CSAT 2023: every question, solved
80 questions from the official UPSC Prelims 2023 paper, in paper order. Tap an option; the answer and the reason appear instantly.
Read the passage
We often hear about conflicts among different States in India over river waters. Of the 20 major river systems, 14 are already water-stressed; 75% of the population lives in water-stressed regions, a third of whom live in water-scarce areas. Climate change, the demands of rising population and the need for agriculture to keep pace, and increased rate of urbanization and industrialization will exacerbate water stress. According to the Constitution of India, water is a State subject and not that of the Union, except for regulation of inter-State rivers. Key to ensuring balance between competing demands of various stakeholders is a basin-based approach to allocate water amongst constituent regions and States. Allocating fair share of water to them requires assessments based on objective criteria, such as specificities of the river basin, size of dependent population, existing water use and demand, efficiency of use, projected future use, etc. while ensuring the environmental needs of the river and aquifers.
Which one of the following statements best reflects the most rational, practical and immediate action required to ensure fair and equitable allocation of water to different stakeholders ?
Answer (a) & explanation
The immediate action is a national, practical legal and policy framework for allocating water (a). Step 1: the passage says water is a State subject, and States often fight over rivers. Step 2: it says the key is a basin-based approach that shares water using objective criteria: basin features, population, current use, efficiency and future needs. That is a framework for allocation, so (a). Step 3: options (b) river linking and (c) new channels are engineering projects the passage never proposes. Option (d), cutting demand, is not the fair-allocation step it asks for.
Tip: under the Constitution, water is in the State List (Entry 17), while inter-State rivers fall under the Union (Entry 56). This is why fair-sharing rules matter.
Answer as per the official UPSC answer key.
Read the passage
More than half of Indian women and almost a quarter of Indian men of working age suffer from anaemia. According to studies, they are anywhere from 5–15% less productive than they could be, as a result thereof. India also has the largest tuberculosis burden in the world, costing 170 million workdays to the country annually. But what is just as important as lost productivity now is lost potential in the future. It is becoming increasingly clear that on many measures of cognitive ability, malnourished Indian children perform two or three times worse than their adequately nourished peers. For an economy that will be more dependent on highly skilled workers, this poses a significant challenge. And it is one that really should be addressed given India’s demographic outlook.
Which one of the following statements best reflects what is implied by the passage ?
Answer (d) & explanation
The implication is that for rapid growth, attention must go to people's health and nutrition (d). Step 1: the passage says anaemia makes workers 5–15% less productive, and TB costs 170 million workdays a year. Step 2: it adds that malnourished children do two or three times worse on cognitive tests, hurting future potential. So health and nutrition drive growth. That is (d). Step 3: option (c) says only skilled workers need attention, which is too narrow. Options (a) and (b) shift to education and skills, which the passage does not discuss.
Tip: the passage links health to productivity both now and in future. Pick the option that covers all the people, not one group.
Answer as per the official UPSC answer key.
Read the passage
In India, a majority of farmers are marginal and small, less educated and possess low adaptive capabilities to climate change, perhaps because of credit and other constraints. So, one cannot expect autonomous adaptation to climate change. Even if it was possible, it would not be sufficient to offset losses from climate change. To deal with this, adaptation to climate change is paramount, alongside a fast mitigation response. Another solution is to have a planned or policy-driven adaptation, which would require the government to come up with policy recommendations. Perception is a necessary pre-requisite for adaptation. Whether farmers are adapting agricultural practices to climate change depends on whether they perceive it or not. However, this is not always enough for adaptation. It is important how a farmer perceives the risks associated with climate change.
Which one of the following statements best reflects the most logical and rational message conveyed by the author of the passage ?
Answer (c) & explanation
The message is that farmers' risk perceptions matter for motivating adaptation (c). Step 1: the passage says perception is a "necessary pre-requisite" for adaptation. Step 2: it ends with its key point: "It is important how a farmer perceives the risks" of climate change. That is (c). Step 3: option (a) says adaptation is only the government's job; the passage mentions policy as just one solution. Option (b) is not discussed. Option (d) says mitigation is not possible, but the passage calls for "a fast mitigation response".
Tip: the last line of a passage often carries its main message. Check options with "only" or "not possible" against the text.
Answer as per the official UPSC answer key.
Raj has ten pairs of red, nine pairs of white and eight pairs of black shoes in a box. If he randomly picks shoes one by one (without replacement) from the box to get a red pair of shoes to wear, what is the maximum number of attempts he has to make?
Answer (d) & explanation
The maximum is 45 attempts (d). Step 1: think of the unluckiest case. First he picks every non-red shoe: 9 + 8 = 17 pairs = 34 shoes. Step 2: still unlucky, he then picks all 10 red shoes of the same foot (say all left shoes): 34 + 10 = 44. Step 3: the next shoe must be a red right shoe, which makes a pair. So 45 picks at most. Option (c), 44, stops one short. Options (a) and (b) ignore the left-right problem.
Tip: shoes have a left and a right, unlike socks. In worst-case questions, always ask what "a pair" really needs.
(c) 44 forgets that ten left shoes still do not make a pair.
Answer as per the official UPSC answer key.
In how many ways can a batsman score exactly 25 runs by scoring single runs, fours and sixes only, irrespective of the sequence of scoring shots?
Answer (b) & explanation
1. Choose how many sixes and fours. The singles then fill up the rest of the 25, so they never cause a problem.
2. 0 sixes: fours can be 0 to 6 (up to 24 runs). That is 7 ways.
3. 1 six: 19 runs left, fours 0 to 4. 5 ways.
4. 2 sixes: 13 left, fours 0 to 3. 4 ways.
5. 3 sixes: 7 left, fours 0 or 1. 2 ways.
6. 4 sixes: 1 left, no four. 1 way.
7. Total: 7 + 5 + 4 + 2 + 1 = 19.
Answer as per the official UPSC answer key.
There are four letters and four envelopes and exactly one letter is to be put in exactly one envelope with the correct address. If the letters are randomly inserted into the envelopes, then consider the following statements :
- 1. It is possible that exactly one letter goes into an incorrect envelope.
- 2. There are only six ways in which only two letters can go into the correct envelopes.
Which of the statements given above is/are correct?
Answer (b) & explanation
Only statement 2 is correct (b). Step 1: if three letters are in the right envelopes, only one envelope is left, and it is the right one for the fourth letter. So exactly one wrong letter is impossible. Statement 1 is false. Step 2: for exactly two right letters, choose which two are right: AB, AC, AD, BC, BD, CD = 6 ways. Step 3: the other two must both be wrong, so they swap, in only 1 way. Total = 6 × 1 = 6. Statement 2 is true. Options (a), (c) and (d) fail.
Tip: in matching problems, "exactly one wrong" can never happen. A wrong letter always pushes at least one other letter out of place.
Answer as per the official UPSC answer key.
What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100 ?
Answer (a) & explanation
The remainder is 0 (a). Step 1: 100 = 4 × 25. Step 2: 85 = 5 × 17 and 95 = 5 × 19 give two 5s, so the product has 5 × 5 = 25. 96 = 32 × 3 gives plenty of 2s, so 4 also divides it. Step 3: so the product is a multiple of 4 × 25 = 100. The remainder is 0. Options (b), (c) and (d) assume something is left over.
Tip: to divide by 100, check for two 5s and two 2s among the factors. If both are there, the remainder is zero.
Answer as per the official UPSC answer key.
What is the unit digit in the expansion of (57242)9×7×5×3×1 ?
Answer (a) & explanation
The unit digit is 2 (a). Step 1: only the last digit of 57242 matters, which is 2. Powers of 2 end in the cycle 2, 4, 8, 6, of length 4. Step 2: the power is 9 × 7 × 5 × 3 × 1 = 945. 945 ÷ 4 leaves a remainder of 1. Step 3: remainder 1 gives the first term of the cycle: 2. Options (b), (c) and (d) are other cycle terms.
Tip: an odd power can never be a multiple of 4. A product of odd numbers is always odd, which quickly rules out the 4th term (6).
Answer as per the official UPSC answer key.
If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F ?
Answer (d) & explanation
The sum of the six digits is 31 (d). Step 1: units place: C + F must end in 1. Two non-zero digits cannot add to 1, so C + F = 11, carrying 1. Step 2: tens place: B + E + 1 must end in 1, so B + E = 10, carrying 1. Step 3: hundreds place: A + D + 1 = 11, so A + D = 10. Total = 11 + 10 + 10 = 31. Options (a), (b) and (c) miss the carries.
Tip: in addition puzzles, work from the units place and track the carry at each step.
don't forget the carries. Without them you get 1 + 1 + 1 per column and land on a wrong option.
Answer as per the official UPSC answer key.
D is a 3-digit number such that the ratio of the number to the sum of its digits is least. What is the difference between the digit at the hundred's place and the digit at the unit's place of D?
Answer (c) & explanation
The difference is 8 (c). Step 1: to make (number ÷ digit sum) small, keep the hundreds digit small and the other two digits large. Step 2: with hundreds digit 1, try the best options: 199 ÷ 19 ≈ 10.47; 189 ÷ 18 = 10.5; 198 ÷ 18 = 11. So 199 is best. With hundreds digit 2, the ratio is at least 299 ÷ 20 ≈ 15. Step 3: D = 199. The difference between hundreds and units digits = 9 − 1 = 8. Options (a), (b) and (d) fail.
