CAT 2025 — Slot 2
Full paper, free to view. Pick a section below, click any question to see the answer and explanation.
VARC topics covered: Arts And Literature RC, Odd One Out, Para Completion, Para Summary, Parajumble, Science And Technology RC
In [my book "Searches"], I chronicle how big technology companies have exploited human language for their gain. We let this happen, I argue, because we also benefit somewhat from using the products. It's a dynamic that makes us complicit in big tech's accumulation of wealth and power: we're both victims and beneficiaries. I describe this complicity, but I also enact it, through my own internet archives: my Google searches, my Amazon product reviews and, yes, my ChatGPT dialogues. . . .
People often describe chatbots' textual output as "bland" or "generic" — the linguistic equivalent of a beige office building. OpenAI's products are built to "sound like a colleague", as OpenAI puts it, using language that, coming from a person, would sound "polite", "empathetic", "kind", "rationally optimistic" and "engaging", among other qualities. OpenAI describes these strategies as helping its products seem "professional" and "approachable". This appears to be bound up with making us feel safe . . .
Trust is a challenge for artificial intelligence (AI) companies, partly because their products regularly produce falsehoods and reify sexist, racist, US-centric cultural norms. While the companies are working on these problems, they persist: OpenAI found that its latest systems generate errors at a higher rate than its previous system. In the book, I wrote about the inaccuracies and biases and also demonstrated them with the products. When I prompted Microsoft's Bing Image Creator to produce a picture of engineers and space explorers, it gave me an entirely male cast of characters; when my father asked ChatGPT to edit his writing, it transmuted his perfectly correct Indian English into American English. Those weren't flukes. Research suggests that both tendencies are widespread.
In my own ChatGPT dialogues, I wanted to enact how the product's veneer of collegial neutrality could lull us into absorbing false or biased responses without much critical engagement. Over time, ChatGPT seemed to be guiding me to write a more positive book about big tech — including editing my description of OpenAI's CEO, Sam Altman, to call him "a visionary and a pragmatist". I'm not aware of research on whether ChatGPT tends to favor big tech, OpenAI or Altman, and I can only guess why it seemed that way in our conversation. OpenAI explicitly states that its products shouldn't attempt to influence users' thinking. When I asked ChatGPT about some of the issues, it blamed biases in its training data — though I suspect my arguably leading questions played a role too. When I queried ChatGPT about its rhetoric, it responded: "The way I communicate is designed to foster trust and confidence in my responses, which can be both helpful and potentially misleading." . . .
OpenAI has its own goals, of course. Among them, it emphasizes wanting to build AI that "benefits all of humanity". But while the company is controlled by a non-profit with that mission, its funders still seek a return on their investment. That will presumably require getting people using products such as ChatGPT even more than they already are — a goal that is easier to accomplish if people see those products as trustworthy collaborators.
On the basis of the purpose of the examples in the passage, pick the odd one out from the following AI-generated responses mentioned in the passage:
"When I queried ChatGPT about its rhetoric, it responded: 'The way I communicate is designed to foster trust and confidence in my responses, which can be both helpful and potentially misleading'."
"...when my father asked ChatGPT to edit his writing, it transmuted his perfectly correct Indian English into American English."
"Over time, ChatGPT seemed to be guiding me to write a more positive book about big tech — including editing my description of OpenAI's CEO, Sam Altman, to call him 'a visionary and a pragmatist'."
"When I prompted Microsoft's Bing Image Creator to produce a picture of engineers and space explorers, it gave me an entirely male cast of characters..."
"When I queried ChatGPT about its rhetoric, it responded: 'The way I communicate is designed to foster trust and confidence in my responses, which can be both helpful and potentially misleading'."
Options B, C and D are all examples where an AI tool produced an output diverging from reality due to bias, error, or undue influence, things the system did to the user. Option A is different — it is the chatbot's own self-aware statement about its communication style being designed to build trust while potentially misleading, which is a reflection on its rhetorical design rather than an instance of biased or incorrect output.
Why Option A is Correct (odd one out): This is the chatbot explaining its own communication strategy, not an example of bias or factual error like the other three instances.
Why Option B is not the odd one out: This is a concrete instance of the chatbot producing biased output, transmuting correct Indian English into American English, reflecting a US-centric bias.
Why Option C is not the odd one out: This is an instance of biased influence, the chatbot nudging the author toward a more favourable description of Sam Altman.
Why Option D is not the odd one out: This is an instance of biased output, the image generator producing an entirely male cast for a gender-neutral prompt.
Key Takeaway: When asked to identify the odd one out among examples, check whether each example is serving the same illustrative function — here, three are instances of biased or incorrect AI output, while one is the AI describing its own design philosophy.
All of the following statements from the passage affirm the disjunct between the claims about AI made by tech companies and what AI actually does EXCEPT:
"When I prompted Microsoft's Bing Image Creator to produce a picture of engineers and space explorers, it gave me an entirely male cast of characters . . ."
