CSAT (GS Paper II) 2025

Answer: (c) Sustenance of livelihood and degradation of ecosystem while being together exacerbate one another, leading to conflicts and imbalance.

The passage is built around a clash between three uses of an ecosystem — livelihoods, commercial gain and conservation — and shows that degrading the forest hurts the very people who live off it. Option (c) captures this: livelihood needs and ecosystem degradation worsen each other and produce conflict and imbalance.

• ✗ (a) The author treats conservation as one of the competing needs, not as something to set aside. Nothing says livelihood dependence makes biodiversity a non-issue.
• ✗ (b) Fundamental rights are never mentioned. The passage speaks of harm to livelihoods, not of a rights violation — the idea comes from outside the passage.
• ✓ (c) Use of the ecosystem leads to deforestation, floods, siltation and microclimatic instability, which in turn damage forest-dependent livelihoods. The author calls these 'conflicts' and says they are acute where dependence on the ecosystem is high — the mutual worsening and imbalance that (c) describes.
• ✗ (d) Too extreme. The author notes that commercial exploitation 'has the potential to boost national income'; the passage describes a trade-off, not a call for a complete stop.

Remember · For 'critical message', choose the option that states the passage's central tension; drop options that add new ideas or use absolutes.

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Answer: (d) Neither I nor II

Neither statement is taken for granted by the author. The passage actually says commercial exploitation of ecosystems can boost national income, which contradicts I, and it never discusses sharing of resources between countries (II).

• ✗ I The last sentence says that in developing countries commercial exploitation 'has the potential to boost national income'. Saying no country needs ecosystems for national income goes against the passage, and 'no country' is an absolute claim.
• ✗ II Sharing of resources between resource-rich and resource-poor countries is not discussed anywhere. It is a policy suggestion brought in from outside the passage.

Remember · An assumption is what the author must take for granted; statements that contradict the text or add new policy ideas are invalid.

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Answer: (d) Timing plays a crucial role in the case of green technology development.

The passage explains that green technology gets cheaper the more it is used, so moving early and gradually brings its benefits without much disruption, while a late, chaotic transition costs more. The conclusion is about when to act, which is why timing is the central idea.

• ✗ (a) Economies of scale is only one of the reasons costs fall; the passage never calls it 'essential' for the transition. Its conclusion is about moving early, not about scale as a precondition.
• ✗ (b) Too broad. The passage speaks only of green innovation and renewable energy, not of 'modern technological progress' in general.
• ✗ (c) The size of economies is never compared. 'Economies' in the passage are those that move early, not large economies with an advantage.
• ✓ (d) The closing lines — moving early and gradually lets economies 'reap the benefits … without much disruption', and a late transition 'is costlier' — make timing the decisive factor.

Remember · The central idea usually sits in the author's concluding advice; examples such as solar prices only support it.

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Answer: (c) Both I and II

Both statements extend the author's own reasoning. The passage holds that early green investment pays off with less disruption and lower cost, which supports I, and it uses the collapse in solar prices as the proof that falling costs make the transition easier, which supports II.

• ✓ I The author says moving early lets economies 'reap the benefits of path-dependent green investment without much disruption' and that a late transition is 'costlier'. Lower cost and less disruption imply gains for growth and for public spending, and the argument is made for economies in general, which includes one like India.
• ✓ II Solar energy is offered as 'the strongest evidence' of the steep learning curve. The author's case assumes that other green technologies can follow the same falling-cost path, which would make the transition smoother.

Remember · Ask: does the author's argument need this to be true? Hedged statements ('most likely') that extend the author's logic are usually valid.

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Answer: (c) 8

Dividing N = p + q + r by N turns the condition into 1 = 1/a + 1/b + 1/c with three different whole numbers, and the only such solution is 1/2 + 1/3 + 1/6. So N must be a multiple of 6, and there are 8 multiples of 6 below 50.

1. Divide N = p + q + r by N: 1 = p/N + q/N + r/N = 1/a + 1/b + 1/c, where a = N/p, b = N/q, c = N/r are whole numbers.
2. Distinct factors mean a, b, c are distinct. Take a < b < c. If a ≥ 3, the sum is at most 1/3 + 1/4 + 1/5 < 1, so a = 2.
3. Then 1/b + 1/c = 1/2 with 2 < b < c: b = 3 gives c = 6; b = 4 gives c = 4 (not distinct); b ≥ 5 is too small. So (a, b, c) = (2, 3, 6).
4. Hence N must be divisible by 2, 3 and 6, i.e. a multiple of 6, and then N = N/2 + N/3 + N/6 always works.
5. Multiples of 6 below 50: 6, 12, 18, 24, 30, 36, 42, 48 — that is 8 numbers.
6. Check: 6 = 3 + 2 + 1 and 48 = 24 + 16 + 8.

Remember · When a number equals a sum of its own factors, divide through by it — the problem becomes unit fractions adding to 1.

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Answer: (d) More than 6

Why not the tempting option · UPSC's key is (d). The item says 'three prime numbers p, q and r', not three different primes, and uses 'distinct' only for the sums — so p = q = r is allowed, every prime below 20 can be the middle term and there are 8 sums. Assuming the primes must differ gives 4 sums, option (a), but that assumption is not in the item. In the exam, do not add 'distinct' where the question has not written it.

Since p − q = q − r, the three primes are in arithmetic progression and p + q + r = 3q. The item does not say the primes must be different, so p = q = r is allowed and each of the 8 primes below 20 can be q, giving 8 distinct sums — more than 6.

1. p − q = q − r gives p + r = 2q, so p + q + r = 3q. The sum depends only on the middle prime q.
2. The item does not require the primes to be different, so p = q = r is allowed (both differences are then 0).
3. So every prime q below 20 gives a value: q = 2, 3, 5, 7, 11, 13, 17, 19 gives p + q + r = 6, 9, 15, 21, 33, 39, 51, 57.
4. That is 8 distinct values, which is more than 6.
5. Trap: if one assumed the primes must be different, only (3, 5, 7), (3, 7, 11), (3, 11, 19), (5, 11, 17) and (7, 13, 19) would work, giving 4 sums — but the item says 'distinct' only of the sums, not of the primes, so equal primes count.

Remember · Equal differences mean an arithmetic progression: the sum of three terms is three times the middle one. Apply 'distinct' only where the question states it.

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Answer: (c) Three

Only three unordered triples satisfy 1/p + 1/q + 1/r = 1: (3, 3, 3), (2, 4, 4) and (2, 3, 6). Their sums 9, 10 and 11 are all different, so p + q + r can take three values.

1. Arrange so that p ≤ q ≤ r. Then 1/p is the largest of the three fractions, so 1/p ≥ 1/3, i.e. p ≤ 3; and p ≥ 2 because 1/p must be less than 1.
2. p = 3: 1/q + 1/r = 2/3 with q, r ≥ 3 forces q = r = 3. Sum = 9.
3. p = 2: 1/q + 1/r = 1/2 with q ≤ r gives (q, r) = (3, 6) or (4, 4). Sums = 11 and 10.
4. Possible values of p + q + r: 9, 10 and 11 — three values.
5. Check: 1/2 + 1/3 + 1/6 = 1, 1/2 + 1/4 + 1/4 = 1, 1/3 + 1/3 + 1/3 = 1.

Remember · For unit-fraction equations, sort the variables and bound the smallest one first; the cases then collapse quickly.

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Answer: (b) July, April

Counting forward, the gap between successive months shrinks by one each time: 12, 11, 10, 9, 8, 7, 6, 5 months. Nine months after October is July, and six months after October is April.

1. Write the months as numbers: 1, 1, 12, 10, X, 3, 10, Y, 9.
2. Count forward from each month to the next: January → January is 12 months, January → December is 11, December → October is 10.
3. The gap falls by one each time: October + 9 months = July (X); July + 8 = March; March + 7 = October.
4. October + 6 months = April (Y); April + 5 = September, which matches the last term.
5. So X = July and Y = April.

Remember · For month or weekday series, convert to numbers and study forward gaps, wrapping around after 12 (or 7).

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Answer: (d) Cannot be determined

The '12 overs' condition follows automatically from the first one: a run rate 1.5 times N/18 is N/12, which reaches N in exactly 12 overs whatever N is. With no independent equation, N cannot be found.

1. 10% fewer than 20 overs is 18 overs, so team Y's run rate was N/18 runs per over.
2. A 50% higher run rate is 1.5 × N/18 = N/12 runs per over.
3. At N/12 runs per over, N runs take N ÷ (N/12) = 12 overs — for every value of N.
4. The last condition is always true and gives no new information, so N cannot be determined.
5. Check: N = 72 and N = 144 both satisfy every condition, so the data cannot pick one value.

