CAT 2025 — Slot 3
Full paper, free to view. Pick a section below, click any question to see the answer and explanation.
VARC topics covered: Culture & Identity RC, Nature And Environment RC, Odd One Out, Para Completion, Para Summary, Parajumble, Philosophy RC, Politics & Governance RC
In 1982, a raging controversy broke out over a forest act drafted by the Government of India. This act sought to strengthen the already extensive powers enjoyed by the forest bureaucracy in controlling the extraction, disposal and sale of forest produce. It also gave forest officials greater powers to strictly regulate the entry of any person into reserved forest areas. While forest officials justified the act on the grounds that it was necessary to stop the continuing deforestation, it was bitterly opposed by representatives of grassroots organisations, who argued that it was a major violation of the rights of peasants and tribals living in and around forest areas. . . .
The debate over the draft forest act fuelled a larger controversy over the orientation of state forest policy. It was pointed out, for example, that the draft act was closely modelled on its predecessor, the Forest Act of 1878. The earlier Act rested on a usurpation of rights of ownership by the colonial state which had little precedent in precolonial history. It was further argued that the system of forestry introduced by the British—and continued, with little modification, after 1947—emphasised revenue generation and commercial exploitation, while its policing orientation excluded villagers who had the most longstanding claim on forest resources. Critics called for a complete overhaul of forest administration, pressing the government to formulate policy and legislation more appropriate to present needs. . . .
That debate is not over yet. The draft act was shelved, though it has not as yet been formally withdrawn. Meanwhile, the 1878 Act (as modified by an amendment in 1927) continues to be in operation. In response to its critics, the government has made some important changes in forest policy, e.g., no longer treating forests as a source of revenue, and stopping ecologically hazardous practices such as the clearfelling of natural forests. At the same time, it has shown little inclination to meet the major demand of the critics of forest policy—namely, abandoning the principle of state monopoly over forest land by handing over areas of degraded forests to individuals and communities for afforestation.
. . . [The] 1878 Forest Act itself was passed only after a bitter and prolonged debate within the colonial bureaucracy, in which protagonists put forward arguments strikingly similar to those being advanced today. As is well known, the Indian Forest Department owes its origin to the requirements of railway companies. The early years of the expansion of the railway network, c. 1853 onwards, led to tremendous deforestation in peninsular India owing to the railway's requirements of fuelwood and construction timber. Huge quantities of durable timbers were also needed for use as sleepers across the newly laid tracks. Inexperienced in forestry, the British called in German experts to commence systematic forest management. The Indian Forest Department was started in 1864, with Dietrich Brandis, formerly a Lecturer at Bonn, as the first Inspector General of Forests. The new department needed legislative backing to function effectively, and in the following year, 1865, the first forest act was passed. . . .
Which one of the following best encapsulates the reason for the "raging controversy" developing into a "larger controversy"?
The 1982 draft forest act replicated colonial measures of control and regulation of forest resources.
The 1982 draft forest act was unjustifiably defended by forest officials in the face of bitter opposition by grassroots organisations.
The 1982 draft forest act violated the rights of tribals and peasants who lived in and around forest areas.
The 1982 draft forest act further enabled the commercial exploitation of forest resources by the forest bureaucracy.
The 1982 draft forest act replicated colonial measures of control and regulation of forest resources.
The passage states that the 1982 act was modelled on the 1878 Forest Act, which itself rested on a colonial usurpation of ownership rights with little precedent in precolonial history. This colonial continuity is what expanded the controversy from being about the specific 1982 act into a larger debate about the orientation of state forest policy as a whole.
Why Option A is Correct: This captures the key link the passage draws, that the 1982 act mirrored colonial control mechanisms from 1878, which is precisely what fuelled the broader debate over the entire orientation of forest policy.
Why Option B is
Incorrect: This describes a detail about the controversy, forest officials defending the act, but it does not explain why the controversy grew into a larger debate about forest policy's broader history and orientation.
Why Option C is
Incorrect: This restates the grassroots organisations' objection to the 1982 act specifically, but does not address why the debate expanded into a larger historical controversy about colonial-era policy.
Why Option D is
Incorrect: This is a specific criticism of forest administration's commercial focus, but it is a sub-point within the larger controversy, not the encompassing reason the controversy grew larger.
Key Takeaway: When asked why a specific controversy expanded into a larger one, find the point in the passage that explicitly connects the specific event to the broader historical or structural issue.
All of the following, if true, would weaken the narrative presented in the passage EXCEPT that:
certain tribal groups in India are responsible for climate change because their sustenance has historically depended on mass scale deforestation.
before British rule, peasants and tribal groups were denied access to forest resources by Indian rulers and their administrations.
the timber requirement for railway works in nineteenth century India was met through import from China, in exchange for spices.
nineteenth century German forestry experts were infamous for violating the rights of indigenous communities that lived in forest regions.
nineteenth century German forestry experts were infamous for violating the rights of indigenous communities that lived in forest regions.
The passage's narrative blames the colonial era forest department, started under German expert Dietrich Brandis, for imposing exploitative forest policies, and broadly sympathises with peasants and tribals' rights being violated. Saying German experts were infamous for violating indigenous rights would actually support the passage's narrative rather than weaken it, since the passage's critique of colonial forestry assumes precisely this kind of rights violation.
Why Option A is
Incorrect (as a candidate for the exception): If tribal groups themselves were responsible for mass deforestation, this would weaken the narrative's framing of villagers as having a legitimate, sustainable claim to forest resources.
Why Option B is
Incorrect (as a candidate for the exception): If precolonial Indian rulers also denied peasants and tribals access to forest resources, this would weaken the passage's implicit suggestion that colonial control over forests had little precedent in precolonial history.
Why Option C is
Incorrect (as a candidate for the exception): If timber needs were met through imports rather than domestic deforestation, this would weaken the narrative's claim that railway expansion caused the deforestation that prompted forest department formation.
Why Option D is Correct: This statement aligns with, rather than weakens, the passage's critical narrative about colonial forestry's harmful, rights-violating practices, so it does not contradict or weaken the passage's argument.
Key Takeaway: For "weaken except" questions, check whether each statement actually opposes the passage's framing, since one option may simply restate or support the passage's existing critique instead of weakening it.
According to the passage, which one of the following is not common to the 1878 Forest Act and the 1982 draft forest act?
Both sparked controversy and debate among the various stakeholders.
Both reflect a colonial mindset.
Both resulted in large scale deforestation.
Both sought to establish the state's monopoly over forest resources.
Both resulted in large scale deforestation.
The passage does not state that the 1878 Act caused large scale deforestation, in fact deforestation is attributed earlier to railway expansion before the act was passed, and the act was a response meant to manage forest resources, not a direct cause of mass deforestation itself.
Why Option A is
Incorrect (as a candidate for not common): Both acts are explicitly described as sparking controversy, the 1982 act among grassroots organisations and forest officials, and the 1878 act within the colonial bureaucracy itself.
Why Option B is
Incorrect (as a candidate for not common): The passage states the 1982 draft act was closely modelled on the 1878 Act, both reflecting the same colonial approach to forest control.
Why Option D is
Incorrect (as a candidate for not common): Both acts are described as strengthening state control and ownership over forest resources, continuing the principle of state monopoly.
Why Option C is Correct: Large scale deforestation is attributed to railway expansion preceding the 1865 and 1878 Acts, not to the acts themselves, making this the feature not common or accurately attributable to both acts.
Key Takeaway: Be careful not to confuse a historical cause mentioned in the passage, like railway-driven deforestation, with the effects of a specific legislative act discussed afterward.
According to the passage, which one of the following reforms is yet to happen in India's forest policies?
Recognising the state's claim to forest land use.
A ban on deforestation.
Recognising the significance of forests to ecology.
Involving local people in cultivating forests.
Involving local people in cultivating forests.
The passage states that the government has stopped treating forests purely as revenue sources and has ended ecologically hazardous practices like clearfelling, but it has shown little inclination to hand over degraded forest areas to individuals and communities for afforestation, meaning involving local people in cultivating forest land remains an unmet demand.
Why Option A is
Incorrect: The passage indicates the opposite has happened, the government still holds onto its monopoly over forest land rather than relinquishing this claim, this option misstates the situation as already reformed.
Why Option B is
Incorrect: The passage does not mention a full ban on deforestation as a reform that has or has not happened, this specific claim is not addressed.
Why Option C is
Incorrect: The passage states the government has stopped ecologically hazardous practices like clearfelling, which reflects movement toward recognising ecological significance, this reform has already happened, unlike what the question asks for.
Why Option D is Correct: This matches the passage's explicit statement that the government has shown little inclination to hand over forest land to individuals and communities, meaning local involvement in cultivation remains an unfulfilled demand.
Key Takeaway: Distinguish between reforms the passage says have already occurred and the one specific demand it explicitly states remains unmet.
Once a society accepts a secular mode of creativity, within which the creator replaces God, imaginative transactions assume a self-conscious form. The tribal imagination, on the other hand, is still to a large extent dreamlike and hallucinatory. It admits fusion between various planes of existence and levels of time in a natural and artless manner. In tribal stories, oceans fly in the sky as birds, mountains swim in water as fish, animals speak as humans and stars grow like plants. Spatial order and temporal sequence do not restrict the narrative. This is not to say that tribal creations have no conventions or rules, but simply that they admit the principle of association between emotion and the narrative motif. Thus stars, seas, mountains, trees, men and animals can be angry, sad or happy.
It might be said that tribal artists work more on the basis of their racial and sensory memory than on the basis of a cultivated imagination. In order to understand this distinction, we must understand the difference between imagination and memory. In the animate world, consciousness meets two immediate material realities: space and time. We put meaning into space by perceiving it in terms of images. The image-making faculty is a genetic gift to the human mind—this power of imagination helps us understand the space that envelops us. With regard to time, we make connections with the help of memory; one remembers being the same person today as one was yesterday.
The tribal mind has a more acute sense of time than the sense of space. Somewhere along the history of human civilization, tribal communities seem to have realized that domination over territorial space was not their lot. Thus, they seem to have turned almost obsessively to gaining domination over time. This urge is substantiated in their ritual of conversing with their dead ancestors: year after year, tribals in many parts of India worship terracotta or carved-wood objects representing their ancestors, aspiring to enter a trance in which they can converse with the dead. Over the centuries, an amazingly sharp memory has helped tribals classify material and natural objects into a highly complex system of knowledge. . .
One of the main characteristics of the tribal arts is their distinct manner of constructing space and imagery, which might be described as 'hallucinatory'. In both oral and visual forms of representation, tribal artists seem to interpret verbal or pictorial space as demarcated by an extremely flexible 'frame'. The boundaries between art and non-art become almost invisible. A tribal epic can begin its narration from a trivial everyday event; tribal paintings merge with living space as if the two were one and the same. And within the narrative itself, or within the painted imagery, there is no deliberate attempt to follow a sequence. The episodes retold and the images created take on the apparently chaotic shapes of dreams. In a way, the syntax of language and the grammar of painting are the same, as if literature were painted words and painting were a song of images.
