Question 1

In the following syllogism problem about electric goods, treat the statements as true and decide which of the given conclusions logically follow. Statements: All radios are electric goods. All table lamps are electric goods. Conclusions: I. Some rad…

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Introduction / Context: This syllogism question deals with two types of items, radios and table lamps, and a larger category, electric goods. You are told that both radios and table lamps belong to the class of electric goods, and you must determine whether any overlap between radios and table lamps is forced by the statements. This tests whether you understand that sharing a common superset does not automatically imply an overlap between the subsets. Given Data / Assumptions: Statement 1: All radios are electric goods. Statement 2: All table lamps are electric goods. Conclusion I: Some radios are table lamps. Conclusion II: Some table lamps are radios. All items are being discussed only with respect to the category of electric goods. Concept / Approach: In set theory terms, the statements say Radios ⊂ ElectricGoods and TableLamps ⊂ ElectricGoods. This means both sets are contained within the larger electric goods set. However, the premises do not mention any direct relationship between radios and table lamps. We must decide whether this limited information forces an intersection between the two subsets. If we can draw a valid Venn diagram in which radios and table lamps do not overlap, then neither conclusion is logically necessary. Step-by-Step Solution: Step 1: Translate the statements into set language: All radios are electric goods: Radios is a subset of ElectricGoods. All table lamps are electric goods: TableLamps is a subset of ElectricGoods. Step 2: Observe that the statements do not say that all electric goods are radios or table lamps. There is no statement such as “all electric goods are radios” or “all electric goods are table lamps.” Step 3: Consider whether some radios must also be table lamps. You can easily imagine a store where radios and table lamps are entirely different products, both belonging to the broader category of electric goods. It is entirely possible that no individual item is simultaneously a radio and a table lamp. Step 4: Construct a Venn diagram. Draw a large circle for ElectricGoods. Inside it, draw two smaller separate circles, one for Radios and one for TableLamps, ensuring they do not overlap. This diagram satisfies both statements, because every radio is inside ElectricGoods and every table lamp is inside ElectricGoods. Step 5: In this diagram, Radios ∩ TableLamps is empty, meaning there is no object that is both a radio and a table lamp. This directly contradicts both Conclusion I and Conclusion II, which require at least some overlap between the two sets. Verification / Alternative check: The example above demonstrates that the given premises allow a situation with no overlap between radios and table lamps. For a conclusion like “some radios are table lamps” or “some table lamps are radios” to follow logically, it must hold in every model that satisfies the premises. Because we have found a valid model with zero overlap, these conclusions are not forced by the data, and hence neither conclusion follows. Why Other Options Are Wrong: Option a: “Only conclusion I follows” is incorrect because we have shown that there can be no overlap at all between the sets. Option b: “Only conclusion II follows” is also incorrect for the same reason; Conclusion II is just a symmetric restatement of I. Option d: “Both conclusion I and conclusion II follow” is clearly wrong; both depend on the existence of at least some overlapping items. Option e: “Either conclusion I or conclusion II follows” is incorrect because neither is logically necessary. Common Pitfalls: Many learners mistakenly think that if two sets are both subsets of a third, they must overlap. This is not true in general. For example, dogs and cats are both subsets of animals, but an individual animal is usually not both a dog and a cat. Always check whether the premises explicitly assert an overlap; if they do not, and you can draw a disjoint diagram, then overlap based conclusions do not follow. Final Answer: Neither conclusion I nor conclusion II follows.

Question 2

Read the following statements and treat them as true, then decide which of the given conclusions logically follow. Statements: All swans are parrots. No parrot is a cuckoo. Conclusions: I. No swan is a cuckoo. II. All parrots are swans.

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Introduction / Context: This is a standard two statement syllogism question involving three sets: swans, parrots, and cuckoos. You must determine which of the conclusions follow logically from the given statements. The statements define a subset relationship and a complete exclusion relationship. The challenge is to correctly propagate the exclusion from parrots to swans and to see whether there is any forced equivalence between parrots and swans. Given Data / Assumptions: Statement 1: All swans are parrots. Statement 2: No parrot is a cuckoo. Conclusion I: No swan is a cuckoo. Conclusion II: All parrots are swans. All statements are to be read as logically precise descriptions of the sets. Concept / Approach: When we say “All swans are parrots,” we are stating that the set of swans is contained inside the set of parrots. When we say “No parrot is a cuckoo,” we say that the sets parrots and cuckoos are disjoint. To determine whether a conclusion follows, we must ask whether it holds in all possible Venn diagrams that satisfy the premises. For Conclusion I, we test whether the disjointness between parrots and cuckoos transfers to swans. For Conclusion II, we test whether the subset relationship can be reversed. Step-by-Step Solution: Step 1: Represent the first statement: All swans are parrots. So the Swans set lies completely within the Parrots set. Step 2: Represent the second statement: No parrot is a cuckoo. This means the Parrots set and the Cuckoos set do not overlap at all. Step 3: Consider Conclusion I: “No swan is a cuckoo.” Since every swan is a parrot (from Statement 1), and no parrot is a cuckoo (from Statement 2), it is impossible for any swan to be a cuckoo. If a swan were a cuckoo, it would also be a parrot, contradicting Statement 2. Therefore, Swans ∩ Cuckoos is empty. Conclusion I is logically valid. Step 4: Now consider Conclusion II: “All parrots are swans.” This asserts that the Parrots set is contained inside the Swans set. However, the first statement only tells us that Swans is a subset of Parrots, not that they are equal. There might be parrots that are not swans. Step 5: Construct a counterexample for Conclusion II. Imagine a world with two birds: one swan and one non swan parrot. Let the swan be a parrot as well, satisfying “All swans are parrots.” Let the non swan parrot simply be a parrot that is not a swan. Also, let there be no cuckoos at all, so “No parrot is a cuckoo” is trivially true. In this arrangement, not all parrots are swans, because one parrot is not a swan. Thus Conclusion II does not necessarily follow. Verification / Alternative check: Using a Venn diagram, draw a large circle for Parrots. Inside it, draw a smaller circle for Swans. For Cuckoos, draw a separate circle that does not intersect Parrots at all. It is visually clear that no swan (inside Parrots) can intersect with Cuckoos, confirming Conclusion I. It is equally clear that Parrots is larger than Swans and need not be entirely inside Swans. There can be parrots outside the Swans circle, which invalidates Conclusion II as a universal claim. Why Other Options Are Wrong: Option a: “Only conclusion II follows” is incorrect because Conclusion II does not follow, while Conclusion I does. Option c: “Either conclusion I or conclusion II follows” is wrong because there is no either or situation; Conclusion I always follows, while Conclusion II does not. Option d: “Both conclusion I and conclusion II follow” is false because Conclusion II does not logically follow. Option e: “Neither conclusion I nor conclusion II follows” is incorrect since Conclusion I is clearly implied by the premises. Common Pitfalls: Learners sometimes misread “All swans are parrots” as “All parrots are swans,” which reverses the subset relationship. They also may think that if a set is a subset of another, they are automatically equal, which is not generally true. Always pay close attention to the direction of “all A are B” statements and remember that they do not guarantee the reverse unless explicitly stated. Final Answer: Only conclusion I follows.