Tip: the hundreds digit adds 100 per unit but only 1 to the digit sum. So keep it as small as possible.
Answer as per the official UPSC answer key.
Read the passage
The emissions humans put into the atmosphere now will affect the climate in the middle of the century and onwards. Technological change, meanwhile, could make a future transition away from fossil fuels cheap or it might not, leaving the world with a terrible choice between sharply reducing emissions at huge cost or suffering through the effects of unabated warming. Businesses that do not hedge against the threat of uncertain outcomes fail. The world cannot afford such recklessness on climate change.
Which one of the following statements best reflects the crucial message conveyed by the author of the passage ?
Answer (c) & explanation
The crucial message is that waiting for better technology before cutting emissions is unwise (c). Step 1: the passage says today's emissions will affect the climate from mid-century. Step 2: technology "could make a future transition cheap or it might not". Betting on it is reckless, like a business that fails to hedge. So act now. That is (c). Step 3: option (b) makes technology the only solution, the very gamble the author warns against. Option (a), closing businesses, and option (d), only renewable industries, are not argued.
Tip: when an author says "it might not", he is warning against relying on it. Pick the option that rejects waiting.
Answer as per the official UPSC answer key.
Read the passage
Environmental problems cause health problems. Substantial changes in lifestyle can reduce environmental or health problems, but this idea appears almost impossible to adopt. With environmental problems, individual efforts can be perceived as having a negligible effect and therefore lead to inertia. With health, on the other hand, individual choices can make the difference between life and death, literally. And yet, barring a few, there seems to be the same collective lethargy towards making their choices.
Which one of the following statements best implies the most rational assumption that can be made from the passage ?
Answer (a) & explanation
The rational assumption is that we are likely to spend more on cure than prevention (a). Step 1: the passage says lifestyle changes could cut health problems. Step 2: but people show "collective lethargy" about making those choices, even when life and death are at stake. If they will not prevent illness, they will later pay for cure. That is (a). Step 3: option (c), health protected while environmental problems go on, goes against the first line. Options (b), government's job, and (d), western values, are not in the passage.
Tip: an assumption often follows from the passage's main complaint. Here the complaint is inaction on prevention.
Answer as per the official UPSC answer key.
Read the passage
Many people are not eating the right food. For some, it is simply a decision to stick with food they enjoy but which is not too healthy. This is leading to an increase in non-communicable diseases. This in turn leads to major burden on our health-care systems that have the potential to derail the economic progress which is essential for the poor to improve their lives. For others, it is about limited access to nutritious food or a lack of affordability, leading to monotonous diets that do not provide the daily nutrients for them to develop fully. Part of the reason nutrition is under threat worldwide is that our food systems are not properly responding to nutritional needs. Somewhere along that long road from farm to fork, there are serious detours taking place.
Which one of the following statements best reflects the crux of the passage ?
Answer (b) & explanation
The crux is that food-based nutrition must be at the centre of policy debate (b). Step 1: the passage says many people eat unhealthy food, which raises non-communicable diseases. Others lack access to nutritious food. Step 2: it says our food systems are not responding to nutritional needs, with "detours" from farm to fork. So nutrition must come first in food policy. That is (b). Step 3: option (a), Universal Basic Income, is not mentioned. Options (c) GM crops and (d) food fortification are specific fixes the passage does not propose.
Tip: the crux is the broad idea the passage builds to. Options naming specific technologies are usually too narrow.
Answer as per the official UPSC answer key.
Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in order). Consider the following :
- 1. p + q + r – s – t is definitely even.
- 2. 2p + q + 2r – 2s + t is definitely odd.
Which of the above statements is/are correct ?
Answer (a) & explanation
Only statement 1 is correct (a). Step 1: subtracting a number has the same odd-even effect as adding it. So p + q + r − s − t has the same parity as the sum of all five. Step 2: the five include two odd numbers, and two odds make an even sum. So statement 1 is always even, and true. Step 3: in 2p + q + 2r − 2s + t, the doubled terms are even. The parity depends on q + t, which depends on which numbers are odd. So it is not definitely odd. Options (b), (c) and (d) fail.
Tip: for odd-even questions, ignore the signs and any doubled terms. Just count the odd numbers left.
Answer as per the official UPSC answer key.
Consider the following in respect of prime number p and composite number c.
- 1. (p + c)/(p – c) can be even.
- 2. 2p + c can be odd.
- 3. pc can be odd.
Which of the statements given above are correct ?
Answer (d) & explanation
All three statements can be true (d). Step 1: "can be" needs only one example each. Step 2: statement 1: p = 11, c = 9. (11 + 9) ÷ (11 − 9) = 20 ÷ 2 = 10, which is even. Statement 2: c = 9. 2p is even, and even + 9 is odd. Step 3: statement 3: p = 3, c = 9. 3 × 9 = 27, odd. So options (a), (b) and (c) fail.
Tip: remember that composite numbers can be odd (9, 15, 21). Many "can be" traps assume composites are even.
Answer as per the official UPSC answer key.
A 3-digit number ABC, on multiplication with D gives 37DD where A, B, C and D are different non-zero digits. What is the value of A + B + C?
Answer (a) & explanation
A + B + C is 18 (a). Step 1: 37DD = 3700 + 11D. Since ABC × D = 37DD, D must divide 3700. So D is 1, 2, 4 or 5. Step 2: D = 1 or 2 makes ABC a 4-digit number. D = 5 gives 3755 ÷ 5 = 751, which repeats the digit 5. Step 3: D = 4: 3744 ÷ 4 = 936. All digits differ. A + B + C = 9 + 3 + 6 = 18. Option (d) is wrong: the data fix the answer.
Tip: split a coded number into a fixed part and a part with the unknown. Divisibility then narrows the choices quickly.
Answer as per the official UPSC answer key.
For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by :
Answer (d) & explanation
XYZXYZ is always divisible by 7, 11 and 13 (d). Step 1: write it out: XYZXYZ = XYZ × 1000 + XYZ = XYZ × 1001. Step 2: 1001 = 7 × 11 × 13. Step 3: so every such number is divisible by all three. Options (a), (b) and (c) each leave one out. For example, 123123 = 123 × 7 × 11 × 13.
Tip: 1001 = 7 × 11 × 13 is worth remembering. Any 3-digit block written twice is a multiple of 1001.
Answer as per the official UPSC answer key.
125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side?
Answer (a) & explanation
27 cubes are surrounded on every side (a). Step 1: 125 = 5 × 5 × 5, so the block is 5 cubes on each side. Step 2: a cube with neighbours on all six sides cannot be on the outer surface. Remove one layer from each face. Step 3: the inside block is (5 − 2) × (5 − 2) × (5 − 2) = 3 × 3 × 3 = 27. Options (b), (c) and (d) miscount.
Tip: for an n × n × n cube, the hidden inner cubes are (n − 2)³. The cubes with paint on just one face are 6(n − 2)².
Answer as per the official UPSC answer key.
How many distinct 8-digit numbers can be formed by rearranging the digits of the number 11223344 such that odd digits occupy odd positions and even digits occupy even positions?
Answer (c) & explanation
1. The odd digits are 1, 1, 3, 3. They go into the 4 odd places. Two places get 1 and two get 3. Choosing which two places get the 1s can be done in 6 ways (list them: 12, 13, 14, 23, 24, 34).
2. The even digits 2, 2, 4, 4 go into the 4 even places in the same way: 6 ways.
3. Each choice on one side goes with each choice on the other: 6 × 6 = 36.
Answer as per the official UPSC answer key.
A, B, C working independently can do a piece of work in 8, 16 and 12 days respectively. A alone works on Monday, B alone works on Tuesday, C alone works on Wednesday; A alone, again works on Thursday and so on.
Consider the following statements :
- 1. The work will be finished on Thursday.
- 2. The work will be finished in 10 days.
Which of the above statements is/are correct ?
Answer (a) & explanation
Only statement 1 is correct (a). Step 1: take total work = 48 units (LCM of 8, 16, 12). A does 6, B does 3 and C does 4 units a day. One 3-day cycle = 13 units. Step 2: three cycles (9 days) = 39 units. Day 10 (A) brings it to 45. Day 11 (B) brings it to 48. The work ends on day 11. Step 3: day 1 is Monday, so day 8 is Monday again and day 11 is a Thursday. Statement 1 is true; statement 2 (10 days) is false.
Tip: in rotating-work problems, use LCM units and count full cycles first, then finish day by day.
the last day is a B day, and B needs a full day for the last 3 units. So it takes 11 days, not 10.
Answer as per the official UPSC answer key.
Read the passage
To tackle the problem of pollution in cities, policy makers think that drastic actions like temporary use of odd-even number scheme for vehicles, closing schools, factories, construction activities, and banning the use of certain type of vehicles are a way forward. Even then the air is not clean. Vehicles more than 15 years old comprise one percent of the total; and taking them off the road will not make any difference. Banning certain fuels and car types arbitrarily is not proper. Diesel engines produce more PM 2·5 and less CO₂ than petrol or CNG engines. On the other hand, both diesel and CNG engines produce more NOₓ than petrol engines. No one has measured the amount of NOₓ that CNG engines are emitting. Arbitrary bans on vehicles that have passed mandated fitness tests and periodic pollution tests are unfair. What is needed is the scientific and reliable information about the source of pollutants on a continuing basis and the technologies that will work to reduce pollution from them.
Which one of the following statements best reflects the most logical and rational implication conveyed by the passage ?
