद्ध (इंटरकनेक्टेड) उपकरणों को सुरक्षा जोखिम हो सकती है और घरेलू उपयोगकर्ताओं को निजता (प्राइवेसी) जोखिम हो सकती है।

Makers of connected devices will gather personal data ‘with or without our agreement’ and turn the home into ‘a corporate storefront’. The practical message is that such devices carry privacy and security risks for their users.

• ✗ (a) Capital expenditure in manufacturing is not discussed.
• ✗ (b) The passage is wary of this data gathering; it does not praise it as a driver of trade or national development.
• ✗ (c) Comfort and quality of life are not mentioned; the passage dwells on the downside.
• ✓ (d) Personal data gathered ‘with or without our agreement’ and constant observation of consumer behaviour point to privacy and security risks for home users.

Remember · Read the tone: phrases like ‘with or without our agreement’ signal a warning, so pick the cautionary option.

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Answer: (c) स्थिर, पर्याप्त और समावेशी विकास प्राप्त करने के लिए उच्च पर्यावरण मानकों को बनाए रखना अब एक पूर्वापेक्षा है।

Green growth means rethinking growth around environmental sustainability and frugal use of resources; otherwise we face an economic dead end or affluence for a small elite amid poverty. So high environmental standards are now a condition for steady and inclusive growth.

• ✗ (a) The passage treats sustainability as necessary for growth to continue, not as opposed to it.
• ✗ (b) The passage warns against resource-intensive growth; it does not say environmental degradation is the price of ending poverty.
• ✓ (c) Resource-frugal strategies are needed ‘if we are to avoid an economic dead end’ (steady, sufficient growth) or a world where ‘only a small elite’ enjoys affluence (inclusive growth).
• ✗ (d) The passage offers a way forward; it does not call degradation and inequality inevitable.

Remember · When the passage proposes a solution, reject options that call the problem inevitable or the solution harmful.

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Answer: (c) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर दिया जा सकता है, किन्तु अकेले किसी भी एक कथन का प्रयोग कर नहीं दिया जा सकता है

‘x/y is odd’ alone allows endless pairs, and xy = 12 alone allows six ordered pairs. Together, of the pairs with product 12, only x = 6, y = 2 gives an odd quotient (3).

1. Statement I alone: x = 3y, 5y, 7y, …, for example (3, 1), (5, 1), (6, 2); not unique.
2. Statement II alone: (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1); not unique.
3. Together, test x/y for each pair: 12/1 = 12 (even), 6/2 = 3 (odd); 4/3, 3/4, 2/6 and 1/12 are not whole numbers.
4. Only x = 6, y = 2 works, so both statements together answer the question.

Remember · When one statement gives a short list, test each item on it against the other statement.

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Answer: (c) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर दिया जा सकता है, किन्तु अकेले किसी भी एक कथन का प्रयोग कर नहीं दिया जा सकता है

Statement I makes X’s share 4/9 of the total and Statement II makes Y’s share 2/9; together Z gets 3/9, so Y received the least. Either statement alone leaves the split between the other two open.

1. Statement I: X = (4/5)(Y + Z), so X : (Y + Z) = 4 : 5 and X = 4/9 of the total. Y and Z share 5/9 in an unknown way, so it is not sufficient.
2. Statement II: Y = (2/7)(X + Z), so Y = 2/9 of the total. X and Z share 7/9 in an unknown way (X could even be less than Y), so it is not sufficient.
3. Together: X = 4/9 and Y = 2/9, so Z = 1 − 6/9 = 3/9.
4. Y received the least.

Remember · ‘A gets p/q of what the others get’ means A’s share is p/(p + q) of the total.

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Answer: (d) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर भी नहीं दिया जा सकता

Statement II only tells us that the highest and lowest marks add up to 120 (twice the average). Statement I confirms 70 + 50 = 120, but neither statement says anything about how many students there are.

1. Let there be n students, with total marks 60n.
2. Statement II: (60n − H − L) ÷ (n − 2) = 60 gives H + L = 120, which is true for any n.
3. Statement I: H = 70 and L = 50, consistent with H + L = 120, but n is still free.
4. Example: marks 50, 60, 70 (n = 3) and 50, 60, 60, 70 (n = 4) both fit everything. So the question cannot be answered.

