CSAT 2025: every question, solved
80 questions from the official UPSC Prelims 2025 paper, in paper order. Tap an option; the answer and the reason appear instantly.
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Read the passage
Maintaining an ecosystem just to conserve biodiversity will affect its commercial potential as well as the livelihoods dependent on the ecosystem. There is also a conflict between using an ecosystem only for livelihoods, for commercial exploitation, or strictly for conservation. Deforestation caused due to commercial exploitation will lead to indirect harm like floods, siltation problems and microclimatic instability, apart from adversely affecting livelihoods dependent on forests. These conflicts are particularly acute in developing countries where the dependence of people on the ecosystem is significant, and commercial exploitation has the potential to boost national income.
Which one of the following statements best reflects the critical message conveyed by the author of the passage?
Answer (c) & explanation
The critical message is that livelihoods and ecosystem damage feed each other, causing conflict and imbalance (c). Step 1: the passage says conservation, livelihoods and commercial use pull against each other. Step 2: it adds that deforestation causes floods, siltation and climate instability, which in turn hurt livelihoods. So each problem worsens the other: (c). Step 3: option (a) dismisses conservation; the passage does not. Option (b) brings in fundamental rights, never mentioned. Option (d) says exploitation should be "completely stopped"; the passage weighs trade-offs and even notes it can boost national income.
Tip: passages about "conflicts" usually want a balanced answer. Reject extreme options like "completely stopped". This trade-off is the core idea of sustainable development.
Answer as per the official UPSC answer key.
Read the passage
Maintaining an ecosystem just to conserve biodiversity will affect its commercial potential as well as the livelihoods dependent on the ecosystem. There is also a conflict between using an ecosystem only for livelihoods, for commercial exploitation, or strictly for conservation. Deforestation caused due to commercial exploitation will lead to indirect harm like floods, siltation problems and microclimatic instability, apart from adversely affecting livelihoods dependent on forests. These conflicts are particularly acute in developing countries where the dependence of people on the ecosystem is significant, and commercial exploitation has the potential to boost national income.
With reference to above passage, the following assumptions have been made:
- I. No country needs to depend on ecosystems to boost national income.
- II. Resource-rich countries need to share their resources with those of scant resources so as to prevent the degradation of ecosystems.
Which of the above assumptions is/are valid?
Answer (d) & explanation
Neither assumption is valid (d). Step 1: assumption I says no country needs ecosystems to boost national income. But the passage says commercial exploitation "has the potential to boost national income" in developing countries. So I contradicts the passage. Step 2: assumption II asks rich countries to share resources with poor ones. The passage never talks about sharing between countries. Step 3: so I goes against the text and II is outside it. Options (a), (b) and (c) fail.
Tip: an assumption that contradicts a line in the passage is invalid at once. One that brings in a new idea (like international sharing) is also invalid. Check each against the text, not against what sounds fair.
Answer as per the official UPSC answer key.
Read the passage
The history of renewable energy suggests there is a steep learning curve, meaning that, as more is produced, costs fall rapidly because of economies of scale and learning by doing. The firms’ green innovation is path-dependent: the more a firm does, the more it is likely to do in the future. The strongest evidence for this is the collapse in the price of solar energy, which became about 90% cheaper during the 2010s, repeatedly beating forecasts. Moving early and gradually gives economies more time to adjust, allowing them to reap the benefits of path-dependent green investment without much disruption. A late, more chaotic transition is costlier.
Which one of the following statements best reflects the central idea of the passage?
Answer (d) & explanation
The central idea is that timing matters greatly in green technology (d). Step 1: the passage explains a learning curve: the more is produced, the faster costs fall, as with solar, which became about 90% cheaper in the 2010s. Step 2: its conclusion: "Moving early and gradually" helps economies adjust, while "a late, more chaotic transition is costlier." That is about timing: (d). Step 3: option (a), economies of scale, and (b), path-dependent innovation, are supporting points, not the main idea. Option (c), large economies, is never mentioned.
Tip: the conclusion of a passage usually gives its central idea; earlier lines are evidence. Ask: what is the author finally recommending?
Answer as per the official UPSC answer key.
Read the passage
The history of renewable energy suggests there is a steep learning curve, meaning that, as more is produced, costs fall rapidly because of economies of scale and learning by doing. The firms’ green innovation is path-dependent: the more a firm does, the more it is likely to do in the future. The strongest evidence for this is the collapse in the price of solar energy, which became about 90% cheaper during the 2010s, repeatedly beating forecasts. Moving early and gradually gives economies more time to adjust, allowing them to reap the benefits of path-dependent green investment without much disruption. A late, more chaotic transition is costlier.
With reference to the above passage, the following assumptions have been made:
- I. Path-dependent green investments will eventually most likely benefit growth as well as public finances in a country like India.
- II. If other green technologies follow the same pattern as that of solar energy, there will most likely be an easy green transition.
Which of the above assumptions is/are valid?
Answer (c) & explanation
Both assumptions are valid (c), as per UPSC's official key. Step 1: the passage says early, path-dependent green investment gives benefits "without much disruption", while a late transition is costlier. Lower costs and less disruption help both growth and public finances, including in a country like India (I). Step 2: solar became about 90% cheaper, beating forecasts. If other green technologies follow the same curve, the transition would likely be easier (II). Step 3: both follow from the passage's logic. Options (a), (b) and (d) fail. Some coaching keys printed (b), but the official answer is (c).
Tip: "most likely" softens an assumption, making it easier to accept. Note that solar's cost fall has made it one of India's cheapest new power sources.
Answer as per the official UPSC answer key.
A natural number N is such that it can be expressed as N = p + q + r, where p, q and r are distinct factors of N. How many numbers below 50 have this property?
Answer (c) & explanation
Eight numbers below 50 have this property (c). Step 1: if p, q, r are factors of N, write them as N/a, N/b, N/c. Then N = N/a + N/b + N/c, so 1/a + 1/b + 1/c = 1 with a, b, c different. Step 2: the only set of different whole numbers that works is 2, 3, 6 (1/2 + 1/3 + 1/6 = 1). So N = N/2 + N/3 + N/6, and N must be divisible by 6. Step 3: multiples of 6 below 50: 6, 12, 18, 24, 30, 36, 42, 48 = 8 numbers. Example: 12 = 6 + 4 + 2. Options (a), (b) and (d) miscount.
Tip: turning factors into fractions of N often reveals a neat rule. Remember 1/2 + 1/3 + 1/6 = 1.
Answer as per the official UPSC answer key.
Three prime numbers p, q and r, each less than 20, are such that p – q = q – r. How many distinct possible values can we get for (p + q + r)?
Answer (d) & explanation
More than 6 sums are possible (d). Step 1: p − q = q − r means p + r = 2q, so the sum p + q + r = 3q. Each different q gives a different sum. Step 2: the question does not say the primes must be different. So p = q = r is allowed (for example, 7 − 7 = 7 − 7). Then every prime below 20 can be q: 2, 3, 5, 7, 11, 13, 17, 19. Step 3: that gives 8 sums, which is more than 6. Options (a), (b) and (c) assume the primes must be distinct.
Tip: read the conditions closely. If a question does not say "distinct", equal values are allowed, and that often changes the answer.
read what is not said. Without "distinct", equal primes are allowed. With distinct primes you would get only 4 (q = 5, 7, 11, 13).
Answer as per the official UPSC answer key.
How many possible values of (p + q + r) are there satisfying 1/p + 1/q + 1/r = 1, where p, q and r are natural numbers (not necessarily distinct)?
Answer (c) & explanation
Three values are possible (c). Step 1: arrange p ≤ q ≤ r. The smallest, p, has the biggest fraction, so 1/p ≥ 1/3. Since 1/p must be less than 1, p is 2 or 3. Step 2: if p = 3, then q = r = 3. If p = 2, then 1/q + 1/r = 1/2, giving (4, 4) or (3, 6). Step 3: the solutions are (3, 3, 3), (2, 4, 4), (2, 3, 6) with sums 9, 10 and 11: three values. Options (a), (b) and (d) fail.
Tip: in unit-fraction equations, order the variables and bound the smallest one first. This keeps the cases few and complete.
Answer as per the official UPSC answer key.
What comes at X and Y respectively in the following sequence?
January, January, December, October, X, March, October, Y, September
Answer (b) & explanation
X is July and Y is April (b). Step 1: write the months as numbers: 1, 1, 12, 10, X, 3, 10, Y, 9. Step 2: each step goes back by one more month than before: back 0 (Jan to Jan), back 1 (Jan to Dec), back 2 (Dec to Oct), back 3 (Oct to July, X = 7), back 4 (July to March) ✓. Step 3: continue: back 5 (March to October) ✓, back 6 (October to April, Y = 4), back 7 (April to September) ✓. Options (a), (c) and (d) break the pattern.
Tip: turn months or letters into numbers, and count around the circle (after 1 comes 12). Then look for steps that grow by one each time.
Answer as per the official UPSC answer key.
Team X scored a total of N runs in 20 overs. Team Y tied the score in 10% less overs. Had team Y's average run rate (runs per over) been 50% higher, the scores would have been tied in 12 overs. How many runs were scored by team X?
Answer (d) & explanation
Team X's runs cannot be determined (d). Step 1: Y tied the score in 10% fewer overs: 18 overs. So Y's run rate = N/18. Step 2: 50% higher rate = 1.5 × N/18 = N/12. At that rate, N runs take N ÷ (N/12) = 12 overs, whatever N is. Step 3: so the second fact is automatically true and gives no new information. N can be any number. Options (a) 72, (b) 144 and (c) 216 all fit equally well.
Tip: if a "condition" turns out to be true for every value, it adds no information. Check whether each fact gives a new equation before solving.
a condition that is always true adds no information. Test it before you start solving.
Answer as per the official UPSC answer key.
The price (p) of a commodity is first increased by k%; then decreased by k%; again increased by k%; and again decreased by k%. If the new price is q, then what is the relation between p and q?