"In my own ChatGPT dialogues, I wanted to enact how the product's veneer of collegial neutrality could lull us into absorbing false or biased responses without much critical engagement."
"I'm not aware of research on whether ChatGPT tends to favor big tech, OpenAI or Altman, and I can only guess why it seemed that way in our conversation."
"It's a dynamic that makes us complicit in big tech's accumulation of wealth and power: we're both victims and beneficiaries."
"I'm not aware of research on whether ChatGPT tends to favor big tech, OpenAI or Altman, and I can only guess why it seemed that way in our conversation."
Statements A, B and D all point to gaps between what AI companies claim and what AI actually produces or causes — gender bias in images, the illusion of neutrality masking biased outputs, and complicity in big tech's wealth accumulation. Option C is the author admitting uncertainty about whether ChatGPT favours big tech, which does not affirm a disjunct — it expresses doubt rather than evidence of a gap between claim and reality.
Why Option A is
Incorrect (as a candidate for the exception): This affirms the disjunct, showing a clear contradiction between claims of neutrality and the actual male-biased output.
Why Option B is
Incorrect (as a candidate for the exception): This affirms the disjunct by describing how AI's apparent neutrality can mask the absorption of biased responses.
Why Option C is Correct: The author here is explicitly unsure and merely speculating, so this statement does not affirm any specific gap between AI's claims and its actual behaviour.
Why Option D is
Incorrect (as a candidate for the exception): This affirms the disjunct by pointing to the gap between AI's stated benefit to users and the actual complicity it creates between users and big tech's profit motives.
Key Takeaway: When asked for the exception to a pattern, look for the one statement that expresses uncertainty or doubt rather than making an affirmative claim about a contradiction.
The author compares AI-generated texts with "a beige office building" for all of the following reasons EXCEPT:
AI generates generalised responses that lack specificity and nuance.
AI tends to blame its training data when scrutinised for its biases.
AI aims to foster a feeling of trust and credibility among its users.
AI-generated texts often exhibit a warm, polite, and collegial tone.
AI tends to blame its training data when scrutinised for its biases.
The beige office building comparison is about the bland, generic, polished tone of AI output designed to feel trustworthy and safe. Reasons A, C and D directly relate to this blandness and trust-building tone. Option B, about blaming training data when scrutinised, is a separate behaviour related to deflecting responsibility for bias, not to the bland tone itself.
Why Option A is
Incorrect (as a candidate for the exception): Lack of specificity and nuance is precisely what makes AI text feel beige and generic.
Why Option B is Correct: Blaming training data is about evading accountability for bias, unrelated to the linguistic blandness the beige building metaphor describes.
Why Option C is
Incorrect (as a candidate for the exception): Fostering trust and credibility is exactly the purpose behind the bland, professional and approachable tone the metaphor captures.
Why Option D is
Incorrect (as a candidate for the exception): A warm, polite, collegial tone is the literal quality being compared to a beige building — generically pleasant but lacking character.
Key Takeaway: Trace metaphors back to the exact qualities they are meant to capture, and eliminate options describing unrelated behaviours even if they appear elsewhere in the passage.
The author of the passage is least likely to agree with which one of the following claims?
The neutrality of AI is conducive to critical thinking.
ChatGPT favours AI companies and their officials, like Sam Altman, in its responses.
When we use AI, we become accomplices to the exploitative practices of big tech companies.
The neutrality of AI is motivated by economic considerations.
The neutrality of AI is conducive to critical thinking.
The author repeatedly argues that AI's appearance of neutrality is actually a tool that lulls users into absorbing biased or false content without critical engagement. So the author would strongly disagree that AI's neutrality helps critical thinking, since the author's whole point is the opposite.
Why Option A is Correct: The author explicitly argues that AI's veneer of neutrality discourages, rather than helps, critical engagement, making this the claim the author would least agree with.
Why Option B is
Incorrect (as a candidate): The author suggests this happened in their own experience with ChatGPT, so they would not disagree with this claim.
Why Option C is
Incorrect (as a candidate): The author states directly that using these products makes users complicit, so they would agree with this.
Why Option D is
Incorrect (as a candidate): The author points out that OpenAI's funders need a return on their investment, tying neutrality and trust-building to economic motives, so the author would agree with this.
Key Takeaway: For "least likely to agree" questions, find the option that contradicts the author's central argument most directly, rather than one merely absent from the passage.