Remember · Test whether a second condition is independent; if it reduces to an identity, the value cannot be determined.

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Answer: (a) p(10⁴ − k²)² = q × 10⁸

Each k% rise followed by a k% fall multiplies the price by (100 + k)(100 − k)/10⁴ = (10⁴ − k²)/10⁴. Two such pairs give q = p(10⁴ − k²)²/10⁸, i.e. p(10⁴ − k²)² = q × 10⁸.

1. A k% increase multiplies the price by (100 + k)/100; a k% decrease multiplies it by (100 − k)/100.
2. One increase and one decrease: (100 + k)(100 − k)/10⁴ = (10⁴ − k²)/10⁴.
3. This pair happens twice, so q = p × (10⁴ − k²)²/10⁸.
4. Cross-multiplying: p(10⁴ − k²)² = q × 10⁸.
5. Check with k = 10: q = p × 1.1 × 0.9 × 1.1 × 0.9 = 0.9801p, and (10⁴ − 100)²/10⁸ = 9900²/10⁸ = 0.9801.

Remember · Successive percentage changes multiply. A k% rise and k% fall together leave the factor (10⁴ − k²)/10⁴ — always a net loss.

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Answer: (d) India needs to formulate effective climate change mitigation and adaptation strategies at the State/region level.

The passage shows how the impacts of warming and the chances to cut emissions differ from coastal States to dry districts to the Himalaya and peninsular wind sites, and says these differences must shape future goals. The practical message is region-specific planning, as in (d).

• ✗ (a) The passage stresses that each State's challenges differ, but it does not make climate action 'essentially the responsibility of each State'. Who is responsible is never discussed.
• ✗ (b) The author treats diversity as something to plan around, not as a reason no strategy can work. This reverses the passage's constructive tone.
• ✗ (c) The Union Government and net zero emissions are not mentioned; the passage's whole point is regional variation.
• ✓ (d) Sea-level rise for coastal States, variable monsoon for drier districts, wind power in peninsular and offshore areas — the author says 'these various aspects need to be considered' for clear goals. That is a case for State- or region-level mitigation and adaptation strategies.

Remember · A 'pragmatic' message proposes a workable approach drawn from the passage; reject options that shift responsibility, declare things impossible or add new actors.

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Answer: (a) I only

The author treats a State's natural resources, such as wind, as opportunities to cut emissions within its response to warming, so linking green energy with climate strategy is assumed. The passage calls regional challenges distinctive, not more severe, so II is not assumed.

• ✓ I The passage pairs each State's warming challenges with 'opportunities for reducing emissions depending on its natural resources', citing wind power in peninsular and offshore areas. Treating green energy as part of the climate response is taken for granted.
• ✗ II Coastal States and Himalayan regions are cited as having their own distinct challenges, not more severe ones. The passage makes no comparison of severity between regions.

Remember · Watch comparatives like 'much more severe': if the passage only says 'distinctive' or 'unique', the comparison is not assumed.

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Answer: (d) Insecurity, more than poverty, is the main economic issue that Government policies must address.

The passage's key claim is that insecurity — felt even by people just above the poverty line — is a more acutely felt economic problem than poverty itself, and that many Government policies aim to ease it. Option (d) restates this.

• ✗ (a) The passage calls social inequality the main social problem and insecurity the main economic problem. It does not merge poverty, inequality and insecurity into one social problem, nor does it address the political executive.
• ✗ (b) No causal link from poverty to social inequality is drawn. The passage in fact ranks insecurity above poverty.
• ✗ (c) 'Unmanageable crisis' is extreme and absent; the passage notes that Government policies already aim at mitigating insecurities.
• ✓ (d) The passage says insecurity, 'more than poverty', is the most acutely felt economic problem, and that many Government policies are 'intended towards mitigating these insecurities'.

Remember · When the author ranks one thing 'more than' another, the right option keeps that ranking; options that reverse or blur it are traps.

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Answer: (a) I only

The passage directly says that even people just above the poverty line face several economic insecurities, so I holds. It never links removing poverty with peace or social equality, so II is an outside claim.

• ✓ I The passage says 'even those just over the poverty line are subject to multiple economic insecurities' arising from wealth or health risks, market fluctuations and job-related uncertainties.
• ✗ II Nothing is said about poverty eradication bringing peace or social equality. The passage actually shifts attention away from poverty towards insecurity.

Remember · An assumption that promises a sweeping result, like 'peace and social equality', from a single cause is usually beyond the passage.

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Answer: (c) Both I and II

The fewest cuts come from dividing the cube into 3 × 4 × 5 slices, which takes 2 + 3 + 4 = 9 cuts. The inner block untouched by paint is (3 − 2) × (4 − 2) × (5 − 2) = 6 pieces, so both statements are correct.

1. If the cube is cut into a × b × c pieces, it needs (a − 1) + (b − 1) + (c − 1) cuts. With a × b × c = 60, make a + b + c as small as possible.
2. Factors closest to one another give the smallest sum: 3 × 4 × 5 = 60 with sum 12 (2 × 5 × 6 gives 13, 2 × 2 × 15 gives 19). Minimum cuts = 2 + 3 + 4 = 9.
3. Unpainted pieces are the ones strictly inside: (3 − 2) × (4 − 2) × (5 − 2) = 1 × 2 × 3 = 6.
4. Both statements are correct.

• ✓ I The 3 × 4 × 5 split needs 2 + 3 + 4 = 9 cuts; every other way of making 60 pieces needs more.
• ✓ II Removing the painted outer layer leaves an inner block of 1 × 2 × 3 = 6 pieces with no paint.

Remember · a × b × c pieces need (a + b + c − 3) cuts; unpainted pieces = (a − 2)(b − 2)(c − 2).

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Answer: (c) 65

Both examples are Pythagorean triples: 7, 24, 25 and 12, 16, 20. So a b means √(a² + b²), and 16 63 = √(256 + 3969) = √4225 = 65.

1. 7² + 24² = 49 + 576 = 625 = 25², and 12² + 16² = 144 + 256 = 400 = 20². So a * b = √(a² + b²).
2. 16 * 63 = √(16² + 63²) = √(256 + 3969) = √4225 = 65.
3. Check: 65² = 4225.

Remember · When results look like hypotenuses, test √(a² + b²); know common triples such as 7-24-25 and 16-63-65.

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Answer: (b) 200 km

With a fixed budget, the distance he can drive is inversely proportional to the petrol price. A 10% price rise brings the distance down to 100/110 of 2200 km, i.e. 2000 km, so he must travel 200 km less.

1. New price = ₹90 × 1.1 = ₹99 per litre.
2. Petrol used now: 2200 ÷ 16 = 137.5 litres, costing 137.5 × 90 = ₹12,375.
3. At ₹99 a litre, ₹12,375 buys 12,375 ÷ 99 = 125 litres, enough for 125 × 16 = 2000 km.
4. Reduction = 2200 − 2000 = 200 km.
5. Check (faster route): distance falls in the ratio 100 : 110, so 2200 × 100/110 = 2000 km; the mileage figure is not even needed.

Remember · Fixed spending: quantity varies inversely with price. A p% price rise cuts quantity by p/(100 + p) — here 10/110 = 1/11.

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Answer: (c) 1078

Each remainder is 2 less than its divisor, so N + 2 is a multiple of LCM(3, 5, 6, 9) = 90. The smallest 4-digit number of the form 90k − 2 is 1080 − 2 = 1078.

1. In every case the remainder is 2 short of the divisor: 3 − 1 = 5 − 3 = 6 − 4 = 9 − 7 = 2.
2. So N + 2 is divisible by 3, 5, 6 and 9, i.e. by their LCM, 90.
3. N = 90k − 2. k = 11 gives 988 (only 3 digits); k = 12 gives 1080 − 2 = 1078.
4. Check: 1078 = 3 × 359 + 1 = 5 × 215 + 3 = 6 × 179 + 4 = 9 × 119 + 7.

Remember · If divisor minus remainder is the same d for every divisor, the number is (LCM × k) − d.

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Answer: (d) Neither I nor II

A comparison is certain only when every sign between the two letters points the same way. B and E are separated by a change of direction at C, and S = T ≤ U = V allows S to equal V, so neither conclusion is always true.

1. Statement I: between B and E the chain reads B > C < D > E. The direction changes at C, so B and E cannot be compared.
2. Example: B = 5, C = 1, D = 10, E = 8 fits the chain, and here B is less than E. So I is not always true.
3. Statement II: between S and V the chain reads S = T ≤ U = V, which gives only S ≤ V.
4. If T = U, then S = V, so S is not always less than V. II is also not always true.
5. Neither statement is correct.

• ✗ I B > C and D > E, but C < D breaks the chain; B can be smaller than, equal to or larger than E.
• ✗ II S = T ≤ U = V gives S ≤ V, so S may equal V; 'always less than' does not follow.