Non-human living forms exhibit human emotions in tribal narratives because tribal narratives:
have a self-conscious form.
accommodate existential fluidity.
abandon all rules and regulations.
are rudimentary and underdeveloped.
accommodate existential fluidity.
The passage describes tribal narratives admitting fusion between different planes of existence and levels of time, allowing oceans to fly and mountains to swim, all driven by an association between emotion and narrative motif. This fluid, boundary-crossing quality, where objects and animals can take on human emotions without being restricted by spatial or temporal logic, is best described as accommodating existential fluidity.
Why Option A is
Incorrect: The passage explicitly contrasts tribal imagination with the self-conscious form that arises once a society adopts secular, individual creativity, so tribal narratives are described as the opposite of self-conscious.
Why Option C is
Incorrect: The passage clarifies that tribal creations do have conventions and rules, just different ones based on emotional association, so saying they abandon all rules misrepresents the passage.
Why Option D is
Incorrect: The passage does not characterise tribal narratives as rudimentary or underdeveloped, it presents their fluid, hallucinatory quality as a distinct and rich mode of imagination, not as deficient.
Why Option B is Correct: This matches the passage's description of tribal narratives freely fusing planes of existence and time, where animals and natural elements take on human emotions without the constraints of ordinary logic.
Key Takeaway: Look for terms that precisely capture a passage's specific descriptive language, rather than options that contradict explicit statements elsewhere in the text.
On the basis of the passage, which one of the following explains the main difference between imagination and memory?
Imagination is a genetic gift to humans whereas memory is central to human consciousness.
Tribal groups value memory over imagination when it comes to creating art and literature.
Imagination needs to be cultivated whereas memory is more intuitive because it is racial and sensory.
Imagination helps humans make sense of space while memory helps them understand time.
Imagination helps humans make sense of space while memory helps them understand time.
The passage explains that imagination helps put meaning into space through images, while memory helps us connect with time, recognising ourselves as the same person across days. This direct mapping, imagination to space and memory to time, is the core distinction the passage draws.
Why Option A is
Incorrect: This restates two separate facts about imagination and memory but does not capture the core functional distinction, that one relates to space and the other to time.
Why Option B is
Incorrect: This describes tribal groups' preference, a separate point made later in the passage, not the general explanation of the difference between imagination and memory itself.
Why Option C is
Incorrect: This refers to a different distinction, between cultivated imagination and racial or sensory memory in tribal artists specifically, not the general definitional difference asked about.
Why Option D is Correct: This precisely reflects the passage's statement that imagination helps us understand space while memory helps us connect across time.
Key Takeaway: When a passage gives a direct definitional contrast, prefer the option that mirrors that exact contrast over options describing related but distinct ideas.
Which one of the following best explains why tribals in India worship their dead ancestors?
For tribals, conversing with the dead becomes a way of seeking control over time.
Tribals show respect to their ancestors through terracotta and carved-wood objects.
Tribals possess a sophisticated knowledge system that is based on memory.
Tribals seek territorial domination over the spaces that they inhabit.
For tribals, conversing with the dead becomes a way of seeking control over time.
The passage states that tribal communities, realising domination over territorial space was not their lot, turned to gaining domination over time instead, and that the ritual of conversing with dead ancestors substantiates this urge for control over time.
Why Option B is
Incorrect: This describes the form the worship takes, terracotta and carved-wood objects, but not the underlying reason the passage gives for why tribals perform this worship.
Why Option C is
Incorrect: This refers to a separate point about tribals' knowledge system built from memory, not the specific explanation for ancestor worship.
Why Option D is
Incorrect: The passage explicitly states tribals turned away from seeking territorial domination, making this the opposite of what the passage argues.
Why Option A is Correct: This matches the passage's explicit reasoning, that the ritual of conversing with dead ancestors is a manifestation of the tribal urge to dominate time, since domination over space was unavailable to them.
Key Takeaway: Trace a described ritual back to the specific motivation the passage assigns to it, rather than to incidental details about how the ritual is performed.
All of the following, if true, would weaken the passage's claims about the hallucinatory tribal imagination EXCEPT that:
tribal art excludes the depiction of the mundane reality of everyday life and objects.
tribal narratives exhibit a chronological beginning, middle, and end.
tribal stories depict the natural world in accordance with rational scientific knowledge.
shamanic rituals involving conversing with the dead often feature in tribal stories.
shamanic rituals involving conversing with the dead often feature in tribal stories.
Options A, B and C all describe scenarios that would directly contradict the passage's claims about tribal art being hallucinatory, fluid, and unbound by sequence or scientific rationality. Option D, mentioning that shamanic rituals of conversing with the dead feature in tribal stories, is consistent with the passage's own description of tribal ancestor worship and trance rituals, so it does not weaken the claims about hallucinatory imagination.
Why Option A is
Incorrect (as a candidate for the exception): If tribal art excluded depictions of everyday life, this would contradict the passage's claim that the boundary between art and non-art is invisible and that epics can begin from trivial everyday events.
Why Option B is
Incorrect (as a candidate for the exception): If tribal narratives followed a clear chronological structure, this would contradict the passage's claim that there is no deliberate attempt to follow a sequence in tribal narratives.
Why Option C is
Incorrect (as a candidate for the exception): If tribal stories depicted nature according to rational scientific knowledge, this would contradict the passage's examples of oceans flying and mountains swimming, which is explicitly dreamlike and not scientifically rational.
Why Option D is Correct: This aligns with the passage's own description of tribal rituals involving trance-like conversations with ancestors, so it does not weaken the hallucinatory characterisation, it actually supports it.
Key Takeaway: In a "weaken except" question, watch for an option that actually restates something the passage already claims, since this will not weaken but rather support the argument.
Over the course of the twentieth century, humans built, on average, one large dam a day, hulking structures of steel and concrete designed to control flooding, facilitate irrigation, and generate electricity. Dams were also lucrative contracts, large-scale employers, and the physical instantiation of a messianic drive to conquer territories and control nature. Some of the results of that drive were charismatic mega-infrastructure—the Hoover on the Colorado River or the Aswan on the Nile—but most of the tens of thousands of dams that dot the Earth's landscape have drawn little attention. These are the smaller, though not inconsequential, barriers that today impede the flow of water on nearly two-thirds of the world's large waterways. Chances are, what your map calls a "lake" is actually a reservoir, and that thin blue line that emerges from it once flowed very differently.
Damming a river is always a partisan act. Even when explicit infrastructure goals—irrigation, flood control, electrification—were met, other consequences were significant and often deleterious. Across the world, river control displaced millions of people, threatening livelihoods, foodways, and cultures. In the western United States, dams were often an instrument of colonialism, used to dispossess Indigenous people and subsidize settler agriculture. And as dams slowed the flow of water, inhibited the movement of nutrients, and increased the amount of toxic algae and other parasites, they snuffed out entire river ecologies. Declining fish populations are the most evident effect, but dams also threaten a host of other animals—from birds and reptiles to fungi and plants—with extinction. Every major dam, then, is also a sacrifice zone, a place where lives, livelihoods, and ways of life are eliminated so that new sorts of landscapes can support water-intensive agriculture and cities that sprout downstream of new reservoirs.
Such sacrifices have been justified as offerings at the temples of modernity. Justified by—and for—whom, though? Over the course of the twentieth century, rarely were the costs and benefits weighed thoughtfully and decided democratically. As Kader Asmal, chair of the landmark 2000 World Commission on Dams, concluded, "There have been precious few, if any, comprehensive, independent analyses as to why dams came about, how dams perform over time, and whether we are getting a fair return from our $2 trillion investment." A quarter-century later, Asmal's words ring ever truer. A litany of dams built in the mid-twentieth century are approaching the end of their expected lives, with worrying prospects for their durability. Droughts, magnified and multiplied by the effects of climate change, have forced more and more to run below capacity. If ever there were a time to rethink the mania for dams, it would be now.
There is some evidence that a combination of opposition, alternative energy sources, and a lack of viable projects has slowed the construction of major dams. But a wave of recent and ongoing construction, from India and China to Ethiopia and Canada, continues to tilt the global balance firmly in favor of water impoundment.
Which one of the following sets of terms is closest to mapping the key arguments of the passage?
Partisan act - Threatened livelihoods - Toxic algae - Quarter century
Lucrative contracts - Sacrifice zone - Expected lives - Global balance
Mega-infrastructure - Sacrifice zone - Worshipping modernity - Water impoundment
Physical instantiation - Partisan act - Decided democratically - Alternative energy
Mega-infrastructure - Sacrifice zone - Worshipping modernity - Water impoundment
The passage discusses dams as mega-infrastructure projects, describes their harmful consequences as creating "sacrifice zones," frames the unquestioned justification for building them as akin to worshipping modernity, and closes on the global trend of continued water impoundment despite emerging opposition.
Why Option A is
Incorrect: While these terms appear in the passage, "quarter century" is a specific time reference rather than a key conceptual term mapping the passage's central argument.
Why Option B is
Incorrect: "Lucrative contracts" and "expected lives" are specific details rather than central concepts capturing the passage's main arguments about justification, harm, and the unresolved global trend.
Why Option D is
Incorrect: "Physical instantiation" and "decided democratically" are specific phrases used incidentally, not the core conceptual terms mapping the passage's overall argument about dams as sacrifice zones justified by modernity's mania.
Why Option C is Correct: These four terms accurately capture the passage's progression, from describing dams as infrastructure, to their human and ecological cost as sacrifice zones, to the unquestioned justification akin to worship, to the continuing global trend of water impoundment.
Key Takeaway: Choose the term set that traces the passage's actual argumentative arc, not just terms that appear verbatim but represent only incidental details.
What does the author wish to communicate by referring to the Hoover and Aswan dams in the first paragraph?
The Colorado and Nile rivers may be seen as thin blue lines on a map.
By building dams like the Hoover and Aswan dams, large-scale employers became messianic figures.
The designers and builders of these mega-structures were highly charismatic individuals.
The drive to control nature is evident not only in mega-infrastructures like the Hoover and Aswan dams, but in smaller dams as well.
The drive to control nature is evident not only in mega-infrastructures like the Hoover and Aswan dams, but in smaller dams as well.
The passage uses the Hoover and Aswan dams as examples of charismatic mega-infrastructure, then pivots to note that most of the tens of thousands of dams worldwide are smaller and have drawn little attention, despite also being products of the same drive to conquer territories and control nature.
Why Option A is
Incorrect: This is a factual detail mentioned later about reservoirs being mistaken for lakes, unrelated to the specific purpose of mentioning Hoover and Aswan.
Why Option B is
Incorrect: This misreads the passage, "messianic drive" describes the broader ideological motive behind dam-building generally, not a claim that large-scale employers became messianic figures through building these specific dams.
Why Option C is
Incorrect: The passage does not describe the designers and builders as charismatic individuals, it describes the dams themselves as "charismatic mega-infrastructure," a different claim about the structures, not their builders.
Why Option D is Correct: This matches the passage's purpose, using Hoover and Aswan as recognisable examples of the broader drive to control nature, which the passage then extends to the much larger, less noticed set of smaller dams worldwide.
Key Takeaway: When an example is given before a broader claim, check what general point the example is meant to support, rather than treating the example as a standalone fact.
