UPSC 2026 CSAT PYQs with Answers

The question cannot be answered due to insufficient data. The boy's earning depends on two variables: number of deliveries and kilometres travelled. The condition gives total earning = 75 x deliveries + 12 x kilometres, but the question does not fix either variable or provide a relationship between them. Therefore the required number cannot be determined uniquely.

Liquid B forms approximately 27 percent of the mixture. Each replacement removes 1 litre from a 10 litre mixture and refills it, so the fraction of liquid A left is multiplied by 9/10 each time. The operation is performed three times. Fraction of A remaining = (9/10)^3 = 729/1000 = 72.9 percent. Therefore liquid B forms 100 - 72.9 = 27.1 percent of the mixture, approximately 27 percent.

12,000. Profit share is proportional to capital x time. A contributes (1/3 capital) for (1/3 duration), giving 1/9. B contributes 1/4 x 1/4 = 1/16. C contributes the remaining capital 1 - 1/3 - 1/4 = 5/12 for the full duration, giving 5/12. Ratio = 1/9 : 1/16 : 5/12 = 16:9:60. C's share = 60/(16+9+60) x 17,000 = 12,000.

Circular letters are standardized letters addressed to a group of people for common information, so statement 1 is correct. Their content is standardized, not customized (statement 2 wrong); personal matters like appraisals and increments are not conveyed by circular letters (statement 3 wrong); and circular letters are far more cost-effective than personalised letters (statement 4 wrong). Hence only statement 1 is correct.

1 and 2 only. The questionnaire is for judging feasibility of a family-oriented vacation resort near a mountain lake destination. Size of family is directly relevant because it indicates accommodation type, room capacity and family facilities. Budget is directly relevant because it indicates price feasibility and expected spending capacity. Number of earners is intrusive and not necessary for deciding resort feasibility. Food allergies and dietary restrictions may matter later for operations, but they are not central to the feasibility objective stated in the question. Hence 1 and 2 only should be included.

Semantic barriers arise from misunderstanding the meaning of a word, so A-2. Cognitive barriers include information overload, so B-4. Organisational barriers include lack of feedback, so C-1. Affective barriers arise from emotions such as fear of social stigma, so D-3. Hence the code A-2, B-4, C-1, D-3.

Encouraging personnel to seek each other's perspectives builds mutual understanding and resolves conflict constructively, so action 3 is appropriate. Forcing cultural-marker activities on personnel can be intrusive, and leaving conflicts to resolve naturally abdicates leadership. Hence only action 3 is the most appropriate directive.

A memorandum is used to intimate a directive (A-3); a flyer disseminates non-targeted information (B-4); Bcc informs confidentially (C-2); minutes record decisions (D-1). Hence the code A-3, B-4, C-2, D-1.

A cricket captain communicating with team members is generally informal but firm, so A-1. Communication between a judge and lawyers in court is formal and firm because it follows courtroom procedure, so B-4. Communication between a Vice Chancellor and Deputy Registrar is formal and may remain open-ended for administrative discussion, so C-3. Communication between peers and coworkers is informal and open-ended, so D-2. This matches option B.

S1 concerns protecting the environment while S3 concerns climate alarmism and extremist views within environmental activism, so S3 is neither a counterpoint to S1 nor unconnected to it; relationships 1 and 2 are incorrect. S2, which says scientific consensus on the extent of human contribution is lacking, could be the reason why the alarmist views of S3 become the focus of climate change deniers. Hence only relationship 3 is correct.

Weighted scores: Candidate 1 = 0.2 x 5 + 0.3 x 4 + 0.5 x 6 = 5.2; Candidate 2 = 0.2 x 4 + 0.3 x 6 + 0.5 x 5 = 5.1; Candidate 3 = 0.2 x 3 + 0.3 x 2 + 0.5 x 8 = 5.2; Candidate 4 = 0.2 x 9 + 0.3 x 4 + 0.5 x 3 = 4.5; Candidate 5 = 0.2 x 8 + 0.3 x 8 + 0.5 x 2 = 5.0. The top three scores are 5.2, 5.2 and 5.1, so the cutoff score for selecting exactly three candidates is 5.1.