Remember · Trace the signs between the two letters: a change of direction means no conclusion; any ≤ or ≥ rules out strict < or >.

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Answer: (c) 5

The product of all odd numbers up to 999 is odd and contains 5 as a factor. An odd multiple of 5 always ends in 5.

1. The product contains the factor 5, so it is a multiple of 5 and its unit digit is 0 or 5.
2. Every factor is odd, so the product is odd and cannot end in 0.
3. Hence the unit digit is 5.
4. Check: 1 × 3 × 5 = 15 already ends in 5, and multiplying any number ending in 5 by an odd number keeps the last digit 5.

Remember · Odd × 5 ends in 5; an even number times 5 ends in 0. Spot a factor of 5 before multiplying anything.

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Answer: (a) The benefit of economic reforms percolates down more slowly to the agriculture sector than in other sectors of the economy.

The passage contrasts agriculture, which peaked at 3.5% growth in the 1980s and then stagnated, with non-agricultural growth that kept rising into the 2000s. The only option that follows from this contrast is that the gains of the reform period reached agriculture more slowly than the rest of the economy.

• ✓ (a) Non-agricultural growth 'consistently increased from the 1980s to 2000s' — a period spanning the economic reforms — while agriculture 'suffered relative stagnation'. The natural corollary is that the benefits of that period reached agriculture more slowly than other sectors.
• ✗ (b) The green revolution is not mentioned; the passage in fact records agriculture's best decade in the 1980s and draws no verdict on any programme.
• ✗ (c) No comparison with other countries or with mechanisation is made. The passage compares agriculture only with India's own non-agricultural economy.
• ✗ (d) Migration is not discussed. The passage describes the stagnation but does not attribute it to any cause.

Remember · A corollary follows from the facts given; reject options that name a cause (migration, green revolution, other countries) the passage never discusses.

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Answer: (d) Neither I nor II

The passage only records the widening gap between agricultural and non-agricultural growth; it neither diagnoses causes nor suggests remedies. Both statements are specific policy prescriptions, so neither is an assumption the author makes.

• ✗ I The passage offers no remedy for the divergence. Commercial crops and corporate farming are solutions the reader brings in, not something the author relies on.
• ✗ II Free crop insurance and subsidies 'at par with developed countries' are detailed policy prescriptions. Nothing in the passage discusses insurance, subsidies or developed countries.

Remember · If a passage only describes a trend, any assumption offering a specific cure is almost always invalid.

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Answer: (a) We need to free the handloom industry from the limited narrative linked to preserving cultural heritage.

The author argues that tying handloom to a single idea of 'tradition' has backfired: that idea is now contested, and many people who do not see it as theirs ignore handloom. The logical message is to move handloom beyond this narrow cultural-heritage narrative.

• ✓ (a) The passage says the notion of tradition 'as a single, linear entity is being strongly contested' and that promoting handloom 'as a traditional industry' leads large sections to ignore it. Freeing handloom from this limited heritage narrative follows directly.
• ✗ (b) This restates the very policy view the author questions. The passage suggests the 'tradition' framing limits handloom's appeal rather than protects it.
• ✗ (c) Modernisation and organised-industry status are not discussed; the passage is about narratives and identity, not production methods.
• ✗ (d) Converting to machine-made products would end handloom itself, and nothing in the passage suggests it.

Remember · When an author critiques a framing, the right answer usually calls for changing that framing, not reinforcing it.

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Answer: (d) Neither I nor II

The passage criticises the narrative used to promote handloom, not State involvement or the products themselves. Both statements draw conclusions the author never makes.

• ✗ I The passage questions the basis on which the State supports handloom (tradition), not State support as such. 'No need … in any manner' is an extreme claim.
• ✗ II The passage explains that large sections ignore handloom because it is promoted as 'traditional' through a contested narrative, not because the products have lost their appeal in a changing world.

Remember · Separate the author's target (a narrative) from the thing described (the sector, its products); assumptions attacking the wrong target are invalid.

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Answer: (c) 22

Divisibility by 9 is already covered by 3, so count the numbers from 1 to 100 that are not divisible by 2, 3, 5 or 7. Below 121 these are just 1 and the primes from 11 to 97, which gives 1 + 21 = 22.

1. 9 is a multiple of 3, so only 2, 3, 5 and 7 matter.
2. Multiples of 2, 3, 5, 7 up to 100: 50 + 33 + 20 + 14 = 117.
3. Multiples of 6, 10, 14, 15, 21, 35: 16 + 10 + 7 + 6 + 4 + 2 = 45.
4. Multiples of 30, 42, 70, 105: 3 + 2 + 1 + 0 = 6; multiples of 210: 0.
5. Divisible by at least one = 117 − 45 + 6 = 78, so not divisible by any = 100 − 78 = 22.
6. Check: such numbers are 1 and the 21 primes from 11 to 97 (the smallest composite without the factors 2, 3, 5, 7 is 11² = 121), so 1 + 21 = 22.

Remember · Drop redundant divisors first (9 with 3). Up to 120, numbers free of 2, 3, 5, 7 are just 1 and primes from 11 onwards.

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Answer: (d) None of the above

The largest x + y is 8 + 7 = 15 and the smallest x − y is 4 − 7 = −3, so the ratio is −5. None of the listed values matches; 15/2 comes from wrongly taking the smallest x − y as 4 − 2.

1. Maximum of (x + y): take both at their largest, 8 + 7 = 15.
2. Minimum of (x − y): take the smallest x and the largest y, 4 − 7 = −3.
3. Ratio = 15 ÷ (−3) = −5.
4. −5 is not 6, 15/2 or −15/2, so the answer is 'None of the above'.
5. Check: the trap 15/2 uses 4 − 2 = 2, but y = 7 makes x − y smaller.

Remember · To minimise a difference, take the smallest first term and the largest second term — and keep track of the sign.

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Answer: (b) 5

If p and k were both odd, p² + k would be even and larger than 2, so one of them must be 2. With p = 2, k can be 3, 7, 13 or 19; with k = 2 (and p = 3, giving 11) k = 2 also works, so k has 5 possible values.

1. If p and k are both odd primes, p² + k is odd + odd = even and greater than 2, so it cannot be prime. Hence p = 2 or k = 2.
2. p = 2: 4 + k must be a prime below 30. k = 3, 7, 13, 19 give 7, 11, 17, 23 (prime); k = 2, 5, 11, 17, 23 give 6, 9, 15, 21, 27 (not prime).
3. k = 2: p² + 2 must be a prime below 30. p = 3 gives 11 (prime); p = 5 gives 27 (not prime). So k = 2 is possible.
4. Possible values of k: 2, 3, 7, 13, 19 — five values.

Remember · Odd + odd is even: when a sum of primes must be prime, one of them is usually 2.

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Answer: (d) More than 8

Every number in such a set is a divisor of 1001 = 7 × 11 × 13. The sets that include 1 alone number 12, already more than 8; a full count gives 32 sets of three different numbers.

1. 1001 = 7 × 11 × 13, so every number must be one of its divisors: 1, 7, 11, 13, 77, 91, 143, 1001.
2. The LCM must contain 7, 11 and 13, and the HCF must be 1, so no prime may divide all three numbers.
3. Sets containing 1 automatically have HCF 1. Those with LCM 1001: {1, 7, 143}, {1, 11, 91}, {1, 13, 77}, {1, 77, 91}, {1, 77, 143}, {1, 91, 143}, and {1, x, 1001} for x = 7, 11, 13, 77, 91, 143 — 12 sets.
4. 12 is already more than 8; sets such as {7, 11, 13} and {7, 11, 143} add still more (32 in all with three different numbers).
5. So n is more than 8.

Remember · With a 'more than' option, you only need to cross the threshold: count the easiest family of cases and stop.

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Answer: (b) 3

Writing the numbers in place-value form gives 30 + R − S − T = 20P, which forces P = 1 and S + T = R + 10. With distinct digits, Q = 3 and T below 6, only (R, S) = (2, 8), (2, 7) and (4, 9) work.

1. PQR − PS = PPT means (100P + 10Q + R) − (10P + S) = 100P + 10P + T.
2. Simplify: 90P + 10Q + R − S = 110P + T, so 10Q + R − S − T = 20P. With Q = 3: 30 + R − S − T = 20P.
3. The left side is at most 30 + 9 − 1 − 1 = 37, so 20P = 20 and P = 1. Then S + T = R + 10.
4. P = 1 and Q = 3 are used, so R, S, T are different digits from 2, 4, 5, 6, 7, 8, 9, with T = 2, 4 or 5.
5. T = 2: S = R + 8 needs R = 1 (not allowed). T = 4: S = R + 6 gives R = 2, S = 8. T = 5: S = R + 5 gives (R, S) = (2, 7) or (4, 9).
6. So (R, S) has 3 possible values.
7. Check: 132 − 18 = 114, 132 − 17 = 115, 134 − 19 = 115.