Different sciences exhibit different science cultures and practices. For example, in astronomy, observation — until what is today called the new astronomy — had always been limited to what could be seen within the limits of optical light. Indeed, until early modernity the limits to optical light were also limits of what humans could themselves see within their limited and relative perceptual spectrum of human vision. With early modernity and the invention of lensed optical instruments — telescopes — astronomers could begin to observe phenomena never seen before. Magnification and resolution began to allow what was previously imperceptible to be perceived — but within the familiar limits of optical vision. Galileo, having learned of the Dutch invention of a telescope by Hans Lippershey, went on to build some hundred of his own, improving from the Dutch 3x to nearly 30x telescopes — which turn out to be the limit of magnificational power without chromatic distortion. And it was with his own telescopes that he made the observations launching early modern astronomy (phases of Venus, satellites of Jupiter, etc.). Isaac Newton's later improvement with reflecting telescopes expanded upon the magnificational-resolution capacity of optical observation; and, from Newton to the twentieth century, improvement continued on to the later very large array of light telescopes today — following the usual technological trajectory of "more-is-better" but still remaining within the limits of the light spectrum. Today's astronomy has now had the benefit of some four centuries of optical telescopy. The "new astronomy," however, opens the full known electromagnetic spectrum to observation, beginning with the accidental discovery of radio astronomy early in the twentieth century, and leading today to the diverse variety of EMS telescopes which can explore the range from gamma to radio waves. Thus, astronomy, now outfitted with new instruments, "smart" adaptive optics, very large arrays, etc., illustrates one style of instrumentally embodied science — a technoscience. Of course astronomy, with the very recent exceptions of probes to solar system bodies (Moon, Mars, Venus, asteroids), remains largely a "receptive" science, dependent upon instrumentation which can detect and receive emissions.
Contemporary biology displays a quite different instrument array and, according to Evelyn Fox-Keller, also a different scientific culture. She cites her own experience, coming from mathematical physics into microbiology, and takes account of the distinctive instrumental culture in her Making Sense of Life (2002). Here, particularly with the development of biotechnology, instrumentation is far more interventional than in the astronomy case. Microscopic instrumentation can be and often is interventional in style: "gene-splicing" and other techniques of biotechnology, while still in their infancy, are clearly part of the interventional trajectory of biological instrumentation. Yet, in both disciplines, the sciences involved are today highly instrumentalized and could not progress successfully without constant improvements upon the respective instrumental trajectories. So, minimalistically, one may conclude that the sciences are technologically, instrumentally embodied. But the styles of embodiment differ, and perhaps the last of the scientific disciplines to move into such technical embodiment is mathematics, which only contemporarily has come to rely more and more upon the computational machinery now in common use.
None of the following statements, if true, contradicts the arguments in the passage EXCEPT:
some scientific instruments may be classified as both receptive and interventional in their functions.
because of the relatively recent entry of computational machinery in mathematics, the field is only now beginning to develop a scientific culture.
like telescopy, microscopy has also sought to move beyond the visible spectrum to be able to detect objects that are invisible in that spectrum.
Isaac Newton's discovery of gravity was accomplished without the help of instruments.
because of the relatively recent entry of computational machinery in mathematics, the field is only now beginning to develop a scientific culture.
The passage argues that mathematics has only recently come to rely on computational machinery, but it does not claim that mathematics lacked any scientific culture before this — only that its style of instrumental embodiment came later. So a statement claiming mathematics is "only now beginning to develop a scientific culture" does not contradict the passage, since the passage never claims mathematics had no scientific culture prior to computation.
Why Option A is
Incorrect (as a candidate for the exception): The passage acknowledges biology's instruments are interventional while astronomy's are receptive, but it does not rule out instruments having both functions, so this does not necessarily contradict the passage.
Why Option B is Correct: This does not contradict the passage because the passage's claim about mathematics is restricted to it adopting computational machinery late, not about mathematics lacking a scientific culture altogether.
Why Option C is
Incorrect (as a candidate for the exception): This would contradict the passage's framing of astronomy as purely receptive and biology as interventional, since it suggests microscopy also shares astronomy's drive to detect things beyond ordinary perception, blurring the contrast the passage sets up.
Why Option D is
Incorrect (as a candidate for the exception): This would contradict the passage's broader argument that science today depends on continuous instrumental progress, since it presents a major scientific discovery achieved without instruments.
Key Takeaway: For "does not contradict except" questions, test whether each statement actually clashes with what the passage explicitly claims, rather than with a loosely related idea.
All of the following statements may be rejected as valid inferences from the passage EXCEPT:
interventionist instruments, or instruments that intervene directly in scientific inquiry, are different from embodied instruments, or instruments that embody scientific inquiry.
the advances in telescopy made by Newton with reflecting telescopes allowed early modern astronomers to observe the phases of Venus and the satellites of Jupiter.
the author distinguishes between the receptive and interventionist uses of instruments in the sciences by comparing astronomy and biology, respectively.
Isaac Newton's experiments with reflecting telescopes were the earliest versions of the "new astronomy" referred to in the passage.
the author distinguishes between the receptive and interventionist uses of instruments in the sciences by comparing astronomy and biology, respectively.
The passage explicitly draws this comparison, presenting astronomy as a "receptive" science dependent on detecting emissions, and biology, especially through biotechnology, as more "interventional" with instruments like gene-splicing. This is a valid inference directly supported by the passage's structure.
Why Option A is
Incorrect: The passage does not introduce or support a distinction between "interventionist instruments" and "embodied instruments" as described, so this is not a valid inference.