Question 3

Read the following statement about written examinations in primary schools and the two arguments. Decide which of the given arguments are strong. Statement: Should written examinations in primary schools be abolished? Arguments: I. Yes, young studen…

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Introduction / Context: This question is from the topic of argument evaluation. You are asked whether written examinations in primary schools should be abolished, and two arguments are provided, one in favour and one against. Your task is to decide which arguments are strong, meaning they are relevant to the question and provide a reasonable, significant consideration for decision making. The focus is not on personal opinion but on the logical strength and relevance of each argument. Given Data / Assumptions: Issue: Should written examinations in primary schools be abolished? Argument I (for abolition): Young students feel burdened and such examinations do not help in the natural thinking process. Argument II (against abolition): Without examinations students will not be serious about education and learning. We assume that any argument directly addressing educational outcomes, child development, or seriousness of learning can be considered for strength. Concept / Approach: A strong argument is directly related to the question, has a reasonable basis, and addresses an important consequence or principle. A weak argument is either irrelevant, based on trivial considerations, or clearly illogical. In questions of education policy, factors like mental burden on children, natural cognitive development, and seriousness towards learning are all significant and relevant. We must evaluate each argument on these criteria. Step-by-Step Solution: Step 1: Evaluate Argument I. It states that young students feel burdened by written examinations and that such exams do not help their natural thinking process. This argument is directly related to the well being and cognitive development of primary school children, which is central to the question. Step 2: The claim that examinations create a burden suggests potential stress, anxiety, or loss of interest at an early age. The additional point that they may not help the natural thinking process questions the educational value of written tests at this level. Together, these points form a serious concern about the negative impact on young learners. Therefore, Argument I is relevant and significant and can be considered strong. Step 3: Now evaluate Argument II. It states that without examinations, students will not be serious about education and learning. This addresses the role of exams as a motivational tool and as a mechanism for accountability. It claims that assessments ensure that students study and take their education seriously. Step 4: This reasoning is also directly connected to the central issue. If examinations are removed, there is a legitimate concern that some students may lose discipline or focus if there is no structured evaluation system. Even if one disagrees with the degree of this effect, the argument is neither trivial nor irrelevant; it highlights a real potential consequence in the teaching learning process. Step 5: Because Argument II addresses seriousness towards learning, which is a core objective of schooling, and it connects that seriousness with the presence of exam based evaluation, it is also a strong argument. It presents a reasonable counterpoint to the first argument. Verification / Alternative check: To verify, imagine a policy discussion in an education board meeting. One expert might argue that early written exams overload young minds and hinder creative thinking, aligning with Argument I. Another expert might caution that without some kind of written evaluation, students and parents may not prioritize regular study, aligning with Argument II. Both positions would be taken seriously in such a debate, showing that both arguments are substantial and relevant. Because strong arguments can exist on both sides of an issue, it is entirely possible and often correct to mark both as strong. Why Other Options Are Wrong: Option b: “Only argument I is strong” undervalues the concern about student seriousness and accountability raised by Argument II. Option c: “Neither argument I nor argument II is strong” is incorrect because both arguments are directly tied to important educational outcomes. Option d: “Only argument II is strong” ignores the significant child development concerns highlighted in Argument I. Option e: Suggesting ambiguity about which argument is weak is unnecessary; both are clearly relevant and weighty. Common Pitfalls: One common error is to think that in such questions only one side can be strong, which is not true. Strong arguments can exist both for and against the same proposal. Another mistake is to judge arguments purely on personal preference rather than on their logical connection to the issue. Always ask: Does this argument directly address the central question, and does it highlight a serious, reasonable consequence or principle? Final Answer: Both arguments I and II are strong.

Question 4

You are given a question and two coding style statements about colour names. Using the information, decide which statement(s) are sufficient to answer the question. Question: What is the colour of jasmine? Statements: I. Blue is called white, white …

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Introduction / Context: This is a data sufficiency style question involving coded colour names. The common background knowledge is that jasmine flowers are naturally white in colour. The two statements describe how actual colours are renamed in a code language. Your task is to judge which statements provide enough information to determine the coded colour word for jasmine. Remember that in data sufficiency, you are not asked to solve in real life, but to determine whether the given data are enough to get a unique answer. Given Data / Assumptions: General knowledge: Jasmine flowers are naturally white. Statement I: Blue is called white, white is called red, and red is called yellow. Statement II: Yellow is called purple, purple is called black, white is called brown, and brown is called orange. Question: What is the colour of jasmine (in the coded naming system)? Concept / Approach: We treat the statements as a renaming scheme or code language. Each statement tells us which word is used to refer to a particular real colour. Because jasmine is known to be white in real life, we only need to know how the real colour white is renamed. If a statement clearly tells us the coded name for white, then that statement alone is sufficient. We check each statement separately for this information and then decide about sufficiency. Step-by-Step Solution: Step 1: Start with the real world fact that jasmine is white. This fact is assumed to be known for the purpose of the question. Step 2: Use Statement I alone. It says: Blue is called white. White is called red. Red is called yellow. Here, the second part directly tells us that the actual colour white will be referred to as “red” in this code language. Step 3: Since jasmine is naturally white, under Statement I's naming scheme, jasmine will be called red. This uniquely answers the question using only Statement I. Hence, Statement I alone is sufficient. Step 4: Now use Statement II alone. It says: Yellow is called purple. Purple is called black. White is called brown. Brown is called orange. Again, we see a direct mapping: the actual colour white is called “brown”. Step 5: Because jasmine is white in real life, under Statement II's naming scheme, jasmine will be called brown. This is again a unique answer using only Statement II. Therefore, Statement II alone is also sufficient to answer the question. Verification / Alternative check: The crucial observation is that each statement independently gives us the code name for the real colour white. For data sufficiency, we do not need both statements together if one alone already tells us exactly how to name jasmine. While the two statements lead to different coded names for jasmine (red under Statement I, brown under Statement II), each is a complete and consistent scheme by itself. The question is framed so that you are evaluating sufficiency under each independent coding system, not combining them into a single system. Why Other Options Are Wrong: Option a: “The data in statement I alone are sufficient” is partially correct but incomplete, because Statement II alone is also sufficient. Option c: “Data in both statements I and II together are not sufficient” misunderstands the goal; each statement individually provides a clear mapping, so together they are certainly not insufficient. Option d: “The data in statement II alone are sufficient” has the same problem as option a; it ignores the sufficiency of Statement I. Option e: “The data in both statements I and II together are required” is incorrect because we do not need both; each alone is enough. Common Pitfalls: A common misunderstanding is to think that the coded names across the two statements must agree, and therefore both are needed. In typical data sufficiency questions of this style, each statement is considered as defining its own mapping. You are checking whether either statement, viewed independently, gives you enough information. Another pitfall is to miss the general knowledge that jasmine is white, which is essential to linking the coding rules back to the flower in question. Final Answer: The data in either statement I alone or statement II alone are sufficient.