Remember · For letter-digit equations, expand in place values, simplify, and use digit limits to fix the leading digit first.

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

The 18-letter series is the block ABBCCC written three times. Filling the five blanks to fit this block gives B, C, B, A, C.

1. The series has 18 places. Split it into three blocks of six: AB_CC_ | A_BCCC | _BBC_C.
2. The middle block A_BCCC suggests the repeating block ABBCCC.
3. Fill every block as ABBCCC: first block needs B and C, second needs B, third needs A and C.
4. In order the blanks are B, C, B, A, C.
5. Check: ABBCCC ABBCCC ABBCCC.

Remember · For letter series with blanks, count the length, try equal blocks, and start from the most complete block.

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Answer: (d) Classroom needs to be reimagined and teaching needs to be re-invented.

The author criticises exam-focused teaching and calls for critical thinking, learning by doing and questioning from many disciplines, ending with 'Teaching has to be re-invented'. Option (d), reimagining the classroom and re-inventing teaching, carries that central idea.

• ✗ (a) Resources and funding are not discussed; the passage is about how teaching and learning are done.
• ✗ (b) The quotation says higher education should not be 'a mere servant of the government policy' — a point about independent thinking, not about removing institutions from Government control. The option overreaches.
• ✗ (c) The author wants teaching re-invented, not the classroom abolished. 'Done away with' is extreme.
• ✓ (d) The passage contrasts learning for the examination with critical thinking, learning through doing and multidisciplinary enquiry, and ends with 'Teaching has to be re-invented' — which (d) restates.

Remember · In a short passage the last line often states the thesis; pick the option that restates it without extremes.

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Answer: (a) I only

The author asks higher education to change because the demands of society and the future of work are changing, so I is taken for granted. Funds for higher education are never discussed, so II is not.

• ✓ I The author says 'the new demands of society and the future of work' require a different kind of learning, and that teaching must be re-invented. This takes for granted that higher education has to keep adapting to changes in society.
• ✗ II Money is never mentioned. The problem the passage identifies is the method of teaching — learning for the examination — not a shortage of funds.

Remember · An assumption must connect to the author's reasons; a new cause such as funding, absent from the passage, is invalid.

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Answer: (b) Animal-based food is far less efficient than grain/plant-based food in terms of production and utilization.

The passage's thrust is that feeding grain to animals wastes most of its calories: only 16% returns to us through chicken and five to seven per cent through large animals. Animal-based food is therefore far less efficient than eating grains and legumes directly.

• ✗ (a) The passage does not call for a public policy or for promoting cereal consumption; it explains an inefficiency.
• ✓ (b) The author says the conversion of plant-based food into animal food 'is far from efficient' and supports it with figures: 16% of calories recovered from chickens and five to seven per cent from large animals.
• ✗ (c) No dietary prescription is made. The author describes the calorie loss but does not say plant protein should replace animal protein.
• ✗ (d) The passage never calls inequality 'inevitable in any fast changing society'; it links the shift in food use to growing wealth.

Remember · The crux is usually the claim the passage backs with data; here the calorie figures support the inefficiency point.

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Answer: (b) II only

The passage links rising wealth to more animal products, and shows that feeding grain to animals loses most of its calories, so II follows. It says nothing about food manufacturing and processing industries, so I is not assumed.

• ✗ I Food manufacturing and processing industries are not discussed at all. This is a prescription from outside the passage.
• ✓ II As societies become wealthier they eat more animal products, so more grain and legumes go to animal feed, and only 16% (chickens) or five to seven per cent (large animals) of those calories come back. That is a large calorie loss through indirect use of farm produce.

Remember · A valid assumption often chains two lines of the passage (wealth → more animal food → calorie loss); advice to new actors is invalid.

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Answer: (b) 4

35ⁿ needs n factors of 5 and n factors of 7. The product has only four 5s (one from 385, three from 1000) but ten 7s, so the maximum n is 4.

1. 35 = 5 × 7, so count the 5s and the 7s in the product; n is the smaller count.
2. Fives: 385 = 5 × 7 × 11 gives one, and 1000 = 2³ × 5³ gives three; 7, 343, 2401 and 77777 have none. Total = 4.
3. Sevens: 7 (one), 343 = 7³ (three), 385 (one), 2401 = 7⁴ (four), 77777 = 7 × 11111 (one). Total = 10.
4. Each 35 needs one 5 and one 7, so n = the smaller of 4 and 10 = 4.

Remember · For divisibility by a power of a composite number, count each prime factor separately; the scarcer prime sets the limit.

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Answer: (b) 12

Each term is the previous one multiplied by 0.5, 1, 1.5, 2 and 2.5 in turn. So X = 24 × 0.5 = 12, and 12 × 1 gives the next term, 12.

1. From 12 onwards: 12 × 1.5 = 18, 18 × 2 = 36, 36 × 2.5 = 90. The multiplier rises by 0.5 each time.
2. Going backwards, the earlier multipliers are 1 and 0.5.
3. So X = 24 × 0.5 = 12, and X × 1 = 12 matches the third term.
4. Check: 24 × 0.5 = 12, 12 × 1 = 12, 12 × 1.5 = 18, 18 × 2 = 36, 36 × 2.5 = 90.

Remember · If differences do not work, try ratios; ratios rising by a fixed step (0.5, 1, 1.5 …) are a common UPSC pattern.

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Answer: (b) 13

Why not the tempting option · UPSC's key is 13. Counting the meeting exactly at 7:00 a.m. would give 14, but 'between 5:20 a.m. and 7:00 a.m.' excludes the end points, and 5:20 itself is not a meeting time. In the exam, read 'between' as excluding the end points, as UPSC does.

Walking in opposite directions, together they cover 8 rounds an hour, so they meet every 7.5 minutes. Between 5:20 and 7:00 they meet at 5:22:30, 5:30, … 6:52:30 — 13 times; the meeting exactly at 7:00 falls on the end point and is not 'between' the two times.

1. In opposite directions, they close the gap at 5 + 3 = 8 rounds per hour, so they meet once every 1/8 hour = 7.5 minutes.
2. Meetings happen 7.5, 15, 22.5, 30, … minutes after 5:00 a.m.
3. 5:20 is 20 minutes and 7:00 is 120 minutes after 5:00. The first meeting after 5:20 is at 22.5 minutes (5:22:30).
4. Meetings from 22.5 to 112.5 minutes: (112.5 − 22.5) ÷ 7.5 + 1 = 13. The next one, at 120 minutes, is exactly 7:00 a.m. — the end point, not between the two times.
5. So they cross each other 13 times.

Remember · Opposite directions on a circular track: meetings per hour = sum of rounds per hour. Then check the end points: 'between' excludes them.

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Answer: (b) QRPS

Replacing the letters as in (b) gives 60 − 15 × 3 + 20 ÷ 4 = 60 − 45 + 5 = 20. The other orders give 60, 61.75 and −73.

1. Test each option, doing ÷ and × before + and − (BODMAS).
2. (a) SPRQ: 60 ÷ 15 + 3 × 20 − 4 = 4 + 60 − 4 = 60.
3. (b) QRPS: 60 − 15 × 3 + 20 ÷ 4 = 60 − 45 + 5 = 20, which matches.
4. (c) QRSP: 60 − 15 × 3 ÷ 20 + 4 = 60 − 2.25 + 4 = 61.75.
5. (d) SPQR: 60 ÷ 15 + 3 − 20 × 4 = 4 + 3 − 80 = −73.

Remember · In symbol-substitution items, test each option with BODMAS; a rough size check quickly rules out most options.

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Answer: (c) 20 m

To pass a walker, the tram covers its own length at the relative speed, so (v − 3) × 8 = (v − 4) × 9, giving v = 12 km/h. Relative to X it moves at 9 km/h = 2.5 m/s, so in 8 seconds it covers 20 m, which is its length.

1. Let the tram's speed be v km/h. Passing a walker completely means covering the tram's length at the relative speed.
2. Length = (v − 3) × 8 = (v − 4) × 9 (same units), so 8v − 24 = 9v − 36 and v = 12 km/h.
3. Relative speed past X = 12 − 3 = 9 km/h = 9 × 5/18 = 2.5 m/s.
4. Length = 2.5 m/s × 8 s = 20 m.
5. Check with Y: (12 − 4) × 5/18 = 20/9 m/s, and 20/9 × 9 s = 20 m.

Remember · Overtaking a moving person: length = speed difference × time; equate two such lengths to find the speed. 1 km/h = 5/18 m/s.