Answer (a) & explanation
1. Put k = 0. Then the price never changes, so q = p. Check each option: (a) gives p × 10⁸ = q × 10⁸, so p = q. Works. (c) also gives p = q. (b) and (d) fail.
2. To choose between (a) and (c): each "up k%, then down k%" round multiplies the price by (10⁴ − k²) ÷ 10⁴. Example: up 10% and down 10% turns 100 into 99, and (10000 − 100) ÷ 10000 = 0·99.
3. There are two such rounds, so the bracket comes twice (squared) and 10⁴ comes twice (10⁸).
4. Answer: (a).
Answer as per the official UPSC answer key.
Read the passage
Each State in India faces a distinctive set of challenges regarding the impact of warming, but also offers its own set of opportunities for reducing emissions depending on its natural resources. For example, coastal States need to take action to protect their shores from sea level rise, districts that are drier need to prepare for variable monsoon precipitation. Himalayan regions have their own unique challenges, and selected parts of peninsular India and offshore areas offer great opportunities for harnessing wind power. These various aspects need to be considered for developing clear and sustainable goals for the future.
Which one of the following statements best reflects the most logical, rational and pragmatic message conveyed by the author of the passage?
Answer (d) & explanation
The message is that India needs climate strategies at the State or regional level (d). Step 1: the passage says each State faces "a distinctive set of challenges" and has its own opportunities. Coasts face sea-level rise; dry districts face erratic monsoons; peninsular areas suit wind power. Step 2: so plans must suit each region: (d). Step 3: option (a) makes it solely the State's duty; the passage does not exclude the Centre. Option (b) is defeatist. Option (c) gives the job only to the Union Government, against the passage.
Tip: prefer options that match the passage's scope without adding "only" or "essentially". India has a National Action Plan on Climate Change, and every State has its own State Action Plan.
Answer as per the official UPSC answer key.
Read the passage
Each State in India faces a distinctive set of challenges regarding the impact of warming, but also offers its own set of opportunities for reducing emissions depending on its natural resources. For example, coastal States need to take action to protect their shores from sea level rise, districts that are drier need to prepare for variable monsoon precipitation. Himalayan regions have their own unique challenges, and selected parts of peninsular India and offshore areas offer great opportunities for harnessing wind power. These various aspects need to be considered for developing clear and sustainable goals for the future.
With reference to the passage, the following assumptions have been made:
- I. Green energy production can be linked to/integrated with the climate change mitigation and adaptation strategies.
- II. Effects of climate change are much more severe in coastal and mountainous regions.
Which of the above assumptions is/are valid?
Answer (a) & explanation
Only assumption I is valid (a). Step 1: the passage lists wind power opportunities alongside adaptation needs like coastal protection, and says all these must be considered in setting goals. So green energy fits into climate strategies (I). Step 2: assumption II says climate effects are "much more severe" in coastal and mountain regions. The passage says each region has its own challenges, including dry districts; it never ranks them. Step 3: so II makes a comparison the passage does not. Options (b), (c) and (d) fail.
Tip: reject comparisons ("more severe", "most affected") unless the passage clearly compares. Mitigation means cutting emissions; adaptation means coping with impacts.
Answer as per the official UPSC answer key.
Read the passage
If the social inequality is the most acutely felt social problem in India, insecurity, more than poverty, is the most acutely felt economic problem. Besides those below the official poverty line, even those just over the poverty line are subject to multiple economic insecurities of various kinds (due to wealth and/or health risks, market fluctuations, job-related uncertainties). Many Government policies are actually intended towards mitigating these insecurities.
Which one of the following statements best reflects the critical message conveyed by the passage?
Answer (d) & explanation
The critical message is that insecurity, more than poverty, is the main economic issue policy must address (d). Step 1: the passage says plainly: "insecurity, more than poverty, is the most acutely felt economic problem." Step 2: it adds that even those just above the poverty line face risks from health, markets and jobs, and many policies aim at "mitigating these insecurities". That is (d). Step 3: option (a) calls insecurity a social problem; the passage calls inequality the social problem. Option (b) makes poverty the primary cause. Option (c) calls it "unmanageable", which is not stated.
Tip: watch for options that mix up labels (social vs economic). Social security schemes like health insurance and pensions target such insecurity.
Answer as per the official UPSC answer key.
Read the passage
If the social inequality is the most acutely felt social problem in India, insecurity, more than poverty, is the most acutely felt economic problem. Besides those below the official poverty line, even those just over the poverty line are subject to multiple economic insecurities of various kinds (due to wealth and/or health risks, market fluctuations, job-related uncertainties). Many Government policies are actually intended towards mitigating these insecurities.
With reference to the above passage, the following assumptions have been made:
- I. People above the poverty line also are prone to suffer from anxiety about economic insecurity.
- II. Eradication of poverty can result in peace and social equality in the country.
Which of the above assumptions is/are valid?
Answer (a) & explanation
Only assumption I is valid (a). Step 1: the passage says "even those just over the poverty line are subject to multiple economic insecurities." So people above the line also worry about economic security (I). Step 2: assumption II says ending poverty will bring peace and social equality. But the passage says insecurity affects people beyond poverty, and treats social inequality as a separate problem. Ending poverty alone would not solve both. Step 3: so II overreaches. Options (b), (c) and (d) fail.
Tip: when a passage separates two problems, do not assume solving one solves the other. Look for the passage's own words to support each assumption.
Answer as per the official UPSC answer key.
A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?
- I. The minimum number of cuts is 9.
- II. The number of smaller pieces which are not painted on any face is 6.
Select the correct answer using the code given below:
Answer (c) & explanation
Both statements are correct (c). Step 1: for the fewest cuts, split the 60 pieces as evenly as possible across three directions: 60 = 3 × 4 × 5. Cuts needed = (3 − 1) + (4 − 1) + (5 − 1) = 2 + 3 + 4 = 9. So I is right. Step 2: pieces with no paint come from the inside: remove one layer from each side in every direction: (3 − 2) × (4 − 2) × (5 − 2) = 1 × 2 × 3 = 6. So II is right. Options (a), (b) and (d) fail.
Tip: for the minimum cuts, keep the three factors as close together as possible. Unpainted inner pieces = (a − 2)(b − 2)(c − 2).
Answer as per the official UPSC answer key.
If 7 * 24 = 25 and 12 * 16 = 20, then what is 16 * 63 equal to?
Answer (c) & explanation
1. Look at the first two: 7 × 7 + 24 × 24 = 49 + 576 = 625 = 25 × 25. And 12 × 12 + 16 × 16 = 144 + 256 = 400 = 20 × 20.
2. So the rule is: square both numbers, add, and find the number whose square is the total.
3. 16 × 16 + 63 × 63 = 256 + 3969 = 4225 = 65 × 65. Answer 65.
Answer as per the official UPSC answer key.
The petrol price shot up by 10% as a result of the hike in crude oil prices. The price of petrol before the hike was ₹ 90 per litre. A person travels 2200 km every month and his car gives a mileage of 16 km per litre. By how many km should he reduce his travel if he wants to maintain his expenditure at the previous level?
Answer (b) & explanation
He must cut his travel by 200 km (b). Step 1: to keep spending the same, price × litres must stay fixed. The price rises by 10%, from ₹90 to ₹99, so the litres he buys must fall to 100/110 = 10/11 of before. Step 2: his travel falls in the same ratio: 2200 × 10/11 = 2000 km. Step 3: so he must reduce by 2200 − 2000 = 200 km. The mileage (16 km/l) is not needed. Option (c), 220 km, wrongly takes 10% of 2200.
Tip: a 10% price rise needs a cut of 10/110 = 1/11 (about 9.1%) in use, not 10%, to keep spending the same.
a 10% rise needs a 1/11 cut (about 9.1%), not a 10% cut of 220 km.
Answer as per the official UPSC answer key.
A 4-digit number N is such that when divided by 3, 5, 6, 9 leaves a remainder 1, 3, 4, 7 respectively. What is the smallest value of N?
Answer (c) & explanation
The smallest value of N is 1078 (c). Step 1: notice the remainders: 3 leaves 1, 5 leaves 3, 6 leaves 4, 9 leaves 7. Each remainder is 2 less than its divisor. So N + 2 is divisible by all of 3, 5, 6 and 9. Step 2: LCM(3, 5, 6, 9) = 90. So N = 90k − 2. Step 3: the smallest 4-digit value: 90 × 12 = 1080, so N = 1078. (90 × 11 − 2 = 988 is only 3 digits.) Options (a), (b) and (d) fail the remainder checks.
Tip: when divisor − remainder is the same in every case, N + (that gap) is a multiple of the LCM. This is a classic shortcut.
Answer as per the official UPSC answer key.
Consider the following statements:
- I. If A ≤ B > C < D > E > F ≥ G = H; then B is always greater than E.
- II. If P > Q = R ≥ S = T ≤ U = V > W; then S is always less than V.
Which of the statements given above is/are correct?
Answer (d) & explanation
Neither statement is correct (d). Step 1: statement I: A ≤ B > C < D > E. The chain breaks at B > C < D: B and D are not linked, so B and E cannot be compared. B need not be greater than E. Step 2: statement II: S = T ≤ U = V gives S ≤ V. S could be equal to V, so "always less than V" fails. Step 3: both claims are not guaranteed. Options (a), (b) and (c) fail.
Tip: a conclusion holds only if all signs between the two letters point the same way. A "≤" in the chain means the result can be "equal", so "always less" is wrong.
a chain proves something only if every sign between the two letters points the same way.
Answer as per the official UPSC answer key.
What is the unit digit in the multiplication of 1 × 3 × 5 × 7 × 9 × . . . × 999?