Answer (b) & explanation
The implication is that knee-jerk reactions cannot fix pollution, but an evidence-based approach can (b). Step 1: the passage says drastic steps like odd-even and closing schools still leave the air unclean. Step 2: it ends: "What is needed is the scientific and reliable information about the source of pollutants". That is (b). Step 3: option (a) says curbs are difficult to implement; the passage calls them unfair, not hard. Option (c), heavy penalties, is not suggested. Option (d), lack of laws, is not discussed.
Tip: "unfair" and "difficult" are different complaints. Match the passage's exact objection.
Answer as per the official UPSC answer key.
Read the passage
Good corporate governance structures encourage companies to provide accountability and control. A fundamental reason why corporate governance has moved onto the economic and political agenda worldwide has been the rapid growth in international capital markets. Effective corporate governance enhances access to external financing by firms, leading to greater investment, higher growth and employment. Investors look to place their funds where the standards of disclosure, of timely and accurate financial reporting, and of equal treatment to all stakeholders are met.
Which of the following statements best reflects the logical inference from the passage given above ?
Answer (b) & explanation
The logical inference is that good corporate governance improves firms' credibility (b). Step 1: the passage says investors put their money where disclosure, accurate reporting and fair treatment are strong. So good governance builds trust. That is (b). Step 2: option (a) says external financing is the world's agenda. But the passage says corporate governance moved onto the agenda. Step 3: option (c) says capital markets ensure good governance; they only explain its rise. Option (d), supply chains, is not mentioned.
Tip: separate cause from result. Capital markets explain why governance became important; they do not guarantee it.
Answer as per the official UPSC answer key.
Read the passage
Elephants are landscape architects, creating clearings in the forest, preventing overgrowth of certain plant species and allowing space for the regeneration of others, which in turn provide sustenance to other herbivorous animals. Elephants eat plants, fruits and seeds, propagating the seeds when they defecate in other places as they travel. Elephant dung provides nourishment to plants and animals and acts as a breeding ground for insects. In times of drought, they access water by digging holes which benefits other wildlife.
Which one of the following statements best reflects the most logical and rational inference that can be drawn from the passage ?
Answer (b) & explanation
The best inference is that elephants are a keystone species that benefit biodiversity (b). Step 1: the passage says elephants make clearings, spread seeds through dung, feed insects and plants with dung, and dig waterholes in drought. Step 2: all these help many other species. A species whose role supports so many others is a keystone species. That is (b). Step 3: option (c) says biodiversity cannot survive without them, which is too absolute. Option (d) gives elephants a purpose, as if they plan the forest. Option (a) talks of home range, which is not discussed.
Tip: a keystone species has a large effect on its ecosystem relative to its numbers. India declared the elephant its National Heritage Animal in 2010.
Answer as per the official UPSC answer key.
If 7 ⊕ 9 ⊕ 10 = 8, 9 ⊕ 11 ⊕ 30 = 5, 11 ⊕ 17 ⊕ 21 = 13, what is the value of 23 ⊕ 4 ⊕ 15 ?
Answer (a) & explanation
The value is 6 (a). Step 1: find the rule from the examples. Add the three numbers, then add the digits of the sum. Step 2: 7 + 9 + 10 = 26 → 2 + 6 = 8. 9 + 11 + 30 = 50 → 5. 11 + 17 + 21 = 49 → 4 + 9 = 13. The rule works for all three. Step 3: 23 + 4 + 15 = 42 → 4 + 2 = 6. Options (b), (c) and (d) do not follow this rule.
Tip: for made-up symbols, test simple operations (sum, product, digit sum) on all examples. The rule must fit every one.
Answer as per the official UPSC answer key.
Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible ?
Answer (c) & explanation
12 values of x are possible (c). Step 1: 7x is always divisible by x. So 7x + 96 is divisible by x only if x divides 96. Step 2: 96 = 2⁵ × 3. Step 3: the number of divisors = (5 + 1) × (1 + 1) = 12. So x can be any of 12 values: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. Options (a), (b) and (d) miscount.
Tip: to count divisors, write the number as a product of prime powers. Add 1 to each power and multiply.
Answer as per the official UPSC answer key.
If p, q, r and s are distinct single digit positive numbers, then what is the greatest value of (p + q)(r + s)?
Answer (b) & explanation
The greatest value is 225 (b). Step 1: to make the product big, use the four largest digits: 9, 8, 7, 6. Their total is 30. Step 2: for a fixed total, a product of two parts is largest when the parts are equal. So make each bracket 15: 9 + 6 and 8 + 7. Step 3: 15 × 15 = 225. Option (a), 230, is impossible. Option (c), 224 = 16 × 14, is slightly less. Option (d), 221 = 17 × 13, is less again.
Tip: for a fixed sum, the product is greatest when the numbers are as close as possible.
Answer as per the official UPSC answer key.
A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13 ?
Answer (a) & explanation
The remainder is 11 (a). Step 1: 999999 = 13 × 76923. So every block of six 9s is a multiple of 13. Step 2: 99 = 6 × 16 + 3. After 16 blocks, only 999 is left over. Step 3: 999 = 13 × 76 + 11. So the remainder is 11. Options (b), (c) and (d) do not match.
Tip: 999999 (six 9s) is divisible by 7, 11, 13 and 37. Use it to cut long strings of 9s into simple blocks.
Answer as per the official UPSC answer key.
Each digit of a 9-digit number is 1. It is multiplied by itself. What is the sum of the digits of the resulting number ?
Answer (c) & explanation
The digit sum is 81 (c). Step 1: 111111111 × 111111111 = 12345678987654321. Step 2: add the digits: 2 × (1 + 2 + ... + 8) + 9 = 2 × 36 + 9 = 81. Step 3: so the answer is (c). Option (a), 64, is 8², the result for eight ones. Options (b) and (d) are guesses.
Tip: for a number made of n ones (n up to 9), the square reads 1, 2, ..., n, ..., 2, 1. Its digit sum is simply n². Here 9² = 81.
Answer as per the official UPSC answer key.
What is the sum of all digits which appear in all the integers from 10 to 100?
Answer (b) & explanation
The sum is 856 (b). Step 1: from 10 to 99, the tens place shows each digit 1 to 9 exactly 10 times. That gives 45 × 10 = 450. Step 2: the units place shows each digit 0 to 9 exactly 9 times (once per ten). That gives 45 × 9 = 405. Step 3: add the digits of 100, which is 1. Total = 450 + 405 + 1 = 856. Option (a), 855, forgets 100. Options (c) and (d) miscount.
Tip: count digit sums place by place. Each place cycles through the digits evenly.
don't drop the 100 at the end. 855 is the trap option.
Answer as per the official UPSC answer key.
ABCD is a square. One point on each of AB and CD; and two distinct points on each of BC and DA are chosen. How many distinct triangles can be drawn using any three points as vertices out of these six points?
Answer (c) & explanation
20 triangles can be drawn (c). Step 1: any 3 of the 6 points make a triangle, unless all 3 lie on one straight line. Step 2: the most points on any one side is 2. So no three chosen points are in a line. Step 3: the number of ways to choose 3 from 6 = (6 × 5 × 4) ÷ (3 × 2 × 1) = 20. Options (a), (b) and (d) wrongly subtract or add cases.
Tip: number of triangles = C(n, 3) minus the groups of 3 points that are in a line. Here there are none to subtract.
Answer as per the official UPSC answer key.
Read the passage
In India, the segregation of municipal waste at source is rare. Recycling is mostly with the informal sector. More than three-fourths of the municipal budget goes into collection and transportation, which leaves very little for processing/resource recovery and disposal. Where does waste-to-energy fit into all this ? Ideally it fits in the chain after segregation (between wet waste and the rest), collection, recycling, and before getting to the landfill. Which technology is most appropriate in converting waste to energy depends on what is in the waste (that is biodegradable versus non-biodegradable component) and its calorific value. The biodegradable component of India’s municipal solid waste is a little over 50 per cent, and biomethanation offers a major solution for processing this.
Based on the above passage, the following assumptions have been made :
- 1. Collection, processing and segregation of municipal waste should be with government agencies.
- 2. Resource recovery and recycling require technological inputs that can be best handled by private sector enterprises.
Which of the assumptions given above is/are correct ?
Answer (d) & explanation
Neither assumption is correct (d). Step 1: the passage describes the problems: little segregation, recycling mostly by the informal sector, and three-fourths of the budget spent on collection and transport. Step 2: it never says who should run collection and segregation, so assumption 1 (government agencies) has no basis. Step 3: it never compares public and private firms on recycling technology, so assumption 2 fails too. Options (a), (b) and (c) fail.
Tip: an assumption must be needed for the author's argument. If the passage never discusses who should do a job, any "who" assumption is out.
Answer as per the official UPSC answer key.
Read the passage
In India, the segregation of municipal waste at source is rare. Recycling is mostly with the informal sector. More than three-fourths of the municipal budget goes into collection and transportation, which leaves very little for processing/resource recovery and disposal. Where does waste-to-energy fit into all this ? Ideally it fits in the chain after segregation (between wet waste and the rest), collection, recycling, and before getting to the landfill. Which technology is most appropriate in converting waste to energy depends on what is in the waste (that is biodegradable versus non-biodegradable component) and its calorific value. The biodegradable component of India’s municipal solid waste is a little over 50 per cent, and biomethanation offers a major solution for processing this.