The word "instantiation" is used in the first paragraph. Which one of the following pairs of terms would be the best substitute for it in the context of its usage in the paragraph?
Exemplification and manifestation
Concreteness and viability
Durability and timeliness
Development and construction
Exemplification and manifestation
The phrase "physical instantiation of a messianic drive" describes dams as the concrete manifestation or embodiment of that drive to conquer nature. The best substitute would be terms meaning embodiment or representation in physical form.
Why Option B is
Incorrect: "Concreteness and viability" do not capture the sense of dams being a manifestation of an abstract drive, viability refers to feasibility, which is unrelated to the meaning of instantiation here.
Why Option C is
Incorrect: "Durability and timeliness" are unrelated to the meaning of instantiation, which is about embodiment or manifestation, not about how long-lasting or well-timed something is.
Why Option D is
Incorrect: "Development and construction" describe the process of building dams, not the specific sense of instantiation as the physical embodiment of an abstract drive or ideology.
Why Option A is Correct: "Exemplification and manifestation" precisely capture the meaning of instantiation in this context, dams serving as a concrete representation of the broader messianic drive to control nature.
Key Takeaway: For vocabulary-in-context questions, test whether the substitute terms preserve the specific sense the word carries in that sentence, not just its general dictionary meaning.
All of the following statements may be considered valid inferences from the passage EXCEPT that:
despite increasing evidence of opposition to dams as well as alternatives to them, they continue to be built.
smaller, though not inconsequential, dams are safer than large dam projects.
processes of colonisation have used dam-building to make people vacate their territories.
dam-building has proved to be an extremely costly enterprise that may not be justifiable.
smaller, though not inconsequential, dams are safer than large dam projects.
The passage explicitly describes smaller dams as "not inconsequential," directly contradicting any claim that they are safer than larger dam projects. The passage gives no comparative safety claim about smaller versus larger dams at all.
Why Option A is
Incorrect (as a candidate for the exception): The passage explicitly states that despite opposition, alternative energy sources, and lack of viable projects slowing construction, a wave of recent construction continues to favour dam building, supporting this inference.
Why Option C is
Incorrect (as a candidate for the exception): The passage explicitly states dams were often an instrument of colonialism used to dispossess Indigenous people, directly supporting this inference.
Why Option D is
Incorrect (as a candidate for the exception): The passage cites the $2 trillion investment and questions whether a fair return was achieved, along with the lack of independent analyses, supporting the inference that dam-building's costliness may not be justifiable.
Why Option B is Correct: The passage explicitly calls smaller dams "not inconsequential," implying they too cause significant harm, contradicting any claim that they are safer than large dam projects.
Key Takeaway: Watch for an option that contradicts an explicit qualifier in the passage, here, the passage takes pains to say smaller dams are not harmless, directly undermining a claim of their relative safety.
Imagine a world in which artificial intelligence is entrusted with the highest moral responsibilities: sentencing criminals, allocating medical resources, and even mediating conflicts between nations. This might seem like the pinnacle of human progress: an entity unburdened by emotion, prejudice or inconsistency, making ethical decisions with impeccable precision. . . .
Yet beneath this vision of an idealised moral arbiter lies a fundamental question: can a machine understand morality as humans do, or is it confined to a simulacrum of ethical reasoning? AI might replicate human decisions without improving on them, carrying forward the same biases, blind spots and cultural distortions from human moral judgment. In trying to emulate us, it might only reproduce our limitations, not transcend them. But there is a deeper concern. Moral judgment draws on intuition, historical awareness and context - qualities that resist formalisation. Ethics may be so embedded in lived experience that any attempt to encode it into formal structures risks flattening its most essential features. If so, AI would not merely reflect human shortcomings; it would strip morality of the very depth that makes ethical reflection possible in the first place.
Still, many have tried to formalise ethics, by treating certain moral claims not as conclusions, but as starting points. A classic example comes from utilitarianism, which often takes as a foundational axiom the principle that one should act to maximise overall wellbeing. From this, more specific principles can be derived, for example, that it is right to benefit the greatest number, or that actions should be judged by their consequences for total happiness. As computational resources increase, AI becomes increasingly well-suited to the task of starting from fixed ethical assumptions and reasoning through their implications in complex situations.
But what, exactly, does it mean to formalise something like ethics? The question is easier to grasp by looking at fields in which formal systems have long played a central role. Physics, for instance, has relied on formalisation for centuries. There is no single physical theory that explains everything. Instead, we have many physical theories, each designed to describe specific aspects of the Universe: from the behaviour of quarks and electrons to the motion of galaxies. These theories often diverge. Aristotelian physics, for instance, explained falling objects in terms of natural motion toward Earth's centre; Newtonian mechanics replaced this with a universal force of gravity. These explanations are not just different; they are incompatible. Yet both share a common structure: they begin with basic postulates - assumptions about motion, force or mass - and derive increasingly complex consequences. . . .
Ethical theories have a similar structure. Like physical theories, they attempt to describe a domain - in this case, the moral landscape. They aim to answer questions about which actions are right or wrong, and why. These theories also diverge and, even when they recommend similar actions, such as giving to charity, they justify them in different ways. Ethical theories also often begin with a small set of foundational principles or claims, from which they reason about more complex moral problems.
Choose the one option below that comes closest to being the opposite of "utilitarianism".
The committee adopted a non-egoist framework, ranking policies by their contribution to overall social welfare and treating self-interest as a derivative concern within institutional evaluation.
The authors advocated an absolutist stance, following exceptionless rules regardless of outcomes and evaluating choices by broadest societal benefit.
The council followed a prioritarian approach, assigning greater moral weight to improvements for the worst-off rather than to maximising total welfare across the affected population.
The policy was cast as deontological ethics, selecting the option that delivered the highest total benefit to citizens while presenting duty as a secondary consideration in public decision-making.
The council followed a prioritarian approach, assigning greater moral weight to improvements for the worst-off rather than to maximising total welfare across the affected population.
Utilitarianism is defined in the passage as the principle of maximising overall wellbeing or total happiness across the greatest number. The opposite of this would be a framework that prioritises improvements for the worst-off specifically, rather than maximising aggregate welfare, which is precisely what prioritarianism does.
Why Option A is
Incorrect: This describes ranking policies by their contribution to overall social welfare, which is essentially restating utilitarianism's core principle rather than opposing it.
Why Option B is
Incorrect: Despite using the word "absolutist," this option still evaluates choices by "broadest societal benefit," which mirrors utilitarianism's logic of aggregate benefit rather than opposing it.
Why Option D is
Incorrect: This option, despite labelling itself "deontological," still selects the option with the "highest total benefit," which is utilitarian reasoning, making it inconsistent and not a true opposite.
Why Option C is Correct: This describes assigning greater moral weight to the worst-off rather than maximising total welfare, which directly contrasts with utilitarianism's core aggregative principle, making it the closest opposite.
Key Takeaway: When asked for an opposite, check the underlying logic of each option rather than just the label used, since some options use contrasting terminology but still apply the same reasoning being asked to oppose.
Which one of the options below best summarises the passage?
The passage rejects formal methods in principle. It holds that moral judgement cannot be expressed in disciplined terms and concludes that AI should not serve in courts, medicine, or diplomacy under any conditions.
The passage weighs the appeal of an impersonal AI judge against doubts about moral grasp. It claims codified schemes retain case nuance at scale and uses a physics analogy to predict convergence on a unified framework.
The passage weighs the appeal of an impersonal AI judge against doubts about moral grasp. It warns that codification can erode case-sensitive judgement, allow axiom-led reasoning at scale, and use a physics analogy to model structured plurality.
The passage highlights administrative gains from automation. It treats reproducing human moral judgement as progress and argues that, as computational resources increase, AI can be responsible for decision-making across varied institutional settings.
The passage weighs the appeal of an impersonal AI judge against doubts about moral grasp. It warns that codification can erode case-sensitive judgement, allow axiom-led reasoning at scale, and use a physics analogy to model structured plurality.
The passage weighs the appeal of an AI moral arbiter against deep doubts about whether it can truly grasp morality, warns that formalising ethics risks losing nuance and context, but also shows how AI could reason from fixed ethical starting points, using the physics analogy to illustrate how different theories can coexist without merging into one framework.
Why Option A is
Incorrect: This claims the passage rejects formal methods entirely and concludes AI should never serve in such roles, but the passage actually explores how ethics has been formalised before and how AI might apply fixed starting points, without reaching such an absolute rejection.
Why Option B is
Incorrect: This claims the passage predicts convergence on a unified ethical framework through the physics analogy, but the passage's physics analogy is used to show that theories diverge and coexist, not that they converge into one framework.
Why Option D is
Incorrect: This claims the passage treats reproducing human moral judgment as progress, but the passage explicitly raises concern that AI replicating human decisions might only reproduce human biases and limitations, not improve on them.
Why Option C is Correct: This captures the passage's balanced exploration, weighing AI's appeal against doubts about moral understanding, warning about the risk of losing nuance through codification, and using the physics analogy to illustrate structured plurality rather than convergence.
Key Takeaway: Watch for summary options that overstate the passage's position into an absolute conclusion, when the actual passage maintains a more exploratory, balanced argument.
The passage compares ethics to physics, where different theories apply to different aspects of a domain and says AI can reason from fixed starting points in complex cases. Which one of the assumptions below must hold for that comparison to guide practice?
There is a principled way to decide which ethical framework applies to which class of cases, so the system can select the relevant starting points before deriving a recommendation.
Real cases never straddle different areas, so a case always fits exactly one framework without any overlap whatsoever.
A single master framework replaces all others after translation into one code, so domain boundaries disappear in application.
Once formalised, all ethical frameworks yield the same recommendation in every case, so selection among them is unnecessary.
There is a principled way to decide which ethical framework applies to which class of cases, so the system can select the relevant starting points before deriving a recommendation.
For an AI system to reason from fixed ethical starting points the way physics applies different theories to different domains, there must be a reliable, principled way to determine which ethical framework or starting point is appropriate for a given class of cases before any reasoning can proceed.
Why Option B is
Incorrect: This assumes real cases never overlap between frameworks, which is a much stronger and unrealistic assumption than what is needed, the comparison does not require this level of clean separation, only a principled way of selecting an applicable framework.
Why Option C is
Incorrect: This assumes all frameworks eventually merge into one master code, which actually contradicts the physics analogy's point about theories remaining distinct and diverging rather than collapsing into a single system.
Why Option D is
Incorrect: This assumes all frameworks always agree on recommendations, removing any need for selection, which contradicts the passage's explicit point that ethical theories diverge and justify similar actions differently.
Why Option A is Correct: This captures the minimal necessary assumption, that the system needs a principled way to pick the relevant ethical starting point for a given case before applying that framework's reasoning, mirroring how physics applies different theories to different domains.
Key Takeaway: For "must hold" assumption questions, identify the minimal condition genuinely required for the analogy to work, rather than a stronger or unrelated claim that goes beyond what's necessary.
