CAT 2023 — Slot 2
Full paper, free to view. Pick a section below, click any question to see the answer and explanation.
VARC topics covered: Culture & Identity RC, History & Civilization RC, Nature And Environment RC, Odd One Out, Para Completion, Para Summary, Parajumble, Politics & Governance RC
The Positivists, anxious to stake out their claim for history as a science, contributed the weight of their influence to the cult of facts. First ascertain the facts, said the positivists, then draw your conclusions from them. . . . This is what may [be] called the common-sense view of history. History consists of a corpus of ascertained facts. The facts are available to the historian in documents, inscriptions, and so on . . . [Sir George Clark] contrasted the "hard core of facts" in history with the surrounding pulp of disputable interpretation forgetting perhaps that the pulpy part of the fruit is more rewarding than the hard core. . . . It recalls the favourite dictum of the great liberal journalist C. P. Scott: "Facts are sacred, opinion is free." . . .
What is a historical fact? . . . According to the common-sense view, there are certain basic facts which are the same for all historians and which form, so to speak, the backbone of history—the fact, for example, that the Battle of Hastings was fought in 1066. But this view calls for two observations. In the first place, it is not with facts like these that the historian is primarily concerned. It is no doubt important to know that the great battle was fought in 1066 and not in 1065 or 1067, and that it was fought at Hastings and not at Eastbourne or Brighton. The historian must not get these things wrong. But [to] praise a historian for his accuracy is like praising an architect for using well-seasoned timber or properly mixed concrete in his building. It is a necessary condition of his work, but not his essential function. It is precisely for matters of this kind that the historian is entitled to rely on what have been called the "auxiliary sciences" of history—archaeology, epigraphy, numismatics, chronology, and so forth. . . .
The second observation is that the necessity to establish these basic facts rests not on any quality in the facts themselves, but on an apriori decision of the historian. In spite of C. P. Scott's motto, every journalist knows today that the most effective way to influence opinion is by the selection and arrangement of the appropriate facts. It used to be said that facts speak for themselves. This is, of course, untrue. The facts speak only when the historian calls on them: it is he who decides to which facts to give the floor, and in what order or context. . . . The only reason why we are interested to know that the battle was fought at Hastings in 1066 is that historians regard it as a major historical event. . . . Professor Talcott Parsons once called [science] "a selective system of cognitive orientations to reality." It might perhaps have been put more simply. But history is, among other things, that. The historian is necessarily selective. The belief in a hard core of historical facts existing objectively and independently of the interpretation of the historian is a preposterous fallacy, but one which it is very hard to eradicate.
According to this passage, which one of the following statements best describes the significance of archaeology for historians?
Archaeology helps historians to locate the oldest civilisations in history.
Archaeology helps historians to ascertain factual accuracy.
Archaeology helps historians to carry out their primary duty.
Archaeology helps historians to interpret historical facts.
Archaeology helps historians to ascertain factual accuracy.
The passage states that the historian is entitled to rely on auxiliary sciences like archaeology, epigraphy and numismatics for establishing basic facts such as dates and locations. This is described as a necessary condition of the historian's work, not his essential function, meaning archaeology helps establish factual accuracy rather than carrying out the historian's primary interpretive duty.
Why Option A is
Incorrect: The passage does not mention locating the oldest civilisations as a function of archaeology in this context, this is a general association with archaeology that the passage does not make.
Why Option B is Correct: The passage explicitly classifies archaeology as an auxiliary science used to establish basic facts, meaning it helps historians confirm accuracy of dates, places and events.
Why Option C is
Incorrect: The passage explicitly distinguishes between establishing facts, for which auxiliary sciences are used, and the historian's primary function, which is interpretation and selection. Archaeology helps with the former, not the latter.
Why Option D is
Incorrect: The passage does not associate archaeology with interpretation, interpretation is what the historian does after the basic facts are established, and it is this interpretive function the author considers the historian's essential role.
Key Takeaway: When a passage distinguishes between a necessary condition and an essential function, the auxiliary tool belongs to the former category and cannot be credited with the latter.
All of the following, if true, can weaken the passage's claim that facts do not speak for themselves, EXCEPT:
the truth value of a fact is independent of the historian who expresses it.
a fact, by its very nature, is objective and universal, irrespective of the context in which it is placed.
facts, like truth, can be relative: what is fact for person X may not be so for person Y.
the order in which a series of facts is presented does not have any bearing on the production of meaning.
facts, like truth, can be relative: what is fact for person X may not be so for person Y.
The passage argues that facts do not speak for themselves because the historian selects, arranges and contextualises them, giving them meaning. Options A, B and D, if true, would all weaken this claim by suggesting facts have objective, independent meaning regardless of how they are presented. Option C, saying facts can be relative and subjective, actually supports the passage's claim that facts are not objective and self-evident, rather than weakening it.
Why Option A is
Incorrect: If the truth value of a fact is independent of the historian expressing it, this would suggest facts have an objective truth that exists outside the historian's framing, weakening the passage's claim that historians give meaning to facts through selection.
Why Option B is
Incorrect: If facts are objective and universal irrespective of context, this would directly contradict the passage's claim that context and arrangement determine what facts mean.
Why Option C is Correct: If facts can be relative and vary between individuals, this reinforces rather than weakens the passage's argument that facts do not carry universal, self-evident meaning, making this the exception.
Why Option D is
Incorrect: If the order in which facts are presented has no bearing on meaning, this would directly contradict the passage's specific point that it is the historian's selection and arrangement of facts that shapes their meaning.
Key Takeaway: In "weaken except" questions, find the option that supports the passage's claim rather than contradicting it, since supporting a claim cannot weaken it.
If the author of the passage were to write a book on the Battle of Hastings along the lines of his/her own reasoning, the focus of the historical account would be on:
exploring the socio-political and economic factors that led to the Battle.
producing a detailed timeline of the various events that led to the Battle.
deriving historical facts from the relevant documents and inscriptions.
providing a nuanced interpretation by relying on the auxiliary sciences.
exploring the socio-political and economic factors that led to the Battle.
The passage argues that the historian's essential function is not factual accuracy but interpretation, selection and arrangement, giving floor to the facts that matter and deciding their context. An account written along these lines would focus on why the battle mattered, exploring the socio-political and economic factors that led to it, rather than cataloguing dates and timelines.
Why Option A is Correct: This matches the author's view that the historian's primary function is to give meaning to facts through interpretation, contextualisation and selection, which in a book about the Battle of Hastings would mean examining the factors that made the battle historically significant.
Why Option B is
Incorrect: Producing a detailed timeline is precisely what the passage associates with the auxiliary sciences and with getting basic facts right, which the author describes as a necessary condition but not the historian's primary role.
Why Option C is
Incorrect: Deriving historical facts from documents and inscriptions is what positivists do, and the author explicitly critiques this as the common-sense view that reduces history to a corpus of ascertained facts rather than meaningful interpretation.
Why Option D is
Incorrect: While the author mentions auxiliary sciences like archaeology, their role is limited to establishing factual accuracy, not providing a nuanced interpretation. The interpretation comes from the historian's own analytical work.
Key Takeaway: When asked how an author would apply their own argument, identify which approach most closely mirrors the method the author explicitly champions rather than the methods they criticise.
All of the following describe the "common-sense view" of history, EXCEPT:
real history can be found in ancient engravings and archival documents.
only the positivist methods can lead to credible historical knowledge.
history is like science: a selective system of cognitive orientations to reality.
history can be objective like the sciences if it is derived from historical facts.
history is like science: a selective system of cognitive orientations to reality.
The common-sense view of history, as described in the passage, is associated with positivism, the belief that facts are objective, found in documents and inscriptions, and form the hard core of history. The passage attributes the description of science as a selective system of cognitive orientations to reality to Professor Talcott Parsons, and uses it to argue that the historian is necessarily selective, which is actually the author's own counter-argument against the common-sense view.
Why Option A is
Incorrect: The passage explicitly states that historical facts are available in documents, inscriptions and so on, directly associating the common-sense view with finding history in such archival sources.
Why Option B is
Incorrect: The passage describes the positivists as contributors to the cult of facts and the common-sense view, so the positivist method is directly associated with this view.
Why Option C is Correct: This describes the author's own counter-position, not the common-sense view. The author uses Parsons' formulation to argue that history, like science, is selective and interpretive, which directly contradicts the common-sense view that history is a corpus of objective facts.
Why Option D is
Incorrect: The common-sense view treats history as something that can be made objective by grounding it in facts, mirroring scientific method, which is why positivists sought to claim history as a science.
Key Takeaway: Be careful not to attribute to the view being critiqued the very counter-argument the author uses to critique it, since these belong to opposite sides of the passage's argument.
Umberto Eco, an Italian writer, was right when he said the language of Europe is translation. Netflix and other deep-pocketed global firms speak it well. Just as the EU employs a small army of translators and interpreters to turn intricate laws or impassioned speeches of Romanian MEPs into the EU's 24 official languages, so do the likes of Netflix. It now offers dubbing in 34 languages and subtitling in a few more. . . .
The economics of European productions are more appealing, too. American audiences are more willing than before to give dubbed or subtitled viewing a chance. This means shows such as "Lupin", a French crime caper on Netflix, can become global hits. . . . In 2015, about 75% of Netflix's original content was American; now the figure is half, according to Ampere, a media-analysis company. Netflix has about 100 productions under way in Europe, which is more than big public broadcasters in France or Germany. . . .
Not everything works across borders. Comedy sometimes struggles. Whodunits and bloodthirsty maelstroms between arch Romans and uppity tribesmen have a more universal appeal. Some do it better than others. Barbarians aside, German television is not always built for export, says one executive, being polite. A bigger problem is that national broadcasters still dominate. Streaming services, such as Netflix or Disney+, account for about a third of all viewing hours, even in markets where they are well-established. Europe is an ageing continent. The generation of teens staring at phones is outnumbered by their elders who prefer to gawp at the box.
In Brussels and national capitals, the prospect of Netflix as a cultural hegemon is seen as a threat. "Cultural sovereignty" is the watchword of European executives worried that the Americans will eat their lunch. To be fair, Netflix content sometimes seems stuck in an uncanny valley somewhere in the mid-Atlantic, with local quirks stripped out. Netflix originals tend to have fewer specific cultural references than shows produced by domestic rivals, according to Enders, a market analyst. The company used to have an imperial model of commissioning, with executives in Los Angeles cooking up ideas French people might like. Now Netflix has offices across Europe. But ultimately the big decisions rest with American executives. This makes European politicians nervous.
They should not be. An irony of European integration is that it is often American companies that facilitate it. Google Translate makes European newspapers comprehensible, even if a little clunky, for the continent's non-polyglots. American social-media companies make it easier for Europeans to talk politics across borders. (That they do not always like to hear what they say about each other is another matter.) Now Netflix and friends pump the same content into homes across a continent, making culture a cross-border endeavour, too. If Europeans are to share a currency, bail each other out in times of financial need and share vaccines in a pandemic, then they need to have something in common—even if it is just bingeing on the same series. Watching fictitious northern and southern Europeans tear each other apart 2,000 years ago beats doing so in reality.
Based on information provided in the passage, all of the following are true, EXCEPT:
European television productions have the potential to become global hits.
Netflix has been able to transform itself into a truly European entity.
only half of Netflix's original programming in the EU is now produced in America.
national broadcasters dominate in the EU in terms of total television viewing hours.
Netflix has been able to transform itself into a truly European entity.
The passage describes Netflix as having offices across Europe but notes that ultimately the big decisions rest with American executives, and that Netflix originals tend to have fewer specific cultural references than domestic rivals. This is the opposite of being a truly European entity.
Why Option A is
Incorrect: The passage explicitly cites shows like Lupin becoming global hits, directly supporting the claim that European productions have the potential to become global hits.
Why Option B is Correct: The passage explicitly says Netflix has American executives making final decisions, that its content is sometimes stripped of local quirks, and that it used to have an imperial commissioning model. None of this supports the idea that Netflix has transformed into a truly European entity.
Why Option C is
Incorrect: The passage explicitly states that now only half of Netflix's original content is American, down from 75% in 2015, directly supporting this claim.
Why Option D is
Incorrect: The passage explicitly states that streaming services account for only about a third of all viewing hours even in well-established markets, with national broadcasters dominating, directly supporting this claim.
Key Takeaway: For "all true except" questions, identify the option that directly contradicts something explicitly stated in the passage rather than the one that is simply not mentioned.
The author sees the rise of Netflix in Europe as:
a unifying force.
filling an entertainment gap.
a looming cultural threat.
an economic threat.
a unifying force.