Answer (c) & explanation
The unit digit is 5 (c). Step 1: the product 1 × 3 × 5 × … × 999 has only odd numbers, and one of them is 5. Step 2: any odd number × 5 ends in 5 (for example 3 × 5 = 15, 7 × 5 = 35). Step 3: to end in 0, the product would need a factor 2, and there is none. So the unit digit stays 5. Options (a), (b) and (d) are wrong.
Tip: odd × 5 always ends in 5; even × 5 always ends in 0. Check for 2s and 5s first in unit-digit questions; they decide the answer quickly.
Answer as per the official UPSC answer key.
Read the passage
One of the dismal realities of the agricultural sector in independent India has been that it never experienced a high-growth phase, unlike the non-agricultural economy. The highest decadal growth (compound annual growth rate or CAGR) for agriculture has been just 3·5% in the 1980s. Also, after experiencing a spurt in decadal growth during the 1980s, agricultural growth suffered relative stagnation thereafter. This is in sharp contrast to non-agricultural growth, which consistently increased from the 1980s to 2000s.
Which one of the following statements best reflects the corollary to the above passage?
Answer (a) & explanation
The corollary is that reform benefits reach agriculture more slowly than other sectors (a). Step 1: a corollary is something that follows naturally from the passage. Step 2: the passage says farm growth peaked at 3.5% in the 1980s and then stagnated, while non-farm growth rose steadily from the 1980s to the 2000s, the reform era. So the gains of reform reached farming more slowly: (a). Step 3: options (b) the Green Revolution, (c) mechanisation and (d) migration are not discussed at all.
Tip: a corollary stays close to the passage's facts. India's economic reforms began in 1991; agriculture still employs over 40% of workers but gives a much smaller share of GDP.
Answer as per the official UPSC answer key.
Read the passage
One of the dismal realities of the agricultural sector in independent India has been that it never experienced a high-growth phase, unlike the non-agricultural economy. The highest decadal growth (compound annual growth rate or CAGR) for agriculture has been just 3·5% in the 1980s. Also, after experiencing a spurt in decadal growth during the 1980s, agricultural growth suffered relative stagnation thereafter. This is in sharp contrast to non-agricultural growth, which consistently increased from the 1980s to 2000s.
With reference to the passage, the following assumptions have been made:
The growing divergence between the fortunes of the agricultural and non-agricultural economy in India could have been reduced/contained by :
- I. adapting large-scale cultivation of commercial crops and viable corporate farming.
- II. providing free insurance for all crops and heavily subsidizing seeds, fertilizers, electricity and farm machinery at par with developed countries.
Which of the above assumptions is/are valid?
Answer (d) & explanation
Neither assumption is valid (d). Step 1: the passage only describes the gap: agriculture stagnated while non-farm growth rose. It suggests no remedy. Step 2: assumption I, corporate farming and commercial crops, and assumption II, free insurance and heavy subsidies, are possible policies, but neither is mentioned or implied. Step 3: they come from outside knowledge, so neither is an assumption of this passage. Options (a), (b) and (c) fail.
Tip: when a passage only describes a problem, any specific solution is outside it. Do not accept a remedy just because it sounds reasonable. India already runs PM Fasal Bima Yojana (crop insurance) and PM-KISAN, but the passage does not discuss them.
Answer as per the official UPSC answer key.
Read the passage
In our country, handlooms are equated with a culture that ensures a continuity of tradition. This idea has become part of the public policy-framing and provides a legitimate basis for the State to support the sector. But the notion of tradition as a single, linear entity is being strongly contested today. The narratives dominant in defining culture/tradition in a particular way are seen to have emerged as the identities and histories of large sections. The discounted and, at times, forcibly stifled identities are fighting for their rightful place in history. Against this backdrop, when we promote handloom as a traditional industry, it is not surprising that large sections of our population choose to ignore it.
Which one of the following statements best reflects the most logical and rational message conveyed by the author of the passage?
Answer (a) & explanation
The message is that handloom must be freed from the narrow "cultural heritage" narrative (a). Step 1: the passage says handloom is promoted as tradition, but the idea of one single tradition is now contested. Step 2: so when handloom is sold as "traditional", "large sections of our population choose to ignore it". The narrow narrative is the problem: (a). Step 3: option (b) repeats the very framing the author questions. Options (c), modernising household units, and (d), machine-made products, are not suggested and (d) is extreme.
Tip: identify what the author criticises; the answer usually removes that problem. India marks National Handloom Day on 7 August, recalling the Swadeshi Movement of 1905.
Answer as per the official UPSC answer key.
Read the passage
In our country, handlooms are equated with a culture that ensures a continuity of tradition. This idea has become part of the public policy-framing and provides a legitimate basis for the State to support the sector. But the notion of tradition as a single, linear entity is being strongly contested today. The narratives dominant in defining culture/tradition in a particular way are seen to have emerged as the identities and histories of large sections. The discounted and, at times, forcibly stifled identities are fighting for their rightful place in history. Against this backdrop, when we promote handloom as a traditional industry, it is not surprising that large sections of our population choose to ignore it.
With reference to the above passage, the following assumptions have been made:
- I. There is no need for the State to be involved in any manner in the handloom sector.
- II. Handloom products are no longer appealing and attractive in the rapidly changing modern world.
Which of the above assumptions is/are valid?
Answer (d) & explanation
Neither assumption is valid (d). Step 1: assumption I says the State should have no role at all. The author only questions how the State frames its support (as tradition), not support itself. So I fails. Step 2: assumption II says handloom products are no longer appealing. The passage says people ignore the narrative, the "traditional" label, not the products. So II fails too. Step 3: options (a), (b) and (c) fail.
Tip: separate the product from the way it is presented. Criticising a policy's framing is not rejecting the policy. Extreme words like "no need ... in any manner" are usually wrong.
Answer as per the official UPSC answer key.
Consider the first 100 natural numbers. How many of them are not divisible by any one of 2, 3, 5, 7 and 9?
Answer (c) & explanation
1. 9 adds nothing new: anything divisible by 9 is already divisible by 3. So we need numbers not divisible by 2, 3, 5 or 7.
2. Any number up to 100 that is not prime has a factor of 2, 3, 5 or 7 (because 11 × 11 is already more than 100).
3. So the numbers we want are just 1 and the primes bigger than 7.
4. There are 25 primes up to 100. Leave out 2, 3, 5 and 7: 21 primes. Add the number 1: 22.
9 is in the list to make you count its multiples twice.
Answer as per the official UPSC answer key.
If 4 ≤ x ≤ 8 and 2 ≤ y ≤ 7, then what is the ratio of maximum value of (x + y) to minimum value of (x − y)?
Answer (d) & explanation
The ratio is −5, which is not listed, so the answer is (d). Step 1: the maximum of x + y uses the largest x and largest y: 8 + 7 = 15. Step 2: the minimum of x − y uses the smallest x and the largest y: 4 − 7 = −3. Step 3: ratio = 15 ÷ (−3) = −5. None of the options show −5. Option (c), −15/2, divides by −2 instead of −3. Options (a) and (b) ignore the minus sign.
Tip: to make a difference smallest, make the first number small and the subtracted number big. Always check "None of the above" before forcing a match.
to make x − y small you take y large, not small.
Answer as per the official UPSC answer key.
Let both p and k be prime numbers such that (p2 + k) is also a prime number less than 30. What is the number of possible values of k?
Answer (b) & explanation
Five values of k are possible (b). Step 1: if p = 2, then p² = 4, and 4 + k must be a prime below 30. k = 3, 7, 13, 19 work (sums 7, 11, 17, 23). k = 2 gives 6, not prime. Step 2: if p is odd, p² is odd, so k must be 2 (the even prime) to keep the sum odd. p = 3 gives 9 + 2 = 11, prime. p = 5 gives 27, not prime. So k = 2 also works. Step 3: possible k: 2, 3, 7, 13, 19 = 5 values. Options (a), (c) and (d) miscount.
Tip: in prime questions, split into "2" and "odd" cases. Odd + odd is even, so the even prime 2 often decides things.
Answer as per the official UPSC answer key.
There are n sets of numbers each having only three positive integers with LCM equal to 1001 and HCF equal to 1. What is the value of n?
Answer (d) & explanation
There are more than 8 such sets (d). Step 1: 1001 = 7 × 11 × 13. Each number in a set must be made from these primes (or be 1). Step 2: LCM 1001 means every prime appears somewhere; HCF 1 means no prime is in all three. Step 3: list some sets: {7, 11, 13}, {1, 77, 91}, {1, 77, 143}, {1, 91, 143}, {7, 11, 1001}, {1, 7, 143}, {1, 11, 91}, {1, 13, 77}, {77, 91, 143}. That is already 9, so n is more than 8. Options (a), (b) and (c) are too small.
Tip: when an option says "more than", you only need to find enough cases to pass it; no full count is needed. 1001 = 7 × 11 × 13 is worth remembering.
Answer as per the official UPSC answer key.
Let PQR be a 3-digit number, PPT be a 3-digit number and PS be a 2-digit number, where P, Q, R, S, T are distinct non-zero digits. Further, PQR − PS = PPT. If Q = 3 and T < 6, then what is the number of possible values of (R, S)?
Answer (b) & explanation
1. Write it as an ordinary subtraction: PQR − PS = PPT. The answer keeps P in front, and its middle digit is also P.
2. Middle column: 3 − P (minus 1 if we borrowed) must give P. Only P = 1 with a borrow works: 3 − 1 − 1 = 1.
3. Last column, after borrowing: (R + 10) − S = T. So S + T = R + 10.
4. T can be 2, 4 or 5 (not 1 or 3, which are taken, and less than 6). Try each, using only unused digits:
5. T = 2: S = R + 8 needs R = 1, which is taken. None.
6. T = 4: S = R + 6 gives R = 2, S = 8. One pair.
7. T = 5: S = R + 5 gives (R, S) = (2, 7) or (4, 9). Two pairs.
8. Total: 3 pairs. Check one: 132 − 18 = 114.
Count pairs (R, S), not separate digits: (2, 8), (2, 7) and (4, 9).