Which one of the following statements best reflects the crux of the passage ?
Answer (c) & explanation
The crux is that segregating waste is the first step to make waste-to-energy plants work (c). Step 1: the passage says waste-to-energy fits "after segregation". The right technology depends on what is in the waste. Step 2: but it opens by saying segregation at source "is rare" in India. So without segregation, these plants cannot succeed. That is (c). Step 3: option (b) calls biomethanation the most ideal way; the passage says only "a major solution". Option (a), cheap energy, is not claimed. Option (d) contradicts the "over 50 per cent" biodegradable share.
Tip: India's Solid Waste Management Rules, 2016 require segregation at source. Biomethanation turns wet waste into biogas.
Answer as per the official UPSC answer key.
Read the passage
There is a claim that organic farming is inherently safer and healthier. The reality is that because the organic farming industry is still young and not well-regulated in India, farmers and consumers, alike, are not only confused about what products are best for them, but sometimes use products in ways that could harm them as well. For example, since organic fertilizers are difficult to obtain on a large scale in India, farmers often use farmyard manure, which may contain toxic chemicals and heavy metals. Certain plant sprays, such as Datura flower and leaf spray, have an element called atropine. If it is not applied in the right dose, it can act on the nervous system of the consumer. Unfortunately, how much and when to use it are not well-researched or regulated issues.
Based on the above passage, the following assumptions have been made :
- 1. Organic farming is inherently unsafe for both farmers and consumers.
- 2. Farmers and consumers need to be educated about eco-friendly food.
Which of the assumptions given above is/are correct ?
Answer (b) & explanation
Only assumption 2 is correct (b). Step 1: the passage says farmers and consumers are confused about which products are best and sometimes use them in harmful ways. So both need to be educated (2). Step 2: it blames harm on poor regulation, contaminated farmyard manure and wrong doses of sprays like Datura. Step 3: it does not say organic farming is inherently unsafe (1). The problem is misuse, not organic farming itself. So options (a), (c) and (d) fail.
Tip: "inherently" means by its very nature. If the passage blames how something is used, it is not inherently bad.
Answer as per the official UPSC answer key.
Read the passage
There is a claim that organic farming is inherently safer and healthier. The reality is that because the organic farming industry is still young and not well-regulated in India, farmers and consumers, alike, are not only confused about what products are best for them, but sometimes use products in ways that could harm them as well. For example, since organic fertilizers are difficult to obtain on a large scale in India, farmers often use farmyard manure, which may contain toxic chemicals and heavy metals. Certain plant sprays, such as Datura flower and leaf spray, have an element called atropine. If it is not applied in the right dose, it can act on the nervous system of the consumer. Unfortunately, how much and when to use it are not well-researched or regulated issues.
Which one of the following statements best reflects the most logical, rational and practical message conveyed by the author of the passage ?
Answer (c) & explanation
The message is that Indian farmers need guidance and help to make organic farming sustainable (c). Step 1: the passage says the organic industry is "young and not well-regulated". Step 2: doses and timing of sprays are "not well-researched or regulated". So farmers need guidance. That is (c). Step 3: option (a) says organic farming should not be promoted, and option (b) says there are no safe alternatives. Both reject organic farming, which the author does not do. Option (d), profits and global markets, is not discussed.
Tip: a critical passage is not always against the idea. Here the author wants organic farming done right, not stopped.
Answer as per the official UPSC answer key.
Read the passage
Food consumption patterns have changed substantially in India over the past few decades. This has resulted in the disappearance of many nutritious foods such as millets. While food grain production has increased over five times since independence, it has not sufficiently addressed the issue of malnutrition. For long, the agriculture sector focussed on increasing food production particularly staples, which led to lower production and consumption of indigenous traditional crops/grains, fruits and other vegetables, impacting food and nutrition security in the process. Further, intensive, monoculture agriculture practices can perpetuate the food and nutrition security problem by degrading the quality of land, water and food derived through them.
Based on the above passage, the following assumptions have been made :
- 1. To implement the Sustainable Development Goals and to achieve zero-hunger goal, monoculture agriculture practices are inevitable even if they do not address malnutrition.
- 2. Dependence on a few crops has negative consequences for human health and the ecosystem.
- 3. Government policies regarding food planning need to incorporate nutritional security.
- 4. For the present monoculture agriculture practices, farmers receive subsidies in various ways and government offers remunerative prices for grains and therefore they do not tend to consider crop diversity.
Which of the above assumptions are valid ?
Answer (b) & explanation
Assumptions 2 and 3 are valid (b). Step 1: the passage says monoculture degrades the quality of land, water and food. So depending on a few crops harms health and the ecosystem (2). Step 2: it says grain output rose five times but did not fix malnutrition. So food planning must include nutritional security (3). Step 3: assumption 1 calls monoculture inevitable, which goes against the passage. Assumption 4, about subsidies and prices, is never mentioned. So options (a), (c) and (d) fail.
Tip: 2023 was the International Year of Millets, led by India at the UN. Millets are exactly the "disappearing nutritious foods" this passage mentions.
Answer as per the official UPSC answer key.
A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. Consider the following statements :
- 1. The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is 175.
- 2. The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is 167.
Which of the above statements is/are correct ?
Answer (c) & explanation
Both statements are correct (c). Step 1: think of the worst case. To avoid completing any colour, take one less than each group: 13 + 19 + 25 + 27 + 37 + 53 = 174. The next ball must complete a group, so n = 175 (statement 1 true). Step 2: to miss a colour as long as possible, take every ball except the smallest group (14 black): 180 − 14 = 166. Step 3: the next ball must be black, so m = 167 (statement 2 true). Options (a), (b) and (d) fail.
Tip: worst-case questions: give the unlucky picker everything except what you need, then add one.
for 'one of each colour', the worst case leaves out the smallest group, not the largest.
Answer as per the official UPSC answer key.
If 'ZERO' is written as 'CHUR', then how is 'PLAYER' written ?
Answer (d) & explanation
PLAYER is written SODBHU (d). Step 1: compare ZERO with CHUR. Z → C, E → H, R → U, O → R. Each letter moves 3 places forward, wrapping from Z back to A. Step 2: apply this to PLAYER: P → S, L → O, A → D, Y → B, E → H, R → U. Step 3: the code is SODBHU. Option (a) gets A and Y wrong. Options (b) and (c) go wrong on the last letters.
Tip: when a letter near Z changes to one near A, the code wraps around. Count forward: Y, Z, A, B is 3 steps.
Answer as per the official UPSC answer key.
Consider the following statements :
- 1. A is older than B.
- 2. C and D are of the same age.
- 3. E is the youngest.
- 4. F is younger than D.
- 5. F is older than A.
How many statements given above are required to determine the oldest person/persons ?
Answer (d) & explanation
All five statements are needed (d). Step 1: put them in a chain: C = D (2), D > F (4), F > A (5), A > B (1), and E is the youngest (3). So C = D > F > A > B, and E is last. Step 2: to call C and D the oldest, we must show they beat everyone. B is beaten only through statement 1. E is placed only by statement 3. Step 3: C is linked to D only through statement 2. Drop any one and someone's position is unknown. So all five are needed. Options (a), (b) and (c) fail.
Tip: to name the oldest, every other person must be placed below. Check each person is covered by at least one statement.
Answer as per the official UPSC answer key.
Consider the following including the Question and the Statements :
There are 5 members A, B, C, D, E in a family.
Question : What is the relation of E to B?
Statement-1 : A and B are a married couple.
Statement-2 : D is the father of C.
Statement-3 : E is D's son.
Statement-4 : A and C are sisters.
Which one of the following is correct in respect of the above Question and Statements?
Answer (c) & explanation
All four statements together are sufficient (c). Step 1: statements 2 and 4: D is C's father, and A is C's sister. So D is A's father too. Step 2: statement 3: E is D's son, so E is A's brother. Step 3: statement 1: B is married to A. So E is B's brother-in-law. Each statement is needed: without 2 or 4 we cannot link A to D; without 3 we know nothing of E; without 1 we know nothing of B. Options (a), (b) and (d) fail.
Tip: build a small family tree as you read each statement. Then test each option by removing one statement.
Option (b) leaves out D being C's father, so A cannot be tied to D and E.
Answer as per the official UPSC answer key.
Choose the group which is different from the others :
Answer (d) & explanation
Group (d) is the odd one out (d). Step 1: check whether each number is prime. Groups (a) 17, 37, 47, 97; (b) 31, 41, 53, 67; and (c) 71, 73, 79, 83 contain only primes. Step 2: in group (d), 91 = 7 × 13, which is not prime. Step 3: so (d) is different. Options (a), (b) and (c) all fit the pattern.
Tip: 91 looks prime but is 7 × 13. Other common traps are 51 (3 × 17), 57 (3 × 19) and 87 (3 × 29).
91 looks prime but is 7 × 13. Learn this one by heart.
Answer as per the official UPSC answer key.
Read the passage
Scientists studied the vernal window — transition period from winter to the growing season. They found that warmer winters with less snow resulted in a longer lag time between spring events and a more protracted vernal window. This change in the spring timetable has ecological, social and economic consequences — for agriculture, fisheries and tourism. As the ice melts earlier, the birds don’t return, causing a delay, or lengthening in springtime ecological events.
With reference to the above passage, the following assumptions have been made :
- 1. Global warming is causing spring to come early and for longer durations.
- 2. Early spring and longer period of spring is not good for bird populations.