Remember · If a statement holds for every value of the unknown, it cannot fix that unknown; test two different cases.

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Answer: (a) प्रश्न का उत्तर, दोनों में से अकेले एक कथन का प्रयोग कर दिया जा सकता है, किन्तु अकेले अन्य कथन का प्रयोग कर नहीं दिया जा सकता है

Three distinct primes with a prime sum must all be odd, because including 2 makes the sum even. With the sum below 23, the only possibility is 3 + 5 + 11 = 19, so Statement I alone suffices. Statement II allows 3, 5, 11 and 5, 7, 11 and more.

1. If 2 is one of them, the sum is 2 + odd + odd = even (and more than 2), so not prime. All three must be odd primes.
2. Statement I: odd-prime triples with sum below 23 are 3 + 5 + 7 = 15, 3 + 5 + 11 = 19, 3 + 5 + 13 = 21 and 3 + 7 + 11 = 21; only 19 is prime. So the numbers are 3, 5 and 11: sufficient.
3. Statement II: triples containing 5 with a prime sum include {3, 5, 11} (19), {5, 7, 11} (23) and {3, 5, 23} (31), so it is not sufficient.
4. So the question can be answered by one statement (I) alone but not by the other.

Remember · A prime sum of three distinct primes cannot include 2 (the sum turns even). Start listing from the smallest odd primes.

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Answer: (d) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर भी नहीं दिया जा सकता

Neither statement alone fixes x + y. Even together they only make 3(x + y) an integer, so x + y can be a fraction such as 2/3 (x = y = 1/3) or an integer (x = y = 1).

1. Statement I alone: x = 0.5, y = 0 gives 2x + y = 1 but x + y = 0.5; x = y = 1 gives x + y = 2. Not sufficient.
2. Statement II alone: the same examples with x and y swapped show it is not sufficient.
3. Together: adding them, 3x + 3y is an integer, so x + y is only known to be an integer divided by 3.
4. x = y = 1/3 gives 2x + y = 1 and x + 2y = 1, yet x + y = 2/3; x = y = 1 gives x + y = 2. Both fit, so the answer is still open.

Remember · In ‘is it an integer?’ questions, try fractions like 1/3 or 1/2; they often break an apparent ‘yes’.

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Answer: (c) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर दिया जा सकता है, किन्तु अकेले किसी भी एक कथन का प्रयोग कर नहीं दिया जा सकता है

p + r = 50 − 16 = 34, with both more than 16. Statement I gives p ≤ 17 and Statement II gives p ≥ 17, so together p = ₹17; either alone leaves a range, since prices need not be whole rupees.

1. p + r = 50 − 16 = 34, and p and r are both more than 16 because q is the least.
2. Statement I (p ≤ r): p ≤ 17, so 16 < p ≤ 17; p could be ₹16.50 or ₹17. Not sufficient.
3. Statement II (r ≤ p): p ≥ 17, so 17 ≤ p < 18. Not sufficient.
4. Together, p ≤ r and r ≤ p give p = r = 34 ÷ 2 = ₹17.
5. Note: only if prices had to be whole rupees would each statement alone give ₹17; the question does not say so.

Remember · In data sufficiency, don’t assume whole numbers unless told. ‘Not more than’ in both directions means equality.

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Answer: (d) प्रश्न का उत्तर, दोनों कथनों का एक-साथ प्रयोग कर भी नहीं दिया जा सकता

Combining the statements gives only P > R and S > Q; nothing compares P with Q directly. Examples with P above Q and with P below Q both satisfy the data.

1. Statement I alone: P + Q = R + S says nothing about P versus Q.
2. Statement II alone: P + S > Q + R also depends on S and R, so it is not sufficient.
3. Together: putting R = P + Q − S into Statement II gives 2S > 2Q, i.e. S > Q; similarly P > R. Neither compares P with Q.
4. Example: P = 5, Q = 1, R = 4, S = 2 (P > Q) and P = 5, Q = 6, R = 4, S = 7 (P < Q) both satisfy I and II. So the question cannot be answered.