The final paragraph explicitly argues that Netflix and similar American companies actually facilitate European integration by pumping the same content into homes across the continent, making culture a cross-border endeavour. The author directly says Europeans should not be worried about Netflix as a cultural hegemon and presents it as a unifying force.
Why Option A is Correct: The final paragraph makes the author's position explicit, that Netflix sharing content across European borders is analogous to sharing a currency or vaccines, framing it as a mechanism for cultural integration and unity.
Why Option B is
Incorrect: The passage does not frame Netflix primarily as filling an entertainment gap, it frames it as facilitating cultural integration across Europe.
Why Option C is
Incorrect: While the passage acknowledges that European politicians see Netflix as a cultural threat, the author explicitly says they should not be worried, indicating that the author does not share this view.
Why Option D is
Incorrect: The passage does not present Netflix as an economic threat to Europe, it focuses on cultural concerns and then explicitly dismisses them in favour of a unification narrative.
Key Takeaway: When an author explicitly contradicts a concern raised earlier in the passage and offers an alternative framing, use the author's own conclusion rather than the concerns they are addressing.
Based only on information provided in the passage, which one of the following hypothetical Netflix shows would be most successful with audiences across the EU?
An Italian comedy show hosted by an international star.
An original German TV science fiction production.
A murder mystery drama set in North Africa and France.
A trans-Atlantic romantic drama set in Europe and America.
A murder mystery drama set in North Africa and France.
The passage states that whodunits and historical dramas featuring Romans and tribesmen have more universal appeal across borders, while comedy sometimes struggles to cross borders. A murder mystery drama set in North Africa and France combines the universally appealing whodunit format with a cross-national European and African setting, maximising cross-border appeal.
Why Option A is
Incorrect: The passage explicitly notes that comedy sometimes struggles to travel across borders, making an Italian comedy show a risky choice for EU-wide success regardless of the host's fame.
Why Option B is
Incorrect: The passage notes that German television is not always built for export, making an original German science fiction production a less likely candidate for EU-wide success.
Why Option C is Correct: This combines two features the passage identifies as universally appealing, the whodunit format and a setting spanning two distinct cultural spaces that are nonetheless within the broad European and Mediterranean world, making it most likely to succeed across EU audiences.
Why Option D is
Incorrect: The passage warns that Netflix content can get stuck in an uncanny mid-Atlantic valley when it tries to appeal too broadly, and a trans-Atlantic romantic drama set across Europe and America seems more likely to fall into this trap of having local quirks stripped out.
Key Takeaway: Apply the passage's specific criteria for cross-border success directly to each option, since the correct answer will satisfy more of those criteria than the others.
Which one of the following research findings would weaken the author's conclusion in the final paragraph?
Research shows there is a wide variance in the popularity and viewing of Netflix shows across different EU countries.
Research shows that Netflix has been gradually losing market share to other streaming television service providers.
Research shows that Netflix hits produced in France are very popular with North American audiences.
Research shows that older women across the EU enjoy watching romantic comedies on Netflix, whereas younger women prefer historical fiction dramas.
Research shows there is a wide variance in the popularity and viewing of Netflix shows across different EU countries.
The author's final paragraph argues that Netflix facilitates European integration by providing common cultural content across borders, implying that people across EU countries are watching the same shows. If research showed wide variance in the popularity of Netflix shows across different EU countries, this would directly undermine the claim that Netflix is creating a shared cultural experience that unifies Europeans.
Why Option A is Correct: If Netflix viewing patterns vary widely across EU countries, the shows are not in fact being shared across a common European audience, directly contradicting the author's claim that Netflix makes culture a cross-border endeavour.
Why Option B is
Incorrect: Netflix losing market share to other streaming services would not weaken the argument that Netflix facilitates European cultural integration, since the argument is about what Netflix does, not whether Netflix is commercially dominant.
Why Option C is
Incorrect: Netflix shows being popular in North America is irrelevant to the author's conclusion about Netflix facilitating European integration specifically.
Why Option D is
Incorrect: Different demographic groups within the EU watching different genres is not the same as different countries watching different content. Even if older women and younger women prefer different genres, they could both be watching Netflix, which still supports the cross-border cultural connection the author describes.
Key Takeaway: To weaken a conclusion about shared cultural experience, find evidence that the sharing is not actually occurring uniformly across the relevant group.
The Second Hand September campaign, led by Oxfam . . . seeks to encourage shopping at local organisations and charities as alternatives to fast fashion brands such as Primark and Boohoo in the name of saving our planet. As innocent as mindless scrolling through online shops may seem, such consumers are unintentionally—or perhaps even knowingly—contributing to an industry that uses more energy than aviation. . . .
Brits buy more garments than any other country in Europe, so it comes as no shock that many of those clothes end up in UK landfills each year: 300,000 tonnes of them, to be exact. This waste of clothing is destructive to our planet, releasing greenhouse gasses as clothes are burnt as well as bleeding toxins and dyes into the surrounding soil and water. As ecologist Chelsea Rochman bluntly put it, "The mismanagement of our waste has even come back to haunt us on our dinner plate."
It's not surprising, then, that people are scrambling for a solution, the most common of which is second-hand shopping. Retailers selling consigned clothing are currently expanding at a rapid rate . . . If everyone bought just one used item in a year, it would save 449 million lbs of waste, equivalent to the weight of 1 million Polar bears. "Thrifting" has increasingly become a trendy practice. London is home to many second-hand, or more commonly coined 'vintage', shops across the city from Bayswater to Brixton.
So you're cool and you care about the planet; you've killed two birds with one stone. But do people simply purchase a second-hand item, flash it on Instagram with #vintage and call it a day without considering whether what they are doing is actually effective?
According to a study commissioned by Patagonia, for instance, older clothes shed more microfibres. These can end up in our rivers and seas after just one wash due to the worn material, thus contributing to microfibre pollution. To break it down, the amount of microfibres released by laundering 100,000 fleece jackets is equivalent to as many as 11,900 plastic grocery bags, and up to 40 per cent of that ends up in our oceans. . . . So where does this leave second-hand consumers? [They would be well advised to buy] high-quality items that shed less and last longer [as this] combats both microfibre pollution and excess garments ending up in landfills. . . .
Luxury brands would rather not circulate their latest season stock around the globe to be sold at a cheaper price, which is why companies like ThredUP, a US fashion resale marketplace, have not yet caught on in the UK. There will always be a market for consignment but there is also a whole generation of people who have been taught that only buying new products is the norm; second-hand luxury goods are not in their psyche. Ben Whitaker, director at Liquidation Firm B-Stock, told Prospect that unless recycling becomes cost-effective and filters into mass production, with the right technology to partner it, "high-end retailers would rather put brand before sustainability."
The central idea of the passage would be undermined if:
customers bought all their clothes online.
clothes were not thrown and burnt in landfills.
second-hand stores sold only high-quality clothes.
Primark and Boohoo recycled their clothes for vintage stores.
second-hand stores sold only high-quality clothes.
The passage raises a problem with second-hand shopping, that older clothes shed more microfibres, and suggests buying high-quality items that shed less and last longer as the best solution. If second-hand stores sold only high-quality clothes, this would resolve the microfibre problem and make second-hand shopping a genuinely effective solution, removing the central complication the passage raises about the limitations of thrifting.
Why Option A is
Incorrect: Customers buying all their clothes online is not specifically what the passage's argument about the limitations of second-hand shopping depends on, this does not undermine the central issue about waste and microfibre pollution.
Why Option B is
Incorrect: The passage's central concern is about the microfibre pollution and environmental harm caused by second-hand clothes specifically, not just landfill burning. If clothes weren't burnt in landfills, the passage's specific concern about second-hand microfibre pollution would still stand.
Why Option C is Correct: If second-hand stores sold only high-quality clothes, the main problem the passage identifies with second-hand shopping, microfibre pollution from worn garments, would be eliminated. This would make thrifting a genuinely effective environmental solution, undermining the passage's central argument about its limitations.
Why Option D is
Incorrect: Primark and Boohoo recycling their clothes for vintage stores would increase the supply of second-hand clothing but would not address the core issue about microfibre shedding from older clothes that the passage raises.
Key Takeaway: To undermine a passage's central idea, find the scenario that removes the main problem or complication the passage is built around.
According to the author, companies like ThredUP have not caught on in the UK for all of the following reasons EXCEPT that:
recycling is currently not financially attractive for luxury brands.
luxury brands want to maintain their brand image.
luxury brands do not like their product to be devalued.
the British don't buy second-hand clothing.
the British don't buy second-hand clothing.
The passage gives three reasons for ThredUP not catching on in the UK, recycling not being cost-effective, luxury brands prioritising brand image over sustainability, and luxury brands not wanting their latest season stock circulated at cheaper prices. The passage does not say the British specifically don't buy second-hand clothing, in fact the passage describes the growth of vintage and second-hand shops across London.
Why Option A is
Incorrect: The passage explicitly quotes Ben Whitaker saying high-end retailers would rather put brand before sustainability unless recycling becomes cost-effective, directly stating that recycling is currently not financially attractive.
Why Option B is
Incorrect: The passage explicitly states that luxury brands would rather not circulate stock at cheaper prices, which is about protecting brand image.
Why Option C is
Incorrect: The passage explicitly states that luxury brands prefer not to have their latest season stock sold at a cheaper price, directly stating that devaluation of the product is a concern.
Why Option D is Correct: The passage describes the growth of second-hand shopping in London and the wider UK, and the reasons given for ThredUP not catching on are about luxury brand reluctance and cost-effectiveness, not about British consumer attitudes toward second-hand goods generally.
Key Takeaway: When asked for a reason not given in the passage, verify each option directly against what the passage explicitly states as the cause, and identify the one that contradicts the passage's description of consumer behaviour.
Based on the passage, we can infer that the opposite of fast fashion, 'slow fashion', would most likely refer to clothes that:
are of high quality and long lasting.
do not bleed toxins and dyes.
are sold by genuine vintage stores.
do not shed microfibres.
are of high quality and long lasting.
The passage argues that buying high-quality items that shed less and last longer is the best approach for second-hand shoppers, because this combats both microfibre pollution and excess garments ending up in landfills. Slow fashion, as the opposite of fast fashion which generates waste and pollution through disposable, low-quality items, would most logically refer to clothes that are high quality and long-lasting.
Why Option A is Correct: High quality and long-lasting is the direct opposite of fast fashion's defining characteristic of low-quality, disposable items that quickly end up in landfills, making this the most accurate description of slow fashion as the passage's argument implies it.
Why Option B is
Incorrect: Not bleeding toxins and dyes is a concern about disposal and landfill contamination, not a defining characteristic of slow fashion as the opposite of fast fashion's core problem of low quality and short lifespan.
Why Option C is
Incorrect: Being sold by genuine vintage stores is about the retail channel, not a characteristic of the clothes themselves. Slow fashion as an opposing concept to fast fashion would relate to the nature of the clothes, not where they are sold.
Why Option D is
Incorrect: Not shedding microfibres is a desirable property related to the specific microfibre pollution problem the passage raises, but it is a consequence of high quality rather than the defining characteristic of slow fashion itself.
Key Takeaway: Derive the meaning of an implied concept by identifying what the passage explicitly positions as the solution to the problem caused by the named concept.
The act of "thrifting", as described in the passage, can be considered ironic because it:
has created environmental problems.
is not cost-effective for retailers.
offers luxury clothing at cut-rate prices.
is an anti-consumerist attitude.
has created environmental problems.
Thrifting is presented in the passage as an environmentally conscious, trendy practice meant to reduce fashion waste. The irony is that it has created its own environmental problems, specifically microfibre pollution, meaning the act intended to help the planet is contributing to polluting rivers and oceans through microfibre shedding from laundered old clothes.
Why Option A is Correct: The irony is that thrifting, intended as an environmentally responsible alternative to fast fashion, has itself created environmental problems through microfibre pollution, meaning the cure has become part of the disease.
Why Option B is
Incorrect: The passage does not present thrifting as not cost-effective for retailers, it is actually described as expanding rapidly and creating a viable market.
Why Option C is
Incorrect: Thrifting offering luxury clothing at cut-rate prices is not described as ironic in the passage, it is simply a feature of the consignment market rather than a contradiction of its purpose.
Why Option D is
Incorrect: Thrifting as an anti-consumerist attitude is its intended purpose, not an ironic outcome. The irony the passage identifies is the unintended harmful consequence, not the act's own stated aim.
Key Takeaway: Irony in a passage is typically found where an action intended to produce one outcome inadvertently produces the opposite or an equally harmful one.