Which of the above assumptions is/are correct ?
Answer (a) & explanation
Only assumption 1 is correct (a). Step 1: the passage says warmer winters with less snow cause a "more protracted vernal window" and earlier ice melt. So global warming makes spring come early and last longer (1). Step 2: it says birds don't return on time, delaying spring events. Step 3: but it does not say bird populations suffer. Late return is not the same as harm to numbers. So assumption 2 fails, and options (b), (c) and (d) fail.
Tip: "vernal" means spring. A change in timing is not proof of a fall in numbers; don't stretch the passage.
Answer as per the official UPSC answer key.
Read the passage
A global analysis of nitrogen use efficiency — a measure of the amount of nitrogen a plant takes in to grow versus what is left behind as pollution — says that using too much fertilizers will lead to increased pollution of waterways and the air. Currently, the global average for nitrogen use efficiency is approximately 0·4, meaning 40 per cent of the total nitrogen added to cropland goes into the harvested crop while 60 per cent is lost to the environment, says a study. More than half of the world’s population is nourished by food grown with fertilizers containing synthetic nitrogen, which is needed to produce high crop yields. Plants take the nitrogen they need to grow, and the excess is left in the ground, water and air. This results in significant emissions of nitrous oxide, a potent greenhouse and ozone depleting gas, and other forms of nitrogen pollution, including eutrophication of lakes and rivers and contamination of river water.
Which one of the following statements best reflects the most logical, rational and crucial message implied by the passage ?
Answer (a) & explanation
The crucial message is that better nitrogen-use efficiency is needed for both food and the environment (a). Step 1: the passage says synthetic nitrogen feeds more than half the world's people. Step 2: but only 40% of it reaches the crop; 60% is lost as pollution, including nitrous oxide. So using nitrogen more efficiently serves both food and environment. That is (a). Step 3: option (b) covers only food. Option (c), alternative crops, is not raised. Option (d) discards fertilisers that the passage calls necessary for high yields.
Tip: nitrous oxide is both a greenhouse gas and ozone-depleting. India promotes neem-coated urea and Nano urea to cut nitrogen loss.
Answer as per the official UPSC answer key.
Read the passage
Along with sustainable lifestyles, climate justice is regarded as a significant principle in environmental parlance. Both the principles have bearings on political and economic choices of the nation. So far, in our climate change summits or compacts, both the principles have eluded consensus among nations. Justice, in the judicial sense, is well defined. However, in the context of climate change, it has scientific as well as socio-political connotations. The crucial question in the next few years will be how resources, technologies and regulations are used to support the victims of climate change. Justice in climate is not confined to actions relating to mitigation, but includes the wider notion of support for adaptation to climate change and compensation for loss and damage.
Which one of the following statements best reflects the most logical, rational and crucial message conveyed by the passage ?
Answer (a) & explanation
The crucial message is that climate justice should be built in detail into the rules of new climate agreements (a). Step 1: the passage says climate justice has "eluded consensus" in climate summits. Step 2: it says justice covers not just mitigation but also adaptation support and loss and damage. These need to be spelled out in agreements. That is (a). Step 3: option (b), uneven resources, and option (c), climate refugees, are not the main point. Option (d) blames developed countries, which the passage does not state.
Tip: a Loss and Damage Fund was agreed at COP27 (2022) and made operational at COP28 (2023). It is a step toward the climate justice this passage discusses.
Answer as per the official UPSC answer key.
A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct ?
Answer (c) & explanation
R is less than S (c). Step 1: at the same rate, half-yearly compounding gives more than annual compounding, because interest earns interest sooner. Step 2: here both give the same amount Q. So the half-yearly rate R must be smaller. Step 3: exactly: (1 + R/200)² = 1 + S/100, which gives S = R + R²/400. So S is always more than R for any positive rate. Option (a) and (d) allow equality, which is impossible. Option (b) reverses it.
Tip: more frequent compounding needs a lower rate to reach the same amount. This is why banks quote both a nominal and an effective rate.
Answer as per the official UPSC answer key.
How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers ?
Answer (d) & explanation
There are 9 such numbers (d). Step 1: if 1186 leaves remainder 31, then the divisor n divides 1186 − 31 = 1155. Also, n must be greater than 31 (a remainder is always smaller than the divisor). Step 2: 1155 = 3 × 5 × 7 × 11, which has 2 × 2 × 2 × 2 = 16 divisors. Step 3: those above 31 are 33, 35, 55, 77, 105, 165, 231, 385 and 1155. That is 9. Options (a), (b) and (c) miss some.
Tip: never forget the rule that the divisor must exceed the remainder. It removes the small divisors.
a remainder of 31 needs a divisor larger than 31. Counting all 16 divisors is the trap.
Answer as per the official UPSC answer key.
Let pp, qq and rr be 2-digit numbers where p < q < r. If pp + qq + rr = tt0, where tt0 is a 3-digit number ending with zero, consider the following statements :
- 1. The number of possible values of p is 5.
- 2. The number of possible values of q is 6.
Which of the above statements is/are correct ?
Answer (c) & explanation
Both statements are correct (c). Step 1: pp = 11p, so 11(p + q + r) = tt0 = 110t. So p + q + r = 10t: either 10 or 20. Step 2: sum 10 with p < q < r: (1, 2, 7), (1, 3, 6), (1, 4, 5), (2, 3, 5). Sum 20: (3, 8, 9), (4, 7, 9), (5, 6, 9), (5, 7, 8). Step 3: p takes 1, 2, 3, 4, 5 (five values). q takes 2, 3, 4, 6, 7, 8 (six values). So options (a), (b) and (d) fail.
Tip: repeated-digit numbers like pp are 11 × p. Spotting this turns the puzzle into a simple sum.
Answer as per the official UPSC answer key.
What is the sum of all 4-digit numbers less than 2000 formed by the digits 1, 2, 3 and 4, where none of the digits is repeated ?
Answer (a) & explanation
The sum is 7998 (a). Step 1: below 2000, the first digit must be 1. The other digits 2, 3 and 4 can be arranged in 3 × 2 × 1 = 6 ways. Step 2: thousands place: 6 numbers × 1000 = 6000. Step 3: each of 2, 3, 4 sits twice in each of the other three places: (2 + 3 + 4) × 2 × 111 = 1998. Total = 6000 + 1998 = 7998. Options (b), (c) and (d) miscount.
Tip: for sums of arranged numbers, handle the fixed digit separately, then use (sum of digits) × (times each appears) × 111.
Answer as per the official UPSC answer key.
What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction ?
Answer (c) & explanation
There are 12 selections (c). Step 1: in a circle, a run of consecutive things is fixed by where it starts. Step 2: any of the 12 things can be the starting point. Going clockwise, each start gives a different run of 10. Step 3: so there are 12 selections. Option (a), 3, applies to a straight line of 12 (12 − 10 + 1). Option (d), 66, counts any 10 things, not consecutive ones.
Tip: in a line, runs of k from n = n − k + 1. In a circle (with k less than n), runs of k = n.
66 = 12C10 counts all selections, consecutive or not.
Answer as per the official UPSC answer key.
If today is Sunday, then which day is it exactly on 1010 th day ?
Answer (a) & explanation
The 10¹⁰th day is a Wednesday (a). Step 1: today (Sunday) is day 1. So the 10¹⁰th day is 10¹⁰ − 1 days later. Step 2: 10 leaves remainder 3 when divided by 7. So 10¹⁰ leaves the same remainder as 3¹⁰. Since 3⁶ leaves 1, 3¹⁰ leaves the same as 3⁴ = 81, which is 4. Step 3: so 10¹⁰ − 1 leaves 3. Sunday + 3 days = Wednesday. Option (b), Thursday, forgets that today is day 1.
Tip: "the nth day" counting today as day 1 means n − 1 days ahead. This off-by-one trap is common.
today is day 1, not day 0. Counting it as day 0 gives Thursday, which is an option.
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.
There are three traffic signals. Each signal changes colour from green to red and then from red to green. The first signal takes 25 seconds, the second signal takes 39 seconds and the third signal takes 60 seconds to change the colour from green to red. The durations for green and red colours are same. At 2:00 p.m., they together turn green. At what time will they change to green next, simultaneously ?
Answer (b) & explanation
They turn green together again at 4:10 p.m. (b). Step 1: green and red last the same time. So each full cycle is double: 50 s, 78 s and 120 s. Step 2: LCM of 50, 78 and 120 = 2³ × 3 × 5² × 13 = 7800 s = 130 minutes. Step 3: 2:00 p.m. + 2 hours 10 minutes = 4:10 p.m. Option (a), 4:00 p.m., uses wrong cycles. Options (c) and (d) miscalculate.
Tip: always use the full cycle (green + red) for "turn green again". Using only the green time is the common trap.
LCM of 25, 39 and 60 gives the next time they all change, not the next time they all turn green.
Answer as per the official UPSC answer key.
Read the passage
Sourcing food from non-agricultural lands (uncultivated systems such as forests, wetlands, pastures, etc) in addition to agricultural lands enables a systemic approach to food consumption. It allows rural and tribal communities to sustain themselves for the whole year and steer clear of natural disasters and season-induced shortfalls of agricultural food. Since the productivity of trees is often more resilient to adverse weather conditions than annual crops, forest foods often provide a safety net during periods of food shortages caused by crop failure; forest foods also make important contributions during seasonal crop production gaps.