Remember · Combine the equations to see what they really fix, then build two examples that give opposite answers.

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Answer: (d) हिंसा, निरंकुशता और कट्टरता की दुनिया में, काव्य गतिशील विरोध का एक रूप है।

The passage sets poetry against the ‘tyranny of truth’ and the ‘arrogance of power, wealth and hierarchy’, answering them with ‘humility and defiance’. Its message is that poetry is a form of resistance.

• ✗ (a) The passage says poetry accepts the contradictions of reality and life; it does not speak of imperfections ‘in humans’, nor claim that only poetry recognises them.
• ✗ (b) The passage does not say that truth is revealed through science or religion, nor that poetry is ‘anathema’ to truth; it says absolute truth is ‘not of much use for poetry’.
• ✗ (c) ‘Romantic, imaginary’ and ‘about feeling’ are not in the passage, which is about plurality and democratic reality.
• ✓ (d) ‘Against the tyranny of truth, poetry remains a partisan of democratic reality. Against the arrogance of power, wealth and hierarchy, poetry proposes both humility and defiance.’ Poetry stands against tyranny.

Remember · Repeated ‘Against …’ lines carry the author’s message; here, poetry as resistance to tyranny and arrogance.

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Answer: (d) न तो 1 और न ही 2

The author says flowers exist for a purpose, to attract insects that carry pollen, and that their beauty is only a by-product and a human invention. Both statements say the opposite.

• ✗ 1. Flowers are not a luxury: their petals and perfume are there ‘to attract an insect’ so that pollen is not wasted.
• ✗ 2. The author sees flowers as useful precisely for pollination; their beauty is ‘merely a by-product and a human invention’.

Remember · If a statement reverses the author’s point, it is invalid, however natural it sounds.

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Answer: (a) प्रश्न का उत्तर, दोनों में से अकेले एक कथन का प्रयोग कर दिया जा सकता है, किन्तु अकेले अन्य कथन का प्रयोग कर नहीं दिया जा सकता है

With ages 10a + b and 10b + a, Statement II gives 11(a + b) = (11/6) × 9|a − b|, i.e. 2(a + b) = 3|a − b|, whose only solution is 51 and 15. The difference is 36 whichever is older, so Statement II alone suffices; Statement I alone does not.

1. Let the ages be 10a + b and 10b + a, with a and b non-zero digits. Sum = 11(a + b); difference = 9|a − b|.
2. Statement I alone only says which one is older, so it is not sufficient.
3. Statement II: 11(a + b) = (11/6) × 9|a − b|, which simplifies to 2(a + b) = 3|a − b|.
4. If a > b: 2a + 2b = 3a − 3b, so a = 5b; b = 1 gives a = 5 (b = 2 would need a = 10). The ages are 51 and 15, in some order.
5. Difference = 36 either way, so Statement II alone answers the question.
6. Check: 51 + 15 = 66 = (11/6) × 36.

Remember · For a number and its reverse: sum = 11(a + b), difference = 9(a − b). Ask only what the question actually needs.

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Answer: (c) SP और RQ दोनों

Under ‘either a Director or a Producer’, ruling out one role forces the other. So ‘not a Producer’ implies ‘a Director’ (SP), and ‘not a Director’ implies ‘a Producer’ (RQ).

• ✓ SP If Pradeep is not a Producer (S), the Main Statement leaves only Director (P). S implies P.
• ✓ RQ If Pradeep is not a Director (R), he must be a Producer (Q). R implies Q.

Remember · For ‘either A or B’, the valid pairs are ‘not A, therefore B’ and ‘not B, therefore A’.

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Answer: (d) −25

Replace every sign at once: 10 + 30 − 100 × 50 ÷ 25 becomes 10 − 30 × 100 ÷ 50 + 25. By BODMAS, 30 × 100 ÷ 50 = 60, so the value is 10 − 60 + 25 = −25.