All of the following can reasonably be inferred from the passage EXCEPT:
by analogy with physics, compact postulates can yield broad predictions across incompatible theories and ethics can likewise share structure while continuing to diverge rather than close on a single comprehensive framework.
encoding ethics into fixed structures risks stripping away intuition, history, and context and, if that occurs, the depth that enables reflective judgement disappears. So, machines would mirror our limits rather than exceed them.
the appeal of an AI judge rests on immunity to bribery, partiality, and fatigue; yet the text questions whether procedural cleanliness amounts to moral understanding without lived context and interpretive depth.
with fixed moral starting points and expanding computational resources, the argument forecasts convergence on one ethical system and treats contextual judgement as unnecessary once formal reasoning scales across domains and cultures.
with fixed moral starting points and expanding computational resources, the argument forecasts convergence on one ethical system and treats contextual judgement as unnecessary once formal reasoning scales across domains and cultures.
The passage does not forecast convergence on a single ethical system, in fact it uses the physics analogy specifically to show that theories can remain structurally similar while still diverging and coexisting as distinct frameworks, never merging into one. The passage also never claims contextual judgment becomes unnecessary, it actually raises this as a central concern throughout.
Why Option A is
Incorrect (as a candidate for the exception): This matches the passage's explicit physics analogy, that theories share structure (postulates leading to consequences) while remaining incompatible and divergent, which the passage extends to ethics as well.
Why Option B is
Incorrect (as a candidate for the exception): This matches the passage's stated concern, that encoding ethics into formal structures risks stripping away intuition, history and context, leaving machines to merely mirror human limitations rather than transcend them.
Why Option C is
Incorrect (as a candidate for the exception): This matches the passage's opening framing, the appeal of an AI free from emotion, prejudice or fatigue, set against the deeper question of whether procedural cleanliness equates to true moral understanding.
Why Option D is Correct: This contradicts the passage, which uses the physics analogy to show theories diverge rather than converge, and which raises ongoing concern about losing contextual judgment rather than treating it as something that becomes safely unnecessary.
Key Takeaway: Watch for an inference that reverses the passage's actual argument, claiming convergence and the irrelevance of context when the passage actually argues the opposite, persistent divergence and continued importance of context.
Five jumbled 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 out and key in the number of that sentence as your answer.
1. About half of all the oxygen we breathe is made near the surface of the ocean by phytoplankton that photosynthesize just like land-dwelling plants.
2. A team of scientists that includes Boston University experts has discovered they also produce oxygen on the seafloor.
3. The research team used deep-sea chambers that land on the seafloor and enclose the seawater, sediment, polymetallic nodules, and living organisms.
4. The discovery is a surprise considering oxygen is typically created by plants and organisms with help from the sun—not by rocks on the ocean floor.
5. The deep-sea rocks, called polymetallic nodules, don't only host a surprising number of sea critters.
Sentences 1, 2, 4 and 5 form a coherent flow describing phytoplankton's known role in oxygen production, the surprising discovery that oxygen is also made on the seafloor, the unexpectedness of this given oxygen normally needs sunlight, and the introduction of the deep-sea rocks called polymetallic nodules. Sentence 3 jumps into a methodological detail about deep-sea chambers used by the research team, which belongs later in a fuller passage but breaks the introductory narrative these four sentences are building before the actual research method would logically be explained.
Key Takeaway: When the four sentences build a narrative of discovery and surprise, a sentence introducing a procedural or methodological detail too early is usually the one that disrupts the flow.
The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
The return to the tailor is the juxtaposition of three key things for the mindful Indian shopper. The first is the conscious shift away from the homogeneity of fast fashion, the idea of a hundred other people owning exactly the same Zara trench coat or H&M pleated skirt. The second is an actual understanding of the waste behind the fast fashion market, and wanting not to contribute to that anymore. The last is the shift toward customisation and fit—the idea of having imaginations brought to life and to have them fit exactly; without paying exorbitant rates for that bespoke tailoring. For the individual with a keen fashion sense and a genuine desire to move away from the waste and uniformity of fast fashion without paying the premium for it that indie brands would invariably demand, the tailor is the perfect crossover.
The mindful Indian shoppers are returning to the tailor with a genuine desire to wear clothes which are less expensive, fit them well and are yet fashionable.
The mindful Indian shopper is shifting away from convenience and uniformity of clothing, and waste in fashion, to customisation and less exorbitantly priced clothing.
All Indian shoppers are opting for customisation and a shift away from homogeneity over expensive clothing brands like Zara and H&M.
In the Indian retail market, people believe that expensive branded clothes are wasteful and, therefore, are returning to the neighbourhood tailor.
The mindful Indian shopper is shifting away from convenience and uniformity of clothing, and waste in fashion, to customisation and less exorbitantly priced clothing.
The passage describes three motivations driving Indian shoppers back to tailors: rejecting fast fashion's homogeneity, awareness of its waste, and a desire for affordable customisation and fit, without paying indie brand premiums.
Why Option A is
Incorrect: This focuses mainly on affordability and fit, but omits the rejection of homogeneity and waste, which are explicitly stated as two of the three core motivations.
Why Option B is Correct: This captures all three drivers, the shift away from uniformity and waste, and the move toward customisation at non-exorbitant prices, matching the passage closely.
Why Option C is
Incorrect: This overgeneralises with "all Indian shoppers" and misrepresents the comparison, the passage is about avoiding indie brand premiums, not about choosing tailors over Zara and H&M specifically on cost grounds alone.
Why Option D is
Incorrect: This narrows the reasoning to only the perception of wastefulness in expensive clothes, ignoring the equally important points about uniformity and customisation.
Key Takeaway: A good summary must reflect all the distinct motivations the passage lists, not just one of them.
The given sentence is missing in the paragraph below. Decide where it best fits among the options 1, 2, 3, or 4 indicated in the paragraph.
Sentence: In each of the affected males, the genetic defect was located to the X chromosome in the region of p11-12.
Paragraph: The first suggested evidence of a human genetic mutation associated with aggressive behaviour came from a study in 1993. _____(1). Genetic and metabolic studies were conducted on a large Dutch family in which several of the males has a syndrome of borderline mental retardation and abnormal behaviour. _____(2). The undesirable behaviour included impulsive aggression, arson and exhibitionism. _____(3). A point mutation was identified in the eighth exon of the monoamine oxidase A (MAOA) structural gene which changes glutamine to a termination codon. _____(4).
Option 3
Option 4
Option 2
Option 1
Option 3
The inserted sentence specifies that the genetic defect was located on the X chromosome in a particular region, in each affected male. This level of specific genetic detail fits right after the description of the undesirable behaviours observed in the Dutch family, at Option 3, just before the passage moves to identifying the exact point mutation found in the MAOA gene.
Why Option B is
Incorrect: Option 4 is at the very end, after the specific point mutation in the MAOA gene has already been described. Placing the broader X chromosome location detail after this more specific mutation detail breaks the natural order of moving from general to specific.
Why Option C is
Incorrect: Option 2 comes right after the introduction of the Dutch family's syndrome of mental retardation and abnormal behaviour, before their specific undesirable behaviours are even listed. Inserting the chromosome location detail here is premature.
Why Option D is
Incorrect: Option 1 is right at the start, before the genetic and metabolic studies on the Dutch family are even introduced. The chromosome detail cannot logically appear before the family and their condition have been described.
Why Option A is Correct: Option 3 comes right after the specific undesirable behaviours are listed, and right before the passage narrows down further to the exact point mutation, making it the natural place for the broader chromosome location detail to appear in this general-to-specific narrowing sequence.
Key Takeaway: In passages describing a scientific discovery, insert a sentence at the point where the narrative moves from general findings to a stage of progressively more specific detail.
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. The effigy of a candidate establishes a personal link between him and the voters; the candidate does not only offer a programme for judgement, he suggests a physical climate, a set of daily choices expressed in a morphology, a way of dressing, a posture.
2. Some candidates for Parliament adorn their electoral prospectus with a portrait; this presupposes that photography has a power to convert which must be analysed.
3. Inasmuch as photography is an ellipse of language and a condensation of an 'ineffable' social whole, it constitutes an anti-intellectual weapon and tends to spirit away 'politics' (that is to say a body of problems and solutions) to the advantage of a 'manner of being', a socio-moral status.
4. Photography tends to restore the paternalistic nature of elections, whose elitist essence has been disrupted by proportional representation and the rule of parties (the Right seems to use it more than the Left).
Sentence 2 introduces the topic, candidates using portraits on prospectuses and photography's power to convert. Sentence 1 follows by explaining how this works, that the effigy of a candidate creates a personal link with voters through implied lifestyle and morphology. Sentence 4 then builds on this personal link to argue that photography restores a paternalistic nature to elections that proportional representation had disrupted. Sentence 3 concludes by explaining the broader implication, that photography acts as an anti-intellectual weapon shifting attention from political substance to personal manner. This gives the sequence 2143.
Key Takeaway: Trace the argument from the specific practice introduced first, through its mechanism, to its political effect, and finally to its broader implication, to build the correct sequence.
Five jumbled 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 out and key in the number of that sentence as your answer.
1. The profound emotional impact of music has inspired ongoing research into its relationship with emotions.
2. Music is a universal phenomenon that utilizes a myriad brain resources.
3. This inherent connection to musical expression is deeply intertwined with human identity and experience.
4. The proclivity to create and appreciate music is ubiquitous among humans, permeating daily life across diverse societies.
5. Engaging with music is among the most cognitively demanding tasks a human can undergo, and it is identified across cultures.
Sentences 2, 3, 4 and 5 build a coherent description of music as a universal, identity-linked, cognitively demanding human phenomenon. Sentence 1 shifts the focus to research being inspired by music's emotional impact, which introduces a new angle about research rather than continuing the direct description of music's universal nature being built by the other four sentences.
Key Takeaway: Watch for a sentence that shifts focus from describing a phenomenon to discussing research or study of that phenomenon, since this shift in subject is often the odd one out.
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. When I ask the distinguished LGBTQ activist and writer Cherie Moraga whether she uses Latinx to refer to herself, she tells me, 'I worked too hard for the "a" in Latina to give it up! I refer to myself as Xicana.'
2. Of our accumulated ethnic population, only a third use Hispanic to identify themselves, a mere 14 percent use Latino, and less than 2 percent recognize Latinx.
3. They have done this, although gender in languages is grammatical, not sociological or sexual, and found in linguistic families throughout the world, from French to Russian to Japanese.
4. More recently, activists seeking to render our name gender neutral, out of respect for our LGBTQ members, have devised yet another name for us: Latinx.
Sentence 4 introduces the topic, the recent coining of "Latinx" as a gender-neutral term by activists. Sentence 3 follows by addressing the linguistic basis for this move, pointing out that grammatical gender exists across many language families. Sentence 1 then gives a specific example, Cherie Moraga's preference to retain "Latina" rather than adopt "Latinx". Sentence 2 closes with statistics showing how few people actually use Latinx compared to Hispanic or Latino. This gives the sequence 4312.
Key Takeaway: Move from the general introduction of a concept, through its linguistic or contextual basis, to a specific example, and finally to broader supporting data, to construct the correct sequence.
The given sentence is missing in the paragraph below. Decide where it best fits among the options 1, 2, 3, or 4 indicated in the paragraph.