Question 5

Which of the following statements about the speed of sound in different media at around 25 degree Celsius is true? A. In oxygen gas, the speed of sound is 316 km/s. B. In distilled water, the speed of sound is 1498 km/s.

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Introduction / Context: This question checks your conceptual understanding of the speed of sound in different media and the approximate numerical values involved. You are given two statements about the speed of sound in oxygen gas and distilled water at around 25 degree Celsius, but the units are shown in kilometres per second. Your task is to identify which of the statements are true. The key is to recall or estimate the realistic order of magnitude of the speed of sound in gases and liquids. Given Data / Assumptions: Statement A: In oxygen gas, the speed of sound is 316 km/s. Statement B: In distilled water, the speed of sound is 1498 km/s. Temperature is approximately 25 degree Celsius. You may use general physics knowledge about typical speeds of sound in air and water. Concept / Approach: The speed of sound in a medium depends on both its elastic properties and its density. In everyday physics, speeds of sound are usually expressed in metres per second, not kilometres per second. The typical speed of sound in air at room temperature is a bit above 300 metres per second, and in water it is about 1500 metres per second. Converting between metres per second and kilometres per second helps quickly show whether the given numerical values are realistic or off by a large factor. Step-by-Step Solution: Step 1: Recall the approximate speed of sound in air at room temperature. It is around 340 metres per second, and in pure gases like oxygen it is of the same order, a few hundred metres per second, not hundreds of kilometres per second. Step 2: Convert the given units. Statement A says 316 km/s. Since 1 km/s is 1000 m/s, 316 km/s is 316000 m/s. This is roughly one thousand times larger than the actual typical value for the speed of sound in a gas like oxygen at room temperature. Step 3: Because the realistic value is around a few hundred metres per second, Statement A's claim of 316 kilometres per second is clearly incorrect. It appears that the intended numbers were probably in metres per second, not kilometres per second. Step 4: Now consider Statement B: 1498 km/s in distilled water. Converting units again, 1498 km/s is 1498000 m/s (one million four hundred ninety eight thousand metres per second). The actual speed of sound in water at room temperature is around 1480 to 1500 metres per second, which is three orders of magnitude smaller. Step 5: Just like Statement A, Statement B seems to have used kilometres per second instead of metres per second. The correct order of magnitude is in metres per second, not kilometres per second. Therefore, Statement B is also numerically incorrect as written. Verification / Alternative check: Ask yourself whether any everyday physical phenomenon can easily reach speeds on the order of hundreds or thousands of kilometres per second. Such speeds are closer to a significant fraction of the speed of light, which is about 300000 km/s. Sound waves in ordinary media like gases and liquids are much slower, nowhere near these extreme speeds. Because both statements give values that are too large by a factor of roughly 1000, they cannot be true in their given unit form. Why Other Options Are Wrong: Option a: “Only statement A is true” is incorrect because Statement A has an unrealistic magnitude for the speed of sound in a gas. Option b: “Both statements A and B are true” is wrong because both statements use kilometres per second where metres per second would be appropriate. Option d: “Only statement B is true” is also incorrect; Statement B suffers from the same unit error. Option e: “The truth of both statements cannot be determined” is not appropriate here, because basic physics knowledge clearly shows that the numerical values are not plausible in the stated units. Common Pitfalls: A common mistake is to overlook the units and focus only on the numbers. Since many students memorize approximate speeds like 330 and 1500 without always attaching the metres per second unit, they might incorrectly accept a value expressed in kilometres per second just because the numbers look familiar. Always pay attention to units when evaluating physical quantities; mixing metres per second with kilometres per second leads to errors of a thousand fold. Final Answer: Neither statement A nor statement B is true.

Question 6

Given the inequality chain T ≥ U ≥ M = N, which of the following relations must be definitely true according to the expression?

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Introduction / Context: This question checks the ability to read a chained inequality correctly and derive relations that must always hold. You are given a relation involving four symbols T, U, M and N and asked to identify which statement is definitely true in every valid case. Given Data / Assumptions: The expression T ≥ U ≥ M = N is given as true. Symbol ≥ means greater than or equal to. Symbol = means equality. No extra information about the actual values of T, U, M and N is provided. We must pick the relation that always holds, not just sometimes holds. Concept / Approach: A chain like T ≥ U ≥ M = N can be split into smaller pieces. From T ≥ U and U ≥ M and M = N we can substitute and combine these to get new relations. Any option that contradicts the chain or is only sometimes true cannot be chosen. Only a relation that follows in every allowed arrangement of the four symbols is definitely true. Step-by-Step Solution: Step 1: Write the chain as three parts: T ≥ U, U ≥ M and M = N. Step 2: From M = N we know M and N have exactly the same value. Step 3: From U ≥ M and M = N, replace M by N to get U ≥ N. Step 4: From T ≥ U and U ≥ N, we get T ≥ N. Thus T can never be less than N in any valid case. Step 5: Any option that claims T < U, N > U or U < M directly contradicts the chain and must be false. Verification / Alternative check: Take an example that satisfies the chain. Let N = 3, M = 3, U = 4 and T = 6. Then T ≥ U ≥ M = N becomes 6 ≥ 4 ≥ 3 = 3, which is true. Here T ≥ N becomes 6 ≥ 3, which is also true. Try another example with equal values, for instance N = M = U = T = 5. The chain still holds and T ≥ N is again true. No valid example can violate T ≥ N, which confirms that it is definitely true. Why Other Options Are Wrong: Any option that says T < U or N > U or U < M conflicts with the directions of ≥ in the original chain. The option claiming M > T also contradicts T ≥ U ≥ M. There is no way to assign values so that the chain is true and those relations are simultaneously true, so they cannot be definitely true statements. Common Pitfalls: A common mistake is to ignore the equality part M = N or to treat ≥ as if it always meant strictly greater than. Another error is to rely on a single example instead of checking what must happen in all valid examples. The safest method is to break the chain into pieces, substitute step by step and then derive the final relation. Final Answer: The relation that must always hold is T ≥ N.