1. Swap the signs: + becomes −, − becomes ×, × becomes ÷, ÷ becomes +.
2. 10 + 30 − 100 × 50 ÷ 25 becomes 10 − 30 × 100 ÷ 50 + 25.
3. 30 × 100 ÷ 50 = 3000 ÷ 50 = 60.
4. 10 − 60 + 25 = −25.

Remember · Rewrite the whole expression with the new signs first, then apply BODMAS; never substitute and calculate at the same time.

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Answer: (d) न तो 1 और न ही 2

Statement 1 gives 2x ≥ 3y and 3x = 4z, which lead to 9y ≤ 8z, so ‘9y > 8z’ fails. Statement 2 gives x < 2y and y ≤ z, which still allow x > z (for example x = 1.5, y = z = 1). Neither is correct.

1. Decode: P is >, Q is <, R is ≤ (not greater than), S is ≥ (not less than), T is =.
2. Statement 1: 2x ≥ 3y gives 9y ≤ 6x; 3x = 4z gives 8z = 6x. So 9y ≤ 8z, which rules out 9y > 8z. Incorrect.
3. Statement 2: x < 2y and y ≤ z. Take x = 1.5, y = 1, z = 1: both conditions hold, but x ≤ z fails. Incorrect.
4. So neither statement is correct.

• ✗ 1. From 2x ≥ 3y and 3x = 4z, 9y ≤ 6x = 8z, so 9y can never be greater than 8z.
• ✗ 2. x < 2y and y ≤ z only give x < 2z; x = 1.5, y = z = 1 satisfies both conditions but has x greater than z.

Remember · Decode the symbols, bring both sides to a common term (here 6x), and use one counter-example to kill an ‘always’ claim.

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Answer: (a) 11880

Each letter stands for its position in the alphabet and the code is the product: 1 × 2 × 3 × 4 = 24 and 5 × 6 × 7 × 8 = 1680. So IJKL = 9 × 10 × 11 × 12 = 11880.

1. ABCD: 1 × 2 × 3 × 4 = 24, which matches.
2. EFGH: 5 × 6 × 7 × 8 = 1680, which matches.
3. IJKL: 9 × 10 × 11 × 12 = 90 × 132 = 11880.

Remember · When letters become numbers, test alphabet positions with sum and product first.

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Answer: (d) AANERA

Each word is reversed and an A is put in front: POT → TOP → ATOP, and TRAP → PART → APART. ARENA reversed is ANERA, so the code is AANERA.

1. POT reversed is TOP; with A in front it is ATOP, which matches.
2. TRAP reversed is PART; with A in front it is APART, which matches.
3. ARENA reversed is ANERA; with A in front it is AANERA.

Remember · When a code looks like a real word, check for reversal plus a fixed added letter.

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Answer: (b) 155

The terms are n³ + n² + n for n = 1, 2, 3, …: 3, 14, 39, 84, 155, 258. So the missing term is 125 + 25 + 5 = 155.

1. First differences: 11, 25, 45. Second differences: 14, 20, rising by 6, so the next second difference is 26.
2. Next first difference = 45 + 26 = 71, so the missing term = 84 + 71 = 155.
3. Check the last term: next second difference 32, so the next first difference is 71 + 32 = 103, and 155 + 103 = 258.
4. Formula view: n³ + n² + n gives 3, 14, 39, 84, 155, 258.

Remember · If first differences don’t settle a series, take second differences; a steady rise in them points to a cubic pattern.

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Answer: (b) 2

Q − S = 2S means Q = 3S, so (Q, S) is (15, 5) or (12, 4). Only (15, 5) lets T = R + S + 3 work, with R = 4 and T = 12, which leaves P = 10. So P + R − T = 10 + 4 − 12 = 2.

1. Q − S = 2S gives Q = 3S. From 4, 5, 10, 12, 15, the options are (Q, S) = (15, 5) or (12, 4).
2. Case Q = 12, S = 4: T = R + 7 with R and T from 5, 10, 15; no pair works.
3. Case Q = 15, S = 5: T = R + 8 with R and T from 4, 10, 12; R = 4, T = 12 works, leaving P = 10.
4. P + R − T = 10 + 4 − 12 = 2.

Remember · Use the tightest equation first to list the cases, then eliminate cases with the second equation.

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