Over the past four centuries liberalism has been so successful that it has driven all its opponents off the battlefield. Now it is disintegrating, destroyed by a mix of hubris and internal contradictions, according to Patrick Deneen, a professor of politics at the University of Notre Dame. . . . Equality of opportunity has produced a new meritocratic aristocracy that has all the aloofness of the old aristocracy with none of its sense of noblesse oblige. Democracy has degenerated into a theatre of the absurd. And technological advances are reducing ever more areas of work into meaningless drudgery. "The gap between liberalism's claims about itself and the lived reality of the citizenry" is now so wide that "the lie can no longer be accepted," Mr Deneen writes. What better proof of this than the vision of 1,000 private planes whisking their occupants to Davos to discuss the question of "creating a shared future in a fragmented world"? . . .
Deneen does an impressive job of capturing the current mood of disillusionment, echoing leftwing complaints about rampant commercialism, right-wing complaints about narcissistic and bullying students, and general worries about atomisation and selfishness. But when he concludes that all this adds up to a failure of liberalism, is his argument convincing? . . . He argues that the essence of liberalism lies in freeing individuals from constraints. In fact, liberalism contains a wide range of intellectual traditions which provide different answers to the question of how to trade off the relative claims of rights and responsibilities, individual expression and social ties. . . . liberals experimented with a range of ideas from devolving power from the centre to creating national education systems.
Mr Deneen's fixation on the essence of liberalism leads to the second big problem of his book: his failure to recognise liberalism's ability to reform itself and address its internal problems. The late 19th century saw America suffering from many of the problems that are reappearing today, including the creation of a business aristocracy, the rise of vast companies, the corruption of politics and the sense that society was dividing into winners and losers. But a wide variety of reformers, working within the liberal tradition, tackled these problems head on. Theodore Roosevelt took on the trusts. Progressives cleaned up government corruption. University reformers modernised academic syllabuses and built ladders of opportunity. Rather than dying, liberalism reformed itself.
Mr Deneen is right to point out that the record of liberalism in recent years has been dismal. He is also right to assert that the world has much to learn from the premodern notions of liberty as self-mastery and self-denial. The biggest enemy of liberalism is not so much atomisation but old-fashioned greed, as members of the Davos elite pile their plates ever higher with perks and share options. But he is wrong to argue that the only way for people to liberate themselves from the contradictions of liberalism is "liberation from liberalism itself". The best way to read "Why Liberalism Failed" is not as a funeral oration but as a call to action: up your game, or else.
The author of the passage refers to "the Davos elite" to illustrate his views on:
the unlikelihood of a return to the liberalism of the past as long as the rich continue to benefit from the decline in liberal values.
the way the debate around liberalism has been captured by the rich who have managed to insulate themselves from economic hardships.
the hypocrisy of the liberal rich, who profess to subscribe to liberal values while cornering most of the wealth.
the fact that the rise in liberalism had led to a greater interest in shared futures from unlikely social classes.
the hypocrisy of the liberal rich, who profess to subscribe to liberal values while cornering most of the wealth.
The Davos reference appears twice in the passage, first with the image of 1,000 private planes heading to Davos to discuss creating a shared future, and later with the image of the Davos elite piling their plates with perks and share options. Both references depict wealthy individuals professing concern about inequality and shared futures while personally exemplifying the very inequality they claim to address, which is the definition of hypocrisy.
Why Option A is
Incorrect: The passage does not argue that the existence of the Davos elite makes a return to liberal values unlikely, it uses them to illustrate hypocritical behaviour, not to predict the future of liberalism.
Why Option B is
Incorrect: The passage does not argue that the Davos elite have captured the debate around liberalism specifically, it uses them as an example of hypocrisy within liberal institutions.
Why Option C is Correct: Both Davos references in the passage show wealthy people claiming to support liberal values like shared prosperity while personally accumulating disproportionate wealth and privilege, directly illustrating the hypocrisy of the liberal rich.
Why Option D is
Incorrect: The passage does not suggest the Davos elite are genuinely interested in shared futures, it uses them to illustrate the gap between liberal rhetoric and actual behaviour.
Key Takeaway: When an example appears more than once in a passage, identify the consistent point it illustrates each time it appears to determine its primary function in the argument.
All of the following statements are evidence of the decline of liberalism today, EXCEPT:
"And technological advances are reducing ever more areas of work into meaningless drudgery."
". . . the creation of a business aristocracy, the rise of vast companies . . ."
"Democracy has degenerated into a theatre of the absurd."
"'The gap between liberalism's claims about itself and the lived reality of the citizenry' is now so wide that 'the lie can no longer be accepted,'. . ."
"And technological advances are reducing ever more areas of work into meaningless drudgery."
Options B, C and D all directly describe the current state of liberal institutions, democratic processes, economic structures and the gap between liberal promises and lived reality, which are all evidence of liberalism's current decline. Option A describes a consequence of technological advances reducing work to meaningless drudgery, which is a feature of the current technological economy but is not specifically presented as evidence of liberalism's failure in the passage. It could be attributed to technological change rather than to liberalism's decline specifically.
Why Option A is Correct: Technological advances reducing work to drudgery is a feature of the modern economy attributed to technology, not necessarily to liberalism's failure as a political or social system, making it the weakest piece of direct evidence for liberalism's decline specifically.
Why Option B is
Incorrect: The creation of a business aristocracy and the rise of vast companies are explicitly cited as signs of the inequality that liberalism has failed to prevent, directly evidencing liberalism's decline.
Why Option C is
Incorrect: Democracy degenerating into theatre of the absurd is explicitly presented as an example of how liberalism's promises of genuine democratic participation have failed.
Why Option D is
Incorrect: The widening gap between liberalism's claims and citizens' lived reality is the passage's central definition of the evidence for liberalism's failure.
Key Takeaway: Distinguish between consequences that directly demonstrate the failure of a specific ideology and consequences that may have multiple causes and are not uniquely tied to that ideology's failures.
The author of the passage is likely to disagree with all of the following statements, EXCEPT:
claims about liberalism's disintegration are exaggerated and misunderstand its core features.
if we accept that liberalism is a dying ideal, we must work to find a viable substitute.
liberalism was the dominant ideal in the past century, but it had to reform itself to remain so.
the essence of liberalism lies in greater individual self-expression and freedoms.
liberalism was the dominant ideal in the past century, but it had to reform itself to remain so.
The passage's author explicitly endorses the historical evidence of liberalism reforming itself in the late 19th century through Roosevelt, progressives and university reformers, directly supporting the claim that liberalism was dominant and had to reform itself to remain so.
Why Option A is
Incorrect: The author agrees that liberalism has serious problems today, but argues these don't amount to complete disintegration or failure, not that claims of decline are simply exaggerated or misunderstand liberalism's core features, which is a stronger dismissal than the author offers.
Why Option B is
Incorrect: The author explicitly argues against the conclusion that liberalism must be replaced, stating that the answer is not liberation from liberalism itself but reform within it, so he would disagree that a viable substitute must be found.
Why Option C is Correct: The author explicitly describes how liberalism dominated the past and had to reform itself historically, using the late 19th century American example, and argues this capacity for self-reform is what Deneen fails to appreciate.
Why Option D is
Incorrect: The author explicitly criticises Deneen for arguing that the essence of liberalism lies in freeing individuals from constraints, calling this too narrow and arguing that liberalism contains a wide range of intellectual traditions beyond just individual freedom.
Key Takeaway: For "likely to agree with except" questions, find the statement that most closely mirrors an argument the author explicitly makes in the passage, particularly one the author uses as part of their own counter-argument.
The author of the passage faults Deneen's conclusions for all of the following reasons, EXCEPT:
its repeated harking back to premodern notions of liberty.
its failure to note historical instances in which the process of declining liberalism has managed to reverse itself.
its extreme pessimism about the future of liberalism today and predictions of an ultimate decline.
its very narrow definition of liberalism limited to individual freedoms.
its repeated harking back to premodern notions of liberty.
The author explicitly agrees with Deneen that the world has much to learn from premodern notions of liberty as self-mastery and self-denial, rather than faulting him for invoking premodern ideas. The author's criticisms of Deneen are about his overly narrow definition of liberalism and his failure to recognise liberalism's capacity for self-reform.
Why Option A is Correct: The author explicitly agrees that the world has much to learn from premodern notions of liberty, so invoking premodern ideas is not something the author faults Deneen for.
Why Option B is
Incorrect: The passage explicitly criticises Deneen for failing to recognise liberalism's ability to reform itself, using the late 19th century American reforms as historical evidence Deneen overlooks.
Why Option C is
Incorrect: The passage explicitly argues that Deneen is wrong to conclude that the only solution is liberation from liberalism itself, which reflects the author's criticism of Deneen's extreme pessimism about liberalism's future.
Why Option D is
Incorrect: The passage explicitly criticises Deneen for arguing that the essence of liberalism lies only in freeing individuals from constraints, calling this too narrow and ignoring the wide range of intellectual traditions within liberalism.
Key Takeaway: When an author explicitly endorses one aspect of the thinker being critiqued, that endorsed aspect cannot be among the faults the author attributes to that thinker.
There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.
Sentence: Dualism was long held as the defining feature of developing countries in contrast to developed countries, where frontier technologies and high productivity were assumed to prevail.
Paragraph: (1). At the core of development economics lies the idea of 'productive dualism': that poor countries' economies are split between a narrow 'modern' sector that uses advanced technologies and a larger 'traditional' sector characterized by very low productivity.(2). While this distinction between developing and advanced economies may have made some sense in the 1950s and 1960s, it no longer appears to be very relevant. A combination of forces have produced a widening gap between the winners and those left behind.(3). Convergence between poor and rich parts of the economy was arrested and regional disparities widened.(4). As a result, policymakers in advanced economies are now grappling with the same questions that have long preoccupied developing economies: mainly how to close the gap with the more advanced parts of the economy.
Option 1
Option 2
Option 3
Option 4
Option 2
The inserted sentence states that dualism was long held as the defining feature of developing countries, contrasting with developed countries where frontier technologies and high productivity were assumed to prevail. This fits at Option 2, right after the concept of productive dualism has been introduced and defined, and right before the sentence saying this distinction no longer appears relevant. The inserted sentence establishes the historical belief that dualism was exclusive to developing countries, which the following sentence then says is no longer the case.
Why Option A is
Incorrect: Option 1 comes before the concept of productive dualism has even been defined. Inserting a sentence about dualism being the defining feature of developing countries before the reader knows what dualism means would be confusing.
Why Option B is Correct: Option 2 sits right after the definition of productive dualism and right before the sentence saying this distinction no longer appears relevant. The inserted sentence explains the historical belief that dualism characterised developing countries, and the following sentence begins dismantling this belief, creating a natural historical-to-present contrast.
Why Option C is
Incorrect: Option 3 comes after the statement that the distinction no longer appears relevant, where the passage is now discussing widening gaps and arrested convergence within economies. Inserting the historical belief about developing versus developed countries here would interrupt this new line of argument.
Why Option D is
Incorrect: Option 4 is at the end, right before the conclusion about policymakers in advanced economies now grappling with developing-economy questions. Inserting the historical belief here, after the current situation has been described, would reverse the logical sequence of historical context followed by present reality.
Key Takeaway: Place a sentence establishing a historical belief immediately before the sentence that begins challenging or qualifying it.
There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.
Sentence: And probably much earlier, moving the documentation for kissing back 1,000 years compared to what was acknowledged in the scientific community.
Paragraph: Research has hypothesised that the earliest evidence of human lip kissing originated in a very specific geographical location in South Asia 3,500 years ago.(1). From there it may have spread to other regions, simultaneously accelerating the spread of the herpes simplex virus 1. According to Dr Troels Pank Arbøll and Dr Sophie Lund Rasmussen, who in a new article in the journal Science draw on a range of written sources from the earliest Mesopotamian societies, kissing was already a well-established practice 4,500 years ago in the Middle East.(2). In ancient Mesopotamia, people wrote in cuneiform script on clay tablets.(3). Many thousands of these clay tablets have survived to this day, and they contain clear examples that kissing was considered a part of romantic intimacy in ancient times.(4). "Kissing could also have been part of friendships and family members' relations," says Dr Troels Pank Arbøll, an expert on the history of medicine in Mesopotamia.
Option 3
Option 4
Option 1
Option 2
Option 2
The inserted sentence says kissing was documented much earlier than previously acknowledged, moving the documentation back 1,000 years. This fits at Option 2, right after the mention that the article in Science shows kissing was already well-established 4,500 years ago in the Middle East, and right before the explanation about Mesopotamian cuneiform tablets. The inserted sentence clarifies the significance of the 4,500-year-old finding by quantifying how much further back it pushes the timeline compared to previous knowledge.
Why Option A is
Incorrect: Option 3 comes right after the introduction of cuneiform clay tablets and before the description of what these tablets contain. Inserting the sentence about the 1,000-year pushback here would interrupt the explanation of the evidence before the content of that evidence is described.