Question 7

In the inequality chain L ≥ K ≥ P = N = J @ U ≤ A % R, which pair of symbols should replace (@) and (%) respectively so that L ≥ U and R > J are definitely true?

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Introduction / Context: This question is about symbol substitution in a chain of inequalities. Some comparison signs in the expression are hidden behind special symbols, and you must choose the correct pair so that two target relations, L ≥ U and R > J, will always be true whenever the whole chain holds. Given Data / Assumptions: Base expression: L ≥ K ≥ P = N = J @ U ≤ A % R. The symbols @ and % must be replaced by one of the standard inequality symbols given in the options. The final chain must make both L ≥ U and R > J definitely true. There is no extra numerical information about the letters. Concept / Approach: The method is to plug in each candidate pair of symbols into the expression and then see whether both required relations necessarily follow. If we can find even one counterexample, then that pair is rejected. The correct pair must force L to be at least as large as U and R to be strictly larger than J in every valid arrangement of the letters. Step-by-Step Solution: Step 1: Try option D with @ = = and % = <. The chain becomes L ≥ K ≥ P = N = J = U ≤ A < R. Step 2: Since J = U and L ≥ K ≥ P = N = J, we obtain L ≥ J and thus L ≥ U. So L ≥ U always holds for this choice. Step 3: From U ≤ A and A < R, and J = U, we get J ≤ A < R. Hence R is strictly greater than J in every possible case, so R > J is forced. Step 4: Check other options quickly. Choices that make J ≤ U or J > U without equality do not guarantee both L ≥ U and R > J together; counterexamples can be built. Only the pair =, < gives both required conclusions in all cases. Verification / Alternative check: Pick sample values that satisfy option D. Let J = U = N = P = 2, K = 3, L = 4, A = 3 and R = 5. Then L ≥ K ≥ P = N = J = U ≤ A < R becomes 4 ≥ 3 ≥ 2 = 2 = 2 = 2 ≤ 3 < 5, which is true. Here L ≥ U becomes 4 ≥ 2 and R > J becomes 5 > 2, both satisfied. Trying to alter any value while respecting the chain cannot break these two relations. Why Other Options Are Wrong: When @ is ≤, J ≤ U is allowed, and L can be less than U even though the chain is true, so L ≥ U is not guaranteed. When @ is >, R > J is not forced because R may only be greater than or equal to A, which can still be less than or equal to J. For these pairs, one of the required conditions fails in some valid arrangements, so they cannot be chosen. Common Pitfalls: Students often verify only one target relation and forget to check the other. Another pitfall is to look at the chain locally without writing it cleanly after substitution. Writing the full expanded version of the chain for each option makes it easier to see whether both requirements are always met. Final Answer: The correct replacement is @ = = and % = <, that is, the pair '=, <'.

Question 8

Two statements describe a new wave of Olympic doping cases; based on them, which of the given conclusions about reports and the commonness of doping logically follow?

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Introduction / Context: This is a statements and conclusions question based on Olympic doping incidents. You must treat the two given statements as absolutely true and then decide whether each conclusion necessarily follows, without using outside information or personal opinions about sports and doping. Given Data / Assumptions: Statement I: Twenty eight athletes are facing legal cases due to doping incidents. Statement II: The new wave of Olympic doping cases is based on evidences. Conclusion I: The report of doping is based on tests that happen. Conclusion II: Doping practice is very common these days among athletes. We must decide which conclusions must hold whenever both statements are true. Concept / Approach: In such questions, a conclusion follows only if it is impossible for the statements to be true and the conclusion to be false at the same time. If we can imagine even one situation where the statements hold but a conclusion fails, that conclusion does not logically follow. We therefore try to construct counterexamples rather than rely on real world impressions. Step-by-Step Solution: Step 1: The statements talk about twenty eight athletes with legal cases and say that the cases are based on evidences. They do not specify the type of evidence. Step 2: Conclusion I introduces the idea that the report is based on tests. However, evidence could also come from witnesses, documents, or digital records. Tests are not mentioned, so this added detail is not guaranteed. Step 3: Conclusion II states that doping practice is very common among athletes. Knowing that twenty eight athletes have cases does not tell us how many athletes there are in total, so we cannot judge whether the practice is common or rare. Step 4: Since both conclusions can be false in some situations where the statements remain true, neither conclusion logically follows. Verification / Alternative check: Imagine there are thousands of athletes and only twenty eight doping cases. Then doping could still be relatively uncommon, and conclusion II would be false. Also imagine that the evidence in these cases is entirely video footage and confessions, with no lab testing. In such a case conclusion I would be false. Yet both situations do not contradict the original statements, confirming that neither conclusion is forced. Why Other Options Are Wrong: Options A, B and D each claim that at least one conclusion follows. As shown, both conclusions can fail without affecting the truth of the statements, so those options overstate what can be inferred. Option E suggests that at least one of the two must follow, which is also not guaranteed. Only the option that says neither conclusion follows matches the logical analysis. Common Pitfalls: A frequent mistake is to let general knowledge about doping scandals influence the answer, for example thinking that doping is common or that tests are always used. Another error is to treat any reasonable sounding statement as a valid conclusion even if it introduces new information not present in the statements. Final Answer: The correct evaluation is that neither conclusion I nor conclusion II follows from the given statements.

Question 9

Some notes are coins and no coin is a card; on that basis, which of the given conclusions about cards and notes logically follows?