Which one of the following statements best reflects the most logical and rational message conveyed by the author of the passage ?
Answer (d) & explanation
The message is that agroecosystems should be developed along with conventional agriculture (d). Step 1: the passage says sourcing food from forests, wetlands and pastures "in addition to agricultural lands" helps communities all year. Step 2: forest foods act as a safety net when crops fail. They supplement farming, not replace it. That is (d). Step 3: option (a) says food trees should replace other trees, and option (b) says conventional farming cannot ensure food security; both overreach. Option (c), converting wastelands, is not mentioned.
Tip: "in addition to" is the key phrase. Reject options that turn "addition" into "replacement".
Answer as per the official UPSC answer key.
Read the passage
While awareness on use/misuse and abuse of antibiotics is common knowledge, as is the impact of dosing poultry with antibiotics, the environmental impact of antibiotics-manufacturing companies not treating their waste has scarcely been discussed at any length or seriousness thus far. Pollution from antibiotics factories is fuelling the rise of drug-resistant infections. The occurrence of drug-resistant bacteria surrounding the pharma manufacturing plants is well known.
Which one of the following statements best reflects the most logical and practical message conveyed by the passage ?
Answer (a) & explanation
The practical message is that proper effluent treatment protocols are needed (a). Step 1: the passage says antibiotic factories "not treating their waste" is fuelling drug-resistant infections. Step 2: the direct fix for untreated waste is proper treatment protocols. That is (a). Step 3: option (b), awareness, is already "common knowledge" as the passage says. Option (c) calls the problem unavoidable, which the passage does not say. Option (d), moving plants to rural areas, only shifts the harm elsewhere.
Tip: the most practical solution attacks the exact cause named. Antimicrobial resistance (AMR) is a key global health threat; India runs a National Action Plan on AMR.
Answer as per the official UPSC answer key.
Read the passage
Benefits of good quality school education accrue only when students complete and leave school after having acquired the gateway skills. Like one learns to walk before running, similarly one picks up advanced skills only after picking the basic foundational skills. The advent of the knowledge economy poses new challenges, and one of the severe consequences of having an uneducated workforce will be our inability to keep pace with the global economy. Without a strong learning foundation at the primary level, there can be no improvement in higher education or skill development.
Which one of the following statements best reflects the crux of the passage ?
Answer (a) & explanation
The crux is that to become a global power, India must invest in universal quality education (a). Step 1: the passage says advanced skills come only after foundational skills, as one walks before running. Step 2: an uneducated workforce would leave India unable to keep pace with the global economy. So universal quality schooling is the base. That is (a). Step 3: option (c) focuses on higher education, which reverses the passage's order. Option (b) says India is unable to become a power, which is too strong. Option (d), illiterate parents, is not mentioned.
Tip: India's NEP 2020 and NIPUN Bharat Mission stress foundational literacy and numeracy by Grade 3, the same idea as this passage.
Answer as per the official UPSC answer key.
40 children are standing in a circle and one of them (say child-1) has a ring. The ring is passed clockwise. Child-1 passes on to child-2, child-2 passes on to child-4, child-4 passes on to child-7 and so on. After how many such changes (including child-1) will the ring be in the hands of child-1 again ?
Answer (b) & explanation
The ring returns after 15 changes (b). Step 1: the jumps grow by one each time: 1, 2, 3 and so on. After k passes, the ring has moved 1 + 2 + ... + k = k(k + 1)/2 places. Step 2: it is back with child-1 when this is a multiple of 40. So k(k + 1) must be a multiple of 80 = 16 × 5. Step 3: the smallest such k is 15, since 15 × 16 = 240 = 80 × 3. Options (a) 14, (c) 16 and (d) 17 do not give a multiple of 80 first.
Tip: in a circle of n, you return to the start when the total steps moved is a multiple of n.
Answer as per the official UPSC answer key.
What is the middle term of the sequence
Z, Z, Y, Y, Y, X, X, X, X, W, W, W, W, W, …, A ?
Answer (b) & explanation
The middle term is I (b). Step 1: Z appears 2 times, Y 3 times, X 4 times, and so on. The kth letter from Z appears k + 1 times, so A appears 27 times. Step 2: total terms = 2 + 3 + ... + 27 = 377. The middle term is the 189th. Step 3: the first m letters fill m(m + 3)/2 places. For m = 18 this is exactly 189. The 18th letter counting back from Z is I. Options (a) H, (c) J and (d) M are off.
Tip: counting back from Z, letter number k is the letter at position 27 − k. Here 27 − 18 = 9, which is I.
Answer as per the official UPSC answer key.
Question : Is p greater than q ?
Statement-1: p × q is greater than zero.
Statement-2: p2 is greater than q2.
Which one of the following is correct in respect of the above Question and the Statements ?
Answer (d) & explanation
The question cannot be answered even with both statements (d). Step 1: Statement-1: p × q is positive, so p and q have the same sign. Step 2: Statement-2: p² > q², so p is farther from zero than q. Step 3: together: both positive, like 3 and 2, gives p > q. Both negative, like −3 and −2, gives p < q. Two different answers, so the data are not enough. Options (a), (b) and (c) fail.
Tip: squares hide signs. Always test negative numbers in data sufficiency questions involving squares.
squaring hides the sign. Negatives reverse the order.
Traps like this come back in new clothes. The PYQ Book collects all of them, topic by topic, so none catch you twice. Try the free sample
Answer as per the official UPSC answer key.
Question : Is (p + q – r) greater than (p – q + r), where p, q and r are integers ?
Statement-1: (p – q) is positive.
Statement-2: (p – r) is negative.
Which one of the following is correct in respect of the above Question and the Statements ?
Answer (c) & explanation
Both statements together are needed (c). Step 1: simplify the question first. p + q − r > p − q + r means 2q > 2r, that is, is q > r? Step 2: Statement-1 says p − q is positive, so q < p. Statement-2 says p − r is negative, so p < r. Neither alone links q and r. Step 3: together: q < p < r, so q < r. The question gets a definite "no". A definite "no" still answers it, so (c). Options (a), (b) and (d) fail.
Tip: always simplify the question before reading the statements. A firm "no" counts as an answer in data sufficiency.
Answer as per the official UPSC answer key.
In a party, 75 persons took tea, 60 persons took coffee and 15 persons took both tea and coffee. No one taking milk takes tea. Each person takes at least one drink.
Question: How many persons attended the party ?
Statement-1: 50 persons took milk.
Statement-2: Number of persons who attended the party is five times the number of persons who took milk only.
Which one of the following is correct in respect of the above Question and the Statements ?
Answer (a) & explanation
Statement-2 alone answers it, but Statement-1 alone does not (a). Step 1: people who took tea or coffee = 75 + 60 − 15 = 120. No milk-taker takes tea, so total = 120 + (people who took milk only). Step 2: Statement-1 says 50 took milk. But some of them may also take coffee, so "milk only" is unknown. Not enough. Step 3: Statement-2: total = 5 × (milk only). So 120 + m = 5m, giving m = 30 and total = 150. Options (b), (c) and (d) fail.
Tip: in Venn questions, ask which region the given number covers. "Took milk" is not the same as "took milk only".
Answer as per the official UPSC answer key.
Consider a 3-digit number.
Question : What is the number ?
Statement-1: The sum of the digits of the number is equal to the product of the digits.
Statement-2: The number is divisible by the sum of the digits of the number.
Which one of the following is correct in respect of the above Question and the Statements ?
Answer (d) & explanation
Even both statements are not enough (d). Step 1: Statement-1: the digit sum equals the digit product. For three digits this works for 1, 2 and 3 (1 + 2 + 3 = 1 × 2 × 3 = 6), in any order: six numbers. Step 2: Statement-2: the number must be divisible by the digit sum, 6. So it must be even. Step 3: both 132 and 312 survive. Two answers remain, so the number is not fixed. Options (a), (b) and (c) fail.
Tip: in "what is the number" questions, keep testing until only one number is left. Two survivors mean insufficient data.
Answer as per the official UPSC answer key.
For five children with ages a < b < c < d < e; any two successive ages differ by 2 years.
Question: What is the age of the youngest child ?
Statement-1: The age of the eldest is 3 times the youngest.
Statement-2: The average age of the children is 8 years.
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: the ages go up by 2 each time: a, a + 2, a + 4, a + 6, a + 8. Step 2: Statement-1: the eldest is 3 times the youngest. So a + 8 = 3a, giving a = 4. Step 3: Statement-2: the average is 8. The average of five equally spaced ages is the middle one, a + 4 = 8, so a = 4. Each works alone. Options (a), (c) and (d) fail.
Tip: for equally spaced numbers, the average is the middle term. This saves time in many sequence questions.
Answer as per the official UPSC answer key.
Read the passage
The paradox of choice is illustrated by the story of Buridan’s ass. Jean Buridan, the 14th century philosopher, wrote about free will and the inability to choose due to numerous choices and uncertainties. In the story, a donkey stands between two equally appealing stacks of hay. Unable to decide which to eat, it starves to death. Changes in technology and innovations such as smart phones and tablets only exacerbate our glut of choices. Constant connectivity and overconsumption of real-time data and social media can leave little room for self-reflection and rest, making decisions more difficult. Life is about choices. Many people are overwhelmed with attractive life choices, yet find themselves unhappy and anxious.
Which one of the following statements best reflects the most logical message implied by the above passage ?