200 km/h. Let the original speed be v km/h. In the first case, the train covers 200 km at v, 400 km at v/2 after the accident, and the final 200 km at v/8 after the defect: time = 200/v + 400/(v/2) + 200/(v/8) = 2600/v. In the second case, it covers 200 km at v, 400 km at v/4 after the defect, and the final 200 km at v/8 after the accident: time = 3400/v. The difference is 800/v hours, given as 4 hours. Hence v = 800/4 = 200 km/h.

20. The area is 3 x 100 and tile lengths along the 100 side can be 3, 7 or 11. To maximise number of tiles, use as many length-3 tiles as possible: 31 tiles of length 3 plus one length-7 tile gives 32 tiles. To minimise number, use longer tiles: 4 tiles of length 11 and 8 tiles of length 7 gives 12 tiles. Difference x-y = 32-12 = 20.

In each pair of letters, the first letter advances by one and the second letter retreats by one: ZE becomes AD (Z wraps to A, E to D) and RO becomes SN (R to S, O to N). Applying the same to STOP: ST becomes TS and OP becomes PO. Hence STOP is encoded as TSPO.

Let A = 10a + b and B = 10b + a with a > b. Then A - B = 9(a - b). For this to be a multiple of 27, a - b must be a multiple of 3, i.e., 3, 6 or 9. Since B must be a two-digit number, b cannot be 0. Counting two-digit A with a > b: difference 3 gives (a, b) = (4,1) to (9,6) - 6 values; difference 6 gives (7,1), (8,2), (9,3) - 3 values; difference 9 would require b = 0 and is not allowed. Hence 6 + 3 = 9 such numbers are possible.

1:3. Alloy P has 20% copper and alloy Q has 60% copper. Required alloy has 50% copper. By alligation, P:Q = (60-50):(50-20)=10:30=1:3.

The passage says the convergence model divides economic eras, not nations, into three phases, so statement 1 is incorrect. The closing of the labour-productivity gap with the technology leader is at the heart of convergence theory, so statement 2 is correct. The breakaway, catch-up and fine-tuning pattern means technology leaders have arrived earlier at different economic eras, so statement 3 is correct. The figure 4012200 years is not the time period mentioned in the passage, so statement 4 is incorrect. Hence statements 2 and 3 are correct.

In the breakaway phase, the per capita income of followers is unchanged - not merely slow - so conclusion 1 is incorrect. In the catch-up phase, followers adopt the new technology and close the gap; leaders do not stagnate, so conclusion 2 is incorrect. In the fine-tuning phase, participants extract remaining benefits from an exhausted technology with similar per capita incomes, so conclusion 3 is correct. The passage lists scavenging (3 MYA), hunting, farming (12 thousand years ago) and industrial technology (about 200 years ago), so industrial technology followed scavenging, which preceded farming, and conclusion 4 is correct. Hence conclusions 3 and 4 are correct.

The passage says Grant watched the men's singles quarterfinals, not the semifinal, so statement 1 is incorrect. The 'unexpected ace' was celebrated ("praise be to Grant"), not viewed with opprobrium, so statement 2 is incorrect. The passage says Grant chose the body's quiet wisdom over society's relentless performance metrics, so statement 3 is correct. Hence only statement 3 is correct.

The passage calls Grant's nap "quietly radical" - an elegant case for surrender, so statement 1 is correct. It distinguishes surrendering to limits from laziness, so statement 2 is correct. It calls Grant "a leading man, not of action, but of rest", so statement 3 is correct. Hence all three statements are correct.

7. For divisibility by 11, the difference between sums of alternate digits must be a multiple of 11. In 4x5y790, alternate sums are 4+5+7+0 = 16 and x+y+9. Thus 16-(x+y+9)=7-(x+y) must be 0, 11 or -11. Valid digit solutions give x+y ≡ 7 mod 11. Therefore the remainder when x+y is divided by 11 is 7.