Sentence: Productivity gains, once expected to feed through to broader living standards, now primarily serve to enhance returns to wealth.
Paragraph: Economists now argue that inequality is no longer a by-product of growth but a condition of it. ____ (1) ____. Unlike wages, wealth reflects not just income but also access to assets, favourable institutional conditions - such as low interest rates - and public policies like low taxes and housing shortages. ____ (2) ____. In other words, wealth depends on political choices in ways that income currently does not. It's not just the inequality itself that is the issue but the erosion of mechanisms that once constrained it. ____ (3) ____. Wealth and income inequality are linked, but where wages have stagnated and collective bargaining has weakened, capital income - derived from profits, rents and interest - has been boosted by design. ____ (4) ____.
Option 1
Option 4
Option 3
Option 2
Option 4
The inserted sentence states that productivity gains now primarily serve to enhance returns to wealth rather than living standards. This fits best at the end, Option 4, following the discussion of how capital income has been boosted by design while wages have stagnated, since the inserted sentence draws the natural concluding link between productivity gains and the resulting boost to wealth rather than wages.
Why Option A is
Incorrect: Option 1 comes right after the general claim that inequality is now a condition of growth, before the passage even explains what wealth depends on. Inserting a sentence about productivity gains here is premature, since the distinction between wealth and wages hasn't been established yet.
Why Option C is
Incorrect: Option 3 comes after the point about wealth depending on political choices, right before the passage discusses the erosion of mechanisms that constrained inequality. Inserting the productivity sentence here disrupts the flow into that specific point about eroding constraints.
Why Option D is
Incorrect: Option 2 falls right after the explanation of what wealth reflects, access to assets and favourable policies, before the passage concludes that wealth depends on political choices unlike income. Inserting the productivity sentence here interrupts this logical step.
Why Option B is Correct: Option 4 comes after the passage explains that capital income has been boosted by design while wages stagnated, making it the perfect place for the inserted sentence to conclude that productivity gains now go toward enhancing wealth rather than improving general living standards.
Key Takeaway: Place a concluding-type sentence at the point in the paragraph where the argument has built up enough context for that conclusion to land naturally.
The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In investigating memory-beliefs, there are certain points which must be borne in mind. In the first place, everything constituting a memory-belief is happening now, not in that past time to which the belief is said to refer. It is not logically necessary to the existence of a memory-belief that the event remembered should have occurred, or even that the past should have existed at all. There is no logical impossibility in the hypothesis that the world sprang into being five minutes ago, exactly as it then was, with a population that "remembered" a wholly unreal past. There is no logically necessary connection between events at different times; therefore nothing that is happening now or will happen in the future can disprove the hypothesis that the world began five minutes ago. Hence the occurrences which are CALLED knowledge of the past are logically independent of the past; they are wholly analysable into present contents, which might, theoretically, be just what they are even if no past had existed.
When we discuss the concept of memory-beliefs, we must understand that it is not logically impossible for the event remembered to have never happened at all; it could just be a figment of our imagination.
Memory-beliefs depend wholly on what is remembered in the present, and not on anything else; just as it is not logically impossible that the world came into being five minutes ago, and that everyone now just remembers a wholly imaginary past for it.
When investigating memory beliefs, we must keep in mind that an actual past event is not a prerequisite for a memory-belief to exist, and that what we know of the past could theoretically not need a past at all.
That which we call 'knowledge of the past' is logically independent of the past, since the act of remembering which forms memory-beliefs happens in the present, and does not need to be based in real past occurrences, or even need a past at all.
That which we call 'knowledge of the past' is logically independent of the past, since the act of remembering which forms memory-beliefs happens in the present, and does not need to be based in real past occurrences, or even need a past at all.
The passage argues that memory-beliefs occur entirely in the present, that the actual past event is not logically necessary for a memory-belief to exist, and uses the hypothesis of a world springing into existence five minutes ago to show that what we call "knowledge of the past" is logically independent of any real past.
Why Option A is
Incorrect: This focuses only on the possibility that a remembered event never happened, but misses the deeper logical claim that knowledge of the past as a whole does not depend on the past existing at all.
Why Option B is
Incorrect: This restates the hypothesis and the idea that memory-beliefs depend on the present, but does not capture the conclusion as precisely, that what is called knowledge of the past is logically independent of the past itself.
Why Option C is
Incorrect: This captures the idea that an actual past is not a prerequisite, but stops short of stating the specific conclusion that what we call "knowledge of the past" is itself logically independent of the past, the passage's most precise closing claim.
Why Option D is Correct: This captures both the conclusion, that knowledge of the past is logically independent of the past, and the reasoning behind it, that memory-beliefs happen in the present and do not require any actual past to have occurred.
Key Takeaway: Choose the summary that includes both the passage's central logical conclusion and the reasoning that supports it, rather than one that only captures part of the argument.
Three countries – Pumpland (P), Xiland (X) and Cheeseland (C) – trade among themselves and with the (other countries in) Rest of World (ROW). All trade volumes are given in IC (international currency). The following terminology is used:
• Trade balance = Exports - Imports
• Total trade = Exports + Imports
• Normalized trade balance = Trade balance / Total trade, expressed in percentage terms The following information is known.
1. The normalized trade balances of P, X and C are 0%, 10%, and -20%, respectively.
2. 40% of exports of X are to P. 22% of imports of P are from X.
3. 90% of exports of C are to P; 4% are to ROW.
4. 12% of exports of ROW are to X, 40% are to P.
5. The export volumes of P, in IC, to X and C are 600 and 1200, respectively. P is the only country that exports to C.
How much is exported from C to X, in IC?
Let the total exports of P, X, C, and ROW be Ep, Ex, Ec, and Er respectively, while their total imports be Ip, Ix, Ic, and Ir respectively.
Using the normalized trade balance formula:
For P:
(Ep − Ip)/(Ep + Ip) = 0
⇒ Ep = Ip
For X:
(Ex − Ix)/(Ex + Ix) = 10%
⇒ Ex − Ix = 0.1(Ex + Ix)
⇒ 0.9Ex = 1.1Ix
⇒ Ex/Ix = 11/9
For C:
(Ec − Ic)/(Ec + Ic) = −20%
⇒ Ec − Ic = −0.2(Ec + Ic)
⇒ 1.2Ec = 0.8Ic
⇒ Ic = 1.5Ec
Imports and Exports of C
According to condition (5), P is the only country that exports to C.
Hence, C receives all its imports from P, and the import volume from P to C is 1200.
Therefore,
Ic = 1200.
Since Ic = 1.5Ec,
Ec = 800.
Thus, C's exports are distributed as follows:
- 90% to P ⇒ 0.9 × Ec = 720
- 4% to ROW ⇒ 0.04 × Ec = 32
- The remaining 6% to X ⇒ C → X = 0.06 × 800 = 48
How much is exported from P to ROW, in IC?
Step 1:
Let the total exports of P, X, C, and ROW be Ep, Ex, Ec, and Er respectively, while their corresponding imports be Ip, Ix, Ic, and Ir.
Using the normalized trade balance equation:
For P: (Ep − Ip)/(Ep + Ip) = 0 ⇒ Ep = Ip
For X: (Ex − Ix)/(Ex + Ix) = 10% ⇒ Ex − Ix = 0.1(Ex + Ix)
⇒ 0.9Ex = 1.1Ix
⇒ Ex/Ix = 11/9
For C: (Ec − Ic)/(Ec + Ic) = −20% ⇒ Ec − Ic = −0.2(Ec + Ic)
⇒ 1.2Ec = 0.8Ic
⇒ Ic = 1.5Ec
Imports and Exports of C
As stated in condition (5), P is the only country exporting to C.
Hence, C imports only from P, and the import value is 1200.
Therefore,
Ic = 1200.
Since Ic = 1.5Ec,
Ec = 800.
Thus, C's exports are:
- 90% to P ⇒ 0.9Ec = 720
- 4% to ROW ⇒ 0.04Ec = 32
- Remaining 6% to X ⇒ C → X = 0.06 × 800 = 48
Step 2:
Using condition (2):
40% of X's total exports to P (X → P) is equal to 22% of P's imports received from X.
Therefore,
0.4Ex = 0.22Ip ⇒ Ex = 11Ip/20
Since Ep = Ip, 0.4Ex = 0.22Ep ⇒ Ex = 0.55Ep
Also, 20Ix = 9Ip.
Imports and Exports of P
P imports goods from X, C, and ROW.
- From X = 0.22Ip
- From C = 720
- From ROW = Ip − (0.22Ip + 720) = 0.78Ip − 720
Since exports from ROW to P constitute 40% of Er,
0.4Er = 0.78Ip − 720 ...(1)
Imports and Exports of X
Given that 12% of ROW's exports are sent to X,
ROW → X = 0.12Er.
Hence, X's imports are:
- From P = 600
- From C = 48
- From ROW = 0.12Er
Therefore,
Ix = 648 + 0.12Er.
Since Ix = 0.45Ep, 0.45Ep = 648 + 0.12Er ...(2)
Step 3:
Using equations (1) and (2):
From (1),
Er = (0.78Ep − 720)/0.4
= 1.95Ep − 1800
Substituting this into equation (2):
0.45Ep = 648 + 0.12(1.95Ep − 1800)
⇒ 0.45Ep = 648 + 0.234Ep − 216
⇒ 0.216Ep = 432
⇒ Ep = 2000
Since Ep = Ip,
Ip = 2000
Hence, the completed trade table is:
Thus, 200 units are exported from P to ROW.
How much is exported from ROW to ROW, in IC?
Step 1:
Let the total exports of P, X, C, and ROW be Ep, Ex, Ec, and Er respectively, while their corresponding imports be Ip, Ix, Ic, and Ir.
Using the normalized trade balance equation:
For P: (Ep − Ip)/(Ep + Ip) = 0 ⇒ Ep = Ip
For X: (Ex − Ix)/(Ex + Ix) = 10% ⇒ Ex − Ix = 0.1(Ex + Ix)
⇒ 0.9Ex = 1.1Ix
⇒ Ex/Ix = 11/9
For C: (Ec − Ic)/(Ec + Ic) = −20% ⇒ Ec − Ic = −0.2(Ec + Ic)
⇒ 1.2Ec = 0.8Ic
⇒ Ic = 1.5Ec
Imports and Exports of C
As stated in condition (5), P is the only country exporting to C.
Hence, C imports only from P, and the import value is 1200.
Therefore,
Ic = 1200.
Since Ic = 1.5Ec,
Ec = 800.
Thus, C's exports are:
- 90% to P ⇒ 0.9Ec = 720
- 4% to ROW ⇒ 0.04Ec = 32
- Remaining 6% to X ⇒ C → X = 0.06 × 800 = 48
Step 2:
Using condition (2):
40% of X's total exports to P (X → P) is equal to 22% of P's imports received from X.
Therefore,
0.4Ex = 0.22Ip ⇒ Ex = 11Ip/20
Since Ep = Ip, 0.4Ex = 0.22Ep ⇒ Ex = 0.55Ep
Also, 20Ix = 9Ip.
Imports and Exports of P
P imports goods from X, C, and ROW.