Answer as per the official UPSC answer key.
Consider the sequence AB_CC_A_BCCC_BBC_C that follows a certain pattern. Which one of the following completes the sequence?
Answer (c) & explanation
The blanks are B, C, B, A, C (c). Step 1: put option (c) into the five gaps: ABBCCCABBCCCABBCCC. Step 2: it reads ABBCCC ABBCCC ABBCCC: one A, two Bs and three Cs, repeated three times. A clear pattern. Step 3: the other options do not give a repeating block. For example, (a) puts A in the last gap, breaking the CCC at the end. So (a), (b) and (d) fail.
Tip: in letter-gap series, first count the total letters (here 18) and try block sizes that divide it. Then test options quickly rather than solving from scratch.
Answer as per the official UPSC answer key.
Read the passage
In our country, regrettably, teaching and learning for the examination have been our forte but the new demands of society and the future of work require critical and independent thinking, learning through doing, asking questions from multiple disciplinary perspectives on the same issue, using evidence for building arguments, and reflecting and articulation. Higher education should not “either be a mere servant of the government policy or a passive respondent to public mood.” Higher learning is all about how to think rather than what to think. Teaching has to be re-invented.
Which one of the following statements best reflects the central idea conveyed by the passage?
Answer (d) & explanation
The central idea is that the classroom must be reimagined and teaching re-invented (d). Step 1: the passage says Indian education has focused on exams, but society now needs critical thinking, learning by doing, questioning and evidence-based argument. Step 2: it ends: higher learning is about "how to think rather than what to think. Teaching has to be re-invented." That is (d). Step 3: option (a), lack of resources, is never discussed. Option (b) misreads "not a mere servant of government policy" as "no government control". Option (c), doing away with classrooms, is extreme; the author wants them changed, not removed.
Tip: the last line often states the central idea. The National Education Policy 2020 also stresses critical thinking over rote learning.
Answer as per the official UPSC answer key.
Read the passage
In our country, regrettably, teaching and learning for the examination have been our forte but the new demands of society and the future of work require critical and independent thinking, learning through doing, asking questions from multiple disciplinary perspectives on the same issue, using evidence for building arguments, and reflecting and articulation. Higher education should not “either be a mere servant of the government policy or a passive respondent to public mood.” Higher learning is all about how to think rather than what to think. Teaching has to be re-invented.
With reference to the above passage, the following assumptions have been made:
- I. Higher education is a constantly evolving subject that needs to align towards new developments in all spheres of society.
- II. In our country, sufficient funds are not allocated for promoting higher education.
Which of the above assumptions is/are valid ?
Answer (a) & explanation
Only assumption I is valid (a). Step 1: the passage says "the new demands of society and the future of work" need new kinds of learning, and teaching must be re-invented. So higher education must keep evolving with society (I). Step 2: assumption II, about insufficient funds for higher education, is never mentioned. The passage is about teaching methods, not money. Step 3: so options (b), (c) and (d) fail.
Tip: reject assumptions that bring in a new topic like funds, staff or buildings when the passage talks only about methods. A valid assumption must link directly to the author's argument.
Answer as per the official UPSC answer key.
Read the passage
If there is inequality in the pattern of population growth, there is greater inequality in food production and utilization. As societies become wealthier, their consumption of animal products increases. This means that a greater proportion of such basic foodstuff as grains and legumes that could feed humans directly is instead being converted into feed for poultry and large farm animals. Yet this conversion of plant-based food into animal food for humans is far from efficient. Only 16% of the calories fed to chickens are recovered by us when we eat them. This conversion rate goes down to five to seven per cent in large animals that are fed grain to add fat and some protein before slaughter.
Which one of the following statements best reflects the crux of the passage?
Answer (b) & explanation
The crux is that animal-based food is far less efficient than plant-based food (b). Step 1: the passage says richer societies eat more animal products, so grain that could feed people is fed to animals. Step 2: it gives numbers: only 16% of calories fed to chickens come back to us, and just 5–7% for large animals. It calls this "far from efficient." That is (b). Step 3: options (a), a public policy, and (c), replacing animal protein, are prescriptions the passage does not make. Option (d) calls inequality "inevitable", which is not said.
Tip: crux questions want what the passage proves, not what someone should do about it. The numbers here prove inefficiency.
Answer as per the official UPSC answer key.
Read the passage
If there is inequality in the pattern of population growth, there is greater inequality in food production and utilization. As societies become wealthier, their consumption of animal products increases. This means that a greater proportion of such basic foodstuff as grains and legumes that could feed humans directly is instead being converted into feed for poultry and large farm animals. Yet this conversion of plant-based food into animal food for humans is far from efficient. Only 16% of the calories fed to chickens are recovered by us when we eat them. This conversion rate goes down to five to seven per cent in large animals that are fed grain to add fat and some protein before slaughter.
With reference to the above passage, the following assumptions have been made:
- I. The food manufacturing and processing industries in every country should align their objectives and processes in accordance with the changing needs of the societies.
- II. Wealthier societies tend to incur great loss of calories of food materials due to indirect utilization of their agricultural produce.
Which of the above assumptions is/are valid?
Answer (b) & explanation
Only assumption II is valid (b). Step 1: the passage says wealthier societies feed grain to animals and recover only 16% (chicken) or 5–7% (large animals) of the calories. So they lose many calories through this indirect use of farm produce (II). Step 2: assumption I talks about food manufacturing and processing industries aligning with society's needs. The passage does not discuss these industries at all. Step 3: so I is outside the passage. Options (a), (c) and (d) fail.
Tip: when an assumption names a group (industries, governments) that the passage never mentions, reject it. Feeding animals grain is why meat has a bigger land and water footprint than plant food.
Answer as per the official UPSC answer key.
What is the maximum value of n such that 7 × 343 × 385 × 1000 × 2401 × 77777 is divisible by 35n?
Answer (b) & explanation
The maximum n is 4 (b). Step 1: 35ⁿ = 5ⁿ × 7ⁿ. So we need to count both the 5s and the 7s and take the smaller count. Step 2: 7s are plentiful: 7, 343 = 7³, 2401 = 7⁴, 385 and 77777 = 7 × 11111 all add 7s. So count the 5s. Step 3: 385 = 5 × 7 × 11 gives one 5; 1000 = 2³ × 5³ gives three. The others have none. Total 5s = 4, so n = 4. Options (a), (c) and (d) miscount.
Tip: for the highest power of a composite number, split it into primes and count the scarcer prime. Here 5 is the limiting factor.
Answer as per the official UPSC answer key.
What is X in the sequence 24, X, 12, 18, 36, 90?
Answer (b) & explanation
X is 12 (b). Step 1: look at the later terms: 12 → 18 is × 1.5, 18 → 36 is × 2, 36 → 90 is × 2.5. The multiplier rises by 0.5 each time. Step 2: go backwards: before 1.5 comes 1, and before that 0.5. Step 3: so 24 × 0.5 = 12 = X, and 12 × 1 = 12 ✓. Options (a) 18, (c) 9 and (d) 6 break the pattern.
Tip: if numbers rise unevenly, try ratios instead of differences. When the start of a pattern is missing, work backwards from the known part.
Answer as per the official UPSC answer key.
P and Q walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. P walks at an average speed of 5 rounds per hour and Q walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m.?
Answer (b) & explanation
They cross 13 times (b). Step 1: walking in opposite directions on a circle, their speeds add: 5 + 3 = 8 rounds per hour. So they cross every 1/8 hour = 7.5 minutes. Step 2: crossings happen at 7.5, 15, 22.5, 30, … minutes after 5:00. The first one after 5:20 is at 22.5 minutes (the 3rd crossing). Step 3: 7:00 is 120 minutes, which is the 16th crossing. Between 5:20 and 7:00 (not counting 7:00), crossings 3 to 15 occur: 13 times. Options (a), (c) and (d) miscount the ends.
Tip: opposite directions add speeds; same direction subtracts them. Count crossings carefully at the start and end of the time window.
Answer as per the official UPSC answer key.
If P = +, Q = –, R = ×, S = ÷, then insert the proper notations between the successive numbers in the equation 60 _ 15 _ 3 _ 20 _ 4 = 20 :
Answer (b) & explanation
The correct order is QRPS (b). Step 1: decode: P = +, Q = −, R = ×, S = ÷. Step 2: test (b) QRPS: 60 − 15 × 3 + 20 ÷ 4. Do × and ÷ first: 15 × 3 = 45 and 20 ÷ 4 = 5. Then 60 − 45 + 5 = 20 ✓. Step 3: test (a) SPRQ: 60 ÷ 15 + 3 × 20 − 4 = 4 + 60 − 4 = 60 ✗. Option (c) QRSP gives 60 − 15 × 3 ÷ 20 + 4 = 61.75, and (d) SPQR gives 4 + 3 − 80 = −73.
Tip: test the options rather than building the answer yourself. Always apply BODMAS: × and ÷ before + and −.
Answer as per the official UPSC answer key.
A tram overtakes 2 persons X and Y walking at an average speed of 3 km/hr and 4 km/hr in the same direction and completely passes them in 8 seconds and 9 seconds respectively. What is the length of the tram?
Answer (c) & explanation
The tram is 20 m long (c). Step 1: the tram and walkers move the same way, so use relative speed = tram speed − walker speed. Length = (v − 3) × 8 = (v − 4) × 9, with speeds in km/hr and time in seconds. Step 2: 8v − 24 = 9v − 36, so v = 12 km/hr. Step 3: relative speed past X = 12 − 3 = 9 km/hr = 9 × 5/18 = 2.5 m/s. Length = 2.5 × 8 = 20 m. Check with Y: 8 km/hr = 20/9 m/s × 9 s = 20 m ✓. Options (a), (b) and (d) fail.
Tip: to change km/hr to m/s, multiply by 5/18. Passing a person (a point) means covering only the tram's own length.
Answer as per the official UPSC answer key.