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Answer: (b) 9

A perfect square with 17 digits has a square root of (17 + 1)/2 = 9 digits. Indeed 111111111² = 12345678987654321, and 111111111 has 9 digits.

1. 12345678987654321 has 17 digits.
2. A square with an odd number of digits d has a root with (d + 1)/2 digits: (17 + 1)/2 = 9.
3. Check: 111111111² = 12345678987654321, and 111111111 has 9 digits.

Remember · Digits in √N: (d + 1)/2 if N has d digits with d odd, d/2 if d is even. Also, 11…1² gives 12…n…21.

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Answer: (b) Low-carbon behaviour in people can be brought about by incentivising them.

The passage says household emissions must fall to almost nothing, that at best half of this can come from people changing their behaviour, and that even this needs companies and governments to make low-carbon options cheaper and easier to get. Its central idea is that incentives bring about low-carbon behaviour.

• ✗ (a) The passage says household emissions will have to be cut, not that net zero is possible 'only' through them. The word 'only' makes the option extreme.
• ✓ (b) Behavioural change by individuals and households 'would require companies and governments to provide more incentives to change' — the core idea is that incentives drive low-carbon behaviour.
• ✗ (c) This reverses the logic. The passage wants investment to make low-carbon options cheaper; it does not claim that low-carbon technology makes goods and services cheaper in general.
• ✗ (d) Subsidies to manufacturing industries are not mentioned; the passage speaks of incentives and supply-side investment in general.

Remember · Central idea = the author's main claim with its condition; reject options that add 'only' or name a tool (subsidies) the passage never names.

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Answer: (a) I only

The author links cheaper, widely available low-carbon options, created through supply-side investment, to changes in household behaviour, so I is assumed. The passage never says people cannot change without Government and companies; it only says change on the required scale needs their incentives.

• ✓ I The passage says supply-side investments should make low-carbon options 'cheaper and more widely available', and that behavioural change requires such incentives. It takes for granted that supply-side investment can shape people's behaviour.
• ✗ II The passage is about how much of the reduction behaviour can deliver ('at best, half') and what would help it; it does not say individuals are incapable of changing on their own. 'Not capable' is an extreme claim.

Remember · Separate 'needed for large-scale change' from 'people are incapable'; assumptions that turn a practical need into an absolute inability are invalid.

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Answer: (c) Circular economy can be beneficial for sustainable growth.

The passage builds a case for recovering materials from old batteries and putting them into new ones, because raw materials are being used up fast, are costly and are hard to source. The measured, practical message is that such circular use of resources can support sustainable growth.

• ✗ (a) 'Not possible without' is extreme, and a 'green economy' is not the passage's subject; the author says reuse 'makes sense', not that it is indispensable.
• ✗ (b) 'Every sector … immediately' goes far beyond the passage, which uses only the example of vehicle batteries.
• ✓ (c) Rising raw-material use, reuse below 9%, pricey and hard-to-get battery minerals and risky supply chains lead the author to say 'Reusing materials makes sense' and to propose recycling batteries into new ones — the circular idea, presented as sensible for sustained growth.
• ✗ (d) 'The only option … for their survival' is too strong; the passage calls reuse sensible, not a matter of survival.

Remember · 'Pragmatic' answers are measured: prefer 'can be beneficial' over 'not possible without', 'every sector immediately' or 'only option'.

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Answer: (b) II only

The author recommends recycling battery materials so that fast-growing demand can be met, which assumes that growth and circular use of minerals can go together. The passage does not say automobile factories are already circular; it says batteries should be sent back to a factory.

• ✗ I The passage says old batteries 'should go back to a factory' where their ingredients can be recovered — a recommendation. It never says automobile factories already work this way.
• ✓ II Raw-material use has nearly quadrupled, and the author's answer is to reuse materials rather than cut production. This takes for granted that economic activity can keep growing on recycled minerals.

Remember · A recommendation ('should go back to a factory') is not a description of current practice; do not read 'should' as 'is'.

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Answer: (d) 16 minutes

Each X-pipe fills 1/400, each Y-pipe fills 1/160 and each Z-pipe empties 1/640 of the tank per minute. With 10 X-pipes, 5 Y-pipes and 16 Z-pipes open, the X and Z effects cancel (1/40 each), leaving 1/32 per minute, so half the tank takes 16 minutes.

1. Take the tank as 1. Set X: 20 pipes fill 0.7 in 14 minutes, so one pipe fills 0.7 ÷ (14 × 20) = 1/400 per minute.
2. Set Y: 10 pipes fill 3/8 in 6 minutes, so one pipe fills (3/8) ÷ 60 = 1/160 per minute.
3. Set Z: 16 pipes empty 1/2 in 20 minutes, so one pipe empties (1/2) ÷ 320 = 1/640 per minute.
4. Open pipes: 10 of X fill 10/400 = 1/40; 5 of Y fill 5/160 = 1/32; all 16 of Z empty 16/640 = 1/40 per minute.
5. Net rate = 1/40 + 1/32 − 1/40 = 1/32 of the tank per minute.
6. Time for 50% = (1/2) ÷ (1/32) = 16 minutes.

Remember · With sets of pipes, find the rate of one pipe first, then multiply by the number open; watch for rates that cancel.

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Answer: (c) 2

1ⁿ is always 1, and 2ⁿ is a multiple of 4 once n ≥ 2. So the remainder is 3 when n = 1 and 1 for every larger n — two distinct remainders.

1. n = 1: 1 + 2 = 3, remainder 3.
2. n ≥ 2: 2ⁿ is divisible by 4 and 1ⁿ = 1, so 1ⁿ + 2ⁿ leaves remainder 1.
3. Only two remainders occur: 3 and 1.
4. Check: n = 2 gives 5 (remainder 1), n = 3 gives 9 (remainder 1).

Remember · Powers of 2 are multiples of 4 from 2² onwards; always check the first case separately.

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Answer: (c) 37

Every number of the form QQQ equals Q × 111 = Q × 3 × 37, and 481 = 13 × 37. They always share the factor 37, and 13 can never divide 3Q for a single digit Q, so the HCF is 37.

1. QQQ = Q × 111 = Q × 3 × 37.
2. 481 = 13 × 37.
3. HCF = 37 × HCF(3Q, 13). For a single digit Q, 3Q is at most 27 and is never 13 or 26, so HCF(3Q, 13) = 1.
4. So the HCF of P and 481 is 37, whatever the digit Q.
5. Check: HCF(222, 481) = 37, since 222 = 6 × 37 and 481 = 13 × 37.

Remember · Repeated-digit numbers: aaa = a × 3 × 37. Knowing 111 = 3 × 37 settles such items at sight.

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Answer: (d) 9

The numbers 1 to 99 use 9 + 180 = 189 digits. The remaining 300 digits cover exactly 100 three-digit numbers, 100 to 199, so the 489th digit is the final 9 of 199.

1. Digits from 1–9: 9. Digits from 10–99: 90 × 2 = 180. Total so far: 189.
2. Digits still needed: 489 − 189 = 300, all from three-digit numbers.
3. 300 ÷ 3 = 100 exactly, so the 489th digit is the last digit of the 100th three-digit number.
4. The 100th three-digit number is 100 + 99 = 199; its last digit is 9.
5. Check: digits 487–489 are 1, 9, 9 (from 199), and digit 490 is the 2 of 200.

Remember · Count digit blocks: 9 one-digit, 180 two-digit, 2700 three-digit. An exact division means the digit is the last one of that number.

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Answer: (d) Cannot be concluded

The item gives no rule about who tells the truth. Even if we assume every innocent person speaks the truth, both 'P stole' and 'Q stole' fit all the statements, so the thief cannot be identified.

1. The item says only one person is guilty but gives no rule on who tells the truth or lies.
2. Test the most helpful assumption: innocent people speak the truth. If R stole, innocent P's claim 'Q stole it' would be false — so R is ruled out.
3. If P stole: innocent Q says R and Q did not steal (true), and innocent R says he did not steal (true). Consistent.
4. If Q stole: innocent P says he did not steal and Q stole it (true), and innocent R's statements are true. Consistent.
5. Both P and Q remain possible, so the thief cannot be concluded.

Remember · In truth-and-lie puzzles, test each suspect as culprit for consistency; if more than one survives, the answer is 'cannot be concluded'.

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Answer: (c) Both I and II

The points force P to have one win and one draw and R one draw and one loss; the goal condition rules out Q winning a match, so Q drew both. P's only goal beat R 1–0, leaving P–Q at 0–0, and the Q–R draw must be 1–1 — so both statements hold.