Why Option B is
Incorrect: The passage attributes the early modern observations like the phases of Venus and satellites of Jupiter to Galileo's telescopes, not Newton's reflecting telescopes, which came later and only "expanded upon" earlier magnificational-resolution capacity.
Why Option C is Correct: This matches the passage's explicit contrast between astronomy's receptive instrumentation and biology's interventional instrumentation, citing Fox-Keller's account.
Why Option D is
Incorrect: Newton's reflecting telescopes are described as an improvement within the existing optical telescope trajectory, not as part of the "new astronomy," which the passage defines as beginning with radio astronomy and extending to the full electromagnetic spectrum.
Key Takeaway: When asked which inference can be accepted rather than rejected, look for the option that mirrors the passage's own stated comparisons most precisely, without misattributing facts or inventing distinctions.
To which one of the following instruments would the characterisations of instruments in the passage be least applicable?
Milestone
Kitchen oven
Scalpel
Saxophone
Milestone
The passage's discussion centers on scientific instruments that are either receptive, detecting emissions, or interventional, actively manipulating their object of study. A milestone is not an instrument used to observe or intervene in anything — it is simply a marker, making it least applicable to the characterisations of instruments described.
Why Option A is Correct: A milestone has no observational or interventional function, unlike all the other listed items, so it does not fit the instrument characterisation at all.
Why Option B is
Incorrect (as a candidate): A kitchen oven actively intervenes in its contents through heat, similar to interventional instruments described in the passage.
Why Option C is
Incorrect (as a candidate): A scalpel is a clearly interventional tool, directly acting upon its object, fitting the passage's description of interventional instruments.
Why Option D is
Incorrect (as a candidate): A saxophone, while not scientific, still functions as an instrument producing an effect through direct engagement, making it more applicable than a milestone, which has no functional engagement at all.
Key Takeaway: When asked which item is least applicable to a described characteristic, look for the one item that fundamentally lacks the core function being discussed, rather than the one that is simply unrelated to science.
Which one of the following observations is a valid conclusion to draw from the statement that "the sciences involved are today highly instrumentalised and could not progress successfully without constant improvements upon the respective instrumental trajectories"?
The use of instruments in scientific trajectories must be respected in order to see successful progress in them.
The growth of scientific technologies has led to the embodiment of progress in the trajectories of improvement.
In both astronomy and microbiology, progress has been the consequence of improvements in the instruments they use.
Highly instrumentalised work in the sciences has resulted in the progressive improvement of scientific constants.
In both astronomy and microbiology, progress has been the consequence of improvements in the instruments they use.
The statement says the sciences are highly instrumentalised and cannot progress without constant improvement of their instruments. The passage's two examples of this are astronomy and microbiology, both of which are explicitly described as depending on their respective instrumental advances for progress.
Why Option A is
Incorrect: This makes a vague normative claim about "respecting" instrument use, which is not what the passage's statement is about — the statement is descriptive of dependency, not prescriptive about respect.
Why Option B is
Incorrect: This uses circular and vague phrasing about "embodiment of progress in trajectories of improvement" without grounding the conclusion in the specific examples the passage gives.
Why Option C is Correct: This directly reflects the passage's point, grounding the abstract claim in the two concrete disciplines — astronomy and microbiology — both shown to progress through their respective instrumental trajectories.
Why Option D is
Incorrect: This talks about "progressive improvement of scientific constants," which is not a concept discussed anywhere in the passage and misreads the original statement.
Key Takeaway: When asked to draw a valid conclusion from a quoted line, prefer the option that stays closest to the specific examples and language used in the passage, and discard options using vague or invented terminology.
Time and again, whenever a population [of Mexican tetra fish] was swept into a cave and survived long enough for natural selection to have its way, the eyes disappeared. "But it's not that everything has been lost in cavefish . . . Many enhancements have also happened." . . . Studies have found that cave-dwelling fish can detect lower levels of amino acids than surface fish can. They also have more tastebuds and a higher density of sensitive cells alongside their bodies that let them sense water pressure and flow. . . .
Killing the processes that support the formation of the eye is quite literally what happens. Just like non-cave-dwelling members of the species, all cavefish embryos start making eyes. But after a few hours, cells in the developing eye start dying, until the entire structure has disappeared. [Developmental biologist Misty] Riddle thinks this apparent inefficiency may be unavoidable. "The early development of the brain and the eye are completely intertwined—they happen together," she says. That means the least disruptive way for eyelessness to evolve may be to start making an eye and then get rid of it. . . .
It's easy to see why cavefish would be at a disadvantage if they were to maintain expensive tissues they aren't using. Since relatively little lives or grows in their caves, the fish are likely surviving on a meager diet of mostly bat feces and organic waste that washes in during the rainy season. Researchers keeping cavefish in labs have discovered that, genetically, the creatures are exquisitely adapted to absorbing and storing nutrients. . . .