Answer (d) & explanation
The message is that too few choices may be bad, but too many can be just as hard (d). Step 1: the passage tells of Buridan's ass, which starved between two equally good stacks of hay because it could not choose. Step 2: it says technology adds to our "glut of choices", leaving people unhappy and anxious, yet "life is about choices". That is the paradox of choice, so (d). Step 3: option (b) calls the choices difficult; the passage calls them attractive. Option (a) blames technology for weakening society. Option (c), lacking courage, is not discussed.
Tip: "paradox" signals two sides. Pick the option that holds both: choices are needed, but too many overwhelm.
Answer as per the official UPSC answer key.
Read the passage
Household finance in India is unique. We have a tendency to invest heavily in physical assets such as gold and property. Steps to encourage the financialization of savings are critical. A populace accustomed to traditional processes will not simply jump into financialization. Hurdles to change include onerous bureaucracy, a scepticism of organized financial institutions, a lack of basic information about which of the myriad services and providers is best for each family, and how (and even if) one can make the transition between them if necessary.
Regarding the financialization of household savings, which of the following statements best reflect the solutions that are implied by the passage ?
- 1. A flexible environment is needed to develop solutions.
- 2. Households need customised solutions.
- 3. Innovations in financial technology are required.
Select the correct answer using the code given below :
Answer (d) & explanation
All three solutions are implied (d). Step 1: the passage lists hurdles to financialising savings. "Onerous bureaucracy" implies the need for a flexible environment (1). Step 2: the lack of information on which service is "best for each family" implies customised solutions (2). Step 3: the lack of information and the difficulty of moving between providers imply the need for fintech innovation (3). So options (a), (b) and (c) fail.
Tip: when a passage lists problems, each problem implies its own solution. Match them one by one. Indians traditionally save in gold and property, which this passage calls physical assets.
Answer as per the official UPSC answer key.
Read the passage
Pharmaceutical patents grant protection to the patentee for the duration of the patent term. The patentees enjoy the liberty to determine the prices of medicines, which is time-limited to the period of monopoly, but could be unaffordable to the public. Such patent protection offered to the patentees is believed to benefit the public over the longer term through innovations and research and development (R&D), although it comes at a cost, in the nature of higher prices for the patented medicine. The patent regime and price protection — through a legally validated high price for the medicine during the currency of the patent — provide the patentee with a legitimate mechanism to get returns on the costs incurred in innovation and research.
Based on the above passage, the following assumptions have been made :
- 1. Patent protection given to patentees puts a huge burden on public’s purchasing power in accessing patented medicines.
- 2. Dependence on other countries for pharmaceutical products is a huge burden for developing and poor countries.
- 3. Providing medicines to the public at affordable prices is a key goal during the public health policy design in many countries.
- 4. Governments need to find an appropriate balance between the rights of patentees and the requirements of the patients.
Which of the above assumptions are valid ?
Answer (b) & explanation
Assumptions 1 and 4 are valid (b). Step 1: the passage says patent prices "could be unaffordable to the public". So patents can burden people's buying power (1). Step 2: it also says patents reward R&D and benefit the public in the long run. So governments must balance patent rights and patients' needs (4). Step 3: assumption 2, dependence on imports, and assumption 3, other countries' policy goals, are outside the passage. So options (a), (c) and (d) fail.
Tip: India's Patents Act Section 3(d) blocks "evergreening" of patents, an example of balancing innovation with access to medicines.
Answer as per the official UPSC answer key.
Read the passage
India should ensure the growth of the digital economy while keeping personal data of citizens secure and protected. No one will innovate in a surveillance-oriented environment or in a place where an individual’s personal information is compromised. The ultimate control of data must reside with the individuals who generate it; they should be enabled to use, restrict or monetise it as they wish. Therefore, data protection laws should enable the right kind of innovation — one that is user-centric and privacy protecting.
Based on the above passage, the following assumptions have been made :
- 1. Protection of privacy is not just a right, but it has value to the economy.
- 2. There is a fundamental link between privacy and innovation.
Which of the above assumptions is/are valid ?
Answer (c) & explanation
Both assumptions are valid (c). Step 1: the passage says India should grow its digital economy while keeping personal data safe. So data protection has economic value, not just rights value (1). Step 2: it says "no one will innovate" where personal data is compromised. So privacy and innovation are closely linked (2). Step 3: so options (a), (b) and (d) fail.
Tip: India passed the Digital Personal Data Protection Act in 2023, and its rules were notified in November 2025. The Supreme Court declared privacy a fundamental right in the Puttaswamy case (2017).
Answer as per the official UPSC answer key.
In an examination, the maximum marks for each of the four papers namely P, Q, R and S are 100. Marks scored by the students are in integers. A student can score 99% in n different ways. What is the value of n ?
Answer (d) & explanation
1. 99% of 400 = 396, so the student loses exactly 4 marks in total across the four papers.
2. Count the ways to share the 4 lost marks:
3. All 4 in one paper: 4 ways. 3 + 1: 4 × 3 = 12 ways. 2 + 2: 6 ways. 2 + 1 + 1: 12 ways. 1 + 1 + 1 + 1: 1 way.
4. Total 4 + 12 + 6 + 12 + 1 = 35.
'no paper above 100' does not cut any case here, because a loss can't be negative.
Answer as per the official UPSC answer key.
A flag has to be designed with 4 horizontal stripes using some or all of the colours red, green and yellow. What is the number of different ways in which this can be done so that no two adjacent stripes have the same colour ?
Answer (c) & explanation
There are 24 ways (c). Step 1: the top stripe can be any of 3 colours. Step 2: each stripe below only has to differ from the one above it, so it has 2 choices. Step 3: 3 × 2 × 2 × 2 = 24. "Some or all" colours means using just two colours (like red-green-red-green) is allowed. Option (d), 36, and the others miscount.
Tip: for "no two adjacent the same", the first item has n choices and each next one has n − 1.
Answer as per the official UPSC answer key.
A rectangular floor measures 4 m in length and 2·2 m in breadth. Tiles of size 140 cm by 60 cm have to be laid such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. What is the maximum number of tiles that can be accommodated on the floor ?
Answer (d) & explanation
1. Floor = 400 cm × 220 cm. One tile = 140 cm × 60 cm.
2. Neat rows (all tiles the same way) fit only 8 tiles.
3. Mixing the two directions in a staggered (pinwheel) pattern, as in the figure, fits 9 tiles.
4. 10 cannot fit: across the 220 cm width the tiles can fill at most 200 cm (140 + 60), so a strip of floor is always left empty and there is room for only 9.
don't stop at the neat grid answer 8. "Any orientation" is a hint to mix them.
Answer as per the official UPSC answer key.
There are five persons P, Q, R, S and T each one of whom has to be assigned one task. Neither P nor Q can be assigned Task-1. Task-2 must be assigned to either R or S. In how many ways can the assignment be done ?
Answer (d) & explanation
There are 24 ways (d). Step 1: Task-2 must go to R or S: 2 ways. Step 2: Task-1 cannot go to P, Q or the person who got Task-2. So only 2 people are left for it: 2 ways. Step 3: the other 3 tasks go to the remaining 3 people in 3 × 2 × 1 = 6 ways. Total = 2 × 2 × 6 = 24. Options (a), (b) and (c) miss a factor.
Tip: fill the most restricted task first. Then each later choice becomes simple counting.
Answer as per the official UPSC answer key.
There are large number of silver coins weighing 2 gm, 5 gm, 10 gm, 25 gm, 50 gm each. Consider the following statements :
- 1. To buy 78 gm of coins one must buy at least 7 coins.
- 2. To weigh 78 gm using these coins one can use less than 7 coins.
Which of the statements given above is/are correct ?
Answer (c) & explanation
Both statements are correct (c). Step 1: buying 78 g: 50 + 10 + 10 + 2 + 2 + 2 + 2 = 78 with 7 coins. 50 + 25 leaves 3 g, which no coin can make. Other mixes need more coins. So at least 7 are needed (1 true). Step 2: weighing allows coins on both pans. Put 50 + 25 + 5 on one pan and 2 on the other. That measures 80 − 2 = 78 g with only 4 coins (2 true). Step 3: so options (a), (b) and (d) fail.
Tip: "buy" means adding coins only; "weigh" lets you subtract by using the other pan. The wording changes the answer.
'buy' and 'weigh' are different. On a balance, a coin can also subtract.
Answer as per the official UPSC answer key.
Consider the following :
- I. A + B means A is neither smaller nor equal to B.
- II. A − B means A is not greater than B.
- III. A × B means A is not smaller than B.
- IV. A ÷ B means A is neither greater nor equal to B.
- V. A ± B means A is neither smaller nor greater than B.
Statement : P × Q, P − T, T ÷ R, R ± S
Conclusion-1 : Q ± T
Conclusion-2 : S + Q
Which one of the following is correct in respect of the above Statement and the Conclusions ?
Answer (b) & explanation
Only Conclusion-2 follows (b). Step 1: decode the symbols: + means >, − means ≤, × means ≥, ÷ means < and ± means =. Step 2: the statement becomes P ≥ Q, P ≤ T, T < R, R = S. So S = R > T ≥ P ≥ Q. Step 3: Conclusion-1 (Q = T) need not be true; T could be bigger. Conclusion-2 (S > Q) is true, since S > T ≥ Q. So options (a), (c) and (d) fail.
Tip: decode every symbol first, then write one chain. A single strict ">" in the chain makes the end-to-end link strict.