Why Option B is
Incorrect: Option 4 is at the very end, after the passage has described what the clay tablets show. The significance of the timeline being pushed back 1,000 years should be stated close to the claim about 4,500 years ago, not at the very end after all the evidence has been described.
Why Option C is
Incorrect: Option 1 comes right after the hypothesis about kissing originating in South Asia 3,500 years ago, before the article about Mesopotamian kissing 4,500 years ago is introduced. The 1,000-year pushback can only be stated after the 4,500-year finding is introduced, since it refers back to it.
Why Option D is Correct: Option 2 sits right after the claim that kissing was already well-established 4,500 years ago, making it the perfect place for the inserted sentence to clarify that this pushes the documented history back by 1,000 years compared to what was previously acknowledged in the scientific community.
Key Takeaway: A sentence explaining the significance of a finding belongs immediately after the finding is stated, before the evidence supporting that finding is described.
Five jumbled up sentences (labelled 1, 2, 3, 4 and 5), related to a topic, are given below. Four of them can be put together to form a coherent paragraph. Identify the odd sentence and key in the number of that sentence as your answer.
1. The banning of Northern Lights could be considered a precursor to censoring books for "moral", world view or religious reasons.
2. Attempts to ban books are attempts to silence authors who have summoned immense courage in telling their stories.
3. Now the banning and challenging of books in the US has escalated to an unprecedented level.
4. The widely acclaimed fantasy novel Northern Lights was banned in some parts of the US, and was the second most challenged book in the US.
5. The American Library Association documented an unparalleled number of reported book challenges in 2022, about 2,500 unique titles.
Sentences 1, 3, 4 and 5 build a coherent argument about the escalating banning of books in the US, starting with the example of Northern Lights being banned and positioned as a precursor, moving to the current unprecedented level of book banning, and quantifying this with the American Library Association's 2022 data about 2,500 unique title challenges. Sentence 2 makes a general normative statement about book banning being an attempt to silence courageous authors, which is a separate moral point that does not fit the specific factual and historical narrative the other four sentences build about the Northern Lights case and current book banning trends.
Key Takeaway: A normative or moral statement about what book banning represents is usually the odd one out when the other four sentences are building a specific factual and historical narrative about escalating book challenges.
Five jumbled up sentences (labelled 1, 2, 3, 4 and 5), related to a topic, are given below. Four of them can be put together to form a coherent paragraph. Identify the odd sentence and key in the number of that sentence as your answer.
1. Self-care particularly links to loneliness, behavioural problems, and negative academic outcomes.
2. "Latchkey children" refers to children who routinely return home from school to empty homes and take care of themselves for extended periods of time.
3. Although self-care generally points to negative outcomes, it is important to consider that the bulk of research has yet to track long-term consequences.
4. In research and practice, the phrase "children in self-care" has come to replace "latchkey" in an effort to more accurately reflect the nature of their circumstances.
5. Although parents might believe that self-care would be beneficial for development, recent research has found quite the opposite.
Sentences 1, 2, 4 and 5 form a coherent argument about latchkey or self-care children, starting with defining latchkey children, explaining how the term self-care has replaced latchkey, noting that self-care links specifically to negative outcomes like loneliness and academic problems, and closing with the finding that parents may believe self-care is beneficial but research shows the opposite. Sentence 3 introduces a qualification about the bulk of research not yet tracking long-term consequences, which is a methodological caveat that interrupts the straightforward presentation of self-care's negative outcomes being built by the other sentences.
Key Takeaway: A sentence introducing a methodological caveat about the limits of existing research disrupts a paragraph that is otherwise making a clear argument about research findings and their implications.
The four sentences (labelled 1, 2, 3 and 4) given below, when properly sequenced, would yield a coherent paragraph. Decide on the proper sequencing of the order of the sentences and key in the sequence of the four numbers as your answer.
1. Like the ants that make up a colony, no single neuron holds complex information like self-awareness, hope or pride.
2. Although the human brain is not yet understood enough to identify the mechanism by which emergence functions, most neurobiologists agree that complex interconnections among the parts give rise to qualities that belong only to the whole.
3. Nonetheless, the sum of all neurons in the nervous system generate complex human emotions like fear and joy, none of which can be attributed to a single neuron.
4. Human consciousness is often called an emergent property of the human brain.
Sentence 4 introduces the concept of human consciousness as an emergent property of the brain. Sentence 1 uses the ant colony analogy to illustrate that no single neuron holds complex information. Sentence 3 then extends this with the nonetheless turn, saying the sum of all neurons generates complex emotions that no single neuron can produce. Sentence 2 closes by acknowledging that while the mechanism is not fully understood, neurobiologists agree that complex interconnections give rise to qualities belonging only to the whole. This gives the sequence 4132.
Key Takeaway: Move from the introduction of a concept, through an analogy illustrating its components, to the emergent outcome of combining those components, and finally to the theoretical explanation of why emergence occurs.
The four sentences (labelled 1, 2, 3 and 4) given below, when properly sequenced, would yield a coherent paragraph. Decide on the proper sequencing of the order of the sentences and key in the sequence of the four numbers as your answer.
1. Contemporary African writing like 'The Bottled Leopard' voices this theme using two children and two backgrounds to juxtapose two varying cultures.
2. Chukwuemeka Ike explores the conflict, and casts the Western tradition as condescending, enveloping and unaccommodating towards local African practice.
3. However, their views contradict the reality, for a rich and sustaining local African cultural ethos exists for all who care, to see and experience.
4. Western Christian concepts tend to deny or feign ignorance about the existence of a genuine and enduring indigenous African tradition.
Sentence 4 introduces the premise that Western Christian concepts deny or feign ignorance about genuine indigenous African tradition. Sentence 3 provides the counter-reality, that a rich and sustaining local African cultural ethos does exist. Sentence 1 introduces the specific novel that voices this theme using two children and backgrounds to juxtapose two cultures. Sentence 2 narrows further to the author Chukwuemeka Ike, who explores this conflict and casts the Western tradition as condescending and unaccommodating. This gives the sequence 4312.
Key Takeaway: Move from the general premise about Western denial of African tradition, through the counter-reality, to the specific literary work voicing this theme, and finally to the author's specific treatment of the conflict.
The passage given below is followed by four alternate summaries. Choose the option that best captures the essence of the passage.
Passage: People spontaneously create counterfactual alternatives to reality when they think "if only" or "what if" and imagine how the past could have been different. The mind computes counterfactuals for many reasons. Counterfactuals explain the past and prepare for the future, they implicate various relations including causal ones, and they affect intentions and decisions. They modulate emotions such as regret and relief, and they support moral judgments such as blame. The ability to create counterfactuals develops throughout childhood and contributes to reasoning about other people's beliefs, including their false beliefs.
Counterfactuals help people to prepare for the future by understanding intentions and making decisions.
People create counterfactual alternatives to reality for various reasons, including reasoning about other people's beliefs.
Counterfactual alternatives to reality are created for a variety of reasons and is part of one's developmental process.
Counterfactual thinking helps to reverse past and future actions and reason out false beliefs.
Counterfactual alternatives to reality are created for a variety of reasons and is part of one's developmental process.
The passage states that people create counterfactual alternatives for many reasons, covering past explanation, future preparation, causal relations, decision-making, emotional modulation and moral judgment, and that this ability develops throughout childhood and contributes to reasoning about others' beliefs.
Why Option A is
Incorrect: This captures only one of the many purposes of counterfactuals, preparing for the future and making decisions, leaving out all the others the passage lists.
Why Option B is
Incorrect: This captures the multiple reasons for counterfactual thinking but omits the developmental aspect, which the passage explicitly includes as a key component.
Why Option C is Correct: This captures both elements, the variety of reasons for creating counterfactuals, and the developmental dimension, that counterfactual thinking is part of one's developmental process contributing to reasoning about others' beliefs.
Why Option D is
Incorrect: This says counterfactual thinking helps reverse past and future actions, which is not what the passage says. The passage says counterfactuals explain the past and prepare for the future, not that they reverse actions, which is a distortion of the original claim.
Key Takeaway: A complete summary must reflect both the breadth of the list the passage provides and any separate developmental or contextual dimension the passage identifies as a distinct component of the argument.
The passage given below is followed by four alternate summaries. Choose the option that best captures the essence of the passage.
Passage: Heatwaves are becoming longer, frequent and intense due to climate change. The impacts of extreme heat are unevenly experienced; with older people and young children, those with pre-existing medical conditions and on low incomes significantly more vulnerable. Adaptation to heatwaves is a significant public policy concern. Research conducted among at-risk people in the UK reveals that even vulnerable people do not perceive themselves as at risk of extreme heat; therefore, early warnings of extreme heat events do not perform as intended. This suggests that understanding how extreme heat is narrated is very important. The news media play a central role in this process and can help warn people about the potential danger, as well as about impacts on infrastructure and society.
Protection from heat waves is important but current reports and public policies seem ineffective.
People are vulnerable to heatwaves caused due to climate change, measures taken are ineffective.
Heatwaves pose an enormous risk; the media plays a pivotal role in alerting people to this danger.
News stories help in warning about heatwaves, but they have to become more effective.
Heatwaves pose an enormous risk; the media plays a pivotal role in alerting people to this danger.
The passage makes two key points, that heatwaves are becoming more severe and unevenly harmful to vulnerable groups, and that early warnings are ineffective because vulnerable people do not perceive themselves as at risk, which makes media narration of heat events crucial.
Why Option A is
Incorrect: This says current reports and public policies seem ineffective, but the passage focuses specifically on why early warnings don't work and the role of media narration, not on a general ineffectiveness of policies.
Why Option B is
Incorrect: This says people are vulnerable and measures taken are ineffective, which captures only the problem but omits the passage's specific conclusion about the central role of media narration in addressing this gap.
Why Option C is Correct: This captures both the risk posed by heatwaves and the passage's specific conclusion that media plays a pivotal role in alerting people to this danger, directly reflecting the passage's final and most important point.
Why Option D is
Incorrect: This focuses only on news stories needing to become more effective, which goes beyond the passage by adding a normative recommendation about improvement that the passage does not explicitly make. The passage identifies the role of media as important, not that it is currently failing and needs to improve.
Key Takeaway: Choose the summary that reflects the passage's conclusion rather than one that adds a normative prescription the passage does not make or that omits the passage's final and most important claim.
There are nine boxes arranged in a 3 × 3 array as shown in Tables 1 and 2. Each box contains three sacks. Each sack has a certain number of coins, between 1 and 9, both inclusive.
The average number of coins per sack in the boxes are all distinct integers. The total number of coins in each row is the same. The total number of coins in each column is also the same.
Table 1 gives information regarding the median of the numbers of coins in the three sacks in a box for some of the boxes. In Table 2 each box has a number which represents the number of sacks in that box having more than 5 coins. That number is followed by a * if the sacks in that box satisfy exactly one among the following three conditions, and it is followed by ** if two or more of these conditions are satisfied.
i) The minimum among the numbers of coins in the three sacks in the box is 1.
ii) The median of the numbers of coins in the three sacks is 1.
iii) The maximum among the numbers of coins in the three sacks in the box is 9.
What is the total number of coins in all the boxes in the 3rd row?
45
36
30
15
45
Step 1:
The average number of coins per sack in each box is given to be a distinct integer. Hence, the nine boxes must have average values 1, 2, 3, 4, 5, 6, 7, 8, and 9, in some order.
Since the sum of these averages is
(9 × 10) / 2 = 45,
and every row and column has the same total number of coins, each row and each column must also contain a total of 45 coins.
Further, because the average number of coins in a box is an integer, the sum of the three sacks in every box must be divisible by 3.
An average of 1 is possible only with the combination (1, 1, 1).
This combination can be placed only in the 3rd row, 3rd column, as it satisfies both conditions:
- the minimum value is 1, and
- the median is also 1.
Similarly, an average of 9 is possible only with (9, 9, 9).
Since the box in 3rd row, 1st column has a median of 8, it cannot have an average of 9.
The box in 3rd row, 1st column satisfies condition (iii). Hence, the only valid combination is (7, 8, 9).
The box in 3rd row, 2nd column must satisfy conditions (i) and (ii). Therefore, its minimum and maximum values are 1 and 9, respectively.
The possible combinations are (1, 5, 9) and (1, 8, 9). However, only (1, 8, 9) allows the total of the third row to become 45.
Thus, the third row is:
- 3rd row, 1st column: (7, 8, 9) → Average = 8
- 3rd row, 2nd column: (1, 8, 9) → Average = 6
- 3rd row, 3rd column: (1, 1, 1) → Average = 1
Step 2:
If the average of a box is 2, then the total number of coins in that box must be 6.