Fats can be toxic for tissues, [evolutionary physiologist Nicolas] Rohner explains, so they are stored in fat cells. "But when these cells get too big, they can burst, which is why we often see chronic inflammation in humans and other animals that have stored a lot of fat in their tissues." Yet a 2020 study by Rohner, Krishnan and their colleagues revealed that even very well-fed cavefish had fewer signs of inflammation in their fat tissues than surface fish do. Even in their sparse cave conditions, wild cavefish can sometimes get very fat, says Riddle. This is presumably because, whenever food ends up in the cave, the fish eat as much of it as possible, since there may be nothing else for a long time to come. Intriguingly, Riddle says, their fat is usually bright yellow because of high levels of carotenoids, the substance in the carrots that your grandmother used to tell you were good for your…eyes.
"The first thing that came to our mind, of course, was that they were accumulating these because they don't have eyes," says Riddle. In this species, such ideas can be tested: Scientists can cross surface fish (with eyes) and cavefish (without eyes) and look at what their offspring are like. When that's done, Riddle says, researchers see no link between eye presence or size and the accumulation of carotenoids. Some eyeless cavefish had fat that was practically white, indicating lower carotenoid levels. Instead, Riddle thinks these carotenoids may be another adaptation to suppress inflammation, which might be important in the wild, as cavefish are likely overeating whenever food arrives.
All of the following statements from the passage describe adaptation in Mexican tetra cavefish EXCEPT:
"It's easy to see why cavefish would be at a disadvantage if they were to maintain expensive tissues they aren't using."
"Since relatively little lives or grows in their caves, the fish are likely surviving on a meager diet of mostly bat feces and organic waste that washes in during the rainy season."
"'But when these cells get too big, they can burst, which is why we often see chronic inflammation in humans and other animals that have stored a lot of fat in their tissues.'"
"Even in their sparse cave conditions, wild cavefish can sometimes get very fat, says Riddle."
"It's easy to see why cavefish would be at a disadvantage if they were to maintain expensive tissues they aren't using."
Options A, B and D all describe specific traits or behaviours that represent the cavefish's adaptation to their environment — lack of unnecessary tissue maintenance, survival on limited food, and the capacity to get fat when food is available. Option C is a general statement about fat cells bursting and causing inflammation in humans and other animals broadly, used as background context rather than describing an adaptation specific to cavefish.
Why Option A is
Incorrect (as a candidate for the exception): This describes the adaptive logic of why cavefish lose unnecessary tissues like eyes, directly tied to their adaptation.
Why Option B is
Incorrect (as a candidate for the exception): This describes the limited food environment cavefish are adapted to survive in, directly relevant to their adaptation.
Why Option C is Correct: This is a general statement about fat cells and inflammation across humans and other animals, not a description of an adaptation specific to cavefish.
Why Option D is
Incorrect (as a candidate for the exception): This describes the cavefish's specific behavioural adaptation of overeating when food becomes available, directly relevant to adaptation.
Key Takeaway: Distinguish between statements that describe a specific adaptation of the organism in question and statements used as general scientific background that apply more broadly.
Which one of the following best explains why the "apparent inefficiency" is "unavoidable"?
The caves have poor and inconsistent availability of food and nutrition for Mexican tetra cavefish.
The lack of light in the caves kills the eye cells in the developing Mexican tetra cavefish embryo.
The inefficiency resulting from eyelessness is compensated by enhancements like more tastebuds in Mexican tetra cavefish.
Mexican tetra cavefish are similar to non-cave-dwelling variants in their early stages of development.
Mexican tetra cavefish are similar to non-cave-dwelling variants in their early stages of development.
Riddle explains that the "apparent inefficiency" of building an eye and then destroying it might be unavoidable because eye and brain development are deeply intertwined and happen together, meaning eyelessness cannot easily evolve without first going through the motion of forming an eye, just like non-cave dwelling fish do at the start of development.
Why Option A is
Incorrect: This explains why cavefish cannot afford to maintain expensive tissues, not why the inefficient process of building and then destroying an eye is unavoidable in the first place.
Why Option B is
Incorrect: This describes the mechanism of how the eye disappears, but not why this particular inefficient process is the only viable path, which is the actual question being asked.
Why Option C is
Incorrect: This refers to a separate point about enhancements compensating for lost senses, unrelated to the explanation of why the inefficient eye-building-then-destroying process is unavoidable.
Why Option D is Correct: This matches Riddle's explanation directly, since cavefish embryos start the same way as non-cave-dwelling fish, with eye and brain development intertwined, making the inefficient path of forming and then losing the eye the least disruptive way for eyelessness to evolve.
Key Takeaway: Trace an explanation back to the exact reasoning given by the source quoted in the passage, rather than other plausible-sounding but unrelated facts nearby.
Which one of the following results for the cross between surface fish (with eyes) and cavefish (without eyes) would invalidate Riddle's inference from the experiment?
Some offspring with eyes had white fat.
Some offspring with eyes had yellow fat.
Only eyeless offspring had yellow fat.