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Introduction / Context: This is a classic syllogism problem involving three sets: notes, coins and cards. The statements describe how these sets overlap or do not overlap. You must examine two proposed conclusions and decide which one logically follows if the statements are taken as perfectly true. Given Data / Assumptions: Statement I: Some notes are coins. Statement II: No coin is a card. Conclusion I: All cards can be notes. Conclusion II: Some notes are neither coins nor cards. The phrase can be in conclusion I expresses possibility, not certainty. Concept / Approach: We interpret the statements using Venn diagrams or set language. Some notes are coins means there is at least one element common to the sets notes and coins. No coin is a card means the set of coins and the set of cards do not overlap. A conclusion based on can be is valid if there exists at least one possible arrangement of the sets that satisfies the statements and also makes that conclusion true. Step-by-Step Solution: Step 1: Draw overlapping circles for notes and coins and mark an overlapping region to represent some notes are coins. Step 2: Draw the set of cards completely outside the set of coins, since no coin is a card. Step 3: Check conclusion I. It says all cards can be notes. This is possible if we place the circle for cards wholly inside the notes set but outside coins. This arrangement satisfies both statements and makes all cards notes. So conclusion I is logically possible. Step 4: Check conclusion II. It claims there are some notes that are neither coins nor cards. The statements do not force this. It is possible that the only notes that exist are exactly those that are also coins, leaving no notes in the region outside coins. Step 5: Therefore conclusion II is not guaranteed, while conclusion I is a valid possibility that follows from the statements. Verification / Alternative check: Build two different models. In model one, let the set of notes be larger than the set of coins, so there are notes outside coins. In model two, let the set of notes coincide exactly with the set of coins. Both models satisfy the statements, but in model two there are no notes outside coins or cards, so conclusion II fails. This proves that conclusion II does not necessarily follow. However, in each model it is still possible to position cards inside notes, so conclusion I remains logically possible under the statements. Why Other Options Are Wrong: Option B selects conclusion II, which is not forced. Option C claims that neither conclusion follows, but conclusion I does follow in the sense of possibility. Option D requires both conclusions to follow, which is too strong because conclusion II is not necessary. Option E suggests that at least one of them must follow without specifying which, but the question asks you to identify the exact conclusion that follows, which is only conclusion I. Common Pitfalls: One common error is to misinterpret can be as must be, or to assume that if some notes are coins then there must also be notes that are not coins. Another mistake is to forget that absence of information does not mean the opposite is true. Because the statements do not say anything about notes outside coins, you cannot force conclusion II. Final Answer: The correct choice is that only conclusion I follows from the given statements.

Question 10

A Swachh Bharat cleanliness campaign by an apartment association received very little response from residents; which of the given assumptions is or are implicit in this argument?

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Introduction / Context: This problem belongs to the topic of arguments and implicit assumptions. You are given a short argument about a Swachh Bharat cleanliness campaign that failed to get much response and then two possible assumptions. An assumption is an unstated idea that must be true for the argument to make sense. Given Data / Assumptions: Argument: The Swachh Bharat campaign started by the apartment association did not bring much response from its residents. Assumption 1: Residents do not wish to keep their apartment clean. Assumption 2: The association had failed in the campaign. The argument only reports low response from residents. Concept / Approach: To test whether an assumption is implicit, imagine that the assumption is false and check whether the given argument still remains meaningful and valid. If the argument can still stand, then that assumption is not essential and therefore not implicit. The key is to separate what the sentence actually says from what we might emotionally infer or guess. Step-by-Step Solution: Step 1: The argument simply states a fact: the campaign did not receive much response. Step 2: Consider assumption 1. Even if residents really want cleanliness but were busy, not informed, or did not like the timing of the campaign, the statement about poor response can still be true. So assumption 1 is not necessary. Step 3: Consider assumption 2. The sentence does not say that the association considers the campaign a failure. It only reports the current response level. The association could still treat it as a partial success or a beginning. Step 4: Since the argument remains meaningful even if assumption 2 is false, it is not implicit either. Verification / Alternative check: Imagine a case where residents are very serious about cleanliness but already run their own cleaning drives, so they did not join this specific campaign. Also assume the association only wanted to raise awareness and did so, even with low participation. In this situation both assumptions are false, yet the original sentence about low response is still accurate. This confirms that neither assumption is built into the argument. Why Other Options Are Wrong: Option A claims both assumptions are implicit, which would require the argument to depend on both lack of desire and failure of the association. Option C and option D each treat one of the assumptions as necessary, which is not the case. Option E suggests that exactly one of them must be implicit, but the argument does not rely on either assumption and is simply factual. Common Pitfalls: Learners often rush to blame residents or organisers when they see the phrase did not bring much response and wrongly treat such blame as hidden assumptions. However, assumption questions demand a stricter test: remove the alleged assumption and check if the original statement still makes sense. Final Answer: The correct conclusion is that neither assumption 1 nor assumption 2 is implicit in the given argument.

Question 11

Compare the fractions 29/6 and 43/12 and identify which of the given comparison statements is definitely true.

Accepted Answer

Introduction / Context: This is a basic arithmetic problem that asks you to compare two fractions and select the correct relationship. Such questions test comfort with equivalent fractions and simple numerical comparisons without a calculator. Given Data / Assumptions: Fractions involved: 29/6, 43/12 and 53/12. We must decide which comparison between 29/6 and the other fractions is definitely correct. No approximation or rounding is allowed; the relation must be exact. Concept / Approach: When comparing fractions with different denominators, it is safest to convert them to a common denominator. Here the denominators are 6 and 12, so the least common denominator is 12. Once the denominators are the same, we only need to compare the numerators to determine whether one fraction is greater than, less than or equal to the other. Step-by-Step Solution: Step 1: Convert 29/6 to an equivalent fraction with denominator 12. Multiply top and bottom by 2. Step 2: 29/6 becomes (29 × 2) / (6 × 2) = 58/12. Step 3: Compare 58/12 with 43/12. Since both have denominator 12, compare numerators: 58 is greater than 43. Step 4: Therefore 29/6 > 43/12. This matches option B. Step 5: Briefly check other options. 29/6 = 43/12 would require 58 = 43, which is false. 29/6 = 53/12 would require 58 = 53, also false. 29/6 < 43/12 contradicts 58 > 43. 29/6 < 53/12 contradicts 58 > 53. Verification / Alternative check: As a quick mental check, convert to decimals. 29/6 is approximately 4.83, while 43/12 is approximately 3.58 and 53/12 is approximately 4.42. Clearly 4.83 is greater than both 3.58 and 4.42, so 29/6 is the largest of the three fractions, which is consistent with the exact comparison above. Why Other Options Are Wrong: Options A and D assert equalities, but equalities require identical numerators once denominators match. This is not the case here. Option C reverses the correct inequality sign, and option E also gives the wrong direction of comparison with 53/12. Only option B expresses the true relationship based on exact arithmetic. Common Pitfalls: Candidates sometimes compare fractions by looking only at denominators or only at numerators, which can be misleading. Another pitfall is relying on rough guesses instead of computing equivalent fractions or simple decimal values. Using a common denominator is a reliable and fast method for such questions. Final Answer: The definitely true statement is 29/6 > 43/12.

Question 12

Between X and Y, who is the better dancer, and which of the given statements is or are sufficient to answer this question?