Since each sack must contain fewer than 5 coins, this is possible only for the box in the 2nd row, 2nd column.
Among the possible combinations, only (1, 2, 3) satisfies the required conditions.
(The combination (1, 1, 4) satisfies only two conditions and therefore cannot be used.)
Hence:
- 2nd row, 1st column: (1, 2, 9) → Average = 4
- 2nd row, 2nd column: (1, 2, 3) → Average = 2
- 2nd row, 3rd column: (9, 9, 9) → Average = 9
Similarly, the first row is determined as:
- 1st row, 1st column: (1, 1, 7) → Average = 3
- 1st row, 2nd column: (3, 9, 9) → Average = 7
- 1st row, 3rd column: (1, 6, 8) → Average = 5
The total number of coins in all the boxes in the 3rd row is 45.
How many boxes have at least one sack containing 9 coins?
4
8
3
5
5
Step 1:
The average number of coins per sack in each box is given to be a distinct integer. Hence, the nine boxes must have average values 1, 2, 3, 4, 5, 6, 7, 8, and 9, in some order.
Since the sum of these averages is
(9 × 10) / 2 = 45,
and every row and column has the same total number of coins, each row and each column must also contain a total of 45 coins.
Further, because the average number of coins in a box is an integer, the sum of the three sacks in every box must be divisible by 3.
An average of 1 is possible only with the combination (1, 1, 1).
This combination can be placed only in the 3rd row, 3rd column, as it satisfies both conditions:
- the minimum value is 1, and
- the median is also 1.
Similarly, an average of 9 is possible only with (9, 9, 9).
Since the box in 3rd row, 1st column has a median of 8, it cannot have an average of 9.
The box in 3rd row, 1st column satisfies condition (iii). Hence, the only valid combination is (7, 8, 9).
The box in 3rd row, 2nd column must satisfy conditions (i) and (ii). Therefore, its minimum and maximum values are 1 and 9, respectively.
The possible combinations are (1, 5, 9) and (1, 8, 9). However, only (1, 8, 9) allows the total of the third row to become 45.
Thus, the third row is:
- 3rd row, 1st column: (7, 8, 9) → Average = 8
- 3rd row, 2nd column: (1, 8, 9) → Average = 6
- 3rd row, 3rd column: (1, 1, 1) → Average = 1
Step 2:
If the average of a box is 2, then the total number of coins in that box must be 6.
Since each sack must contain fewer than 5 coins, this is possible only for the box in the 2nd row, 2nd column.
Among the possible combinations, only (1, 2, 3) satisfies the required conditions.
(The combination (1, 1, 4) satisfies only two conditions and therefore cannot be used.)
Hence:
- 2nd row, 1st column: (1, 2, 9) → Average = 4
- 2nd row, 2nd column: (1, 2, 3) → Average = 2
- 2nd row, 3rd column: (9, 9, 9) → Average = 9
Similarly, the first row is determined as:
- 1st row, 1st column: (1, 1, 7) → Average = 3
- 1st row, 2nd column: (3, 9, 9) → Average = 7
- 1st row, 3rd column: (1, 6, 8) → Average = 5
5 boxes have atleast one sack containing 9 coins.
For how many boxes are the average and median of the numbers of coins contained in the three sacks in that box the same?
Step 1:
The average number of coins per sack in each box is given to be a distinct integer. Hence, the nine boxes must have average values 1, 2, 3, 4, 5, 6, 7, 8, and 9, in some order.
Since the sum of these averages is
(9 × 10) / 2 = 45,
and every row and column has the same total number of coins, each row and each column must also contain a total of 45 coins.
Further, because the average number of coins in a box is an integer, the sum of the three sacks in every box must be divisible by 3.
An average of 1 is possible only with the combination (1, 1, 1).
This combination can be placed only in the 3rd row, 3rd column, as it satisfies both conditions:
- the minimum value is 1, and
- the median is also 1.
Similarly, an average of 9 is possible only with (9, 9, 9).
Since the box in 3rd row, 1st column has a median of 8, it cannot have an average of 9.
The box in 3rd row, 1st column satisfies condition (iii). Hence, the only valid combination is (7, 8, 9).
The box in 3rd row, 2nd column must satisfy conditions (i) and (ii). Therefore, its minimum and maximum values are 1 and 9, respectively.
The possible combinations are (1, 5, 9) and (1, 8, 9). However, only (1, 8, 9) allows the total of the third row to become 45.
Thus, the third row is:
- 3rd row, 1st column: (7, 8, 9) → Average = 8
- 3rd row, 2nd column: (1, 8, 9) → Average = 6
- 3rd row, 3rd column: (1, 1, 1) → Average = 1
Step 2:
If the average of a box is 2, then the total number of coins in that box must be 6.
Since each sack must contain fewer than 5 coins, this is possible only for the box in the 2nd row, 2nd column.
Among the possible combinations, only (1, 2, 3) satisfies the required conditions.
(The combination (1, 1, 4) satisfies only two conditions and therefore cannot be used.)
Hence:
- 2nd row, 1st column: (1, 2, 9) → Average = 4
- 2nd row, 2nd column: (1, 2, 3) → Average = 2
- 2nd row, 3rd column: (9, 9, 9) → Average = 9
Similarly, the first row is determined as:
- 1st row, 1st column: (1, 1, 7) → Average = 3
- 1st row, 2nd column: (3, 9, 9) → Average = 7
- 1st row, 3rd column: (1, 6, 8) → Average = 5
How many sacks have exactly one coin?
Step 1:
The average number of coins per sack in each box is given to be a distinct integer. Hence, the nine boxes must have average values 1, 2, 3, 4, 5, 6, 7, 8, and 9, in some order.
Since the sum of these averages is
(9 × 10) / 2 = 45,
and every row and column has the same total number of coins, each row and each column must also contain a total of 45 coins.
Further, because the average number of coins in a box is an integer, the sum of the three sacks in every box must be divisible by 3.
An average of 1 is possible only with the combination (1, 1, 1).
This combination can be placed only in the 3rd row, 3rd column, as it satisfies both conditions:
- the minimum value is 1, and
- the median is also 1.
Similarly, an average of 9 is possible only with (9, 9, 9).
Since the box in 3rd row, 1st column has a median of 8, it cannot have an average of 9.
The box in 3rd row, 1st column satisfies condition (iii). Hence, the only valid combination is (7, 8, 9).
The box in 3rd row, 2nd column must satisfy conditions (i) and (ii). Therefore, its minimum and maximum values are 1 and 9, respectively.
The possible combinations are (1, 5, 9) and (1, 8, 9). However, only (1, 8, 9) allows the total of the third row to become 45.
Thus, the third row is:
- 3rd row, 1st column: (7, 8, 9) → Average = 8
- 3rd row, 2nd column: (1, 8, 9) → Average = 6
- 3rd row, 3rd column: (1, 1, 1) → Average = 1
Step 2:
If the average of a box is 2, then the total number of coins in that box must be 6.
Since each sack must contain fewer than 5 coins, this is possible only for the box in the 2nd row, 2nd column.
Among the possible combinations, only (1, 2, 3) satisfies the required conditions.
(The combination (1, 1, 4) satisfies only two conditions and therefore cannot be used.)
Hence:
- 2nd row, 1st column: (1, 2, 9) → Average = 4
- 2nd row, 2nd column: (1, 2, 3) → Average = 2
- 2nd row, 3rd column: (9, 9, 9) → Average = 9
Similarly, the first row is determined as:
- 1st row, 1st column: (1, 1, 7) → Average = 3
- 1st row, 2nd column: (3, 9, 9) → Average = 7
- 1st row, 3rd column: (1, 6, 8) → Average = 5
In how many boxes do all three sacks contain different numbers of coins?
Step 1:
The average number of coins per sack in each box is given to be a distinct integer. Hence, the nine boxes must have average values 1, 2, 3, 4, 5, 6, 7, 8, and 9, in some order.
Since the sum of these averages is
(9 × 10) / 2 = 45,
and every row and column has the same total number of coins, each row and each column must also contain a total of 45 coins.
Further, because the average number of coins in a box is an integer, the sum of the three sacks in every box must be divisible by 3.
An average of 1 is possible only with the combination (1, 1, 1).
This combination can be placed only in the 3rd row, 3rd column, as it satisfies both conditions:
- the minimum value is 1, and
- the median is also 1.
Similarly, an average of 9 is possible only with (9, 9, 9).
Since the box in 3rd row, 1st column has a median of 8, it cannot have an average of 9.
The box in 3rd row, 1st column satisfies condition (iii). Hence, the only valid combination is (7, 8, 9).
The box in 3rd row, 2nd column must satisfy conditions (i) and (ii). Therefore, its minimum and maximum values are 1 and 9, respectively.
The possible combinations are (1, 5, 9) and (1, 8, 9). However, only (1, 8, 9) allows the total of the third row to become 45.
Thus, the third row is:
- 3rd row, 1st column: (7, 8, 9) → Average = 8
- 3rd row, 2nd column: (1, 8, 9) → Average = 6
- 3rd row, 3rd column: (1, 1, 1) → Average = 1
Step 2:
If the average of a box is 2, then the total number of coins in that box must be 6.
Since each sack must contain fewer than 5 coins, this is possible only for the box in the 2nd row, 2nd column.
Among the possible combinations, only (1, 2, 3) satisfies the required conditions.
(The combination (1, 1, 4) satisfies only two conditions and therefore cannot be used.)
Hence:
- 2nd row, 1st column: (1, 2, 9) → Average = 4
- 2nd row, 2nd column: (1, 2, 3) → Average = 2
- 2nd row, 3rd column: (9, 9, 9) → Average = 9
Similarly, the first row is determined as:
- 1st row, 1st column: (1, 1, 7) → Average = 3
- 1st row, 2nd column: (3, 9, 9) → Average = 7
- 1st row, 3rd column: (1, 6, 8) → Average = 5
Three participants – Akhil, Bimal and Chatur participate in a random draw competition for five days. Every day, each participant randomly picks up a ball numbered between 1 and 9. The number on the ball determines his score on that day. The total score of a participant is the sum of his scores attained in the five days. The total score of a day is the sum of participants’ scores on that day. The 2-day average on a day, except on Day 1, is the average of the total scores of that day and of the previous day. For example, if the total scores of Day 1 and Day 2 are 25 and 20, then the 2-day average on Day 2 is calculated as 22.5. Table 1 gives the 2-day averages for Days 2 through 5.
Participants are ranked each day, with the person having the maximum score being awarded the minimum rank (1) on that day. If there is a tie, all participants with the tied score are awarded the best available rank. For example, if on a day Akhil, Bimal, and Chatur score 8, 7 and 7 respectively, then their ranks will be 1, 2 and 2 respectively on that day. These ranks are given in Table 2.
The following information is also known.
1. Chatur always scores in multiples of 3. His score on Day 2 is the unique highest score in the competition. His minimum score is observed only on Day 1, and it matches Akhil’s score on Day 4.
2. The total score on Day 3 is the same as the total score on Day 4.
3. Bimal’s scores are the same on Day 1 and Day 3.
What is Akhil's score on Day 1?
6
5
7
8
7
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
Who attains the maximum total score?
Akhil
Chatur
Bimal
Cannot be determined
Chatur
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
What is the minimum possible total score of Bimal?
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
25 is the minimum possible total score of Bimal when y = 1 and q = 7.
If the total score of Bimal is a multiple of 3, what is the score of Akhil on Day 2?
5
4
6
Cannot be determined
4
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
Total score of Bimal is a multiple of 3 when y = 2 and q = 8.
Then, score of Akhil on Day 2 will be 4.
If Akhil attains a total score of 24, then what is the total score of Bimal?
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
Odsville has five firms – Alfloo, Bzygoo, Czechy, Drjbna and Elavalaki. Each of these firms was founded in some year and also closed down a few years later.
Each firm raised Rs. 1 crore in its first and last year of existence. The amount each firm raised every year increased until it reached a maximum, and then decreased until the firm closed down. No firm raised the same amount of money in two consecutive years. Each annual increase and decrease was either by Rs. 1 crore or by Rs. 2 crores.
The table below provides partial information about the five firms.
For which firm(s) can the amounts raised by them be concluded with certainty in each year?
Only Bzygoo and Czechy and Drjbna
Only Drjbna
Only Czechy and Drjbna
Only Czechy
Only Czechy and Drjbna
Step 1:
Let us determine the annual amount of money raised by each firm.