If N2 = 12345678987654321, then how many digits does the number N have?
Answer (b) & explanation
N has 9 digits (b). Step 1: recall the pattern: 11² = 121, 111² = 12321, 1111² = 1234321, and so on. The square rises to the number of 1s, then falls. Step 2: 12345678987654321 rises to 9, so N = 111111111, which has 9 digits. Step 3: quick check: the square has 17 digits. A number with d digits has a square of 2d − 1 or 2d digits, so d = 9. Options (a), (c) and (d) fail.
Tip: learn this palindrome pattern of repunits (numbers made of 1s). It works up to nine 1s; after that, carries spoil it.
Answer as per the official UPSC answer key.
Read the passage
Over the next 30 years, many countries are promising to move to net-zero carbon, implying that household emissions will have to be cut to close to nothing. A leading climate scientist reckons that, at best, half the reduction might be achieved through demand-side measures, such as behavioural changes by individuals and households. And even that would require companies and governments to provide more incentives to change through supply-side investments to make low-carbon options cheaper and more widely available.
Which one of the following statements best reflects the central idea conveyed by the passage?
Answer (b) & explanation
The central idea is that low-carbon behaviour can be encouraged through incentives (b). Step 1: the passage says household emissions must fall almost to zero, and at best half can come from changes in people's behaviour. Step 2: even that needs "companies and governments to provide more incentives to change", by making low-carbon options cheaper and widely available. That is (b). Step 3: option (a) says net zero is possible "only" through households, too strong. Option (c) reverses the logic: investment makes low-carbon options cheaper, not the other way. Option (d), subsidies to industry, is not stated.
Tip: look for the "even that would require..." line; it often holds the key. India's Mission LiFE (2022) promotes such climate-friendly lifestyles.
Answer as per the official UPSC answer key.
Read the passage
Over the next 30 years, many countries are promising to move to net-zero carbon, implying that household emissions will have to be cut to close to nothing. A leading climate scientist reckons that, at best, half the reduction might be achieved through demand-side measures, such as behavioural changes by individuals and households. And even that would require companies and governments to provide more incentives to change through supply-side investments to make low-carbon options cheaper and more widely available.
With reference to the above passage, the following assumptions have been made:
- I. Supply-side investments in companies can result in low-carbon behaviour in people.
- II. People are not capable of adapting low-carbon behaviour without the involvement Government and Companies.
Which of the above assumptions is/are valid?
Answer (a) & explanation
Only assumption I is valid (a). Step 1: the passage says supply-side investments make low-carbon options "cheaper and more widely available", which encourages people to change. So such investments can lead to low-carbon behaviour (I). Step 2: assumption II says people are "not capable" of changing without government and companies. The passage only says incentives help; it never says people cannot change on their own. Step 3: "not capable" overstates it. Options (b), (c) and (d) fail.
Tip: "needs help to do more" is not the same as "cannot do at all". Strong words like "not capable" or "impossible" usually make an assumption invalid.
Answer as per the official UPSC answer key.
Read the passage
In only 50 years, the world’s consumption of raw materials has nearly quadrupled, to more than 100 billion tons. Less than 9% of this is reused. Batteries of old vehicles contain materials such as lithium, cobalt, manganese and nickel that are pricey and can be hard to obtain. Supply chains are long and complicated. Buyers’ risks are being aggravated by their suppliers’ poor environmental and labour standards. Reusing materials makes sense. Once batteries reach the ends of their lives, they should go back to a factory where their ingredients can be recovered and put into new batteries.
Which one of the following statements best reflects the most logical, rational and pragmatic message conveyed by the passage?
Answer (c) & explanation
The message is that a circular economy can support sustainable growth (c). Step 1: the passage says raw material use has nearly quadrupled in 50 years, but less than 9% is reused. Battery metals like lithium and cobalt are costly and supply chains are risky. Step 2: it concludes "reusing materials makes sense", sending old batteries back to factories for new ones. Reuse in a loop is a circular economy: (c). Step 3: option (a) says green economy is "not possible" without it. Option (b) says "every sector immediately". Option (d) says "only option". All three are extreme.
Tip: "can be beneficial" is mild and safe; prefer it over absolute options. India's Battery Waste Management Rules, 2022, make producers responsible for collecting and recycling batteries.
Answer as per the official UPSC answer key.
Read the passage
In only 50 years, the world’s consumption of raw materials has nearly quadrupled, to more than 100 billion tons. Less than 9% of this is reused. Batteries of old vehicles contain materials such as lithium, cobalt, manganese and nickel that are pricey and can be hard to obtain. Supply chains are long and complicated. Buyers’ risks are being aggravated by their suppliers’ poor environmental and labour standards. Reusing materials makes sense. Once batteries reach the ends of their lives, they should go back to a factory where their ingredients can be recovered and put into new batteries.
With reference to the above passage, the following assumptions have been made:
- I. Automobile factories are examples of the circular economy.
- II. Economic growth is compatible with circular use of mineral resources.
Which of the above assumptions is/are valid?
Answer (b) & explanation
Only assumption II is valid (b). Step 1: the passage says recovering battery materials and putting them into new batteries makes sense, keeping production going with fewer raw materials. So growth and the circular use of minerals can go together (II). Step 2: assumption I says automobile factories are examples of the circular economy. The passage says batteries should go back to factories, a suggestion, not a description of what already happens. Step 3: so I fails. Options (a), (c) and (d) fail.
Tip: watch the difference between "should" (a recommendation) and "is" (a fact). An assumption that turns a suggestion into a present fact is invalid.
Answer as per the official UPSC answer key.
A set (X) of 20 pipes can fill 70% of a tank in 14 minutes. Another set (Y) of 10 pipes fills 3/8th of the tank in 6 minutes. A third set (Z) of 16 pipes can empty half of the tank in 20 minutes. If half of the pipes of set X are closed and only half of the pipes of set Y are open, and all pipes of the set (Z) are open, then how long will it take to fill 50% of the tank?
Answer (d) & explanation
It takes 16 minutes to fill half the tank (d). Step 1: work out the rate of each full set per minute. X fills 70% in 14 min: 5% a minute; half of X = 2.5%. Y fills 37.5% in 6 min: 6.25%; half of Y = 3.125%. Z empties 50% in 20 min: 2.5% a minute (all of Z is open). Step 2: net filling = 2.5 + 3.125 − 2.5 = 3.125% a minute. Step 3: time for 50% = 50 ÷ 3.125 = 16 minutes. Options (a), (b) and (c) forget to halve a set or ignore the emptying pipes.
Tip: convert everything into "percent of tank per minute". Add filling pipes, subtract emptying ones, then divide.
Answer as per the official UPSC answer key.
If n is a natural number, then what is the number of distinct remainders of (1n + 2n) when divided by 4?
Answer (c) & explanation
There are two distinct remainders (c). Step 1: try n = 1: 1 + 2 = 3, which leaves remainder 3 when divided by 4. Step 2: for n = 2 or more, 2ⁿ is a multiple of 4 (4, 8, 16, …), so it leaves remainder 0. And 1ⁿ is always 1. So 1ⁿ + 2ⁿ leaves remainder 1. Step 3: the only remainders are 3 and 1: two values. Options (a), (b) and (d) fail.
Tip: test small values of n first; patterns usually settle after one or two steps. Remember any power of 2 from 2² onward is divisible by 4.
don't stop at n ≥ 2. The single case n = 1 adds a second remainder.
Answer as per the official UPSC answer key.
Let P = QQQ be a 3-digit number. What is the HCF of P and 481?
Answer (c) & explanation
The HCF is 37 (c). Step 1: a number like QQQ (for example 555) equals Q × 111, and 111 = 3 × 37. So every QQQ has 37 as a factor. Step 2: 481 = 13 × 37. Step 3: the common part is 37. Could the HCF be bigger? Only if QQQ also had 13, but Q is a single digit (1–9) and 3 × Q cannot contain 13. So the HCF is always 37. Options (a), (b) and (d) fail.
Tip: remember 111 = 3 × 37, so every three-same-digit number (111, 222, …, 999) is divisible by 37. Also 1001 = 7 × 11 × 13.
Answer as per the official UPSC answer key.
What is the 489th digit in the number 123456789101112...?
Answer (d) & explanation
The 489th digit is 9 (d). Step 1: the numbers 1 to 9 use 9 digits. The numbers 10 to 99 use 90 × 2 = 180 digits. Total so far: 189. Step 2: 489 − 189 = 300 digits remain. Each three-digit number uses 3 digits, so that is exactly 100 numbers: 100 to 199. Step 3: the 489th digit is the last digit of 199: 9. Options (a), (b) and (c) come from miscounting the blocks.
Tip: count digits block by block: 1-digit (9), 2-digit (180), 3-digit (2700). If the leftover divides exactly, the answer is the last digit of that number.
Answer as per the official UPSC answer key.
A mobile phone has been stolen. There are 3 suspects P, Q and R. They were questioned knowing that only one of them is guilty. Their responses are as follows :
P : I did not steal. Q stole it.
Q : R did not steal. I did not steal.
R : I did not steal. I do not know who did it.
Who stole the mobile phone?
Answer (d) & explanation
The thief cannot be concluded (d). Step 1: we know only one is guilty, but we are not told who speaks the truth or how many lie. Step 2: test each case. If P stole it, P lies, and Q and R can both be telling the truth. If Q stole it, Q lies, and P and R can be truthful. If R stole it, R lies, and Q's "R did not steal" is false, while P's "Q stole it" is false too. Step 3: without a rule like "only one lies", several cases fit. So no one can be named. Options (a), (b) and (c) fail.
Tip: in truth-teller puzzles, look for a rule about who lies. If none is given, the answer is often "cannot be concluded".
don't assume the innocent always tell the truth unless the question says so.
UPSC repeats traps like this year after year. The PYQ Book gathers every trap in one place, so you see them coming. Try the free sample
Answer as per the official UPSC answer key.