Answer as per the official UPSC answer key.
Read the passage
In India, while the unemployment rate is a frequently used measure of poor performance of the economy, under conditions of rising school and college enrolment, it paints an inaccurate picture. The reported unemployment rate is dominated by the experience of younger Indians who face higher employment challenges and exhibit greater willingness to wait for the right job than their older peers. The unemployment challenge is greater for people with secondary or higher education, and rising education levels inflate unemployment challenges.
Which one of the following statements most likely reflects as to what the author of the passage intends to say ?
Answer (b) & explanation
The author says unemployment should be seen in light of rising education and aspirations of young Indians (b). Step 1: the passage says the unemployment rate paints an inaccurate picture when school and college enrolment are rising. Step 2: young, educated Indians are more willing to "wait for the right job", and rising education "inflates" unemployment. So it reflects education and aspirations. That is (b). Step 3: option (a), quality of education, and option (c), labour-intensive industry, are not discussed. Option (d), self-employment, is not suggested.
Tip: this is "aspirational unemployment". Educated youth may wait for a suitable job rather than take any job.
Answer as per the official UPSC answer key.
Read the passage
“Science by itself is not enough, there must be a force and discipline outside the sciences to coordinate them and point to a goal. It is not possible to run a course aright when the goal itself has not been rightly placed. What science needs is philosophy — the analysis of scientific method and the coordination of scientific purposes and results; without this, any science must be superficial. Government suffers, precisely like science, for lack of philosophy. Philosophy bears to science the same relationship which statesmanship bears to politics : movement guided by total knowledge and perspective, as against aimless and individual seeking. Just as the pursuit of knowledge becomes scholasticism when divorced from the actual needs of men and life, so the pursuit of politics becomes a destructive bedlam when divorced from science and philosophy.”
Which one of the following statements best reflects the most rational, logical and practical message conveyed by the passage ?
Answer (b) & explanation
The message is that government should not be run only by practical statesmen, but with people of learning and wisdom (b). Step 1: the passage says science needs philosophy to give it a goal. Step 2: it says government suffers "like science, for lack of philosophy". Politics without science and philosophy becomes "destructive bedlam". So practical statesmen need learned, wise people alongside. That is (b). Step 3: options (a) and (c) prescribe training schemes the passage never proposes. Option (d), all scientists as philosophers, is too absolute.
Tip: this passage is from Will Durant on Plato. Pick the option that applies the passage's comparison (science-philosophy, politics-statesmanship) to government.
Answer as per the official UPSC answer key.
Read the passage
“The last end of the state is not to dominate men, nor to restrain them by fear; rather it is so to free each man from fear that he may live and act with full security and without injury to himself or his neighbour. The end of the state, I repeat, is not to make rational beings into brute beasts and machines. It is to enable their bodies and their minds to function safely. It is to lead men to live by, and to exercise, a free reason; that they may not waste their strength in hatred, anger and guile, nor act unfairly toward one another.”
Based on the above passage, which one of the following terms best expresses the ultimate goal of the state ?
Answer (d) & explanation
The ultimate goal of the state is liberty (d). Step 1: the passage says the state's end is "not to dominate men" but to "free each man from fear". Step 2: it adds that the state should lead men to live by and use "a free reason". Freedom runs through every line. That is (d). Step 3: option (a), personal safety, and option (b), health of body and mind, are only parts of this freedom. Option (c), communal harmony, is a side effect at most.
Tip: this is Spinoza's view: "the end of the state is really liberty". Pick the term that sums up all the lines, not one detail.
Answer as per the official UPSC answer key.
What is the remainder if 2192 is divided by 6 ?
Answer (d) & explanation
The remainder is 4 (d). Step 1: look at remainders of powers of 2 divided by 6: 2¹ = 2 → 2; 2² = 4 → 4; 2³ = 8 → 2; 2⁴ = 16 → 4. They alternate 2, 4, 2, 4. Step 2: odd powers leave 2 and even powers leave 4. Step 3: 192 is even, so the remainder is 4. Option (c), 2, is for odd powers. Options (a) and (b) are impossible, since a power of 2 is never a multiple of 3.
Tip: for remainders of large powers, list the first few. A short repeating cycle almost always appears.
Answer as per the official UPSC answer key.
Consider the sequence ABC_ _ABC_DABBCD_ABCD that follows a certain pattern. Which one of the following completes the sequence?
Answer (d) & explanation
The sequence is completed by DDCA (d). Step 1: put D, D, C, A into the gaps. The sequence becomes ABCDDABCCDABBCDAABCD. Step 2: split it into blocks of four: ABCD, DABC, CDAB, BCDA, ABCD. Step 3: each block moves the last letter to the front of the previous block. The pattern holds throughout, so (d). Options (a), (b) and (c) break it.
Tip: in letter-series gaps, try each option and split the result into equal blocks. A rotation pattern is very common.
Answer as per the official UPSC answer key.
AB and CD are 2-digit numbers. Multiplying AB with CD results in a 3-digit number DEF. Adding DEF to another 3-digit number GHI results in 975. Further A, B, C, D, E, F, G, H, I are distinct digits. If E = 0, F = 8, then what is A + B + C equal to?
Answer (a) & explanation
A + B + C is 6 (a). Step 1: AB × CD = D08, a 3-digit number. So both AB and CD are small, and B × D must end in 8. Step 2: try 12 × 34 = 408. Here D = 4, E = 0 and F = 8. Then GHI = 975 − 408 = 567. Step 3: all nine digits (1, 2, 3, 4, 0, 8, 5, 6, 7) are distinct. So A + B + C = 1 + 2 + 3 = 6. Options (b), (c) and (d) do not fit.
Tip: in such puzzles, a small product means small factors. Start with the smallest two-digit numbers.
Answer as per the official UPSC answer key.
Consider the following statements in respect of five candidates P, Q, R, S and T. Two statements are true and one statement is false.
True Statement : One of P and Q was selected for the job.
False Statement : At least one of R and S was selected for the job.
True Statement : At most two of R, S and T were selected for the job.
Which of the following conclusions can be drawn ?
- 1. At least four were selected for the job.
- 2. S was selected for the job.
Select the correct answer using the code given below :
Answer (d) & explanation
Neither conclusion follows (d). Step 1: the false statement is "at least one of R and S was selected". If it is false, then neither R nor S was selected. So conclusion 2 (S was selected) is wrong. Step 2: only one of P and Q was selected, and T may or may not be. So at most 2 people were selected. Step 3: so conclusion 1 (at least four selected) is wrong too. Options (a), (b) and (c) fail.
Tip: the opposite of "at least one" is "none". Turning the false statement into its opposite is the key step.
the negation of 'at least one' is 'none', not 'only one'.
Answer as per the official UPSC answer key.
Let P, Q, R, S and T be five statements such that :
- I. If P is true, then both Q and S are true.
- II. If R and S are true, then T is false.
Which of the following can be concluded ?
- 1. If T is true, then at least one of P and R must be false.
- 2. If Q is true, then P is true.
Select the correct answer using the code given below :
Answer (a) & explanation
Only conclusion 1 follows (a). Step 1: suppose T is true. By rule II, R and S cannot both be true. If S is false, then by rule I, P must be false (P true would make S true). If R is false, we are done. Either way, P or R is false. So conclusion 1 follows. Step 2: rule I says "if P then Q". This does not give "if Q then P". So conclusion 2 does not follow. Step 3: options (b), (c) and (d) fail.
Tip: "If A then B" also gives "if not B then not A". It never gives "if B then A".
reversing an 'if … then' (Q → P) is the classic mistake. Only the contrapositive (not Q → not P) is safe.
Answer as per the official UPSC answer key.
A cuboid of dimensions 7 cm × 5 cm × 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm × 5 cm, 5 cm × 3 cm, 7 cm × 3 cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1 cm. Which of the following statements is/are correct ?
- 1. There are exactly 15 small cubes with no paint on any face.
- 2. There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.
Select the correct answer using the code given below :
Answer (a) & explanation
Only statement 1 is correct (a). Step 1: unpainted cubes form the inner block: (7 − 2) × (5 − 2) × (3 − 2) = 5 × 3 × 1 = 15. So statement 1 is true. Step 2: blue faces (7 × 3) and green faces (5 × 3) meet along the four edges of length 3 cm. Step 3: each such edge has 3 − 2 = 1 middle cube with exactly those two colours. Four edges give 4 cubes, not 6. So statement 2 is false. Options (b), (c) and (d) fail.
Tip: two coloured faces meet along an edge. Cubes with exactly two colours on an edge = edge length − 2 (the corners have three colours).
corner cubes have three painted faces, so leave them out when you count 'exactly two'.
Answer as per the official UPSC answer key.
The letters of the word "INCOMPREHENSIBILITIES" are arranged alphabetically in reverse order. How many positions of the letter/letters will remain unchanged ?
Answer (c) & explanation
Two positions stay the same (c). Step 1: the 21 letters in reverse alphabetical order are: T S S R P O N N M L I I I I I H E E E C B. Step 2: the original word is I N C O M P R E H E N S I B I L I T I E S. Step 3: compare place by place. Only places 13 and 15 match, both with the letter I. So two positions stay unchanged. Options (a), (b) and (d) miscount.
Tip: write both orders in two rows, one under the other. Count the letters carefully first; repeated letters like I (5 times) decide the answer.
write both rows one above the other and compare letter by letter.
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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.