Accepted Answer

Introduction / Context: This is a data sufficiency question. You are not asked to actually decide who is the better dancer, but only whether the given statements provide enough information to answer that question logically and definitely. Given Data / Assumptions: Main question: Between X and Y who is the better dancer? Statement 1: X had given more dance auditions. Statement 2: Y had given more stage performances. We must judge sufficiency, not make assumptions about talent. Concept / Approach: In data sufficiency, a statement is sufficient only if it allows you to answer the main question with certainty, without needing additional information or assumptions. More auditions or more performances may indicate higher activity but do not necessarily prove greater skill or quality. Step-by-Step Solution: Step 1: Consider statement 1 alone. X has given more auditions, but this does not guarantee that X is the better dancer. X may be less talented and therefore needs more attempts, or may simply be more active. Step 2: Consider statement 2 alone. Y has given more stage performances, but this also does not prove that Y is better. Y could have more opportunities, or the performances could be of a lower level. Step 3: Consider both statements together. Knowing that X has more auditions and Y has more performances still does not allow us to rank their dancing ability with certainty. We do not know judges' opinions, quality ratings, or any objective measure of skill. Step 4: Therefore neither statement alone, nor both together, is sufficient to decide who is the better dancer. Verification / Alternative check: Construct two scenarios. In one scenario, X is highly talented but shy and auditions rarely, while Y auditions and performs many times with average skill. In another scenario, Y is extremely talented and therefore performs a lot, while X keeps auditioning but does not get selected. Both scenarios fit the statements but lead to different answers about who is better, showing insufficiency of the data. Why Other Options Are Wrong: Option A assumes that having more auditions and performances together is enough, which is not logically justified. Options C and D each assume one of the statements is sufficient, which ignores many possibilities. Option E suggests that together the statements are enough, but even combined they do not give a clear criterion for dancing skill. Common Pitfalls: A common error is to equate quantity of activity with quality of performance. In data sufficiency, you must resist such assumptions and rely strictly on what the statements guarantee. Final Answer: The correct evaluation is that neither statement 1 nor statement 2 is sufficient to determine who is the better dancer.

Question 13

Is the expression X − 5 an even integer, and which of the given statements about X is or are sufficient to answer this question?

Accepted Answer

Introduction / Context: This is a data sufficiency question involving parity (even or odd) of an expression. The main task is not to find the value of X, but to decide which statement gives enough information to determine whether X − 5 is an even integer. Given Data / Assumptions: Main question: Is X − 5 even? X is a real number. Statement 1: X − 15 is an integer. Statement 2: X − 10 is an odd integer. Even and odd are considered only for integers. Concept / Approach: To decide whether X − 5 is even, we must first be sure it is an integer and then know its parity. We examine each statement to see whether it fixes X in a way that allows us to deduce the parity of X − 5. If multiple integer values with different parity for X − 5 are possible under a statement, that statement is not sufficient. Step-by-Step Solution: Step 1: From statement 1, X − 15 is an integer. Let X − 15 = k, where k is any integer. Then X = k + 15 and X − 5 = k + 10. Step 2: Because k can be any integer, k + 10 can be even or odd depending on k. So we cannot decide the parity of X − 5. Statement 1 alone is insufficient. Step 3: From statement 2, X − 10 is an odd integer. Let X − 10 = 2n + 1, where n is an integer. Then X = 2n + 11. Step 4: Now X − 5 = (2n + 11) − 5 = 2n + 6 = 2(n + 3), which is clearly an even integer. Step 5: Under statement 2, X − 5 is always even, so statement 2 alone is sufficient. Verification / Alternative check: Choose a sample value under statement 2. If X − 10 = 3, an odd integer, then X = 13 and X − 5 = 8, which is even. If X − 10 = 5, then X = 15 and X − 5 = 10, again even. Any odd integer used in place of X − 10 will always lead to an even value for X − 5. Why Other Options Are Wrong: Option A wrongly claims that statement 1 is sufficient, even though it leaves parity undecided. Option B says neither is sufficient, but we have seen statement 2 alone is enough. Option D claims both are needed, but statement 2 alone already answers the question. Option E suggests that the combination is required, which overcomplicates the situation. Common Pitfalls: One common mistake is to ignore the requirement that even and odd apply only to integers and to overlook that statement 1 allows many different integer values. Another error is failing to express X in algebraic form from each statement, which makes parity reasoning less clear. Final Answer: The correct conclusion is that statement 2 alone is sufficient while statement 1 alone is insufficient to determine whether X − 5 is even.

Question 14

Is Raju fond of travelling, and which of the given statements is or are sufficient to answer this question?

Accepted Answer

Introduction / Context: This data sufficiency question asks whether we can decide if a person, Raju, is fond of travelling based on the given statements. We must judge sufficiency, not write a biography of Raju. Given Data / Assumptions: Main question: Is Raju fond of travelling? Statement 1: Raju has seen East, West, North and Southern parts of India. Statement 2: Raju likes to be in the company of his friends. Fondness for travelling is a behavioural preference related to travel. Concept / Approach: In many exam styles, if a person has travelled extensively across all regions of a country without any other constraints mentioned, it is treated as strong evidence that the person is fond of travelling. Liking company of friends alone does not say anything specific about travel. The question is framed with this typical exam convention in mind. Step-by-Step Solution: Step 1: Consider statement 1 alone. Raju has seen East, West, North and Southern parts of India, which indicates extensive journeys across the country. Step 2: Under usual reasoning in such questions, covering all major regions of a country is taken as enough to conclude that a person is fond of travelling, unless there is contradictory data. Step 3: Consider statement 2 alone. Liking to be in the company of friends relates to social preference and tells nothing specific about travel. People who like friends may or may not like travelling. Step 4: Therefore statement 1 alone is treated as sufficient to answer the question, while statement 2 alone is not. Verification / Alternative check: Think in terms of exam logic rather than real world exceptions. If a question designer wants to show that someone likes travel, they usually write that the person has visited many places or all parts of a country. By contrast, information about liking friends is often used for questions about social behaviour, not travel. Hence the pattern supports statement 1 as sufficient. Why Other Options Are Wrong: Option A reverses the roles of the statements and incorrectly treats statement 2 as sufficient. Option C claims that both are needed, but statement 2 adds nothing to what is already clear from statement 1. Option D says data are insufficient, which ignores the standard interpretation of extensive travel. Option E also overstates the need for combining statements. Common Pitfalls: Some students become over cautious and feel that even extensive travel may happen for compulsion, so they mark data insufficient. However, competitive exam questions usually expect you to use the most natural and straightforward interpretation of such statements rather than hunt for rare exceptions. Final Answer: The correct assessment is that statement 1 alone is sufficient while statement 2 alone is insufficient to decide whether Raju is fond of travelling.

Question 15

The population in metro cities is increasing and causing shortage of living space; which of the given conclusions about government action logically follows?