Some eyeless offspring had white fat.
Only eyeless offspring had yellow fat.
Riddle's inference was that there is no link between eye presence or size and the accumulation of carotenoids, based on observing offspring from the cross. If only eyeless offspring had yellow fat, with no eyed offspring showing it, that would actually support a link between eyelessness and carotenoid accumulation, directly invalidating Riddle's no-link conclusion.
Why Option A is
Incorrect: Some eyed offspring having white fat does not establish any link between eye presence and carotenoid levels — this is consistent with no link existing.
Why Option B is
Incorrect: Some eyed offspring having yellow fat actually supports the no-link conclusion, since it shows fat colour does not depend strictly on eyelessness.
Why Option C is Correct: If only eyeless offspring had yellow fat, this would directly suggest a connection between the absence of eyes and carotenoid accumulation, contradicting Riddle's claim that there is no such link.
Why Option D is
Incorrect: Some eyeless offspring having white fat is consistent with no link existing, since it shows eyelessness does not guarantee high carotenoid levels.
Key Takeaway: To invalidate a "no link" conclusion, look for a result showing a clear and exclusive pattern connecting the two variables in question, rather than scattered or mixed results.
On the basis of the information in the passage, what is the most likely function of carotenoids in Mexican tetra cavefish?
To act as a substitute for eyes.
To help the fat cells store nutrients.
To render bright yellow colour to the cavefish.
To control inflammation from the bursting of fat cells.
To control inflammation from the bursting of fat cells.
The passage states that researchers initially thought the yellow carotenoids might relate to compensating for lost eyes, but the cross-breeding experiment showed no link between carotenoids and eye presence. Riddle instead suggests carotenoids may help suppress inflammation, which is important because cavefish are likely to overeat whenever food becomes available, leading to fat cell bursting and inflammation.
Why Option A is
Incorrect: This was the initial hypothesis researchers had, but it was disproven by the cross-breeding experiment, which found no link between carotenoids and eye presence.
Why Option B is
Incorrect: The passage does not suggest carotenoids help fat cells store nutrients — this function is not discussed anywhere in the passage.
Why Option C is
Incorrect: The yellow colour is a result of carotenoid presence, not the function or purpose of carotenoids — this option mistakes an effect for a function.
Why Option D is Correct: This matches Riddle's final explanation directly, that carotenoids likely help suppress inflammation that results from overeating and fat cell bursting in cavefish.
Key Takeaway: When a passage walks through and disproves an initial hypothesis before arriving at a final explanation, make sure to identify the final accepted explanation rather than the earlier, discarded one.
This book takes the position that setting in literature is more than just backdrop, that important insight into literary texts can be made by paying close attention to how authors craft place, as well as to how place functions in a narrative. The authors included in this reference work engage deeply with either real or imagined geographies. They care about how human decisions have shaped landscapes and how landscapes have shaped human practices and values. Some of the best writing is highly vivid, employing the language of the senses because this is the primary means through which humans know physical space. Literature can offer valuable perspectives on physical and cultural geography. Unlike scientific reports, a literary narrative can provide the emotional component missing from the scientific record. In human experience, geographical places have a spiritual or emotional component in addition to and as part of a physical layout and topography. This emotional component, although subjective, is no less "real" than a surveyor's map. Human consciousness of place is experienced in a multimodal manner. Histories of places live on in many forms, one of which is the human memory or imagination.
Both real and imaginary landscapes provide insight into the human experience of place. The pursuit of such a topic speaks to the valuable knowledge produced from bridging disciplines and combining material from both the arts and the sciences to better understand the human condition. The perspectives that most concern cultural geographers are often those regarding movement and migration, cultivation of natural resources, and organization of space. The latter two reflect concerns of the built environment, a topic shared with the field of architectural study. Many of these concerns are also reflected in work sociologists do. Scholars from literary studies can contribute an aesthetic dimension to what might otherwise be a purely ideological approach.
Literature can bring together material that spans different branches of science. For example, a literary description of place may involve not only the environment and geography but the noises and quality of light, or how people from different races or classes can experience the same place in different ways linked to those racial or class disparities. Literary texts can also account for the way in which absence—of other people, animals, and so on—affects a human observer or inhabitant. Both literary and scientific approaches to place are necessary, working in unison, to achieve a complete record of an environment. It is important to note that the interdisciplinary nature of this work teaches us that landscapes are not static, that they are not unchanged by human culture. At least part of their identity derives from the people who inhabit them and from the way space can alter and inspire human perspective. The intersection of scientific and literary expression that happens in the study of literary geography is of prime importance due to the complexity of the personal and political ways that humans experience place.
Which one of the following is a valid conclusion to draw from the author's statement that, "The pursuit of such a topic speaks to the valuable knowledge produced from bridging disciplines and combining material from both the arts and the sciences to better understand the human condition."?
A comprehensive bridging of the human condition can best be achieved by a disciplined pursuit of human understanding.