1. Three matches at 2 points each give 6 points (3 + 2 + 1 checks). Each team scored exactly one goal, so there were 3 goals in all.
2. P's 3 points = one win + one draw. R's 1 point = one draw + one loss. Q's 2 points = one win + one loss, or two draws.
3. If Q had a win and a loss: P's draw and R's draw must be the same match, P vs R, so P beat Q and Q beat R. P's only goal came against Q, so P–R was 0–0; Q's only goal came against R, so R scored in neither match — but R must score one goal. Impossible.
4. So Q drew both of its matches, and P beat R.
5. P's only goal was needed to beat R (1–0), so P–Q was 0–0. Q's only goal came in the drawn Q–R match, so it ended 1–1 and R's goal came against Q.
6. Both statements are correct.

• ✓ I P scored only once and needed that goal to beat R, so P–Q was a 0–0 draw.
• ✓ II Q's single goal came in the Q–R match, which was a draw, so it ended 1–1 and R's one goal was against Q.

Remember · In tournament puzzles, list the win/draw/loss patterns from the points, then use goal totals to eliminate cases.

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Answer: (b) Our total dependence on genetically honed crops entails possible food insecurity.

The author warns that climate change may spread plant diseases, and that crops bred for high yields have lost the genetic variety needed to cope with change. Depending wholly on these genetically honed crops therefore carries a risk to food security, as (b) says.

• ✗ (a) The passage says the effect of climate change on crop diseases is 'hard to predict'; it speaks of a disease risk, not a definite fall in productivity.
• ✓ (b) Selective breeding has made high-value crops 'spectacular' today but stripped out genetic variation, so a minor or unknown disease 'could suddenly rampage' through them. Relying entirely on such crops puts the food supply at risk.
• ✗ (c) The passage does not discuss what else food security should rest on; its concern is the fragility of genetically honed crops.
• ✗ (d) Replacing honed crops with wild varieties is a drastic prescription the author never makes; the passage even grants that breeding 'is good for yields'.

Remember · The central idea joins the problem to its cause; reject options that overstate certainty or leap to drastic remedies.

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Answer: (c) Both I and II

The passage warns that climate change may make some plant infections 'more common in all climatic zones', which implies diseases reaching areas where they were rare, so I is assumed. It then blames crops' vulnerability on genetic variations that 'have been bred out'; the argument takes for granted that this lost variation, which survives in the crops' unrefined wild relatives, is what would let crops cope with change, so II is also valid: (c).

• ✓ I The passage warns that climate change may make some plant infections 'more common in all climatic zones', which implies diseases spreading into areas where they were not common before.
• ✓ II The author's claim that breeding out variation is 'bad for coping with changes' assumes that such variation is what gives resilience. The variation bred out of refined crops survives in their wild relatives, so the argument presupposes that understanding them would help crops withstand disease, which is what protecting food security means here.

Remember · When a passage blames a problem on something lost (genetic variation), UPSC treats an assumption about recovering it from its source as valid, even if the source is named only in the option.

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Answer: (d) In a democracy, the opposition is indispensable for the balance of political power and good governance.

The author argues that a wise leader learns more from opponents than from supporters, who can push him to disaster, and that the unity of free people rests on an even balance of political power. The message is that opposition is indispensable for that balance and for good governance.

• ✗ (a) This reverses the passage: the author says a statesman should pray 'never to be left without opponents'.
• ✗ (b) The passage wants a balance that makes it impracticable for the opposition 'to be revolutionary and irreconcilable' — the opposite of (b).
• ✗ (c) Opponents keep a ruler 'on the path of reason and good sense' by showing him the dangers; the passage does not say he needs their support to survive.
• ✓ (d) Opponents show where the dangers are, and national unity depends on a balance of power that stops the administration being arbitrary. Opposition is thus indispensable for balance and good governance.

Remember · Check the author's stance word by word; options that reverse it ('without opposition') or exaggerate it ('revolutionaries') are traps.

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Answer: (d) Neither I nor II

The passage values opposition at all times and wants a balance that keeps it from becoming revolutionary. Statement I adds a condition the author never sets, and II praises the aggression the author warns against.

• ✗ I The author says a statesman should pray 'never to be left without opponents' — opposition is always needed, not 'only if' the Head of Government is indifferent.
• ✗ II The passage wants an even balance so that the opposition is not 'revolutionary and irreconcilable'; it does not say a more aggressive opposition means better governance.

Remember · 'Only if' and 'the more …, the better' are red flags; check whether the author sets any such condition or scale.

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Answer: (c) Only four

As siblings, P, Q and R share S as mother and T as father, so S and T are a couple. Only P's gender is known, so the statements that need Q to be male or R to be female are not definite; the other four are.

1. P, Q and R are siblings, so R's mother S and P's father T are the parents of all three — and a couple.
2. P is male (a brother). The genders of Q and R are not given.
3. Definitely true: I (S and T are a couple), III (T is Q's father), IV (S is P's mother), VI (P is S's son).
4. Not definite: II (Q may be a daughter) and V (R may be a son).
5. So four statements are definitely true.

• ✓ I S is the mother and T the father of the same siblings, so they are a couple.
• ✗ II Q's gender is not given; Q could be T's daughter.
• ✓ III T is P's father and Q is P's sibling, so T is Q's father.
• ✓ IV S is R's mother and R is P's sibling, so S is P's mother.
• ✗ V R's gender is not given; R could be T's son.
• ✓ VI P is male and S is his mother, so P is S's son.

Remember · Note every gender actually given; a statement that needs an unknown gender is never 'definitely true'.

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Answer: (a) 399000

The code multiplies the alphabet positions of the letters: 14 × 15 = 210, then × 20 = 4200, then × 5 = 21000. Adding S (19) gives 21000 × 19 = 399000.

1. Alphabet positions: N = 14, O = 15, T = 20, E = 5, S = 19.
2. NO = 14 × 15 = 210. NOT = 210 × 20 = 4200. NOTE = 4200 × 5 = 21000. The code is the product of the positions.
3. NOTES = 21000 × 19 = 399000.
4. Check: 4200 ÷ 210 = 20 (T) and 21000 ÷ 4200 = 5 (E), confirming the rule.

Remember · When a code grows as letters are added, divide consecutive codes: the ratio gives each new letter's value (here its alphabet position).

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Answer: (d) 71813

Each letter keeps a fixed digit across both words: Y = 7, E = 1, M = 8 and N = 3. So YEMEN is 7-1-8-1-3, i.e. 71813.

1. From FRANCE = 654321: F = 6, R = 5, A = 4, N = 3, C = 2, E = 1.
2. From GERMANY = 9158437: G = 9, E = 1, R = 5, M = 8, A = 4, N = 3, Y = 7 (E, R, A, N agree with the first word).
3. YEMEN = Y, E, M, E, N = 7, 1, 8, 1, 3 = 71813.

Remember · For letter-to-digit codes, build a lookup table from the given words and read off the new word; check shared letters agree.

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Answer: (b) 4

P = 3T allows (T, P) = (1, 3), (2, 6) or (3, 9), and S = R + 7 allows (R, Q, S) = (0, 3, 7), (1, 4, 8) or (2, 5, 9). Keeping all five digits different leaves 35291, 63072, 64182 and 94183 — four numbers.

1. P = 3T with T ≠ 0 and P a single digit: (T, P) = (1, 3), (2, 6) or (3, 9).
2. Q = R + 3 and S = Q + 4 = R + 7 ≤ 9, so R = 0, 1 or 2: (R, Q, S) = (0, 3, 7), (1, 4, 8) or (2, 5, 9).
3. T = 1, P = 3: only (2, 5, 9) avoids repeats → 35291.
4. T = 2, P = 6: (0, 3, 7) and (1, 4, 8) work → 63072 and 64182.
5. T = 3, P = 9: only (1, 4, 8) works → 94183.
6. Total: 4 numbers.

Remember · Tie the chained conditions to one variable (here R), list the few cases for each part, then strike out repeated digits.

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Answer: (b) 4

Alone, X, Y and Z would take 18, 24 and 16 days, i.e. 8, 6 and 9 units a day out of 144 units. Y's last 8 2/3 days cover 52 units, leaving 92 units done together at 23 units a day, so n = 4.

1. Full-work times: X = 6 × 3 = 18 days, Y = 8 × 3 = 24 days, Z = 12 × 4/3 = 16 days.
2. Take the work as 144 units (LCM of 18, 24, 16). Daily work: X = 8, Y = 6, Z = 9 units; together 23 units a day.
3. Y alone for 8 2/3 = 26/3 days does 6 × 26/3 = 52 units.
4. Work done by all three together = 144 − 52 = 92 units, so n = 92 ÷ 23 = 4 days.
5. Check: 4 × 23 + 52 = 92 + 52 = 144.

Remember · Turn partial-work data into full-work days, take the LCM as total units, then work in units per day.