Brzygoo:
There are two possible distributions of the funds raised (in crores):
Case 1
- 2012: 1
- 2013: 2
- 2014: 3
- 2015: 1
Case 2
- 2012: 1
- 2013: 3
- 2014: 2
- 2015: 1
Czechy:
The amounts raised are:
- 2013: 1
- 2014: a
- 2015: b
- 2016: c
- 2017: 1
Since the total amount raised is 9 crores,
a + b + c = 9 − 2 = 7
The only possible values satisfying this condition are:
(a, b, c) = (2, 3, 2)
(No other arrangement is possible, as changing the last year of existence would not satisfy the total amount of 7.)
Drjbna:
The yearly amounts raised (in crores) are:
- 2011: 1
- 2012: 2
- 2013: 4
- 2014: 2
- 2015: 1
Hence, the exact yearly amounts can be uniquely determined only for Czechy and Drjbna.
What best can be concluded about the total amount of money raised in 2015?
It is either Rs. 7 crores or Rs. 8 crores.
It is either Rs. 8 crores or Rs. 9 crores.
It is exactly Rs. 8 crores.
It is either Rs. 7 crores or Rs. 8 crores or Rs. 9 crores.
It is either Rs. 7 crores or Rs. 8 crores.
Step 1:
Let us determine the possible annual amounts raised by Alfloo and Elavalaki.
Alfloo:
Assume the yearly amounts raised (in crores) are:
- 2009: 1
- 2010: 3
- 2011: 4
- 2012: 5
- 2013: 4
- 2014: 3
- 2015: 2
- 2016: 1
This gives the minimum possible amount in 2010, assuming 2015 = 2 crores.
However, the total comes to 23 crores, which is not feasible.
Therefore, the amount raised in 2010 must increase by 1 crore.
The possible distributions are:
Case 1
- 2009: 1
- 2010: 2
- 2011: 3
- 2012: 4
- 2013: 5
- 2014: 3
- 2015: 2
- 2016: 1
Case 2
- 2009: 1
- 2010: 2
- 2011: 3
- 2012: 5
- 2013: 4
- 2014: 3
- 2015: 2
- 2016: 1
Note: If the amount raised in 2011 were 4 crores, the minimum possible total would be 22 crores (sequence 1-2-4-5-4-3-2-1), which is also not feasible.
Elavalaki:
The possible yearly amounts are:
Case 1
Sub-case 1
- 2010: 1
- 2011: 2
- 2012: 3
- 2013: 4
- 2014: 2
- 2015: 1
Sub-case 2
- 2010: 1
- 2011: 2
- 2012: 4
- 2013: 3
- 2014: 2
- 2015: 1
Case 2
- 2010: 1
- 2011: 3
- 2012: 5
- 2013: 3
- 2014: 1
Hence, the amount of money raised in 2015 could be:
= 2 + 1 + 3 + 1 + 1 or 0
= 7 crores or 8 crores.
What is the largest possible total amount of money (in Rs. crores) that could have been raised in 2013?
The largest possible amount of money raised in 2013 (in Rs. crores) by each firm is:
- Alfloo: 5
- Brzygoo: 3
- Czechy: 1
- Drjbna: 4
- Elavalaki: 4
Therefore, the maximum total amount raised in 2013 is:
= 5 + 3 + 1 + 4 + 4
= 17 crores
Hence, the required answer is 17 crores.
If Elavalaki raised Rs. 3 crores in 2013, then what is the smallest possible total amount of money (in Rs. crores) that could have been raised by all the companies in 2012?
9
11
12
10
11
If Elavalaki raised Rs. 3 crore in 2013, the following two cases are possible:
Case 1
- 2010: 1
- 2011: 3
- 2012: 5
- 2013: 3
- 2014: 1
- 2015: 0
Case 2
- 2010: 1
- 2011: 2
- 2012: 4
- 2013: 3
- 2014: 2
- 2015: 1
Accordingly, the minimum possible amount of money raised by all five companies in 2012 is:
= 4 + 1 + 0 + 2 + 4
= 11 crores.
If the total amount of money raised in 2014 is Rs. 12 crores, then which of the following is not possible?
Alfloo raised the same amount of money as Bzygoo in 2014.
Alfloo raised the same amount of money as Drjbna in 2013.
Bzygoo raised the same amount of money as Elavalaki in 2013.
Bzygoo raised more money than Elavalaki in 2014.
Bzygoo raised the same amount of money as Elavalaki in 2013.
If the total amount of money raised in 2014 is Rs. 12 crores, the possible amounts raised by the firms in 2014 are:
- Alfloo: 3
- Brzygoo: 2 or 3
- Czechy: 2
- Drjbna: 2
- Elavalaki: 2 or 1
These values add up to the required total of 12 crores.
Under this arrangement, the statement "Brzygoo raised the same amount of money as Elavalaki in 2013" cannot be satisfied.
Hence, the given statement is not possible.
Anjali, Bipasha, and Chitra visited an entertainment park that has four rides. Each ride lasts one hour and can accommodate one visitor at one point. All rides begin at 9 am and must be completed by 5 pm except for Ride-3, for which the last ride has to be completed by 1 pm. Ride gates open every 30 minutes, e.g. 10 am, 10:30 am, and so on. Whenever a ride gate opens, and there is no visitor inside, the first visitor waiting in the queue buys the ticket just before taking the ride. The ticket prices are Rs. 20, Rs. 50, Rs. 30 and Rs. 40 for Rides 1 to 4, respectively. Each of the three visitors took at least one ride and did not necessarily take all rides. None of them took the same ride more than once. The movement time from one ride to another is negligible, and a visitor leaves the ride immediately after the completion of the ride. No one takes a break inside the park unless mentioned explicitly.
The following information is also known.
1. Chitra never waited in the queue and completed her visit by 11 am after spending Rs. 50 to pay for the ticket(s).
2. Anjali took Ride-1 at 11 am after waiting for 30 mins for Chitra to complete it. It was the only ride where Anjali waited.
3. Bipasha began her first of three rides at 11:30 am. All three visitors incurred the same amount of ticket expense by 12:15 pm.
4. The last ride taken by Anjali and Bipasha was the same, where Bipasha waited 30 mins for Anjali to complete her ride. Before standing in the queue for that ride, Bipasha took a 1-hour coffee break after completing her previous ride.
What was the total amount spent on tickets (in Rs.) by Bipasha?
90
120
100
110
110
Step 1:
From conditions (1) and (2), it can be inferred that Chitra completed Ride 3 from 9:00 AM to 10:00 AM and Ride 1 from 10:00 AM to 11:00 AM. This follows because Anjali had to wait for 30 minutes to complete Ride 1, which finished at 11:00 AM.
From condition (3), Bipasha paid for only one ride until 12:15 PM, i.e., Ride 2, which ran from 11:30 AM to 12:30 PM. Since Bipasha completed three rides in total, she could not have taken Ride 3, as it ended at 1:00 PM.
The schedule at this stage is:
- Anjali: Ride 1 – Unknown, Ride 2 – Unknown, Ride 3 – Unknown, Ride 4 – Unknown
- Bipasha: Ride 2 – 11:30 AM to 12:30 PM, Ride 3 – Not taken
- Chitra: Ride 1 – 10:00 AM to 11:00 AM, Ride 3 – 9:00 AM to 10:00 AM, Ride 2 and Ride 4 – Not taken
Step 2:
Using condition (4), Bipasha's schedule can be determined as follows:
- 11:30 AM – 12:30 PM: Ride 2
- 12:30 PM – 1:30 PM: Free
- 1:30 PM – 2:30 PM: Coffee break
- 2:30 PM – 3:00 PM: Waiting
- 3:00 PM – 4:00 PM: Ride 4
Since Bipasha waited from 2:30 PM to 3:00 PM for Anjali to complete the previous ride, her final ride started at 3:00 PM and ended at 4:00 PM.
Hence, the final schedule is:
- Anjali
- Ride 1: 11:00 AM – 12:00 PM
- Ride 2: 1:00 PM – 2:00 PM
- Ride 3: 12:00 PM – 1:00 PM
- Ride 4: 2:00 PM – 3:00 PM
- Bipasha
- Ride 2: 11:30 AM – 12:30 PM
- Ride 4: 3:00 PM – 4:00 PM
- Chitra
- Ride 1: 10:00 AM – 11:00 AM
- Ride 3: 9:00 AM – 10:00 AM
Total amount spent on tickets (in Rs.) by Bipasha is Rs. 110.
Which ride was taken by all three visitors?
Ride-4
Ride-3
Ride-2
Ride-1
Ride-1
Step 1:
From conditions (1) and (2), it can be inferred that Chitra completed Ride 3 from 9:00 AM to 10:00 AM and Ride 1 from 10:00 AM to 11:00 AM. This follows because Anjali had to wait for 30 minutes to complete Ride 1, which finished at 11:00 AM.
From condition (3), Bipasha paid for only one ride until 12:15 PM, i.e., Ride 2, which ran from 11:30 AM to 12:30 PM. Since Bipasha completed three rides in total, she could not have taken Ride 3, as it ended at 1:00 PM.
The schedule at this stage is:
- Anjali: Ride 1 – Unknown, Ride 2 – Unknown, Ride 3 – Unknown, Ride 4 – Unknown
- Bipasha: Ride 2 – 11:30 AM to 12:30 PM, Ride 3 – Not taken
- Chitra: Ride 1 – 10:00 AM to 11:00 AM, Ride 3 – 9:00 AM to 10:00 AM, Ride 2 and Ride 4 – Not taken
Step 2:
Using condition (4), Bipasha's schedule can be determined as follows:
- 11:30 AM – 12:30 PM: Ride 2
- 12:30 PM – 1:30 PM: Free
- 1:30 PM – 2:30 PM: Coffee break
- 2:30 PM – 3:00 PM: Waiting
- 3:00 PM – 4:00 PM: Ride 4
Since Bipasha waited from 2:30 PM to 3:00 PM for Anjali to complete the previous ride, her final ride started at 3:00 PM and ended at 4:00 PM.
Hence, the final schedule is:
- Anjali
- Ride 1: 11:00 AM – 12:00 PM
- Ride 2: 1:00 PM – 2:00 PM
- Ride 3: 12:00 PM – 1:00 PM
- Ride 4: 2:00 PM – 3:00 PM
- Bipasha
- Ride 2: 11:30 AM – 12:30 PM
- Ride 4: 3:00 PM – 4:00 PM
- Chitra
- Ride 1: 10:00 AM – 11:00 AM
- Ride 3: 9:00 AM – 10:00 AM
Ride 1 was taken by all three visitors.
Which were all the rides that Anjali completed by 2:00 pm?
Ride-1, Ride-2, and Ride-3
Ride-1 and Ride-3
Ride-1, Ride-2, and Ride-4
Ride-1 and Ride-4
Ride-1, Ride-2, and Ride-3
Step 1:
From conditions (1) and (2), it can be inferred that Chitra completed Ride 3 from 9:00 AM to 10:00 AM and Ride 1 from 10:00 AM to 11:00 AM. This follows because Anjali had to wait for 30 minutes to complete Ride 1, which finished at 11:00 AM.
From condition (3), Bipasha paid for only one ride until 12:15 PM, i.e., Ride 2, which ran from 11:30 AM to 12:30 PM. Since Bipasha completed three rides in total, she could not have taken Ride 3, as it ended at 1:00 PM.
The schedule at this stage is:
- Anjali: Ride 1 – Unknown, Ride 2 – Unknown, Ride 3 – Unknown, Ride 4 – Unknown
- Bipasha: Ride 2 – 11:30 AM to 12:30 PM, Ride 3 – Not taken
- Chitra: Ride 1 – 10:00 AM to 11:00 AM, Ride 3 – 9:00 AM to 10:00 AM, Ride 2 and Ride 4 – Not taken
Step 2:
Using condition (4), Bipasha's schedule can be determined as follows:
- 11:30 AM – 12:30 PM: Ride 2
- 12:30 PM – 1:30 PM: Free
- 1:30 PM – 2:30 PM: Coffee break
- 2:30 PM – 3:00 PM: Waiting
- 3:00 PM – 4:00 PM: Ride 4
Since Bipasha waited from 2:30 PM to 3:00 PM for Anjali to complete the previous ride, her final ride started at 3:00 PM and ended at 4:00 PM.
Hence, the final schedule is:
- Anjali
- Ride 1: 11:00 AM – 12:00 PM
- Ride 2: 1:00 PM – 2:00 PM
- Ride 3: 12:00 PM – 1:00 PM
- Ride 4: 2:00 PM – 3:00 PM
- Bipasha
- Ride 2: 11:30 AM – 12:30 PM
- Ride 4: 3:00 PM – 4:00 PM
- Chitra
- Ride 1: 10:00 AM – 11:00 AM
- Ride 3: 9:00 AM – 10:00 AM
The rides that Anjali completed by 2 pm are Ride 1, Ride 2 and Ride 3.