HCF x LCM = the product of the two numbers. If this product is the cube of one of them, say a^3, then the other number is a^2. So the numbers are a and a^2. Their difference a^2 - a = a(a - 1) is the product of two consecutive integers, hence even (I correct), and one number is a perfect square (II correct). Hence both statements are correct.

23:24. Let female workers in both companies be 10k and 5m respectively. Since female workers are equal, choose common female count 10 units for A and 10 units for B. Then A has males 13 and females 10, total 23. B's ratio 7:5 means females 10 correspond to males 14, total 24. Ratio of total workers A:B = 23:24.

47. The sequence starts 4, 7. Each later term is the sum of the two immediately preceding terms: third = 11, fourth = 18, fifth = 29, sixth = 47.

6 km, North. Start facing east. Move 4 km east, turn right and move 3 km south, turn left and move 2 km east, then turn left and move 3 km north. Net displacement is 6 km east and 0 km north-south, so the distance is 6 km and the final facing direction is north.

The passage says the grazing land was reserved for grazing by Rai Sahib and was being used by village cattle, so statement 1 is not correct. Rai Sahib was explicitly NOT like the selfish Zamindars, so statement 2 is not correct. Farmers had offered a good price, so it can be ascertained that he could have made money, making statement 3 not correct. The passage says he held that a landlord who did not look after his tenants was less than human, so statement 4 is correct. Hence statements 1, 2 and 3 are not correct.

The passage describes Rai Sahib specifically, not all landlords, so conclusion 1 is not valid. It describes cattle grazing on the common ground but does not establish that the ground is intended for the cattle of that village, so conclusion 2 is not valid. Rai Sahib "held to the old values" and cared for his tenants, supporting conclusion 3. The passage says the rest of the day would be scorching hot ("loo" wind), so conclusion 4 is not valid. Hence only conclusion 3 is valid.

The passage says previous waves of customer-service technology stoked concerns of job losses that failed to materialise, and that AI could free human agents for more creative and rewarding tasks. Hence conclusions 1 and 2 follow; conclusion 3 (inevitable stress) is not supported.

The passage suggests that if AI takes over routine customer service, humans could be redirected to more creative, rewarding tasks. Hence the value of human intervention in the workplace affected by AI might be enhanced through redirection towards more fulfilling tasks.

Weigh three cubes against three. If they balance, the cube left out is the light one. If not, take the lighter group of three: weigh one cube against another; if they balance, the third cube is light, otherwise the lighter of the two is identified. Hence 2 attempts suffice, and with 7 cubes fewer attempts cannot guarantee certainty.

6. X's time = 1/5 + 2/10 + 3/4 hours = 0.2 + 0.2 + 0.75 = 1.15 hours. Y's time = 1/4 + 2/10 + 3/5 hours = 0.25 + 0.2 + 0.6 = 1.05 hours. Difference = 0.10 hour = 6 minutes.

19. In two-digit positive integers from 10 to 99, digit 5 appears 10 times in the tens place: 50 to 59. It appears 9 times in the units place: 15,25,35,45,55,65,75,85,95. The number 55 contributes two appearances and is already counted once in each place, as it should be. Total appearances = 10 + 9 = 19.

Since every red is blue, every blue is green and every green is yellow, by transitivity every blue is yellow (P), every red is green (Q) and every red is yellow (R). Hence P, Q and R are all correct.

The passage says thermal power plants would "clean up or shut down", not summarily shut down, so statement 1 is incorrect. It says CNG supply had to be assured when diesel buses were stopped, so statement 2 is correct. Coal was banned by the Delhi Government (later extended to the NCR), not by the Supreme Court, so statement 3 is incorrect. Hence only statement 2 is correct.

The opening line says the key source of the battle for clean skies is the fuel we burn, so inference 1 is correct. The passage does not say bans work where the will is strong and people are convinced, so inference 2 is not correct. The Supreme Court agreed that fiscal measures were needed to keep clean fuel cheaper, so inference 3 is correct. Hence inferences 1 and 3 are correct.