- From X = 0.22Ip
- From C = 720
- From ROW = Ip − (0.22Ip + 720) = 0.78Ip − 720
Since exports from ROW to P constitute 40% of Er,
0.4Er = 0.78Ip − 720 ...(1)
Imports and Exports of X
Given that 12% of ROW's exports are sent to X,
ROW → X = 0.12Er.
Hence, X's imports are:
- From P = 600
- From C = 48
- From ROW = 0.12Er
Therefore,
Ix = 648 + 0.12Er.
Since Ix = 0.45Ep, 0.45Ep = 648 + 0.12Er ...(2)
Step 3:
Using equations (1) and (2):
From (1),
Er = (0.78Ep − 720)/0.4
= 1.95Ep − 1800
Substituting this into equation (2):
0.45Ep = 648 + 0.12(1.95Ep − 1800)
⇒ 0.45Ep = 648 + 0.234Ep − 216
⇒ 0.216Ep = 432
⇒ Ep = 2000
Since Ep = Ip,
Ip = 2000
Hence, the completed trade table is:
1008 IC is exported from ROW to ROW.
What is the trade balance of ROW?
100
0
-200
200
200
Step 1:
Let the total exports of P, X, C, and ROW be Ep, Ex, Ec, and Er respectively, while their corresponding imports be Ip, Ix, Ic, and Ir.
Using the normalized trade balance equation:
For P: (Ep − Ip)/(Ep + Ip) = 0 ⇒ Ep = Ip
For X: (Ex − Ix)/(Ex + Ix) = 10% ⇒ Ex − Ix = 0.1(Ex + Ix)
⇒ 0.9Ex = 1.1Ix
⇒ Ex/Ix = 11/9
For C: (Ec − Ic)/(Ec + Ic) = −20% ⇒ Ec − Ic = −0.2(Ec + Ic)
⇒ 1.2Ec = 0.8Ic
⇒ Ic = 1.5Ec
Imports and Exports of C
As stated in condition (5), P is the only country exporting to C.
Hence, C imports only from P, and the import value is 1200.
Therefore,
Ic = 1200.
Since Ic = 1.5Ec,
Ec = 800.
Thus, C's exports are:
- 90% to P ⇒ 0.9Ec = 720
- 4% to ROW ⇒ 0.04Ec = 32
- Remaining 6% to X ⇒ C → X = 0.06 × 800 = 48
Step 2:
Using condition (2):
40% of X's total exports to P (X → P) is equal to 22% of P's imports received from X.
Therefore,
0.4Ex = 0.22Ip ⇒ Ex = 11Ip/20
Since Ep = Ip, 0.4Ex = 0.22Ep ⇒ Ex = 0.55Ep
Also, 20Ix = 9Ip.
Imports and Exports of P
P imports goods from X, C, and ROW.
- From X = 0.22Ip
- From C = 720
- From ROW = Ip − (0.22Ip + 720) = 0.78Ip − 720
Since exports from ROW to P constitute 40% of Er,
0.4Er = 0.78Ip − 720 ...(1)
Imports and Exports of X
Given that 12% of ROW's exports are sent to X,
ROW → X = 0.12Er.
Hence, X's imports are:
- From P = 600
- From C = 48
- From ROW = 0.12Er
Therefore,
Ix = 648 + 0.12Er.
Since Ix = 0.45Ep, 0.45Ep = 648 + 0.12Er ...(2)
Step 3:
Using equations (1) and (2):
From (1),
Er = (0.78Ep − 720)/0.4
= 1.95Ep − 1800
Substituting this into equation (2):
0.45Ep = 648 + 0.12(1.95Ep − 1800)
⇒ 0.45Ep = 648 + 0.234Ep − 216
⇒ 0.216Ep = 432
⇒ Ep = 2000
Since Ep = Ip,
Ip = 2000
Hence, the completed trade table is:
The trade balance of ROW = Er – Ir = 2100 – 1900 = 200 IC.
Which among the countries P, X, and C has/have the least total trade?
Only P
Only C
Both X and C
Only X
Both X and C
Step 1:
Let the total exports of P, X, C, and ROW be Ep, Ex, Ec, and Er respectively, while their corresponding imports be Ip, Ix, Ic, and Ir.
Using the normalized trade balance equation:
For P: (Ep − Ip)/(Ep + Ip) = 0 ⇒ Ep = Ip
For X: (Ex − Ix)/(Ex + Ix) = 10% ⇒ Ex − Ix = 0.1(Ex + Ix)
⇒ 0.9Ex = 1.1Ix
⇒ Ex/Ix = 11/9
For C: (Ec − Ic)/(Ec + Ic) = −20% ⇒ Ec − Ic = −0.2(Ec + Ic)
⇒ 1.2Ec = 0.8Ic
⇒ Ic = 1.5Ec
Imports and Exports of C
As stated in condition (5), P is the only country exporting to C.
Hence, C imports only from P, and the import value is 1200.
Therefore,
Ic = 1200.
Since Ic = 1.5Ec,
Ec = 800.
Thus, C's exports are:
- 90% to P ⇒ 0.9Ec = 720
- 4% to ROW ⇒ 0.04Ec = 32
- Remaining 6% to X ⇒ C → X = 0.06 × 800 = 48
Step 2:
Using condition (2):
40% of X's total exports to P (X → P) is equal to 22% of P's imports received from X.
Therefore,
0.4Ex = 0.22Ip ⇒ Ex = 11Ip/20
Since Ep = Ip, 0.4Ex = 0.22Ep ⇒ Ex = 0.55Ep
Also, 20Ix = 9Ip.
Imports and Exports of P
P imports goods from X, C, and ROW.
- From X = 0.22Ip
- From C = 720
- From ROW = Ip − (0.22Ip + 720) = 0.78Ip − 720
Since exports from ROW to P constitute 40% of Er,
0.4Er = 0.78Ip − 720 ...(1)
Imports and Exports of X
Given that 12% of ROW's exports are sent to X,
ROW → X = 0.12Er.
Hence, X's imports are:
- From P = 600
- From C = 48
- From ROW = 0.12Er
Therefore,
Ix = 648 + 0.12Er.
Since Ix = 0.45Ep, 0.45Ep = 648 + 0.12Er ...(2)
Step 3:
Using equations (1) and (2):
From (1),
Er = (0.78Ep − 720)/0.4
= 1.95Ep − 1800
Substituting this into equation (2):
0.45Ep = 648 + 0.12(1.95Ep − 1800)
⇒ 0.45Ep = 648 + 0.234Ep − 216
⇒ 0.216Ep = 432
⇒ Ep = 2000
Since Ep = Ip,
Ip = 2000
Hence, the completed trade table is:
Total trade:
P = Ep + Ip = 4000
X = Ex + Ix = (1100 + 900) = 2000
C = Ec + Ic = (800 + 1200) = 2000
Hence, the least total trade = X and C.
Seven children, Aarav, Bina, Chirag, Diya, Eshan, Farhan, and Gaurav, are sitting in a circle facing inside (not necessarily in the same order) and playing a game of 'Passing the Buck'.
The game is played over 10 rounds. In each round, the child holding the Buck must pass it directly to a child sitting in one of the following positions:
• Immediately to the left;
• Immediate to the right;
• Second to the left; or
• Second to the right.
The game starts with Bina passing the Buck and ends with Chirag receiving the Buck. The table below provides some information about the pass types and the child receiving the Buck. Some information is missing and labelled as '?'.
Who is sitting immediately to the right of Bina?
Eshan
Aarav
Farhan
Chirag
Eshan
In Round 8, Diya receives the Buck. In Round 9, she passes it in such a way that Farhan receives it.
If Farhan sits to the immediate left of Diya, then there is no way Farhan passes it in a way receives it.
Therefore, Diya must pass second to the right and then Farhan must pass immediate to the right and Chirag received it.
Final arrangement looks like:
Eshan is sitting immediately to the right of Bina.
Who is sitting third to the left of Eshan?
Chirag
Aarav
Divya
Gaurav
Chirag
In Round 8, Diya receives the Buck. In Round 9, she passes it in such a way that Farhan receives it.
If Farhan sits to the immediate left of Diya, then there is no way Farhan passes it in a way receives it.
Therefore, Diya must pass second to the right and then Farhan must pass immediate to the right and Chirag received it.
Final arrangement looks like:
Chirag is sitting third to the left of Eshan.
For which of the following pass types can the total number of occurrences be uniquely determined?
Immediately to the left
Second to the right
Immediately to the right
Second to the left
Immediately to the right
In Round 8, Diya receives the Buck. In Round 9, she passes it in such a way that Farhan receives it.
If Farhan sits to the immediate left of Diya, then there is no way Farhan passes it in a way receives it.
Therefore, Diya must pass second to the right and then Farhan must pass immediate to the right and Chirag received it.
Final arrangement looks like:
For Immediately to the right pass type the total number of occurrences can be uniquely determined.
For which of the following children is it possible to determine how many times they received the Buck?
Eshan
Gaurav
Bina
Farhan
Gaurav
In Round 8, Diya receives the Buck. In Round 9, she passes it in such a way that Farhan receives it.
If Farhan sits to the immediate left of Diya, then there is no way Farhan passes it in a way receives it.
Therefore, Diya must pass second to the right and then Farhan must pass immediate to the right and Chirag received it.
Final arrangement looks like:
For Gaurav it is possible to determine the number of times he received the Buck due to round 4.
Anu, Bijay, Chetan, Deepak, Eshan, and Faruq are six friends. Each of them uses a mobile number from exactly one of the two mobile operators - Xitel and Yocel. During the last month, the six friends made several calls to each other. Each call was made by one of these six friends to another. The table below summarizes the number of minutes of calls that each of the six made to (outgoing minutes) and received from (incoming minutes) these friends, grouped by the operators. Some of the entries are missing.
It is known that the duration of calls from Faruq to Eshan was 200 minutes.
Also, there were no calls from:
i. Bijay to Eshan,
ii. Chetan to Anu and Chetan to Deepak,
iii. Deepak to Bijay and Deepak to Faruq,
iv. Eshan to Chetan and Eshan to Deepak.
What was the duration of calls (in minutes) from Bijay to Anu?
What was the total duration of calls (in minutes) made by Anu to friends having mobile numbers from Operator Yocel?
What was the total duration of calls (in minutes) made by Faruq to friends having mobile numbers from Operator Yocel?
What was the duration of calls (in minutes) from Deepak to Chetan?
50
125
0
100
100
Anirbid, Chandranath, Koushik, and Suranjan participated in a puzzle solving competition. The competition comprised 10 puzzles that had to be solved in the same sequence, i.e., a competitor got access to a puzzle as soon as they solved the previous puzzle. Some of the puzzles were visual puzzles and the others were number-based puzzles. The winner of the competition was the one who solved all puzzles in the least time.
The following charts describe their progress in the competition. The chart on the left shows the number of puzzles solved by each competitor at a given time (in minutes) after the start of the competition. The chart on the right shows the number of visual puzzles solved by each competitor at a given time (in minutes) after the start of the competition.
Who had solved the largest number of puzzles by the 20-th minute from the start of the competition?