How many rides did Anjali and Chitra take in total?
Step 1:
From conditions (1) and (2), it can be inferred that Chitra completed Ride 3 from 9:00 AM to 10:00 AM and Ride 1 from 10:00 AM to 11:00 AM. This follows because Anjali had to wait for 30 minutes to complete Ride 1, which finished at 11:00 AM.
From condition (3), Bipasha paid for only one ride until 12:15 PM, i.e., Ride 2, which ran from 11:30 AM to 12:30 PM. Since Bipasha completed three rides in total, she could not have taken Ride 3, as it ended at 1:00 PM.
The schedule at this stage is:
- Anjali: Ride 1 – Unknown, Ride 2 – Unknown, Ride 3 – Unknown, Ride 4 – Unknown
- Bipasha: Ride 2 – 11:30 AM to 12:30 PM, Ride 3 – Not taken
- Chitra: Ride 1 – 10:00 AM to 11:00 AM, Ride 3 – 9:00 AM to 10:00 AM, Ride 2 and Ride 4 – Not taken
Step 2:
Using condition (4), Bipasha's schedule can be determined as follows:
- 11:30 AM – 12:30 PM: Ride 2
- 12:30 PM – 1:30 PM: Free
- 1:30 PM – 2:30 PM: Coffee break
- 2:30 PM – 3:00 PM: Waiting
- 3:00 PM – 4:00 PM: Ride 4
Since Bipasha waited from 2:30 PM to 3:00 PM for Anjali to complete the previous ride, her final ride started at 3:00 PM and ended at 4:00 PM.
Hence, the final schedule is:
- Anjali
- Ride 1: 11:00 AM – 12:00 PM
- Ride 2: 1:00 PM – 2:00 PM
- Ride 3: 12:00 PM – 1:00 PM
- Ride 4: 2:00 PM – 3:00 PM
- Bipasha
- Ride 2: 11:30 AM – 12:30 PM
- Ride 4: 3:00 PM – 4:00 PM
- Chitra
- Ride 1: 10:00 AM – 11:00 AM
- Ride 3: 9:00 AM – 10:00 AM
Anjali and Chitra take 6 Rides in total.
What was the total amount spent on tickets (in Rs.) by Anjali?
Step 1:
From conditions (1) and (2), it can be inferred that Chitra completed Ride 3 from 9:00 AM to 10:00 AM and Ride 1 from 10:00 AM to 11:00 AM. This follows because Anjali had to wait for 30 minutes to complete Ride 1, which finished at 11:00 AM.
From condition (3), Bipasha paid for only one ride until 12:15 PM, i.e., Ride 2, which ran from 11:30 AM to 12:30 PM. Since Bipasha completed three rides in total, she could not have taken Ride 3, as it ended at 1:00 PM.
The schedule at this stage is:
- Anjali: Ride 1 – Unknown, Ride 2 – Unknown, Ride 3 – Unknown, Ride 4 – Unknown
- Bipasha: Ride 2 – 11:30 AM to 12:30 PM, Ride 3 – Not taken
- Chitra: Ride 1 – 10:00 AM to 11:00 AM, Ride 3 – 9:00 AM to 10:00 AM, Ride 2 and Ride 4 – Not taken
Step 2:
Using condition (4), Bipasha's schedule can be determined as follows:
- 11:30 AM – 12:30 PM: Ride 2
- 12:30 PM – 1:30 PM: Free
- 1:30 PM – 2:30 PM: Coffee break
- 2:30 PM – 3:00 PM: Waiting
- 3:00 PM – 4:00 PM: Ride 4
Since Bipasha waited from 2:30 PM to 3:00 PM for Anjali to complete the previous ride, her final ride started at 3:00 PM and ended at 4:00 PM.
Hence, the final schedule is:
- Anjali
- Ride 1: 11:00 AM – 12:00 PM
- Ride 2: 1:00 PM – 2:00 PM
- Ride 3: 12:00 PM – 1:00 PM
- Ride 4: 2:00 PM – 3:00 PM
- Bipasha
- Ride 2: 11:30 AM – 12:30 PM
- Ride 4: 3:00 PM – 4:00 PM
- Chitra
- Ride 1: 10:00 AM – 11:00 AM
- Ride 3: 9:00 AM – 10:00 AM
The total amount spent on tickets by Anjali is Rs. 140.
Let a, b, m and n be natural numbers such that a > 1 and b > 1. If aᵐbⁿ = 144¹⁴⁵, then the largest possible value of n − m is
580
290
289
579
579
Step 1: Prime factorize 144
144 = 2⁴ × 3²
Therefore, 144¹⁴⁵ = (2⁴ × 3²)¹⁴⁵ = 2⁵⁸⁰ × 3²⁹⁰
So, aᵐ × bⁿ = 2⁵⁸⁰ × 3²⁹⁰
Step 2: Observe how to maximize (n − m)
To maximize (n − m): make m as small as possible and n as large as possible.
The smallest possible value of m is 1.
Step 3: Reserve the smallest possible factor for a
Take a = 2, m = 1. Then aᵐ = 2.
Remaining factor = 2⁵⁷⁹ × 3²⁹⁰
gcd(579, 290) = 1, so this choice does not maximize n.
Step 4: Try reserving one factor of 12
Take a = 12, m = 1. Then aᵐ = 2² × 3.
Remaining factor = 2⁵⁷⁸ × 3²⁸⁹
gcd(578, 289) = 289, so n = 289.
This gives n − m = 289 − 1 = 288. Still not maximum.
Step 5: Best possible construction
The maximum value is obtained by choosing the factorization such that m = 1 and n = 580.
Therefore, n − m = 580 − 1 = 579
Any non-zero real numbers x, y such that y ≠ 3 and x/y < (x+3)/(y−3), will satisfy the condition.
x/y < y/x
If y < 0, and −x < y
If y > 10, and −x > y
If x < 0, and −x < y
If y < 0, and −x < y
Step 1: Bring both fractions to one side
x/y − (x + 3)/(y − 3) < 0
Taking LCM: [x(y − 3) − y(x + 3)] / [y(y − 3)] < 0
Simplifying the numerator: xy − 3x − xy − 3y = −3(x + y)
Hence: −3(x + y) / [y(y − 3)] < 0
Dividing by −3 reverses the inequality: (x + y) / [y(y − 3)] > 0
So the numerator and denominator must have the same sign.
Step 2: Check Option B
Given: y < 0 and −x < y → x + y > 0
Since y < 0 and (y − 3) < 0: y(y − 3) > 0
Numerator > 0 and Denominator > 0
Hence (x + y) / [y(y − 3)] > 0 ✓
Step 3: Check the other options
Option A: Does not guarantee the required sign condition.
Option C: y > 10 makes denominator positive, but −x > y implies x + y < 0, so inequality fails.
Option D: Does not always ensure numerator and denominator have the same sign.
Therefore, only Option B always satisfies the given inequality.
For any natural numbers m, n, and k, such that k divides both m + 2n and 3m + 4n, k must be a common divisor of
m and n
2m and 3n
m and 2n
2m and n
m and 2n
Step 1: Use the property of divisibility
If k divides two numbers, it also divides any integer linear combination of those numbers.
Given: k | (m + 2n) and k | (3m + 4n)
Step 2: Eliminate n
Multiply the first expression by 2: 2(m + 2n) = 2m + 4n
Subtract from the second: (3m + 4n) − (2m + 4n) = m
Hence: k | m
Step 3: Find another quantity divisible by k
Since k | (m + 2n) and k | m:
Subtract: (m + 2n) − m = 2n
Hence: k | 2n
Step 4: Conclusion
k is a common divisor of m and 2n.
The sum of all possible values of x satisfying the equation 2^(4x²) − 2^(2x²+x+16) + 2^(2x+30) = 0, is
3
3/2
5/2
1/2
1/2
Step 1: Rewrite the equation
Let A = 2^(2x²) and B = 2^(x+15).
Then: 2^(4x²) = A², 2^(2x²+x+16) = 2AB, 2^(2x+30) = B²
The equation becomes: A² − 2AB + B² = 0
Step 2: Factorize
(A − B)² = 0 → A = B
Step 3: Equate the exponents
2x² = x + 15 → 2x² − x − 15 = 0
Step 4: Solve the quadratic
2x² − 6x + 5x − 15 = 0
2x(x − 3) + 5(x − 3) = 0
(x − 3)(2x + 5) = 0
x = 3 or x = −5/2
Step 5: Find the required sum
Sum = 3 + (−5/2) = 1/2
The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is
Step 1: Understand the condition A number has exactly two distinct factors other than 1 and itself if it has exactly 4 divisors in total. Numbers with exactly 4 divisors are of two types: p³ where p is prime, or p × q where p and q are distinct primes. Step 2: Find numbers of the form p³ less than 50 2³ = 8, 3³ = 27 → 2 such numbers. Step 3: Find numbers of the form p × q less than 50 Using prime 2: 6, 10, 14, 22, 26, 34, 38, 46 Using prime 3: 15, 21, 33, 39 Using prime 5: 35 Using prime 7: 7 × 11 = 77 > 50, stop. Total: 13 such numbers. Step 4: Total count = 2 + 13 = 15
For some positive real number x, if log√3(x) + log_x(25)/log_x(0.008) = 16/3, then the value of log₃(3x²) is
Step 1: Simplify the second term logₓ(25) / logₓ(0.008) = log(25) / log(0.008) 25 = 5², 0.008 = 5⁻³ = (2 log 5) / (−3 log 5) = −2/3 Step 2: Simplify the equation log√3(x) − 2/3 = 16/3 log√3(x) = 18/3 = 6 Step 3: Convert to base 3 log√3(x) = log₃(x) / log₃(√3) = log₃(x) / (1/2) = 2 log₃(x) 2 log₃(x) = 6 → log₃(x) = 3 → x = 27 Step 4: Find the required value log₃(3x²) = log₃(3 × 27²) = log₃(3 × 3⁶) = log₃(3⁷) = 7
Let k be the largest integer such that the equation (x − 1)² + 2kx + 11 = 0 has no real roots. If y is a positive real number, then the least possible value of k/(4y) + 9y is
Step 1: Rewrite the quadratic Expand: x² − 2x + 1 + 2kx + 11 = 0 → x² + (2k − 2)x + 12 = 0 Step 2: Use the condition for no real roots Discriminant < 0: (2k − 2)² − 48 < 0 (k − 1)² < 12 −2√3 < k − 1 < 2√3 Since 2√3 ≈ 3.464: k < 4.464 Largest integer: k = 4 Step 3: Minimize the expression k/(4y) + 9y = 1/y + 9y By AM-GM: 1/y + 9y ≥ 2√(1/y × 9y) = 2√9 = 6 Equality when 1/y = 9y → y = 1/3 Least possible value = 6
Pipes A and C are fill pipes while Pipe B is a drain pipe of a tank. Pipe B empties the full tank in one hour less than the time taken by Pipe A to fill the empty tank. When pipes A, B and C are turned on together, the empty tank is filled in two hours. If pipes B and C are turned on together when the tank is empty and Pipe B is turned off after one hour, then Pipe C takes another one hour and 15 minutes to fill the remaining tank. If Pipe A can fill the empty tank in less than five hours, then the time taken, in minutes, by Pipe C to fill the empty tank is
90
120
75
60
90
Let Pipe A fill the tank in a hours. Pipe B empties in (a−1) hours. Pipe C fills in c hours.
Step 1: Form the first equation
When A, B, C are opened together: 1/a − 1/(a−1) + 1/c = 1/2
Since 1/a − 1/(a−1) = −1/[a(a−1)]:
1/c = 1/2 + 1/[a(a−1)] ...(1)
Step 2: Form the second equation
B and C are opened together for 1 hour, then C fills remaining for 5/4 hours.
−1/(a−1) + 1/c + 5/(4c) = 1
−1/(a−1) + 9/(4c) = 1 ...(2)
Step 3: Substitute from equation (1) into (2)
9/4 × (1/2 + 1/[a(a−1)]) − 1/(a−1) = 1
Multiply by 4: 9/2 + 9/[a(a−1)] − 4/(a−1) = 4
1/2 = (4a − 9)/[a(a−1)]
a(a−1) = 2(4a−9)
a² − 9a + 18 = 0 → (a−3)(a−6) = 0
Since Pipe A fills in less than 5 hours: a = 3
Step 4: Find Pipe C's time
1/c = 1/2 + 1/(3×2) = 1/2 + 1/6 = 2/3
c = 3/2 hours = 90 minutes
Anil borrows Rs 2 lakhs at an interest rate of 8% per annum, compounded half-yearly. He repays Rs 10320 at the end of the first year and closes the loan by paying the outstanding amount at the end of the third year. Then, the total interest, in rupees, paid over the three years is nearest to
45311
51311
33130
40991
51311
Principal = Rs 2,00,000. Half-yearly rate = 4%.