3. Three-digit powers of 2 are 128, 256 and 512. The next one, 1024, is four-digit, and the previous one, 64, is two-digit. Hence there are 3 such numbers.

S is the mother of T, so S is female and R, the spouse of S, is male. Since Q and R are the children of P and P has a son and a daughter, R is the son and Q is the daughter. Hence Q is the sister of R, i.e., the sister of the husband of S.

Take the salary as 100: saving = 10, expenditure = 90. New salary = 112, new expenditure = 90 x 1.1 = 99, so new saving = 112 - 99 = 13. The saving rises from 10 to 13, a 30% increase.

8. The units digit of 6^129 is always 6. Powers of 7 have units cycle 7,9,3,1; since 307 leaves remainder 3 when divided by 4, 7^307 has units digit 3. Product units digit = 6 x 3 = 18, so the units digit is 8.

4:58:42 PM. Let the distance of the explosion spot from the camp be d metres. The second drone starts at 5:02 PM and its picture is received at 5:05 PM, so it takes 3 minutes = 180 seconds to reach the spot. At 30 m/s, d = 30 x 180 = 5400 m. Sound from the explosion takes 5400/300 = 18 seconds to reach the camp. The first drone then takes 180 seconds to reach the spot. Therefore the first picture is received 18+180 = 198 seconds after the explosion. Since that receipt time is 5:02 PM, the explosion occurred at 5:02:00 - 3:18 = 4:58:42 PM.

The JJ Act, 2015 defines a heinous offence as one with a minimum punishment of seven years, so an offence with no minimum prescribed punishment cannot be considered heinous (statement 1 correct). The 2021 amendment categorises an offence with no minimum sentence but a maximum sentence of seven years or more as a serious offence (statement 2 correct). Hence both statements are correct.

The passage says activists hold that the new "serious offence" category does not merit the transfer of a case to the adult criminal justice system, so conclusion 1 is not valid. The JJ Act, 2021 does categorise an offence as serious based on the maximum sentence (seven years or more) when there is no minimum sentence, so conclusion 2 is valid. Hence only conclusion 2 is valid.

81. Let c be the number of correct answers and w be the number of wrong answers. From the first marking scheme, 5c - 2w = 69. From the second, 4c - w = 84. Doubling the second equation gives 8c - 2w = 168. Subtracting the first equation gives 3c = 99, so c = 33. Then 4 x 33 - w = 84, so w = 48. Total questions attempted = 33 + 48 = 81.

Let the time for X to run 7 rounds be one unit, so X's speed is 7 and Y's is 5 rounds per unit. In the first unit, X gains 2 rounds on Y and overtakes him twice (at relative distances of 1 and 2 rounds). After Y reverses and doubles his speed, the relative speed becomes 7 + 10 = 17 rounds per unit over the remaining 2 units, i.e., 34 relative rounds, with meetings at relative distances 1/17 to 34/17 rounds. The 34th meeting coincides exactly with the final stop, so 33 meetings occur before stopping. Total meetings = 2 + 33 = 35.

If X is correct, Y cannot be incorrect: were Y incorrect, W would be correct, which would make X incorrect - a contradiction. So I is correct. If Z is correct, then X is correct (contrapositive of the first condition), and then Y is correct by I - so it IS necessary that Y is correct, making II incorrect. Hence only I is correct.

Car X carries A with two others. B cannot be in X (A and B do not travel by the same car) and B does not travel by Y, so B is in Z; C is also in Z, so only C travels with B (III). E and G travel together, and the remaining persons D and F fill car X with A, so no one travels alone (I). D and F share car X with A, so it is not true that only D travels with F (II). Hence I and III are correct.