Accepted Answer

Introduction / Context: This is a statements and conclusions question about urban population growth and living conditions. You must accept the given statement as true and then test which policy suggestions logically follow, without inserting your own policy preferences. Given Data / Assumptions: Statement: Population in metro cities is increasing tremendously, leading to shortage of living space and a drop in living conditions. Conclusion 1: Government should re-plan city development. Conclusion 2: Government should demolish illegal buildings and build cottages for people. We look for conclusions that are reasonably suggested by the problem described. Concept / Approach: A conclusion in such questions should be a logical or reasonable step suggested by the given problem description. The statement points to a broad issue (unplanned growth and poor conditions) rather than a very specific remedy. A general call to re-plan city development is a natural conclusion, while a very detailed solution may or may not be implied. Step-by-Step Solution: Step 1: The statement clearly links rapid population growth with shortage of space and worse living conditions. Step 2: Re-planning city development is a broad policy tool that directly targets this kind of structural problem. It is natural to conclude that the government should reconsider planning strategies. Step 3: Demolishing illegal buildings and building cottages is only one specific kind of action. The statement does not say that illegal buildings are the main cause or that cottages are the chosen solution. Step 4: Therefore conclusion 1 follows reasonably from the problem description, but conclusion 2 is too specific and not directly supported. Verification / Alternative check: Imagine a city where population growth is managed by vertical development and mass transit. There may still be shortages or other issues, but the key need is careful planning. The government could improve conditions by revising housing, transport and zoning policies without necessarily demolishing illegal buildings or building cottages. Hence conclusion 2 does not universally follow from the statement. Why Other Options Are Wrong: Option A treats both conclusions as following, which wrongly elevates one specific policy to the same level as the general planning suggestion. Option B claims that neither follows, ignoring the clear link between the problem and the need for re-planning. Option D selects only conclusion 2, which is not implied. Option E suggests that at least one of the two follows without specifying which, but the question expects the exact correct one. Common Pitfalls: Candidates sometimes read their own views on urban policy into the question, for example assuming that illegal buildings are always the main cause, or feeling that cottages are desirable. However, logical reasoning questions need you to stick closely to what is actually stated. Final Answer: The correct choice is that only conclusion 1 follows from the given statement.

Question 16

Some fruits are vegetables and some vegetables are not edible; based on this, which of the given conclusions logically follows?

Accepted Answer

Introduction / Context: This is a syllogism style reasoning question involving three sets: fruits, vegetables and edible items. You must analyse the statements and decide which of the two conclusions necessarily follows if the statements are taken as true. Given Data / Assumptions: Statement 1: Some fruits are vegetables. Statement 2: Some vegetables are not edible. Conclusion 1: Some fruits are not edible. Conclusion 2: Some vegetables are fruits. We consider only the logical content of the statements, not general world knowledge. Concept / Approach: The statement some fruits are vegetables means that there is at least one object that is both a fruit and a vegetable. The statement some vegetables are not edible tells us that within the set of vegetables there is at least one non edible element. We must check if these facts force either of the two conclusions in all possible diagrams consistent with the statements. Step-by-Step Solution: Step 1: Interpret statement 1. Some fruits are vegetables means that the sets fruits and vegetables overlap. So some vegetables are also fruits. Step 2: This is exactly what conclusion 2 says: some vegetables are fruits. So conclusion 2 directly restates the overlap given in statement 1 and therefore definitely follows. Step 3: Interpret statement 2. Some vegetables are not edible. This gives a region inside the vegetables set that lies outside the edible set. However, this region may or may not overlap with fruits. Step 4: It is possible that the vegetables that are also fruits are all edible, while the non edible vegetables are a different group. In that case, no fruit is non edible, and conclusion 1 would be false. Step 5: Because such a diagram satisfies both statements but makes conclusion 1 false, conclusion 1 does not necessarily follow. Verification / Alternative check: Construct an example. Let fruits be {apple, mango}, vegetables be {apple, mango, bitter_leaf}, edible items be {apple, mango}. Here apple and mango are both fruits and vegetables and are edible. Bitter_leaf is a vegetable but not edible. Both statements are true, yet there is no fruit that is not edible, so conclusion 1 fails. However, some vegetables (apple and mango) are fruits, so conclusion 2 holds in every such construction that respects statement 1. Why Other Options Are Wrong: Option A claims that neither conclusion follows, which ignores the direct overlap between fruits and vegetables given by statement 1. Option B chooses only conclusion 1, which is not forced. Option D says both follow, but we have shown that conclusion 1 can fail. Option E suggests an either or condition that is not relevant here, because only conclusion 2 is certain. Common Pitfalls: Students often assume that if some vegetables are not edible and some fruits are vegetables, then there must be a fruit that is not edible. This is not logically necessary, because the overlapping part of fruits and vegetables may fall entirely inside the edible region while the non edible part remains separate. Final Answer: The correct evaluation is that only conclusion 2 follows from the given statements.

Question 17

All hill stations have a sunset point and X is a hill station; which of the given conclusions about sunset points logically follows?

Accepted Answer

Introduction / Context: This question is a straightforward syllogism involving hill stations and sunset points. You must use the given universal statement and a specific instance to derive which conclusion is logically valid. Given Data / Assumptions: Statement I: All hill stations have a sunset point. Statement II: X is a hill station. Conclusion I: X has a sunset point. Conclusion II: Places other than hill stations do not have sunset points. Concept / Approach: All hill stations have a sunset point is a universal statement about hill stations. It does not say anything about non hill stations. From such a statement and an example of a hill station, we can safely conclude properties of that example. However, we cannot infer properties for all other places not mentioned in the statement. Step-by-Step Solution: Step 1: From statement I, every hill station belongs to the set of places that have a sunset point. Step 2: From statement II, X is a member of the set of hill stations. Step 3: Combining these, X must also be a member of the set of places that have a sunset point. This directly supports conclusion I. Step 4: There is no information given about places that are not hill stations. They may or may not have sunset points. Therefore conclusion II, which states that all non hill stations lack sunset points, is not justified. Verification / Alternative check: Imagine a sea beach that is not a hill station but has a beautiful sunset point. This does not contradict the statement that all hill stations have a sunset point, because the original statement never claimed that only hill stations have sunset points. Hence conclusion II can be false while both statements remain true, which shows that it does not follow. Why Other Options Are Wrong: Option B selects conclusion II, which makes an unjustified claim about all other places. Option C denies both conclusions, ignoring the clear link between hill stations and sunset points for X. Option D treats both conclusions as true, which again overextends the original statement. Option E suggests an either or situation that is not appropriate here, because exactly one conclusion is correct. Common Pitfalls: A common mistake is to misread all hill stations have a sunset point as only hill stations have a sunset point. That extra word only completely changes the meaning but does not appear in the given statement. Always be careful with such subtle distortions in reasoning questions. Final Answer: The correct conclusion is that only conclusion I follows from the given statements.