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Answer: (c) 29

The gaps between terms are consecutive prime numbers: 2, 3, 5, 7, 11, 13. So X = 18 + 11 = 29, and 29 + 13 = 42 confirms it.

1. Differences: 3 − 1 = 2, 6 − 3 = 3, 11 − 6 = 5, 18 − 11 = 7 — consecutive primes.
2. The next primes are 11 and 13, so X = 18 + 11 = 29.
3. Check: 29 + 13 = 42, the last term.

Remember · Try differences first; gaps of 2, 3, 5, 7 signal a prime-number pattern.

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Answer: (a) Corporate capitalism is important for economic growth of a State and also for democracy.

The author claims no democracy has grown economically without corporate capitalism, then lists what it brings — modernisation, jobs, competitiveness and State revenue for welfare and fairer opportunities. The critical message is that it matters both for economic growth and for democracies.

• ✓ (a) The opening line ties democracy's economic growth to corporate capitalism, and the rest lists its gains — jobs, competitiveness, revenue for welfare and a more level playing field. It is presented as important for growth and for democracies.
• ✗ (b) Political objectives are not discussed; the passage is about growth, jobs and welfare.
• ✗ (c) The passage speaks of democracies that have 'grown economically', not of any State's economic 'survival'. The claim is widened and made absolute.
• ✗ (d) The passage says democracies have needed corporate capitalism to grow, not that capitalism depends on democracy. 'Mutual dependence' goes beyond the text.

Remember · Check the direction of a link: 'X needed Y' does not mean 'Y needs X'. Reject options that make a one-way link mutual or absolute.

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Answer: (c) Both I and II

Why not the tempting option · UPSC's key accepts both assumptions. The tempting objection to II is that the passage's route to the poor is State revenue spent on welfare, not a market 'trickle-down'; but the statement is testing whether the author assumes the marginalised benefit from corporate capitalism, and the passage plainly does, through jobs and through revenue-funded welfare. In the exam, read a named 'effect' in its broad sense unless the passage contradicts it.

The author credits corporate capitalism with generating jobs and with producing State revenue that funds welfare for the marginalised. Both the employment benefit (I) and the benefit that reaches the poor from corporate growth (II) are built into the argument, so UPSC's key accepts both.

• ✓ I The passage says corporate capitalism 'generates jobs', shifts workers from agriculture to industry and services, and is the only way to fix unemployment; a growing, better-employed workforce is taken for granted.
• ✓ II A vibrant corporate base yields 'additional revenues for the State' that can fund 'greater welfare for the marginalized'. The author assumes the gains of corporate growth reach the poor, which is the trickle-down idea in its broad sense.

Remember · When an author lists the benefits of something, assumptions that restate those benefits in other words are usually valid.

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Answer: (c) It takes individuals out of a state of isolation and gives them a chance to participate in the common endeavour.

The line pictures voluntary associations standing between a weak individual and a strong State. They play this 'buffer' role by drawing isolated individuals into joint action, giving them a strength they lack alone — the explanation in (c).

• ✗ (a) The passage calls the State 'potentially powerful'; it does not say the State cannot enforce its will. The option shifts the focus to the State's weakness.
• ✗ (b) This describes what a cooperative does for its members' welfare; it does not explain the buffer role between the individual and the State.
• ✓ (c) Alone, the individual is 'relatively powerless' before the State. Voluntary associations bring individuals together to act in common, taking them out of isolation — that collective strength is what makes them a buffer.
• ✗ (d) Multiple loyalties may follow from membership of many groups, but the quotation is about protection between the individual and the State, not divided allegiance.

Remember · For a one-line quotation, explain its key image (here the 'buffer'); options about side effects or other institutions miss it.

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Answer: (c) Both I and II

'There exists' needs only one example. The number 2 becomes 3 after a 50% increase and 5 after a 150% increase; all three are primes with exactly two factors, so both statements are correct.

1. Statement I: 2 has 2 factors. Increased by 50%, it becomes 3, which also has 2 factors. So I is correct.
2. Statement II: 2 increased by 150% becomes 2 × 2.5 = 5, which also has 2 factors. So II is correct.
3. Both statements are correct.
4. Check: other examples exist too, e.g. 10 → 15 (4 factors each) for I and 6 → 15 (4 factors each) for II.

• ✓ I 2 → 3: both are primes with exactly 2 factors.
• ✓ II 2 → 5: both are primes with exactly 2 factors.

Remember · 'There exists' statements need just one example — try small primes first.

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Answer: (c) 31 km

Plotting the places on a grid shows two road routes from D to F: via C, A and B (6 + 10 + 6 + 9 km) and via E and G (5 + 14 + 12 km). Both total 31 km.

1. Put A at (0, 0), with east = +x and north = +y.
2. B = (0, 6); C = (10, 0); D = 6 km south of C = (10, −6); E = 5 km west of D = (5, −6); F = 9 km west of B = (−9, 6); G = 12 km south of F = (−9, −6).
3. Route D → C → A → B → F: 6 + 10 + 6 + 9 = 31 km.
4. Route D → E → G → F: 5 + 14 + 12 = 31 km (E to G runs from x = 5 to x = −9).
5. Either way the distance is 31 km.

Remember · For direction-and-road items, fix one place as the origin and write coordinates; route lengths then follow by simple addition.

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Answer: (b) 1331

Both examples turn a cube n³ into (n + 3)³: 4³ becomes 7³ and 6³ becomes 9³. So 512 = 8³ becomes 11³ = 1331.

1. 64 = 4³ and 343 = 7³; 216 = 6³ and 729 = 9³.
2. The rule: n³ is written as (n + 3)³.
3. 512 = 8³, so it is written as 11³ = 1331.

Remember · Recognise squares and cubes at sight (up to 20² and 12³); number codes often shift the base.

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Answer: (a) 0

Every power of 9 leaves remainder 3 when divided by 6, and there are 98 terms. 98 × 3 = 294 is a multiple of 6, so the remainder is 0.

1. 9 leaves remainder 3 when divided by 6, and 3 × 3 = 9 again leaves 3. So every power 9ᵏ leaves remainder 3.
2. Number of terms from 9³ to 9¹⁰⁰: 100 − 3 + 1 = 98.
3. Sum of remainders = 98 × 3 = 294 = 6 × 49, so the remainder is 0.
4. Check: each term is odd and a multiple of 3; 98 odd terms add to an even number, and an even multiple of 3 is divisible by 6.

Remember · For the remainder of a sum, reduce each term first; count terms carefully (3 to 100 is 98 terms).

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Answer: (d) The Question can be answered even without using any of the Statements.

Among one-digit numbers only 6 (factors 1, 2, 3, 6) and 8 (factors 1, 2, 4, 8) have exactly four factors, so the smallest is 6. The question answers itself; the statements add nothing.

1. List factor counts for 1–9: only 6 (1, 2, 3, 6) and 8 (1, 2, 4, 8) have exactly 4 factors.
2. The smallest such number is 6 — found from the question alone.
3. The statements (2 and 3 are factors) are true of 6 but are not needed.

• ✗ Statement I Not needed: the question already fixes the answer as 6.
• ✗ Statement II Not needed: 6 is found without it.

Remember · In data sufficiency, first try the question alone; if it is already settled, choose 'even without using any of the Statements'.

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Answer: (d) The Question can be answered even without using any of the Statements.

Dividing both sides by 11 gives P × PQ = 100R + S, which must be at least 100. With P ≤ 3 and Q ≤ 4, only 33 × 34 = 1122 works, so Q = 4 without using either statement.

1. PP = 11 × P and RRSS = 1100R + 11S = 11 × (100R + S). So P × PQ = 100R + S.
2. R is non-zero, so P × PQ must be at least 100. P = 1 gives at most 1 × 14 = 14; P = 2 gives at most 2 × 24 = 48. Too small.
3. P = 3: 3 × (30 + Q) = 90 + 3Q reaches 100 only for Q = 4, giving 102, so R = 1 and S = 2.
4. Check: 33 × 34 = 1122, and the digits 3, 4, 1, 2 are distinct.
5. Q = 4 follows from the question alone; neither statement is needed.

• ✗ Statement I Not needed: R = 1 already follows from the question itself.
• ✗ Statement II Not needed: S = 2 also follows from the question itself.

Remember · PP and RRSS are multiples of 11 — divide it out first. Digit limits often settle the question on their own.

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Answer: (d) The Question cannot be answered even using any of the Statements.

Together the statements show that Q is a brother or sister of P's father, but Q's gender is never given. Q could be P's uncle or aunt, so the relation cannot be fixed even with both statements.