Three teams P, Q, R participated in a tournament in which the teams play with one another exactly once. A win fetches a team 2 points and a draw 1 point. A team gets no point for a loss. Each team scored exactly one goal in the tournament. The team P got 3 points, Q got 2 points and R got 1 point. Which of the following statements is/are correct?
- I. The result of the match between P and Q is a draw with the score 0-0.
- II. The number of goals scored by R against Q is 1.
Which of the statements given above is/are correct?
Answer (c) & explanation
Both statements are correct (c). Step 1: points: P = 3 (a win and a draw), Q = 2, R = 1 (a draw and a loss). There are three matches: P–Q, P–R and Q–R. Step 2: suppose P beat Q and drew R. Then Q must beat R for 2 points. P has only one goal, so P–Q was 1–0 and P–R 0–0; R's one goal comes against Q, so Q needs 2 goals to win, but Q has only one. Impossible. So P drew Q, P beat R, Q drew R. Step 3: each team scored exactly one goal. P's single goal beat R 1–0, so P–Q was 0–0 (I true). Q's goal came against R in a 1–1 draw, so R scored 1 against Q (II true). Options (a), (b) and (d) fail.
Tip: in points tables, list each team's possible results first, then match goals.
Answer as per the official UPSC answer key.
Read the passage
It is hard to predict how changes in the climate and the atmosphere’s chemistry will affect the prevalence and virulence of agricultural diseases. But there is a risk that such changes will make some plant infections more common in all climatic zones, perhaps catastrophically so. Part of the problem is that centuries of selective breeding have refined the genomes of most high-value crops. They are spectacular at growing in today’s conditions but genetic variations that are not immediately useful to them have been bred out. This is good for yields but bad for coping with changes. A minor disease or even an unknown one could suddenly rampage through a genetically honed crop.
Which one of the following statements best reflects the central idea conveyed by the passage?
Answer (b) & explanation
The central idea is that relying fully on genetically honed crops risks food insecurity (b). Step 1: the passage says selective breeding has made crops great at growing today, but has bred out genetic variety. Step 2: so they cope badly with change: "a minor disease or even an unknown one could suddenly rampage through a genetically honed crop". That threatens food supply: (b). Step 3: option (a) talks about productivity, but the passage is about disease risk. Option (c) is vague. Option (d), replacing crops with wild varieties, is an extreme remedy the passage never suggests.
Tip: genetic uniformity is a known danger: the Irish potato famine (1840s) wiped out a crop with little variety. Seed banks protect crop diversity.
Answer as per the official UPSC answer key.
Read the passage
It is hard to predict how changes in the climate and the atmosphere’s chemistry will affect the prevalence and virulence of agricultural diseases. But there is a risk that such changes will make some plant infections more common in all climatic zones, perhaps catastrophically so. Part of the problem is that centuries of selective breeding have refined the genomes of most high-value crops. They are spectacular at growing in today’s conditions but genetic variations that are not immediately useful to them have been bred out. This is good for yields but bad for coping with changes. A minor disease or even an unknown one could suddenly rampage through a genetically honed crop.
With reference to the above passage, the following assumptions have been made
- I. Global climate change can result in the migration of several plant diseases to new areas.
- II. Scientific understanding of the wild relatives of our present crops would enable us to strengthen food security.
Which of the above assumptions is/are valid?
Answer (c) & explanation
Both assumptions are valid (c). Step 1: the passage says climate change may make plant infections "more common in all climatic zones". So diseases can spread to new areas (I). Step 2: it says useful genetic variety has been bred out of modern crops. That variety still exists in wild relatives of crops, so studying them can help breed hardier crops and protect food security (II). Step 3: both fit the passage's argument. Options (a), (b) and (d) fail.
Tip: wild relatives of crops carry genes for disease and drought resistance. India's National Bureau of Plant Genetic Resources in New Delhi runs one of the world's largest gene banks.
Answer as per the official UPSC answer key.
Read the passage
“A good statesman, like any other sensible human being, learns more from his opponents than from his fervent supporters. For his supporters will push him to disaster unless his opponents show him where the dangers are. So if he is wise he will often pray to be delivered from his friends, because they will ruin him. But, though it hurts, he ought also to pray never to be left without opponents; for they keep him on the path of reason and good sense. The national unity of free people depends upon a sufficiently even balance of political power to make it impracticable for the administration to be arbitrary and for opposition to be revolutionary and irreconcilable.”
Which one of the following statements best reflects the critical message conveyed by the author of the passage?
Answer (d) & explanation
The message is that opposition is indispensable for balance of power and good governance in a democracy (d). Step 1: the passage says a statesman learns more from opponents than from supporters, because opponents "keep him on the path of reason and good sense". Step 2: it ends: free people's unity needs "a sufficiently even balance of political power". So opposition is essential: (d). Step 3: option (a) says government improves without opposition, the opposite. Option (b) wants revolutionary opposition; the passage warns against that. Option (c) says rulers need opposition's support, a misreading.
Tip: the last sentence often holds the political message. In India, the Leader of the Opposition holds an official, recognised role in Parliament.
Answer as per the official UPSC answer key.
Read the passage
“A good statesman, like any other sensible human being, learns more from his opponents than from his fervent supporters. For his supporters will push him to disaster unless his opponents show him where the dangers are. So if he is wise he will often pray to be delivered from his friends, because they will ruin him. But, though it hurts, he ought also to pray never to be left without opponents; for they keep him on the path of reason and good sense. The national unity of free people depends upon a sufficiently even balance of political power to make it impracticable for the administration to be arbitrary and for opposition to be revolutionary and irreconcilable.”
With reference to the above passage, the following assumptions have been made
- I. In a democracy, a strong opposition is required only if the Head of Government is indifferent.
- II. The more aggressive the opposition, the better is the governance in a democracy.
Which of the above assumptions is/are valid?
Answer (d) & explanation
Neither assumption is valid (d). Step 1: assumption I says a strong opposition is needed "only if" the head of government is indifferent. But the passage says even a wise statesman should "pray never to be left without opponents". So opposition is always needed; I fails. Step 2: assumption II says the more aggressive the opposition, the better. The passage wants balance, and warns against opposition that is "revolutionary and irreconcilable". So II fails. Step 3: options (a), (b) and (c) fail.
Tip: "only if" and "the more ... the better" are traps. Passages about balance reject extremes on both sides.
Answer as per the official UPSC answer key.
P is the brother of Q and R. S is R's mother. T is P's father. How many of the following statements are definitely true ?
- I. S and T are a couple.
- II. Q is T's son.
- III. T is Q's father.
- IV. S is P's mother.
- V. R is T's daughter.
- VI. P is S's son.
Select the correct answer using the code given below:
Answer (c) & explanation
Only four statements are definitely true (c). Step 1: P is the brother of Q and R, so P, Q and R are siblings. S is R's mother, so S is mother of all three. T is P's father, so T is father of all three. Step 2: so I (S and T are a couple), III (T is Q's father), IV (S is P's mother) and VI (P is S's son; P is a brother, so male) are true. Step 3: II says Q is T's son, and V says R is T's daughter. The genders of Q and R are not given, so these are not certain. That makes four. Options (a), (b) and (d) miscount.
Tip: in blood-relation questions, never assume gender from a name or letter. Only words like brother, mother or son fix gender.
Answer as per the official UPSC answer key.
If NO is coded as 210, NOT is coded as 4200 and NOTE is coded as 21000, then how is NOTES coded?
Answer (a) & explanation
NOTES is coded as 399000 (a). Step 1: use alphabet positions: N = 14, O = 15. 14 × 15 = 210 = NO ✓. Step 2: T = 20: 210 × 20 = 4200 = NOT ✓. E = 5: 4200 × 5 = 21000 = NOTE ✓. The rule is the product of positions. Step 3: S = 19: 21000 × 19 = 399000. Options (b), (c) and (d) multiply by 20, 21 or 30 instead of 19.
Tip: when each longer word's code is a multiple of the shorter one's, the rule is likely multiplication by the new letter's position. Know positions by heart: N = 14, S = 19.
Answer as per the official UPSC answer key.
If FRANCE is coded as 654321 and GERMANY is coded as 9158437, then how is YEMEN coded ?
Answer (d) & explanation
YEMEN is coded as 71813 (d). Step 1: match letters to digits from FRANCE = 654321: F = 6, R = 5, A = 4, N = 3, C = 2, E = 1. Step 2: from GERMANY = 9158437: G = 9, E = 1 ✓, R = 5 ✓, M = 8, A = 4 ✓, N = 3 ✓, Y = 7. Step 3: YEMEN = Y E M E N = 7, 1, 8, 1, 3 = 71813. Option (c), 71913, takes M as 9 (that is G). Options (a) and (b) are wrong.
Tip: in letter-to-digit codes, line up each word with its code and check shared letters (here E, R, A, N) to confirm the mapping before answering.
Answer as per the official UPSC answer key.
The 5-digit number PQRST (all distinct digits) is such that T ≠ 0, P is thrice T. S is greater than Q by 4, while Q is greater than R by 3. How many such 5-digit numbers are possible?
Answer (b) & explanation
Four such numbers are possible (b). Step 1: P = 3T with T not 0: (T, P) can be (1, 3), (2, 6), (3, 9). Step 2: Q = R + 3 and S = Q + 4 = R + 7. S is a digit, so R can be 0, 1 or 2: (R, Q, S) = (0, 3, 7), (1, 4, 8), (2, 5, 9). Step 3: combine and keep only numbers with all digits distinct: 35291 ✓, 63072 ✓, 64182 ✓, 94183 ✓. Others repeat a digit (for example 93073 repeats 3; 65292 repeats 2). So four. Options (a), (c) and (d) miscount.
Tip: list the few cases for each condition, then combine and strike out repeats. "All distinct" is the condition people forget.
Answer as per the official UPSC answer key.