Question 18

Given K > R = L, P > L and R ≥ S, which of the following conclusions about the relation between S and L logically follows?

Accepted Answer

Introduction / Context: This inequality reasoning question gives a chain of relations among K, R, L, P and S, and then two possible conclusions about S and L. You must decide whether one, both, either, or neither of the conclusions logically follows. Given Data / Assumptions: Given: K > R = L, P > L, and R ≥ S. Conclusion I: S < L. Conclusion II: L = S. All inequalities and equalities are about the same ordered quantity, such as height or value. Concept / Approach: From K > R = L we know R and L are equal and K is greater than both. From R ≥ S we know S is less than or equal to R. Because R equals L, we can translate R ≥ S into a relation between L and S. We then check whether S must be strictly less than L, exactly equal to L, or could be either. Step-by-Step Solution: Step 1: From R = L and R ≥ S, substitute L for R to get L ≥ S. Step 2: L ≥ S means S ≤ L. Therefore S can be less than L or equal to L. Both possibilities fit the given information. Step 3: Conclusion I claims that S is strictly less than L. This is possible if S < L. Step 4: Conclusion II claims that L equals S. This is also possible if S = L. Step 5: Since the data allow either S < L or S = L without contradiction, one of the two conclusions must be true, but we cannot say which one for sure. Verification / Alternative check: Take example 1: let L = R = 5 and S = 4, with K and P chosen larger than 5. Then K > R = L, P > L and R ≥ S are all satisfied, and S < L holds, so conclusion I is true and conclusion II is false. Take example 2: let L = R = S = 5. The given relations still hold, but now conclusion II is true and conclusion I is false. These examples show that either conclusion can follow, but not both at the same time. Why Other Options Are Wrong: Option A and option B each insist only one specific conclusion follows. Since we can construct valid cases where either one is true, neither option is correct. Option D says neither follows, which is wrong because at least one must hold in every case (S cannot be greater than L). Option E says both follow, which is impossible because S cannot be both strictly less than and equal to L simultaneously. Common Pitfalls: Many students quickly assume that R ≥ S must mean R is strictly greater than S, forgetting that equality is also allowed. Others forget to use the equality R = L when translating the inequality. Always rewrite inequalities step by step and consider all allowed possibilities. Final Answer: The correct evaluation is that either conclusion I or conclusion II follows, but not both together.

Question 19

All razors are blades and all blades are metals; which of the given conclusions about metals, razors and blades logically follows?

Accepted Answer

Introduction / Context: This is a basic syllogism problem involving three sets: razors, blades and metals. You are given two universal statements and must determine which conclusion correctly describes the relationship among these sets. Given Data / Assumptions: Statement 1: All razors are blades. Statement 2: All blades are metals. Conclusion 1: All metals are razors. Conclusion 2: Some metals are blades. Concept / Approach: The two statements can be read as a chain of subsets: razors ⊂ blades ⊂ metals. That is, every razor is a blade, and every blade is a metal. From this chain, we can deduce certain subset relations but must be careful not to reverse them. We test each conclusion against this structure. Step-by-Step Solution: Step 1: From all razors are blades and all blades are metals, we know every razor is also a metal. However, the set of metals can be much larger than the set of razors. Step 2: Conclusion 1 claims that all metals are razors. This would mean metals ⊂ razors, which reverses the actual subset direction and is not implied. Step 3: Conclusion 2 claims that some metals are blades. Since every blade is a metal and we assume blades exist, there is at least one item that is both a blade and a metal. Step 4: Therefore conclusion 2 is necessarily true whenever the statements hold. Verification / Alternative check: Construct a simple example. Let blades be the set {b1, b2} and razors be {b1}, with metals containing {b1, b2, m1}. All razors (b1) are blades, and both blades are metals. Clearly, some metals (b1 and b2) are blades. At the same time, not all metals are razors because m1 is a metal but not a razor. This confirms that conclusion 2 is correct and conclusion 1 is not. Why Other Options Are Wrong: Option A says all conclusions follow, which is wrong because conclusion 1 does not. Option B says neither conclusion follows, ignoring the direct fact that blades are metals. Option D selects only conclusion 1, which reverses the direction of the subset. Option E suggests an either or condition, but the situation here is clear: only conclusion 2 is supported. Common Pitfalls: Students sometimes confuse all A are B with all B are A, which is not logically equivalent. The direction of the all statement matters. Another pitfall is to forget that if all blades are metals, then at least some metals (those blades) must exist as long as blades are not an empty set. Final Answer: The correct evaluation is that only conclusion 2 follows from the given statements.

Question 20

All blue are colours and all colours are shades; which of the given conclusions about blue, colours and shades logically follow?

Accepted Answer

Introduction / Context: This syllogism question deals with three sets: blue, colours and shades. Two universal statements relate these sets, and you must judge which conclusions follow logically if both statements are true. Given Data / Assumptions: Statement 1: All blue are colours. Statement 2: All colours are shades. Conclusion 1: All blue are shades. Conclusion 2: Some shades are colours. Concept / Approach: We can interpret the statements as a chain of subsets: blue ⊂ colours ⊂ shades. That is, everything that is blue is a colour, and everything that is a colour is a shade. From this chain we can deduce transitive relations and existence of overlapping elements. Step-by-Step Solution: Step 1: From statement 1, every blue object is a colour. Step 2: From statement 2, every colour is a shade. Step 3: Combining these, every blue object is a colour and every colour is a shade, so every blue object must also be a shade. This proves conclusion 1. Step 4: Since colours form a subset of shades, and we assume colours exist, there must be at least some shades that are colours. Therefore conclusion 2 is also true. Verification / Alternative check: Take a simple example. Let the set of blue be {b1}, the set of colours be {b1, c1} and the set of shades be {b1, c1, s1}. Then all blue are colours and all colours are shades. Here, all blue elements (b1) are indeed shades, and some shades (b1 and c1) are colours. This concrete model matches both conclusions. Why Other Options Are Wrong: Option B claims only conclusion 1 follows and ignores the obvious existence of colours inside shades. Option C claims only conclusion 2, which is incomplete. Option D says no conclusions follow, which directly contradicts the transitive nature of the given statements. Option E suggests an either or situation, but both conclusions are clearly valid together. Common Pitfalls: A common error is to focus only on the first conclusion and forget to test the second. Another mistake is to treat some as meaning many or most, when it only means at least one. As long as there is at least one colour, there is at least one shade that is a colour. Final Answer: The correct evaluation is that all the conclusions follow from the given statements.

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