Both I and II. Treat matching/adjacency as positional alignment when the rods are kept side by side. The conditions can be satisfied with Q3 in the middle: for example, arrange the first rod positions as P2, P3, P1, P4, P5 and the second rod positions as Q1, Q5, Q3, Q2, Q4. Here Q3 is in the middle and the stated alignments hold. The conditions can also be satisfied with P2 at an extreme: for example, arrange the first rod as P1, P3, P4, P5, P2 and the second rod as Q3, Q5, Q1, Q4, Q2. Hence both statements are possible.

With P = 4 and R = 1, the remaining ranks are 2, 3 and 5 for Q, S and T. If S = 3, then Q cannot be 5, so Q = 2 and T = 5 - not 2. Hence I is not correct. If Q = 3, then S cannot be 2, so S = 5 and T = 2 - not 5. Hence II is not correct. Therefore neither statement is correct.

1 and 3 only are not correct. Statement 1 is not supported because the passage says current penetration is limited to tech-savvy farmers; it does not say they will drive future AgTech companies. Statement 2 is correct because the passage says AI-based advisory services through government programmes would be helpful. Statement 3 is not correct because the passage says AI tools have started offering precision farming in the US and Europe, not that they have replaced traditional agricultural practices. Statement 4 is correct because the passage refers to integrating large datasets to provide real-time agronomic insights.

2 and 3 only. Statement 1 is not a valid assumption because the passage says AI tools have started offering precision farming at scale in the US and Europe, but it does not assume that Western agricultural productivity has marched ahead because of economies of scale. Statement 2 is valid because the passage recommends Indian-language AI tools and affordable solutions for smallholder farmers, assuming that such access can help AI-based solutions reach more small farmers. Statement 3 is also valid because the passage states that India is starting to deploy AI for weather forecasting, pest detection and crop-yield optimisation, though penetration is limited.

1/5. Since y is directly proportional to x^2, y=kx^2. Given x=5 and y=50, k=2. Thus y=2x^2. If y=98, then x^2=49 and x=7. Also x is inversely proportional to z, so xz is constant. Given x=5 and z=7/25, constant = 5 x 7/25 = 7/5. Therefore z = (7/5)/7 = 1/5.

To maximise x*y + z with distinct values from {2, 3, 4, 5}, use x*y = 4 x 5 = 20 and z = 3, giving M = 23. To minimise, use x*y = 2 x 3 = 6 and z = 4, giving N = 10. Hence M - N = 13.

The passage says HMPV has been studied for decades while SARS-CoV-2 was novel, so inference 1 follows. The two viruses belong to very different families with different characteristics, so inference 2 follows. Their long-term impacts are dissimilar, so inference 3 follows. Seasonality is strong in HMPV unlike SARS-CoV-2, so inference 4 is wrong. Hence inferences 1, 2 and 3.

The passage is an informational comparison of the two viruses; it does not aim to evolve methodologies for objective analysis, nor does it discuss remedies of HMPV in conjunction with SARS-CoV-2. Even the comparison of similarities and differences is incidental rather than the writer's stated intent. Hence none of the options reflects the intent of the writer.

The passage says the two viruses belong to very different families, so conclusion 1 (calling them similar viruses) is not valid. It says the viruses have fundamentally different characteristics, epidemiology and impacts, so they should not be dealt with in a similar manner, and conclusion 2 is valid. Hence only conclusion 2 is valid.

3. The first correction changes a score from 0 to 100, so the total increases by 100. Since the class average increases by 4, number of students = 100/4 = 25. The second correction changes 81 to 56, reducing total by 25; this reduces the average by 25/25 = 1. Thus final corrected average y is 3 more than x, so y - x = 3.

16. After three cuts, the pieces are identical and remain unmoved. Typical identical partitions of a cube after three cuts can be slabs, a grid of six pieces, or eight smaller cuboidal pieces. A fourth plane cut can increase the number of pieces only by the number of existing pieces it actually intersects. It cannot intersect and split all eight cuboidal pieces produced by three mutually perpendicular cuts, so 16 pieces is not possible. Values such as 5, 8 and 12 can be obtained from suitable slab/grid arrangements. Therefore 16 is the value that is not possible.