X can complete one-third of a certain work in 6 days, Y can complete one-third of the same work in 8 days and Z can complete three-fourth of the same work in 12 days. All of them work together for n days and then X and Z quit and Y alone finishes the remaining work in 8 2/3 days. What is n equal to?
Answer (b) & explanation
n is 4 days (b). Step 1: find each person's full time. X does 1/3 in 6 days, so the whole work in 18 days. Y: 1/3 in 8 days, so 24 days. Z: 3/4 in 12 days, so 16 days. Step 2: together per day: 1/18 + 1/24 + 1/16 = 8/144 + 6/144 + 9/144 = 23/144. Step 3: Y alone works 8⅔ = 26/3 days: (26/3) × (1/24) = 26/72 = 52/144. The rest, 92/144, was done together: 23n/144 = 92/144, so n = 4. Options (a), (c) and (d) fail.
Tip: turn part-work into full-work days first. Use the LCM (here 144) as the total work to avoid fractions.
Answer as per the official UPSC answer key.
What is X in the sequence 1, 3, 6, 11, 18, X, 42?
Answer (c) & explanation
X is 29 (c). Step 1: find the gaps: 3 − 1 = 2, 6 − 3 = 3, 11 − 6 = 5, 18 − 11 = 7. Step 2: the gaps are the prime numbers in order: 2, 3, 5, 7, so next come 11 and 13. Step 3: X = 18 + 11 = 29, and 29 + 13 = 42 ✓. Options (a) 26, (b) 27 and (d) 30 do not give 42 with a prime gap.
Tip: when gaps grow unevenly, check whether they are primes, squares or Fibonacci numbers. Always confirm with the last given term.
Answer as per the official UPSC answer key.
Read the passage
There has been no democracy that has grown economically without corporate capitalism. It helps in modernizing the economy and enabling the transition from rural to urban, and agriculture to industry and services, which are inevitable with growth. It generates jobs - and there is no other way to fix a country’s unemployment challenge without a further impetus to private business. Big companies can operate on a large scale and become competitive both domestically and externally. A vibrant corporate capitalist base also leads to additional revenues for the State - which in turn, can be used for greater welfare for the marginalized and creating a more level playing field in terms of opportunities.
Which one of the following statements best reflects the critical message conveyed by the author of the passage?
Answer (a) & explanation
The critical message is that corporate capitalism matters for economic growth and also for democracy (a). Step 1: the passage opens: "no democracy has grown economically without corporate capitalism." Step 2: it lists benefits: modernisation, jobs, global competitiveness, and revenue for welfare and a level playing field. So it supports both growth and democratic goals: (a). Step 3: option (b), political objectives, is not the focus. Option (c) widens the claim to "no State", though the passage speaks of democracies. Option (d) invents a two-way mutual dependence.
Tip: match the scope of the passage exactly. If it talks about democracies, reject options that talk about all States.
Answer as per the official UPSC answer key.
Read the passage
There has been no democracy that has grown economically without corporate capitalism. It helps in modernizing the economy and enabling the transition from rural to urban, and agriculture to industry and services, which are inevitable with growth. It generates jobs - and there is no other way to fix a country’s unemployment challenge without a further impetus to private business. Big companies can operate on a large scale and become competitive both domestically and externally. A vibrant corporate capitalist base also leads to additional revenues for the State - which in turn, can be used for greater welfare for the marginalized and creating a more level playing field in terms of opportunities.
With reference to the above passage, the following assumptions have been made:
- I. Corporate capitalism promotes the growth of labour force and provides more employment opportunities.
- II. Poor and marginalized sections of population are benefited by corporate capitalism due to trickle-down effect.
Which of the above assumptions is/are valid?
Answer (c) & explanation
Both assumptions are valid (c). Step 1: the passage says corporate capitalism "generates jobs", and that there is no other way to fix unemployment without private business. So it expands employment (I). Step 2: it adds that a strong corporate base brings extra revenue to the State, which can fund "greater welfare for the marginalized". So the poor benefit as gains flow down (II). Step 3: both follow from the text, so options (a), (b) and (d) fail.
Tip: "trickle-down" means benefits at the top reaching the poor over time, here through taxes funding welfare. Check that each assumption has a matching line in the passage.
Answer as per the official UPSC answer key.
Read the passage
“A network of voluntary associations stands as a ‘buffer’ between the relatively powerless individual and the potentially powerful State.”
Which one of the following statements reflects the best explanation of the above passage?
Answer (c) & explanation
The best explanation is that associations take individuals out of isolation and let them join common efforts (c). Step 1: the passage calls voluntary associations a "buffer" between the "relatively powerless individual" and the "potentially powerful State". Step 2: alone, a person is weak. By joining groups, people act together and gain a voice. That is (c). Step 3: option (a) says the State cannot enforce its will, but the passage calls the State powerful, so (a) reverses it. Options (b) and (d) describe other features of associations, not the buffer idea.
Tip: unpack the metaphor ("buffer") first. Civil society groups like unions, NGOs and cooperatives play this role; Article 19(1)(c) protects the right to form associations.
Answer as per the official UPSC answer key.
Consider the following statements:
- I. There exists a natural number which when increased by 50% can have its number of factors unchanged.
- II. There exists a natural number which when increased by 150% can have its number of factors unchanged.
Which of the statements given above is/are correct?
Answer (c) & explanation
Both statements are correct (c). Step 1: "there exists" needs just one example. Step 2: statement I: increase by 50% means × 1.5. Take 2 → 3. Both are primes with exactly 2 factors. So the factor count is unchanged: true. Step 3: statement II: increase by 150% means × 2.5. Take 2 → 5. Again both are primes with 2 factors: true. Options (a), (b) and (d) fail.
Tip: for "there exists" statements, hunt for one simple example; for "always" statements, hunt for one counter-example. Primes are handy because they all have exactly two factors.
Answer as per the official UPSC answer key.
There are 7 places A, B, C, D, E, F, and G in a city connected by various roads AB, CD, DE, BF, EG. A is 6 km south of B. A is 10 km west of C. D is 5 km east of E. C is 6 km north of D. F is 9 km west of B. F is 12 km north of G. A person travels from D to F through these roads. What is the distance covered by the person?
Answer (c) & explanation
The distance covered is 31 km (c). Step 1: place A at (0, 0). B is 6 km north: (0, 6). C is 10 km east of A: (10, 0). D is 6 km south of C: (10, −6). E is 5 km west of D: (5, −6). F is 9 km west of B: (−9, 6). G is 12 km south of F: (−9, −6). Step 2: the roads are AB, CD, DE, BF and EG. From D, the only route to F is D → E → G → F. Step 3: DE = 5, EG = 5 − (−9) = 14, GF = 12. Total = 5 + 14 + 12 = 31 km. Options (a), (b) and (d) take routes that do not exist.
Tip: draw coordinates, then follow only the given roads, not straight lines.
Answer as per the official UPSC answer key.
In a certain code if 64 is written as 343 and 216 is written as 729, then how is 512 written in that code?
Answer (b) & explanation
512 is written as 1331 (b). Step 1: notice the cubes: 64 = 4³ and 343 = 7³. 216 = 6³ and 729 = 9³. Step 2: the rule: add 3 to the cube root, then cube again: 4 → 7, 6 → 9. Step 3: 512 = 8³, so 8 + 3 = 11, and 11³ = 1331. Option (a), 1000 = 10³, adds only 2. Options (c) 12³ and (d) 13³ add too much.
Tip: learn cubes up to 15 (1331 = 11³, 1728 = 12³, 2197 = 13³). Spotting squares and cubes solves many number codes quickly.
Answer as per the official UPSC answer key.
What is the remainder when 93 + 94 + 95 + 96 + … + 9100 is divided by 6?
Answer (a) & explanation
The remainder is 0 (a). Step 1: every power of 9 leaves remainder 3 when divided by 6. (9 = 6 + 3; 81 = 78 + 3; and so on, since 3 × 9 leaves 3 again.) Step 2: the terms run from 9³ to 9¹⁰⁰: that is 100 − 3 + 1 = 98 terms. Total remainder = 98 × 3 = 294. Step 3: 294 ÷ 6 = 49 exactly, so the remainder is 0. Options (b), (c) and (d) miscount the terms or the remainder.
Tip: count terms carefully: from a to b there are b − a + 1 terms. Then multiply by the common remainder and reduce.
Answer as per the official UPSC answer key.
A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: What is the smallest 1-digit number having exactly 4 distinct factors?
Statement I: 2 is one of the factors.
Statement II: 3 is one of the factors.
Which one of the following is correct in respect of the above Question and the Statements?
Answer (d) & explanation
The question can be answered without either statement (d), as per the key: the statements are not needed. Step 1: one-digit numbers with exactly 4 factors: 6 (1, 2, 3, 6) and 8 (1, 2, 4, 8). The others have fewer (primes have 2; 4 and 9 have 3). Step 2: the smallest is 6, found directly from the question. Step 3: so the statements add nothing; UPSC marks this as (d). Options (a), (b) and (c) suggest the statements are needed.
Tip: in data sufficiency, first try to answer from the question alone. Numbers with exactly 4 factors are either p × q (two different primes) or p³.
read option (d) carefully. It is not the usual "cannot be answered" option.
Answer as per the official UPSC answer key.
A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Let P, Q, R, S be distinct non-zero digits. If PP × PQ = RRSS, where P ≤ 3 and Q ≤ 4, then what is Q equal to?
Statement I : R = 1.
Statement II : S = 2.
Which one of the following is correct in respect of the above Question and the Statements?
Answer (d) & explanation
Q can be found without either statement (d), as per the key. Step 1: P can be 1, 2 or 3, and Q at most 4. Step 2: if P = 1 or 2, then PP × PQ is at most 22 × 24 = 528, too small for a 4-digit number RRSS. So P = 3. Step 3: try 33 × 31 = 1023, 33 × 32 = 1056, 33 × 34 = 1122 (Q cannot be 3, as digits are distinct). Only 1122 has the RRSS shape. So Q = 4, found from the question alone. The statements (R = 1, S = 2) only confirm it.