1. Statement I alone tells us about P's sisters but nothing about Q.
2. Statement II alone links Q to R's father, but not to P.
3. Together: R is P's sister, so R's father is P's father, and Q is a sibling of P's father.
4. Q's gender is not given, so Q could be P's paternal uncle or aunt.
5. The question cannot be answered even using both statements.

• ✗ Statement I Says nothing about Q.
• ✗ Statement II Links Q only to R's father; even combined with I, Q may be P's uncle or aunt.

Remember · In relation-based sufficiency items, check gender: a relation that depends on an unknown gender is not determined.

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Answer: (b) The present calculation of inflation rate does not correctly measure price rise of individual item/commodity.

The passage explains that a single inflation figure is a weighted average across goods and services whose prices rise at different rates, with weights that differ from class to class. Such an aggregate cannot show the actual price rise of any one item, which is the point of (b).

• ✗ (a) Extreme and unsupported: the passage explains how WPI and CPI are built; it never says CPI should be scrapped.
• ✓ (b) The single number is 'an aggregate' of items whose 'price rise differs', arrived at by weighting, and baskets differ for the poor, middle classes and rich. A single weighted figure therefore does not reflect the price rise of individual items.
• ✗ (c) The passage never says services are under-represented; it mentions commodities and services together.
• ✗ (d) Extreme: the passage explains how inflation is measured, not that the figure is useless.

Remember · When a passage explains how an average is built, its key message is usually what the average hides.

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Answer: (d) Trust is important though it entails risk.

The passage presents trust as essential — to entrepreneurship, to global markets, even as a 'super-commodity' — while defining it as accepting vulnerability and the chance of betrayal. The crux is that trust matters despite its risk.

• ✗ (a) The passage says entrepreneurship implies trust in others; it does not say trustworthiness cannot be expected.
• ✗ (b) The passage says trusting means being vulnerable; it says nothing about trustworthy people being 'the most vulnerable'. The option confuses 'trusting' with 'trustworthy' and adds a superlative.
• ✗ (c) 'No economic activity' is too sweeping; the passage speaks of entrepreneurship and trust in expert systems.
• ✓ (d) Trust means 'being vulnerable' and a 'willingness to expose oneself to betrayal', yet it is 'the essence of globalizing behaviour' and a 'super-commodity'. Trust is valuable even though it carries risk.

Remember · When a passage both praises something and names its cost, the crux usually balances the two ('important though risky').

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Answer: (d) The Question cannot be answered even using any of the Statements.

Even combined, the statements allow two outcomes. If Q had 3 goals with 10 minutes left, the match ends 4–4; if Q already had 4, P wins 5–4. So whether P won cannot be decided.

1. Let Q have g goals with 10 minutes left; P then had g − 3.
2. Statement I alone: P ends with g + 1 goals, but Q may also have scored in those 10 minutes. Not sufficient.
3. Statement II alone: Q ends with 4 goals, but P's final score is unknown. Not sufficient.
4. Together: P ends with g + 1 and Q with 4, where g is 3 or 4 (P had g − 3 ≥ 0, and Q's final 4 ≥ g).
5. g = 3 gives a 4–4 draw; g = 4 gives a 5–4 win for P. Both fit, so the question cannot be answered.

• ✗ Statement I P's 4 late goals could be matched by Q's own late goals.
• ✗ Statement II Q's total is known but P's is not; even with I, the result can be 4–4 or 5–4.

Remember · Before choosing 'both together', build two cases that fit both statements; if they give different answers, the data are insufficient.

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Answer: (b) The Question can be answered by using either Statement alone.

The expression simplifies to (p − q)², which is positive unless p = q. Each statement on its own rules out p = q, so the question can be answered using either statement alone.

1. (p + q)² − 4pq = p² + 2pq + q² − 4pq = (p − q)².
2. (p − q)² is positive exactly when p ≠ q; it is 0 when p = q.
3. Without the statements, p = q is possible, so the question cannot be answered alone.
4. Statement I (p < q) gives p ≠ q, so the expression is positive. Statement II (p > q) also gives p ≠ q.
5. Either statement alone is sufficient.

• ✓ Statement I p < q means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.
• ✓ Statement II p > q also means p ≠ q, so (p − q)² > 0 — the answer is 'yes'.

Remember · Simplify before judging sufficiency: (p + q)² − 4pq = (p − q)², so only whether p ≠ q matters.

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Answer: (d) Cannot be determined due to insufficient data

The condition only says X, Y and Z are in geometric progression with a total of 37. Both 16, 12, 9 and 9, 12, 16 satisfy it, giving opposite orders, so the comparison cannot be determined.

1. X : Y = Y : Z means Y² = X × Z, with X + Y + Z = 37.
2. Try Y = 12: then X × Z = 144 and X + Z = 25, so X and Z are 16 and 9.
3. Both (X, Y, Z) = (16, 12, 9) and (9, 12, 16) fit: 16 × 9 = 144 = 12², and the total is 37.
4. The first gives Value-III < Value-II < Value-I; the second gives Value-I < Value-II < Value-III.
5. The order cannot be fixed from the data.

Remember · For a 'cannot be determined' option, try to build two valid cases with different orders; one such pair settles it.

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Answer: (c) Value-I = Value-II

For a fixed sum of 10, the product p × q is largest when p = q = 5, giving 25. Allowing some negative values only adds negative products, so both maxima are 25 and the two values are equal.

1. Value-I: with p + q = 10 and both positive, p × q = p(10 − p) is largest at p = q = 5, giving 25.
2. Value-II: p ≥ −6 and q ≥ −4 allow p from −6 to 14. The product p(10 − p) still peaks at p = 5 (25).
3. The extra cases give negative products, e.g. (−6) × 16 = −96 and 14 × (−4) = −56.
4. So Value-II = 25 = Value-I.

Remember · For a fixed sum, the product is greatest when the two numbers are equal (or as close as possible).

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Answer: (c) Value-I = Value-II

The lowest allowed set, −5 to 5, is balanced around 0, so its average is 0. The product of 0, 1, …, 10 is 0, the smallest possible for non-negative integers. Both values are 0.

1. Value-I: the lowest possible set is −5, −4, …, 4, 5. Its average is 0; any later set has a larger average. So Value-I = 0.
2. Value-II: the set 0, 1, …, 10 has product 0; any set starting above 0 has a positive product. So Value-II = 0.
3. Value-I = Value-II.

Remember · Consecutive integers balanced around 0 average 0; a product of non-negative numbers is smallest (0) when it includes 0.

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Answer: (a) k

p is above the average by exactly the amount q is below it, so p and q together total 2k. Since all three total 3k, r must equal k.

1. Let p = k + d. Then q is below k by the same amount: q = k − d.
2. p + q = 2k, and p + q + r = 3k because the average is k.
3. So r = 3k − 2k = k.
4. Check: p = 7, q = 3, r = 5 has average 5 and r equals the average.

Remember · Deviations from an average always sum to zero; equal and opposite deviations leave the third number at the average.

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Answer: (a) I only

For a number between 0 and 1, higher powers get smaller and square roots get larger. So x² > x³ is true, but x > √x is false.

1. Take x = 0.25 as a test value.
2. I: x² = 0.0625 and x³ = 0.015625, so x² > x³. In general x² − x³ = x²(1 − x) > 0 for 0 < x < 1. True.
3. II: √0.25 = 0.5, which is larger than 0.25, so x > √x is false. For 0 < x < 1, √x is always larger than x.
4. Only statement I is correct.

• ✓ I x² − x³ = x²(1 − x) is positive when 0 < x < 1, so x² > x³.
• ✗ II For a fraction between 0 and 1 the square root is larger: √0.25 = 0.5 > 0.25.

Remember · Between 0 and 1: x³ < x² < x < √x. Test with x = 0.25 to confirm quickly.

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Answer: (c) There can be more than one such number.

Two natural numbers 10 apart have nine numbers strictly between them. These contain one multiple of 5 when both ends are multiples of 5 (5 and 15) and two otherwise (3 and 13). So there can be more than one such number, as (c) says.

1. Take 5 and 15: the only multiple of 5 strictly between them is 10 — one number.
2. Take 3 and 13: the multiples of 5 between them are 5 and 10 — two numbers.
3. The nine numbers between the two always include at least one multiple of 5, so the count is 1 or 2, never 0.
4. Only (c), 'There can be more than one such number', holds in general.

• ✗ (a) True only when both numbers are multiples of 5 (e.g. 5 and 15); 3 and 13 have two such numbers between them.
• ✗ (b) True only when neither number is a multiple of 5; 5 and 15 have just one (10) between them.
• ✓ (c) The count can be 2 (3 and 13 give 5 and 10), so more than one such number is possible.
• ✗ (d) Nine consecutive numbers always include a multiple of 5, so at least one always exists.

Remember · When a count depends on the case, test a boundary case and a general case; pick the option true in every case.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error