8. Let the number of 1 kg weights be between 8 and 11. Count non-negative solutions of a + 2b + 5c + 10d = 20 with a in {8,9,10,11}. The valid combinations are 8 in total: (8,1,0,1), (8,1,2,0), (8,6,0,0), (9,3,1,0), (10,0,0,1), (10,0,2,0), (10,5,0,0), and (11,2,1,0).

5. To get 298 litres with 100, 25, 6 and 1 litre measures, use three 100-litre measurements to get 300 litres, then remove two 1-litre measurements. Total measurements = 3 + 2 = 5. Four measurements cannot make 298 from the available capacities.

Consider f(t) = t + 1/t. It decreases on (0, 1) and increases on (1, 2), so with 1/3 < x < y < 2, statement I is not always true - for example, x = 1/2, y = 3/2 gives 2.5 > 2.17. Statement II: sqrt(1 + y^2)/y = sqrt(1 + 1/y^2), and since y > x > 0, 1/y^2 < 1/x^2, so the left side is always smaller. Hence only statement II is correct.

13.85. When the person moves in the same direction, relative speed = 100-4 = 96 km/hr = 26.67 m/s. Since crossing takes 15 s, train length = 26.67 x 15 = 400 m. In the opposite direction, relative speed = 100+4 = 104 km/hr = 28.89 m/s. Time = 400/28.89 = 13.85 s approximately.

111. This is a trailing-zero/factorisation question. 75^25 = (3 x 5^2)^25 gives 5^50, 25^32 gives 5^64, and 32^75 = (2^5)^75 gives 2^375. Total powers are 2^375 and 5^(50+64) = 5^114. Thus the product contains 10^114. Since the expression already separates a factor 1000 = 10^3, the remaining 10^m has m = 114 - 3 = 111.

The given a's and b's (positions 2, 4, 6, 9, 11, 13, 14 and 15) match the pattern abbab repeated three times: abbab-abbab-abbab. Filling the blank positions accordingly, the 7th and 8th positions from the left are both b. Hence the answer is bb.

The passage shows that using the same deflator for inputs and outputs makes a manufacturer look more productive when input prices fall, so assumption 1 is valid. Double deflation means deflating input and output prices separately, not deflating both against a single input price, so assumption 2 is not valid. Hence only assumption 1 is valid.

The passage says real value added - not "real value" - should not change when output prices and quantities are static; the statement's use of "output cost" also conflates cost with price, so statement 1 is not correct. Double deflation is used by most countries, not universally - India uses the same deflator for inputs and outputs - so statement 2 is not correct. Hence neither statement is correct.

4. Each blue rectangle is 2.5 ft x 8 ft on a 6 ft x 10 ft tabletop and must share exactly two sides with two edges of the tabletop. Therefore each rectangle must occupy a corner, with its 8 ft side along a 10 ft edge and its 2.5 ft side along a 6 ft edge. There are four corner placements. Choosing two corner placements gives six pairs, but the two pairs lying on the same long edge overlap because 8+8 exceeds 10. The remaining four pairs are possible. Hence 4 different patterns are possible.

Arrange the eight persons clockwise as P, Q, R, S, T, U, V, W. Then Q has P and R adjacent, S has T and R adjacent, V has U and W adjacent, and S is opposite W. Going clockwise from P, one meets R (third position) before T (fifth position). From Q, going clockwise, one crosses R, S, T, U and V - five persons - before meeting W.

9. Let forward jumps be f and backward jumps be b. Then f+b=31 and net movement is 5f-2b=15. Substitute b=31-f: 5f - 2(31-f)=15, so 7f=77 and f=11. Then b=20. Difference = 20-11 = 9.

The passage says parents report that their children are inactive and rarely leave the house, not that parents do not encourage sport, so statement 1 is incorrect. It says sport teaches time management, fitness, teamwork and coordination, so statement 2 is correct. It says sport as a way of life builds a healthier, happier society, so statement 3 is correct. Hence statements 2 and 3 are correct.