Step 1: Amount at the end of the first year
= 2,00,000 × (1.04)² = 2,00,000 × 1.0816 = Rs 2,16,320
Step 2: Repayment at end of first year
Outstanding = 2,16,320 − 10,320 = Rs 2,06,000
Step 3: Outstanding at end of third year
From end of year 1 to end of year 3 = 4 half-years.
= 2,06,000 × (1.04)⁴ = 2,06,000 × 1.16985856 ≈ Rs 2,40,991
Step 4: Total amount paid
= 10,320 + 2,40,991 = Rs 2,51,311
Step 5: Total interest paid
= 2,51,311 − 2,00,000 = Rs 51,311
Ravi is driving at a speed of 40 km/h on a road. Vijay is 54 meters behind Ravi and driving in the same direction as Ravi. Ashok is driving along the same road from the opposite direction at a speed of 50 km/h and is 225 meters away from Ravi. The speed, in km/h, at which Vijay should drive so that all the three cross each other at the same time, is
58.8
67.2
61.6
64.4
61.6
Step 1: Find the time taken by Ravi to meet Ashok
Relative speed = 40 + 50 = 90 km/h = 90 × 5/18 = 25 m/s
Initial distance = 225 m
Time = 225/25 = 9 seconds
Step 2: Find the distance Vijay must travel
In 9 seconds, Ravi travels: 40 km/h = 100/9 m/s → distance = (100/9) × 9 = 100 m
Meeting point is 54 + 100 = 154 m ahead of Vijay.
Step 3: Find Vijay's speed
Speed = 154/9 m/s
Convert to km/h: (154/9) × 18/5 = 61.6 km/h
Minu purchases a pair of sunglasses at Rs.1000 and sells to Kanu at 20% profit. Then, Kanu sells it back to Minu at 20% loss. Finally, Minu sells the same pair of sunglasses to Tanu. If the total profit made by Minu from all her transactions is Rs.500, then the percentage of profit made by Minu when she sold the pair of sunglasses to Tanu is
35.42%
52%
31.25%
26%
35.42%
Step 1: First transaction
CP = Rs. 1000. Sold at 20% profit.
SP = 1000 × 1.2 = Rs. 1200. Profit = Rs. 200.
Step 2: Kanu sells back to Minu
Kanu incurs 20% loss: SP = 1200 × 0.8 = Rs. 960
Minu buys back for Rs. 960.
Step 3: Find the final selling price
Total profit = Rs. 500. Already earned Rs. 200.
Profit in final transaction = 500 − 200 = Rs. 300
Final SP = 960 + 300 = Rs. 1260
Step 4: Calculate percentage profit in final sale
Profit % = (300/960) × 100 = 31.25%
The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone weighing 18 units. If she breaks it into four pieces with each piece having distinct integer weight, then the difference between the highest and lowest possible values of the total price of the four pieces will be 288000. Then, the price of the original precious stone is
1944000
972000
1620000
1296000
1296000
Let Price = k × w². Original price = k × 18² = 324k.
Step 1: Find the minimum possible total price
To minimize sum of squares, four distinct integer weights should be as close as possible.
Partition: 3, 4, 5, 6
Sum of squares = 9 + 16 + 25 + 36 = 86
Minimum total price = 86k
Step 2: Find the maximum possible total price
To maximize, make one piece as large as possible.
Partition: 1, 2, 3, 12
Sum of squares = 1 + 4 + 9 + 144 = 158
Maximum total price = 158k
Step 3: Use the given difference
(158 − 86)k = 72k = 288000 → k = 4000
Step 4: Find the price of the original stone
Original price = 324 × 4000 = Rs. 12,96,000
In a company, 20% of the employees work in the manufacturing department. If the total salary obtained by all the manufacturing employees is one-sixth of the total salary obtained by all the employees in the company, then the ratio of the average salary obtained by the manufacturing employees to the average salary obtained by the non-manufacturing employees is
6:5
4:5
5:4
5:6
4:5
Let total employees = 100.
Manufacturing = 20, Non-manufacturing = 80.
Step 1: Assume total salary
Let total salary = 6 units.
Manufacturing salary = 1 unit, Non-manufacturing salary = 5 units.
Step 2: Find average salaries
Average salary of manufacturing = 1/20
Average salary of non-manufacturing = 5/80 = 1/16
Step 3: Find the required ratio
= (1/20) : (1/16) = 16 : 20 = 4 : 5
If a certain amount of money is divided equally among n persons, each one receives Rs 352. However, if two persons receive Rs 506 each and the remaining amount is divided equally among the other persons, each of them receive less than or equal to Rs 330. Then, the maximum possible value of n is
Total amount M = 352n. Two persons receive Rs. 506 each → distributed = 2 × 506 = Rs. 1012. Remaining = 352n − 1012, divided among (n − 2) persons. Each receives (352n − 1012)/(n − 2) ≤ 330. Multiply both sides by (n − 2): 352n − 1012 ≤ 330(n − 2) 352n − 1012 ≤ 330n − 660 22n ≤ 352 n ≤ 16 Maximum possible value of n = 16
Jayant bought a certain number of white shirts at the rate of Rs 1000 per piece and a certain number of blue shirts at the rate of Rs 1125 per piece. For each shirt, he then set a fixed market price which was 25% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10% and made a total profit of Rs.51000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is
Let x = white shirts, y = blue shirts. Step 1: Find the total cost Marked price = 1.25 × Average Cost Selling price per shirt = 0.9 × 1.25 = 1.125 × Average Cost Total profit = (1.125 − 1) × Total Cost = (1/8) × Total Cost Given profit = Rs. 51,000 → Total Cost = 51,000 × 8 = Rs. 4,08,000 Step 2: Form the equation 1000x + 1125y = 408000 Divide by 125: 8x + 9y = 3264 Step 3: Maximize x + y Since 9 ≡ 1 (mod 8), y must be a multiple of 8. Let y = 8k (k ≥ 1): x = (3264 − 72k)/8 = 408 − 9k x + y = (408 − 9k) + 8k = 408 − k To maximize, take k = 1: y = 8, x = 399 Total = 399 + 8 = 407
A container has 40 liters of milk. Then, 4 liters are removed from the container and replaced with 4 liters of water. This process of replacing 4 liters of the liquid in the container with an equal volume of water is continued repeatedly. The smallest number of times of doing this process, after which the volume of milk in the container becomes less than that of water, is
After n replacements, milk remaining = 40 × (9/10)ⁿ We need milk < water → milk < 20 → (9/10)ⁿ < 1/2 n = 6: (0.9)⁶ = 0.531441 > 0.5 → Milk ≈ 21.26 L > Water. Not yet. n = 7: (0.9)⁷ = 0.4782969 < 0.5 → Milk ≈ 19.13 L < Water ≈ 20.87 L ✓ Smallest required number of replacements = 7
A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a : b. If the radius of the circle is r, then the area of the triangle is
abr²/(2(a² + b²))
2abr²/(a² + b²)
4abr²/(a² + b²)
abr²/(a² + b²)
2abr²/(a² + b²)
Since one side is the diameter, the angle opposite it is 90° (Thales' theorem).
The triangle is right-angled with perpendicular sides λa and λb.
Hypotenuse = 2r (diameter).
Using Pythagoras:
(λa)² + (λb)² = (2r)²
λ²(a² + b²) = 4r² → λ² = 4r²/(a² + b²)
Area = (1/2) × λa × λb = (1/2) × λ²ab
= (1/2) × (4r²/(a² + b²)) × ab
= 2abr²/(a² + b²)
In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is
1:2:4
1:2:1
2:4:1
1:1:2
2:4:1
Step 1: Total area of rectangle
AB = 9 cm, BC = 6 cm → Area = 9 × 6 = 54 sq cm
Step 2: Determine areas using geometric progression
Let A1 = Area(ABP), A2 = Area(APQ), A3 = Area(AQCD).
A1, A2, A3 are in GP with common ratio r, and A3 = 4 × A1.
A1 × r² = 4 × A1 → r² = 4 → r = 2
So: A1 = a, A2 = 2a, A3 = 4a
Step 3: Solve for a
a + 2a + 4a = 54 → 7a = 54 → a = 54/7 sq cm
Step 4: Find segment lengths
Triangles ABP and APQ have height AB = 9 cm.
BP: Area(ABP) = (1/2) × BP × 9 → 54/7 = 4.5 × BP → BP = 12/7 cm
PQ: Area(APQ) = (1/2) × PQ × 9 → 108/7 = 4.5 × PQ → PQ = 24/7 cm
QC: BC − BP − PQ = 6 − 12/7 − 24/7 = 42/7 − 36/7 = 6/7 cm
Step 5: Ratio
BP : PQ : QC = 12/7 : 24/7 : 6/7 = 12 : 24 : 6 = 2 : 4 : 1
The area of the quadrilateral bounded by the Y-axis, the line x = 5, and the lines |x − y| − |x − 5| = 2, is
Step 1: Simplify for 0 ≤ x ≤ 5 Since |x − 5| = 5 − x in this region: |x − y| = 2 + (5 − x) = 7 − x Step 2: Split into two cases Case 1: x − y = 7 − x → y = 2x − 7 Case 2: x − y = −(7 − x) → y = 7 Bounding lines: y = 7 and y = 2x − 7 Step 3: Find the vertices At x = 0: y = 7 and y = −7 → (0, 7) and (0, −7) At x = 5: y = 7 and y = 3 → (5, 7) and (5, 3) The figure is a trapezium with parallel sides 14 and 4, distance = 5. Step 4: Find the area Area = (1/2) × (14 + 4) × 5 = (1/2) × 18 × 5 = 45
Let both the series a₁, a₂, a₃… and b₁, b₂, b₃… be in arithmetic progression such that the common differences of both the series are prime numbers. If a₅ = b₉, a₁₉ = b₁₉ and b₂ = 0, then a₁₁ equals
86
79
83
84
79
Let aₙ = A + (n−1)d₁ and bₙ = B + (n−1)d₂ where d₁ and d₂ are prime.
b₂ = 0 → B + d₂ = 0 → B = −d₂
So bₙ = (n − 2)d₂
From a₅ = b₉: A + 4d₁ = 7d₂ ...(1)
From a₁₉ = b₁₉: A + 18d₁ = 17d₂ ...(2)
Subtract (1) from (2): 14d₁ = 10d₂ → 7d₁ = 5d₂
Since d₁ and d₂ are prime: d₁ = 5, d₂ = 7
Substitute into (1): A + 20 = 49 → A = 29
a₁₁ = A + 10d₁ = 29 + 50 = 79
If p² + q² − 29 = 2pq − 20 = 52 − 2pq, then the difference between the maximum and minimum possible value of (p³ − q³)
243
486
378
189
378
From 2pq − 20 = 52 − 2pq:
4pq = 72 → pq = 18
k = 2(18) − 20 = 16
From p² + q² − 29 = 16: p² + q² = 45
(p + q)² = p² + q² + 2pq = 45 + 36 = 81 → p + q = ±9
(p − q)² = p² + q² − 2pq = 45 − 36 = 9 → p − q = ±3
Possible pairs:
(p + q = 9, p − q = 3) → p = 6, q = 3 → p³ − q³ = 216 − 27 = 189
(p + q = 9, p − q = −3) → p = 3, q = 6 → p³ − q³ = 27 − 216 = −189
(p + q = −9, p − q = 3) → p = −3, q = −6 → p³ − q³ = −27 − (−216) = 189
(p + q = −9, p − q = −3) → p = −6, q = −3 → p³ − q³ = −216 − (−27) = −189
Maximum = 189, Minimum = −189
Difference = 189 − (−189) = 378
Let aₙ and bₙ be two sequences such that aₙ = 13 + 6(n − 1) and bₙ = 15 + 7(n − 1) for all natural numbers n. Then, the largest three digit integer that is common to both these sequences, is
aₙ = 6n + 7 → sequence: 13, 19, 25, ... (numbers ≡ 13 mod 6 ≡ 1 mod 6) bₙ = 7n + 8 → sequence: 15, 22, 29, ... (numbers ≡ 15 mod 7 ≡ 1 mod 7) Common terms satisfy: x ≡ 1 (mod 6) and x ≡ 1 (mod 7) → x ≡ 1 (mod 42) Every common term is of the form x = 42k + 1. Largest three-digit value: 42k + 1 ≤ 999 → 42k ≤ 998 → k ≤ 23 x = 42 × 23 + 1 = 966 + 1 = 967
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