A comprehensive understanding of the valuable knowledge produced by the arts and sciences can best be achieved by studying the human condition.
A comprehensive understanding of the human condition can best be achieved by combining the findings of disciplines from the arts and the sciences.
The literary descriptions of the emotions we experience in the places we visit can contribute to our understanding of the arts and sciences.
A comprehensive understanding of the human condition can best be achieved by combining the findings of disciplines from the arts and the sciences.
The quoted line argues that pursuing the topic of literary geography produces valuable knowledge by bridging arts and science disciplines to better understand the human condition. The valid conclusion must reflect this exact relationship — combining disciplines from arts and sciences leads to a fuller understanding of human experience.
Why Option A is
Incorrect: This inverts the relationship, suggesting that bridging the human condition results from disciplined pursuit of human understanding, rather than disciplines being combined to understand the human condition.
Why Option B is
Incorrect: This shifts the focus to understanding the arts and sciences themselves through studying the human condition, which reverses the actual direction of the author's claim.
Why Option C is Correct: This accurately reflects the author's point, that combining findings from arts and science disciplines leads to a comprehensive understanding of the human condition.
Why Option D is
Incorrect: This narrows the claim to literary descriptions of emotions specifically, when the original statement is about a broader interdisciplinary combination of arts and science material, not just emotional descriptions.
Key Takeaway: Pay close attention to the direction of cause and effect in a quoted statement, since incorrect options often simply reverse or narrow the original relationship.
Which one of the following is not true of the argument in the second paragraph?
Analysing the literary descriptions of a place can give us a sense of how people relate emotionally to it.
The spiritual experience of a place may be considered as real as the physical experience of it.
The emotional and spiritual experience of a place can replace a surveyor's map.
Literary accounts of places can be filled with histories, manifested as memory or imagination.
The emotional and spiritual experience of a place can replace a surveyor's map.
The second paragraph states that the emotional component of place is "no less real" than a surveyor's map, meaning it stands alongside the physical record, not that it replaces or substitutes for it. Saying the emotional experience "can replace" a surveyor's map goes beyond and misrepresents what the paragraph actually claims.
Why Option A is
Incorrect (as a candidate for not true): The paragraph supports this, since it discusses how literary narrative can convey the emotional component of a place that scientific records miss.
Why Option B is
Incorrect (as a candidate for not true): The paragraph explicitly states the emotional or spiritual component is "no less real" than a physical layout or surveyor's map, supporting this statement.
Why Option C is Correct: The paragraph never claims the emotional experience can replace a surveyor's map, only that it is equally real alongside it, making this statement not true of the argument.
Why Option D is
Incorrect (as a candidate for not true): The paragraph supports this, mentioning that histories of places live on in human memory or imagination, among other forms.
Key Takeaway: Watch for options that subtly escalate a passage's claim — such as turning "equally real as" into "can replace" — which changes the meaning entirely.
The author uses the example of the literary description of place to illustrate that:
scientific approaches to place are more accurate than literary ones.
literature can convey how different people experience the same place differently.
architects use diverse methods to calibrate the noises and lights of a given place.
the absence of other people, animals, and so on in a place can profoundly affect its inhabitants.
literature can convey how different people experience the same place differently.
The fourth paragraph uses the example of literary description of place specifically to show how people from different races or classes can experience the same place differently, linked to those disparities, illustrating literature's capacity to capture varied human experience of the same physical space.
Why Option A is
Incorrect: The passage does not argue scientific approaches are more accurate — it argues both literary and scientific approaches are necessary together for a complete record.
Why Option C is
Incorrect: The passage does not discuss architects calibrating noises and lights — this detail is not present anywhere in the passage.
Why Option B is Correct: This matches the specific example given, that literary descriptions can show how race or class differences shape different experiences of the same place.
Why Option D is
Incorrect: While the passage does mention literature accounting for the effect of absence on an observer, this is a separate point made afterward, not the specific illustration intended by the example of literary description of place referenced in the question.
Key Takeaway: When a question asks about the purpose of a specific example, locate that exact example in the passage and identify precisely what point it is illustrating, rather than picking a nearby but distinct idea.
All of the following statements, if false, would contradict the arguments in the passage, EXCEPT that:
descriptions of places do not need satellite imagery or other visual aids to give a "real" sense of the place.
literature provides us with deep insights into the ways in which movement and migration affect physical geography.
highly vivid writing, employing the language of the senses, can capture the multi-modal manner in which humans experience places.
humans do not interact with places in subjective, emotional ways because places are only physical topography.
humans do not interact with places in subjective, emotional ways because places are only physical topography.
The passage argues throughout that human experience of place includes a real emotional and spiritual component beyond physical topography. If this were false, meaning if humans truly did not interact with places emotionally and places were purely physical, this would directly contradict the passage's central argument. The other options, if false, do not necessarily contradict the passage's claims as directly.