Koushik
Anirbid
Chandranath
Suranjan
Koushik
How many minutes did Suranjan take to solve the third visual puzzle in the competition?
At what number in the sequence was the fourth number-based puzzle?
Which of the following is the closest to the average time taken by Anirbid to solve the number-based puzzles in the competition?
4.0 minutes
3.8 minutes
2.5 minutes
3.3 minutes
4.0 minutes
Aurevia, Brelosia, Cyrenia and Zerathania are four countries with their currencies being Aurels, Brins, Crowns, and Zentars, respectively. The currencies have different exchange values. Crown's currency exchange rate with Zentars = 0.5, i.e., 1 Crown is worth 0.5 Zentars.
Three travelers, Jano, Kira, and Lian set out from Zerathania visiting exactly two of the countries. Each country is visited by exactly two travelers. Each traveler has a unique Flight Cost, which represents the total cost of airfare in traveling to both the countries and back to Zerathania. The Flight Cost of Jano was 4000 Zentars, while that of the other two travelers were 5000 and 6000 Zentars, not necessarily in that order. When visiting a country, a traveler spent either 1000, 2000 or 3000 in the country's local currency. Each traveler had different spends (in the country's local currency) in the two countries he/she visited. Across all the visits, there were exactly two spends of 1000 and exactly one spend of 3000 (in the country's local currency).
The total "Travel Cost" for a traveler is the sum of his/her Flight Cost and the money spent in the countries visited.
The citizens of the four countries with knowledge of these travels made a few observations, with spends measured in their respective local currencies:
i. Aurevia citizen: Jano and Kira visited our country, and their Travel Costs were 3500 and 8000, respectively.
ii. Brelosia citizen: Kira and Lian visited our country, spending 2000 and 3000, respectively. Kira's Travel Cost was 4000.
iii. Cyrenia citizen: Lian visited our country and her Travel Cost was 36000.
What is the sum of Travel Costs for all travelers in Zentars?
How many Zentars did Lian spend in the two countries he visited?
What was Jano's total spend in the two countries he visited, in Aurels?
One Brin is equivalent to how many Crowns?
0.5
0.125
4
8
4
Which of the following statements is NOT true about money spent in the local currency?
Jano spent 2000 in Aurevia
Lian spent 2000 in Cyrenia
Jano spent 2000 in Cyrenia
Kira spent 1000 in Aurevia
Jano spent 2000 in Aurevia
The monthly sales of a product from January to April were 120, 135, 150 and 165 units, respectively. The cost price of the product was Rs. 240 per unit, and a fixed marked price was used for the product in all the four months. Discounts of 20%, 10% and 5% were given on the marked price per unit in January, February and March, respectively, while no discounts were given in April. If the total profit from January to April was Rs. 138825, then the marked price per unit, in rupees, was
520
525
510
515
525
Let the marked price per unit be M. Cost Price (CP) per unit = Rs. 240.
The selling prices in the four months are:
January: 80% of M = 0.8M
February: 90% of M = 0.9M
March: 95% of M = 0.95M
April: M
Step 2: Find the total revenue
Revenue in January = 120 × 0.8M = 96M
Revenue in February = 135 × 0.9M = 121.5M
Revenue in March = 150 × 0.95M = 142.5M
Revenue in April = 165 × M = 165M
Total revenue = 96M + 121.5M + 142.5M + 165M = 525M
Step 3: Find the total cost
Total units sold = 120 + 135 + 150 + 165 = 570
Total cost = 570 × 240 = Rs. 136800
Step 4: Use the given total profit
Profit = Total Revenue − Total Cost
138825 = 525M − 136800
525M = 275625
M = 275625 ÷ 525 = 525
Teams A, B, and C consist of five, eight, and ten members, respectively, such that every member within a team is equally productive. Working separately, teams A, B, and C can complete a certain job in 40 hours, 50 hours, and 4 hours, respectively. Two members from team A, three members from team B, and one member from team C together start the job, and the member from team C leaves after 23 hours. The number of additional member(s) from team B, that would be required to replace the member from team C, to finish the job in the next one hour, is
4
2
1
3
2
Step 1: Find the work rate of each member
Assume the total work is 200 units (LCM of 40, 50 and 4).
Team A completes 200 units in 40 hours → rate = 5 units/hour → each member = 1 unit/hour.
Team B completes 200 units in 50 hours → rate = 4 units/hour → each member = 1/2 unit/hour.
Team C completes 200 units in 4 hours → rate = 50 units/hour → each member = 5 units/hour.
Step 2: Find the work done in the first 23 hours
2 members from Team A: 2 × 1 = 2 units/hour
3 members from Team B: 3 × (1/2) = 3/2 units/hour
1 member from Team C: 5 units/hour
Total rate = 2 + 3/2 + 5 = 17/2 units/hour
Work done in 23 hours = 23 × 17/2 = 391/2 units
Step 3: Find the remaining work
Remaining work = 200 − 391/2 = 9/2 units
This work has to be completed in the next 1 hour.
Step 4: Find the required additional members from Team B
After the Team C member leaves, existing rate = 2 + 3/2 = 7/2 units/hour
Let x = additional members from Team B (each contributing 1/2 unit/hour).
7/2 + x/2 = 9/2 → 7 + x = 9 → x = 2
− In a school with 1500 students, each student chooses any one of the streams out of science, arts, and commerce, by paying a fee of Rs 1100, Rs 1000, and Rs 800, respectively. The total fee paid by all the students is Rs 15,50,000. If the number of science students is not more than the number of arts students, then the maximum possible number of science students in the school is
Step 1: Let S, A, C = number of students in science, arts, commerce.
S + A + C = 1500
1100S + 1000A + 800C = 15,50,000
Step 2: Eliminate C
C = 1500 − S − A
Substituting: 1100S + 1000A + 800(1500 − S −
A) = 15,50,000
300S + 200A = 3,50,000
Dividing by 100: 3S + 2A = 3500
Step 3: Express A in terms of S
A = (3500 − 3S)/2
Since A must be an integer, S must be even.
Given S ≤ A: S ≤ (3500 − 3S)/2 → 5S ≤ 3500 → S ≤ 700
Step 4: Verify
S = 700 → A = 700, C = 100 ✓ (all non-negative integers)
− In an arithmetic progression, if the sum of fourth, seventh and tenth terms is 99, and the sum of the first fourteen terms is 497, then the sum of first five terms is
Step 1: Let first term = a, common difference = d.
4th term = a+3d, 7th term = a+6d, 10th term = a+9d
Step 2: First condition
(a+3d) + (a+6d) + (a+9d) = 99 → 3a + 18d = 99 → a + 6d = 33
Step 3: Sum of first 14 terms
14/2 × [2a + 13d] = 497 → 7(2a + 13d) = 497 → 2a + 13d = 71
Step 4: Solve
From a + 6d = 33 → 2a + 12d = 66
Subtract: d = 5
a + 30 = 33 → a = 3
Step 5: Sum of first five terms
= 5/2 × [2(3) + 4(5)] = 5/2 × 26 = 65
− Ankita walks from A to C through B, and runs back through the same route at a speed that is 40% more than her walking speed. She takes exactly 3 hours 30 minutes to walk from B to C as well as to run from B to A. The total time, in minutes, she would take to walk from A to B and run from B to C, is
Step 1: Let walking speed = v km/h. Running speed = 1.4v = 7v/5 km/h.
Let AB = x km, BC = y km.
Step 2: Given conditions
Walking from B to C: y/v = 3.5 → y = 7v/2
Running from B to A: x/(7v/5) = 3.5 → x = 49v/10
Step 3: Find the required time
Walking A to B: x/v = 49/10 hours
Running B to C: y/(7v/5) = (7v/2) × (5/7v) = 5/2 hours
Total time = 49/10 + 25/10 = 74/10 = 7.4 hours
Step 4: Convert to minutes
7.4 × 60 = 444 minutes
For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is
811
3289
735
4078
811
Step 1: Let the four-digit number be ABCD.
A + B + C = 15
B + C + D = 16
C = D + 6
Step 2: Subtract 1st equation from 2nd: D − A = 1 → A = D − 1
Since A ≥ 1 and C = D + 6 ≤ 9, we get D ≤ 3.
Also A = D − 1 ≥ 1 → D ≥ 2. So D ∈ {2, 3}.
Step 3: Case D = 2 → A = 1, C = 8, B = 6 → Number: 1682
Step 4: Case D = 3 → A = 2, C = 9, B = 4 → Number: 2493
Step 5: Difference = 2493 − 1682 = 811
Rahul starts on his journey at 5 pm at a constant speed so that he reaches his destination at 11 pm the same day. However, on his way, he stops for 20 minutes, and after that, increases his speed by 3 km per hour to reach on time. If he had stopped for 10 minutes more, he would have had to increase his speed by 5 km per hour to reach on time. His initial speed, in km per hour, was
12
15
18
20
15
Step 1: Total journey time = 6 hours. Let initial speed = x km/h → Total distance = 6x km.
Step 2: Verify x = 15 km/h.
Total distance = 90 km.
After 20-min stop, travelling time = 17/3 hours.
Let t = time before stop.
15t + 18(17/3 − t) = 90 → 15t + 102 − 18t = 90 → t = 4 hours.
Rahul travels 60 km at 15 km/h, then 30 km at 18 km/h (1 hr 40 min).
Total = 4 hrs + 1h40m + 20min stop = 6 hours ✓
Step 3: Check second condition (30-min stop).
Remaining travelling time = 6 − 4 − 0.5 = 1.5 hours.
Remaining distance = 30 km.
Required speed = 30 ÷ 1.5 = 20 km/h.
Increase = 20 − 15 = 5 km/h ✓
The rate of water flow through three pipes A, B and C are in the ratio 4 : 9 : 36. An empty tank can be filled up completely by pipe A in 15 hours. If all the three pipes are used simultaneously to fill up this empty tank, the time, in minutes, required to fill up the entire tank completely is nearest to
73
78
76
71
73
Step 1: Pipe A fills tank in 15 hours → rate = 1/15 tank/hour.
A : B : C = 4 : 9 : 36 → 1 part = (1/15) ÷ 4 = 1/60 tank/hour.
Step 2: Rates
Pipe B = 9 × (1/60) = 3/20 tank/hour
Pipe C = 36 × (1/60) = 3/5 tank/hour
Step 3: Combined rate
= 1/15 + 3/20 + 3/5 = 4/60 + 9/60 + 36/60 = 49/60 tank/hour
Step 4: Time = 1 ÷ (49/60) = 60/49 hours
= (60/49) × 60 = 3600/49 ≈ 73.47 minutes ≈ 73 minutes
If f(x) = (x² + 3x)(x² + 3x + 2) then the sum of all real roots of the equation √(f(x) + 1) = 9701, is
−6
6
3
−3
−3
Step 1: Square both sides: f(x) + 1 = 9701²
f(x) = 9701² − 1 = 9700 × 9702
Step 2: Let y = x² + 3x.
y(y + 2) = 9700 × 9702
y² + 2y + 1 = 9701²
(y + 1)² = 9701²
y + 1 = ±9701 → y = 9700 or y = −9702
Step 3: Solve for x
Case 1: x² + 3x − 9700 = 0 → Discriminant = 9 + 38800 = 38809 = 197²
x = (−3 ± 197)/2 → x = 97 or x = −100
Case 2: x² + 3x + 9702 = 0 → Discriminant = 9 − 38808 < 0 (no real roots)
Step 4: Sum of real roots = 97 + (−100) = −3
For real values of x, the range of the function f(x) = (2x − 3) / (2x² + 4x − 6) is
(−∞, 1/8] ∪ [1, ∞)
(−∞, 1/4] ∪ [1, ∞)
(−∞, 1/8] ∪ [1/2, ∞)
(−∞, 1/4] ∪ [1/2, ∞)
(−∞, 1/8] ∪ [1/2, ∞)
Step 1: Let y = (2x − 3)/(2x² + 4x − 6).