The passage says sport is not just about medals or quota jobs but a way of life, so conclusions 1 and 3 are valid. Sport gets inactive, house-bound young people engaged, mitigating seclusion, so conclusion 2 is valid. It teaches exactly the skills (time management, teamwork) needed in the corporate sector, so conclusion 4 is not valid. Hence conclusions 1, 2 and 3 are valid.

14. X, Y and Z land on marks separated by 4, 6 and 5 respectively. A mark common to all three must differ from 199 by a multiple of lcm(4,6,5)=60. Between 195 and 1000, the common marks are 199, 259, 319, ..., 979. This is 14 marks.

Treat QU as one block since Q must always be followed by U. We then arrange the four items QU, E, U, E, where the two Es are identical: 4!/2! = 12. Hence 12 words can be formed.

From A + C = 6 and A + B + C + D = 17, we get B + D = 11. Using P = R and Q = S, the equations become P + 2Q + D = 15 and 2P + Q + B = 17. Adding them, 3(P + Q) + (B + D) = 32, so 3(P + Q) = 21 and P + Q = 7, which is more than A + C = 6. The other comparisons are not forced. Hence option B is correct.

The passage mentions paintings from Bengal (Pattachitra) and Tamil Nadu (Thanjavur); Assam's bamboo crafts and Madhya Pradesh's tribal sculpture are not paintings. So paintings from four States were not mentioned, making option A the statement that is not correct. Stepping through the portal was like entering another world (B), the Angans were living galleries (C), and the zones were inspired by iconic temples (D) - all correct.

The passage states that Kalagram unfolded as a mosaic of India's diverse regions, each represented by seven Sanskriti Angans (statement 1 valid), and that these thematic zones were inspired by iconic temples such as the Dakshineshwar Kali Temple and the Brahma Mandir and served as treasure troves of regional artistry (statement 2 valid). The passage credits 230 master artisans with using age-old traditional techniques, not "contemporary craftsmanship" (statement 3 invalid), and it does not claim that art forms from all regions of India were showcased (statement 4 invalid). Hence 1 and 2 only.

Statement I: xy^2 = 116 = 2^2 x 29. Since y > 1, the only factorisation with a perfect square is y = 2 and x = 29, so x is prime - Statement I alone is sufficient. Statement II: xz = 261 = 3^2 x 29 allows x = 3, 9, 29 or 87, so it is not sufficient. Hence the question can be answered using one of the statements alone, but not the other.

Statement I: if the sum of any two numbers of T is even, then T is either all even or all odd - it does not tell us whether every number is from Y. Statement II: (p - 1)q is even for all p, q in T only when T is uniform in parity (all even or all odd), again without distinguishing X from Y. Even together, both possibilities remain. Hence the question cannot be answered even using any of the statements.

Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone. The question asks whether x is even. Statement I alone only indicates an even parity condition involving x and y, but does not fix x's parity in every case. Statement II alone also allows more than one parity possibility. Together, the two parity conditions force x to be even, so both statements together are sufficient but neither alone is sufficient.

Statement I: x^2 < y < 1. For x = 0.5, y = 0.3 satisfies it with y < x, while x = 0.2, y = 0.3 satisfies it with y > x - so it does not determine the bigger number. Statement II: y < sqrt(x) < 1 similarly allows both orderings. Together, x^2 < y < sqrt(x) < 1 is satisfied by both (x = 0.5, y = 0.3) and (x = 0.2, y = 0.3). Hence the question cannot be answered even using any of the statements.

The sums of three distinct coins that do NOT include a 5 coin are 13, 23, 31 and 32, whose digit sums are 4, 5, 4 and 5 - none greater than 5. Every sum whose digit sum exceeds 5 includes a coin of denomination 5, so Statement II alone answers the question. Statement I alone does not: 32 = 2 + 10 + 20 is not prime and contains no 5. Hence one of the statements alone (Statement II) answers the question.