Tip: when the conditions leave very few cases, check them all before reading the statements. If the answer comes out, neither statement is needed.
Do not jump to a statement before testing the question on its own. Small limits like P ≤ 3 often settle it.
Answer as per the official UPSC answer key.
A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: How is Q related to P?
Statement I: P has two sisters, R and S.
Statement II: R's father is the brother of Q.
Which one of the following is correct in respect of the above Question and the Statements?
Answer (d) & explanation
Even both statements together are not enough (d). Step 1: statement I says P has two sisters, R and S. So R is P's sibling. Step 2: statement II says R's father is Q's brother. R's father is also P's father. So Q is a sibling of P's father. Step 3: Q could be P's uncle (if male) or aunt (if female). Q's gender is not given, so the exact relation cannot be fixed. Options (a), (b) and (c) fail.
Tip: in relation questions, check that the gender of the person asked about is known. "Brother of Q" tells us about the brother, not about Q.
Answer as per the official UPSC answer key.
Read the passage
A single number for inflation is aggregate across different commodities and Services the price rise differs for different items of consumption. So, the single number is arrived at by assigning weights to different commodities and services. For WPI, the weights in production are used; for CPI, the consumption basket is used. But people are not homogeneous.
The consumption basket is vastly different for the poor, the middle classes, and the rich. Hence, the CPI is different for each of these classes and a composite index requires averaging the baskets.
Which one of the following statements best reflects the most logical, rational and crucial message conveyed by the passage?
Answer (b) & explanation
The message is that the present single inflation number does not correctly show the price rise of individual items (b). Step 1: the passage says one inflation number is an average of many items, each given a weight. Prices of different items rise differently. Step 2: it adds that consumption baskets differ for the poor, middle class and rich, so one composite CPI averages them too. So the single number hides how individual prices move: (b). Step 3: option (a), scrap CPI and use only WPI, and (d), inflation data is useless, are extreme. Option (c), services under-represented, is not discussed.
Tip: WPI uses production weights; CPI uses consumption weights. Since 2014 the RBI targets CPI inflation at 4% (with a band of 2–6%).
Answer as per the official UPSC answer key.
Read the passage
Trust stands commonly defined as being vulnerable to others. Entrepreneurship implies trust in others and willingness to expose oneself to betrayal. Trust in expert systems is the essence of globalizing behaviour; trust itself emerges as a super-commodity in the social market and defines the characteristics of goods and services in a global market. Trusting conduct also means holding others in good esteem, and an optimism that they are, or will be, competent in certain respects.
Which one of the following statements best reflects the crux of the passage?
Answer (d) & explanation
The crux is that trust is important even though it carries risk (d). Step 1: the passage defines trust as "being vulnerable to others", and says entrepreneurship means a "willingness to expose oneself to betrayal". That is the risk. Step 2: it also calls trust "a super-commodity" and "the essence of globalizing behaviour". That is the importance. Both together give (d). Step 3: option (a) says trust cannot be expected in entrepreneurship, the opposite. Option (b) mixes up those who trust with those who are trustworthy. Option (c) says "no economic activity", too sweeping.
Tip: a good crux often joins two sides: "important but risky". Watch for word swaps like trusting versus trustworthy.
Answer as per the official UPSC answer key.
A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: In a football match, team P playing against Q was behind by 3 goals with 10 minutes remaining. Does team P win the match?
Statement I: Team P scored 4 goals in the last 10 minutes.
Statement II: Team Q scored a total of 4 goals in the match.
Which one of the following is correct in respect of the above Question and the Statements?
Answer (d) & explanation
Even both statements together are not enough (d). Step 1: with 10 minutes left, P was 3 goals behind. Statement I: P scored 4 more goals. Statement II: Q finished with 4 goals in all. Step 2: Q's score with 10 minutes left could be 3 or 4 (it may or may not have scored again). If Q had 4 then, P had 1, ends with 5: P wins 5–4. If Q had 3 then and scored one more, P had 0, ends with 4: a 4–4 draw. Step 3: both cases fit all the facts, so we cannot say whether P won. Options (a), (b) and (c) fail.
Tip: when one team's late goals are unknown, test the extreme cases. If they give different results, the data is insufficient.
Answer as per the official UPSC answer key.
A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is (p + q)2 − 4pq, where p, q are natural numbers, positive?
Statement I: p < q.
Statement II: p > q.
Which one of the following is correct in respect of the above Question and the Statements?
Answer (b) & explanation
Either statement alone is enough (b). Step 1: simplify: (p + q)² − 4pq = p² + 2pq + q² − 4pq = (p − q)². Step 2: a square is positive whenever p and q are different, and zero only when p = q. Step 3: statement I (p < q) and statement II (p > q) each say p ≠ q. So each alone answers "yes, it is positive". Options (a), (c) and (d) fail.
Tip: always simplify the expression before reading the statements. (p + q)² − 4pq = (p − q)² is a standard identity worth remembering.
Answer as per the official UPSC answer key.
In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by X to the number of runs scored by Y is equal to ratio of number of runs scored by Y to number of runs scored by Z.
Value-I = Runs scored by X
Value-II = Runs scored by Y
Value-III = Runs scored by Z
Which one of the following is correct?
Answer (d) & explanation
The order cannot be determined (d). Step 1: X : Y = Y : Z means Y² = XZ: the three scores are in continued proportion (a geometric pattern). Step 2: with a total of 37, the whole-number answer is 9, 12, 16 (12² = 9 × 16 = 144). Step 3: but X = 9, Z = 16 and X = 16, Z = 9 both satisfy X : Y = Y : Z. So the scores could rise or fall. Since the order is not fixed, the answer cannot be determined. Options (a), (b) and (c) each pick one order.
Tip: a ratio condition like X : Y = Y : Z is symmetric in X and Z. Check whether swapping values still fits before choosing an order.
Answer as per the official UPSC answer key.
Let p + q = 10, where p, q are integers.
Value-I = Maximum value of p × q when p, q are positive integers.
Value-II = Maximum value of p × q when p ≥ −6, q ≥ −4.
Which one of the following is correct?
Answer (c) & explanation
The two values are equal (c). Step 1: Value-I: p + q = 10 with positive whole numbers. A product is largest when the two numbers are equal: 5 × 5 = 25. Step 2: Value-II allows p down to −6 and q down to −4, so p can range from −6 to 14. The new pairs, like (−6, 16) or (14, −4), have negative products. The best is still 5 × 5 = 25. Step 3: so Value-I = Value-II. Options (a), (b) and (d) fail.
Tip: for a fixed sum, the product is greatest when the numbers are as equal as possible. Allowing negative numbers only adds smaller (negative) products here.
Answer as per the official UPSC answer key.
Consider a set of 11 numbers :
Value-I = Minimum value of the average of the numbers of the set when they are consecutive integers ≥ −5.
Value-II = Minimum value of the product of the numbers of the set when they are consecutive non-negative integers.
Which one of the following is correct?
Answer (c) & explanation
The two values are equal (c). Step 1: Value-I: eleven consecutive integers, all at least −5. To make the average smallest, start as low as allowed: −5 to 5. These cancel in pairs, so the average is 0. Step 2: Value-II: eleven consecutive non-negative integers. Start at 0: 0, 1, …, 10. Anything times 0 is 0, so the product is 0, the smallest possible. Step 3: both values are 0, so they are equal. Options (a), (b) and (d) fail.
Tip: for consecutive numbers, the average is the middle term (here 0). Any set that includes 0 has product 0.
Answer as per the official UPSC answer key.
The average of three numbers p, q and r is k. p is as much more than the average as q is less than the average. What is the value of r?
Answer (a) & explanation
r equals k (a). Step 1: "p is as much more than the average as q is less" means p = k + d and q = k − d for some amount d. Step 2: then p + q = 2k. Step 3: the average of all three is k, so their sum is 3k. Hence r = 3k − 2k = k. Options (b) k − 1, (c) k + 1 and (d) k/2 do not follow. For example, p = 7, q = 3, r = 5 gives average 5 = r.
Tip: values that are equally above and below the average cancel out. The remaining value must then sit exactly at the average.
Answer as per the official UPSC answer key.
Let x be a real number between 0 and 1. Which of the following statements is/are correct?
- I. x2 > x3
- II. x > √x
Select the correct answer using the code given below:
Answer (a) & explanation
Only statement I is correct (a). Step 1: for a number between 0 and 1, multiplying by it makes things smaller. Test x = 1/4. Step 2: statement I: x² = 1/16 and x³ = 1/64. Since 1/16 > 1/64, x² > x³: true. Step 3: statement II: √x = √(1/4) = 1/2, which is bigger than 1/4. So x > √x is false. Options (b), (c) and (d) fail.
Tip: between 0 and 1, higher powers get smaller and roots get bigger: √x > x > x² > x³. Above 1 the order flips. Test with an easy fraction like 1/4.
between 0 and 1 the usual "squaring makes it bigger" instinct runs backwards.
Answer as per the official UPSC answer key.
The difference between any two natural numbers is 10. What can be said about the natural numbers which are divisible by 5 and lie between these two numbers?
Answer (c) & explanation
There can be more than one such number (c). Step 1: take two numbers 10 apart, such as 5 and 15. Strictly between them, only 10 is divisible by 5. That is one number. Step 2: now take 1 and 11. Between them, 5 and 10 are divisible by 5. That is two numbers. Step 3: the count can be one or two depending on the pair. So "only one" (a) and "only two" (b) are not always true, and (d) is false. The safe statement is (c): there can be more than one.
Tip: test both an "edge" case (ends are multiples of 5) and a general case. When answers vary, pick the option that allows for it.
Answer as per the official UPSC answer key.
All years by topic
Every question from 2011 to 2026, grouped by syllabus topic, with year-wise trends.