Rearrange: 2yx² + (4y − 2)x + (3 − 6y) = 0
Step 2: For real x, discriminant ≥ 0.
(4y − 2)² − 4(2y)(3 − 6y) ≥ 0
16y² − 16y + 4 − 24y + 48y² ≥ 0
64y² − 40y + 4 ≥ 0
16y² − 10y + 1 ≥ 0
Step 3: Factor: (8y − 1)(2y − 1) ≥ 0
This holds when y ≤ 1/8 or y ≥ 1/2.
Step 4: Range = (−∞, 1/8] ∪ [1/2, ∞)
The sum of all the digits of the number (10⁵⁰ + 10²⁵ − 123), is
212
221
324
255
221
Step 1: Compute 10²⁵ − 123.
= 99999999999999999999999877
(22 nines followed by 877)
Step 2: Add 10⁵⁰.
The full number has:
- One leading 1
- Then 24 zeros
- Then 1
- Then 21 nines followed by 99877
Step 3: Sum of digits
= 1 + 1 + (21 × 9) + 9 + 9 + 8 + 7 + 7
= 2 + 189 + 40
= 221
− A triangle ABC is formed with AB = AC = 50 cm and BC = 80 cm. Then, the sum of the lengths, in cm, of all three altitudes of the triangle ABC is
Step 1: Altitude from A (to BC)
AB = AC = 50 cm, BC = 80 cm → altitude bisects BC (BD = DC = 40).
Altitude from A = √(50² − 40²) = √(2500 − 1600) = √900 = 30 cm
Step 2: Area = (1/2) × 80 × 30 = 1200 cm²
Step 3: Altitude from B (to AC = 50 cm)
1200 = (1/2) × 50 × h₁ → h₁ = 48 cm
By symmetry (AB = AC), altitude from C = 48 cm also.
Step 4: Sum = 30 + 48 + 48 = 126 cm
If (x² + 1/x²) = 25 and x > 0, then the value of (x⁷ + 1/x⁷) is
44853√3
44856√3
44859√3
44850√3
44853√3
Step 1: Find x + 1/x
x² + 1/x² = (x + 1/x)² − 2 = 25 → (x + 1/x)² = 27 → x + 1/x = 3√3
Step 2: x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)
= (3√3)³ − 3(3√3) = 81√3 − 9√3 = 72√3
Step 3: x⁵ + 1/x⁵ = (x³ + 1/x³)(x² + 1/x²) − (x + 1/x)
= 72√3 × 25 − 3√3 = 1800√3 − 3√3 = 1797√3
Step 4: x⁷ + 1/x⁷ = (x⁵ + 1/x⁵)(x² + 1/x²) − (x³ + 1/x³)
= 1797√3 × 25 − 72√3 = 44925√3 − 72√3 = 44853√3
In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is
30
35
40
55
35
Step 1: Let Physics∩Chemistry = x, Chemistry∩Mathematics = x, Physics∩Mathematics = 2x.
Let t = number who chose all three (t ≥ 1).
Step 2: Inclusion-Exclusion
75 + 111 + 40 − (2x + x + x) + t = 150
226 − 4x + t = 150 → t = 4x − 76
Step 3: Constraints
t ≥ 1 → x ≥ 20 (approximately)
t ≤ x → 4x − 76 ≤ x → 3x ≤ 76 → x ≤ 25
So 20 ≤ x ≤ 25.
Step 4: Maximize Physics but not Mathematics = 75 − 2x.
Minimize x: take x = 20 → 75 − 40 = 35.
Verification: t = 4(20) − 76 = 4 ≥ 1 and t ≤ x ✓
The sum of all possible real values of x for which log₍ₓ₋₃₎(x² − 9) = log₍ₓ₋₃₎(x + 1) + 2, is
−3
√33
3
(3 + √33) / 2
(3 + √33) / 2
Step 1: Domain conditions
x − 3 > 0 and x − 3 ≠ 1 → x > 3 and x ≠ 4.
Step 2: Rewrite using 2 = log₍ₓ₋₃₎(x−3)²
log₍ₓ₋₃₎(x² − 9) = log₍ₓ₋₃₎[(x+1)(x−3)²]
→ x² − 9 = (x+1)(x−3)²
Step 3: Factor and simplify
(x−3)(x+3) = (x+1)(x−3)²
Cancel (x−3) [valid since x > 3]: x + 3 = (x+1)(x−3)
Step 4: Expand and solve
x + 3 = x² − 2x − 3 → x² − 3x − 6 = 0
x = (3 ± √33)/2
Step 5: Check domain
(3 − √33)/2 < 3 → rejected.
Valid solution: x = (3 + √33)/2
The average salary of 5 managers and 25 engineers in a company is 60000 rupees. If each of the managers received 20% salary increase while the salary of the engineers remained unchanged, the average salary of all 30 employees would have increased by 5%. The average salary, in rupees, of the engineers is
45000
50000
54000
40000
54000
Step 1: Total salary = 30 × 60,000 = Rs. 18,00,000
Step 2: After 5% increase in average,
New average = 63,000 → Total increase = 30 × 3,000 = Rs. 90,000
Step 3: This increase comes only from managers' 20% raise.
20% of M = 90,000 → M (total managers' salary) = Rs. 4,50,000
Step 4: Total engineers' salary = 18,00,000 − 4,50,000 = Rs. 13,50,000
Average engineers' salary = 13,50,000 ÷ 25 = Rs. 54,000
ABCD is a trapezium in which AB is parallel to DC, AD is perpendicular to AB, and AB = 3DC. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area, in sq. cm, of the trapezium is
48
30√3
36√2
54
48
Step 1: Let DC = x, AB = 3x.
AD ⊥ AB and incircle radius = 3 cm → AD = 6 cm (= diameter).
Step 2: BC = √((2x)² + 6²) = √(4x² + 36)
Step 3: Tangential quadrilateral property: AB + DC = AD + BC
4x = 6 + √(4x² + 36)
√(4x² + 36) = 4x − 6
Step 4: Square both sides:
4x² + 36 = 16x² − 48x + 36
12x² − 48x = 0 → x = 4
DC = 4 cm, AB = 12 cm
Step 5: Area = (1/2) × (12 + 4) × 6 = 48 sq. cm
− If 12^(12x) × 4^(24x+12) × 5^(2y) = 8^(4z) × 20^(12x) × 243^(3x−6), where x, y and z are natural numbers, then x + y + z equals
Step 1: Express in prime factors.
12 = 2²×3, 4 = 2², 8 = 2³, 20 = 2²×5, 243 = 3⁵
Step 2: Compare powers of 2
LHS: 2^(24x) × 2^(48x+24) = 2^(72x+24)
RHS: 2^(12z) × 2^(24x) = 2^(12z+24x)
72x + 24 = 12z + 24x → 48x + 24 = 12z → z = 4x + 2
Step 3: Compare powers of 3
LHS: 3^(12x), RHS: 3^(15x−30)
12x = 15x − 30 → 3x = 30 → x = 10
z = 4(10) + 2 = 42
Step 4: Compare powers of 5
LHS: 5^(2y), RHS: 5^(12x)
2y = 12x → y = 6x = 60
Step 5: x + y + z = 10 + 60 + 42 = 112
In △ABC, AB = AC = 12 cm and D is a point on side BC such that AD = 8 cm. If AD is extended to a point E such that ∠ACB = ∠AEB, then the length, in cm, of AE is
20
16
18
14
18
Step 1: △ABC is isosceles with AB = AC = 12 cm. Let BC = b.
Midpoint of BC is M; AM² = 144 − b²/4.
Step 2: AD = 8 cm.
Using the condition ∠AEB = ∠ACB, points B, C, D, E are concyclic
(since ∠AEB = ∠ACB means E lies on the circumcircle of △ABC, or by
power of a point / similar triangles).
By Power of a Point: AD × AE = AB × AC
8 × AE = 12 × 12
8 × AE = 144
AE = 18 cm
− Vessels A and B contain 60 litres of alcohol and 60 litres of water, respectively. A certain volume is taken out from A and poured into B. After stirring, the same volume is taken out from B and poured into A. If the resultant ratio of alcohol and water in A is 15 : 4, then the volume, in litres, initially taken out from A is
Step 1: Let x litres be transferred from A to
B.
After transfer: Alcohol in A = 60−x; Vessel B has x alcohol + 60 water (total 60+x).
Step 2: Transfer x litres back from B to
A.
Alcohol returned = x²/(60+x)
Water returned = 60x/(60+x)
Step 3: Final contents of vessel A
Alcohol in A = (60−x) + x²/(60+x) = 3600/(60+x)
Water in A = 60x/(60+x)
Step 4: Ratio condition
3600/(60+x) : 60x/(60+x) = 15 : 4
3600 : 60x = 15 : 4
3600 × 4 = 60x × 15 → 14400 = 900x → x = 16
− The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins that needs to be shifted further from A to B, to make this ratio 1:1, is
Step 1: Let initial coins in A = 17x, B = 7x.
After shifting 108: (17x−108)/(7x+108) = 37/20
Step 2: Cross-multiply
20(17x−108) = 37(7x+108)
340x − 2160 = 259x + 3996
81x = 6156 → x = 76
Step 3: After 108 coins shifted:
A = 17(76) − 108 = 1184, B = 7(76) + 108 = 640
Step 4: Let y more coins be shifted for 1:1 ratio.
1184 − y = 640 + y → 544 = 2y → y = 272
− Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose value lies between 150 and 500. If p is not less than 0.3q and not more than 0.7q, then the sum of the maximum and minimum possible values of p is
Step 1: p + q + r = 900, 150 < r < 500, r is a perfect square.
Let S = 900 − r, so p + q = S.
Step 2: Constraints on p
p ≥ 0.3q = 0.3(S−p) → 1.3p ≥ 0.3S → p ≥ 3S/13
p ≤ 0.7q = 0.7(S−p) → 1.7p ≤ 0.7S → p ≤ 7S/17
Step 3: Maximize p → maximize S → minimize r.
Smallest perfect square > 150: r = 169 → S = 731
Maximum p = ⌊7×731/17⌋ = ⌊5117/17⌋ = 301
Step 4: Minimize p → minimize S → maximize r.
Largest perfect square < 500: r = 484 → S = 416
Minimum p = ⌈3×416/13⌉ = ⌈1248/13⌉ = 96
Step 5: Sum = 301 + 96 = 397
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