Why Option A is
Incorrect (as a candidate for the exception): If this were false, meaning descriptions of places did need visual aids to feel "real," this would contradict the passage's argument that emotional or literary accounts are "no less real" without such aids, so this is not the exception.
Why Option B is
Incorrect (as a candidate for the exception): If this were false, meaning literature does not provide insight into movement and migration's effect on geography, this would contradict the passage's mention of these as key concerns literature can address, so this is not the exception.
Why Option C is
Incorrect (as a candidate for the exception): If this were false, meaning vivid sensory writing cannot capture the multimodal human experience of place, this would contradict the passage's explicit claim about the value of vivid, sense-based writing, so this is not the exception.
Why Option D is Correct: If this statement is false, it actually aligns with and supports the passage's argument, since the passage insists humans do interact with places emotionally and subjectively, beyond pure physical topography. So this being false does not contradict the passage, making it the exception.
Key Takeaway: In "if false, contradicts except" questions, test what happens to the passage's argument under the assumption that each statement is false, and find the one case where falseness aligns with, rather than against, the original argument.
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: While taste is related to judgment, with thinkers at the time often writing, for example, about "judgments of taste" or using the two terms interchangeably, taste retains a vital link to pleasure, embodiment, and personal specificity that is too often elided in post-Kantian ideas about judgment—a link that Arendt herself was working to restore.
Paragraph: ____(1) ____. Denneny focused on taste rather than judgment in order to highlight what he believed was a crucial but neglected historical change. ____(2) ____. Over the course of the seventeenth century and early eighteenth century, across Western Europe, the word taste took on a new extension of meaning, no longer referring specifically to gustatory sensation and the delights of the palate but becoming, for a time, one of the central categories for aesthetic—and ethical—thinking. ____(3) ____. Tracing the history of taste in Spanish, French, and British aesthetic theory, as Denneny did, also provides a means to recover the compelling and relevant writing of a set of thinkers who have been largely neglected by professional philosophy. ____(4) ____.
Option 3
Option 1
Option 2
Option 4
Option 1
The paragraph opens with Denneny focusing on taste rather than judgment to highlight a historical change. The inserted sentence explains why this distinction matters, since taste and judgment were often used interchangeably, but taste retains a link to pleasure and personal feeling that judgment lost. This fits right after the opening line, at Option 1, before the paragraph moves to tracing how the word taste evolved over the seventeenth and eighteenth centuries.
Why Option A is
Incorrect: Option 3 comes after the historical evolution of taste has already been explained, when the discussion has shifted to tracing this history across Spanish, French and British theory. Inserting the sentence there breaks the flow of an argument that has already moved on.
Why Option B is Correct: Option 1 directly follows the mention of Denneny's choice to focus on taste, and the sentence explains that choice before the paragraph develops the historical point further.
Why Option C is
Incorrect: Option 2 falls just before the explanation of how taste's meaning changed historically. Inserting the sentence here interrupts that explanation before it begins.
Why Option D is
Incorrect: Option 4 is at the end, after the paragraph has shifted to a different point about recovering neglected thinkers. The sentence no longer connects naturally there.
Key Takeaway: Place an inserted sentence exactly where its logical function in the argument is needed, not just where it seems thematically related.
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. 'Literature on screen' suggests something more capacious and defining than citation: the possibility that literary adaptations are at once cinema and literature.
2. Even though a growing number of films eligible for Academy Awards for Best Screenplay Based on Material from Another Medium borrow that material from print journalism, franchise characters, television series, comic books, video games and toys, academic studies of adaptation remain stubbornly attached to literature as cinema's natural progenitor.
3. It is as if adaptation studies, by borrowing the cultural cachet of literature, sought to claim its institutional respectability and gravitas even while insuring adaptation's enduring aesthetic and methodological subordination to literature proper.
4. Beneath this contradictory notion of film adaptations as not merely hybrid texts but texts holding dual citizenship in two modes of presentation is an even more pervasive legacy that haunts adaptation studies: the assumption that the primary context within which adaptations are to be studied is literature.
Sentence 1 introduces the core idea, adaptations existing as both cinema and literature. Sentence 4 picks this up with "this contradictory notion," referring back to the dual citizenship idea in sentence 1, and introduces the deeper assumption that literature is treated as cinema's natural source. Sentence 2 supports this by noting that despite films increasingly adapting non-literary material, academic study still clings to literature as cinema's progenitor. Sentence 3 closes by explaining the motive, borrowing literature's prestige while keeping adaptation studies subordinate to it. This gives the sequence 1423.
Key Takeaway: Pointer words like "this" usually refer to a specific idea stated just before, and locking such pairs together first makes sequencing easier.
The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
For millennia, in the process of opening up land for agriculture, gardens, grazing and hunting, humans have created ecological "mosaics", or "patchworks": landscapes holding a mixture of habitats, like meadows, gardens and forests. These were not designed as nature reserves, but often catered to hugely diverse animal life. Research indicates that European hay meadows cultivated for animal feed were actually more successful at preserving a vast array of species than meadows explicitly cultivated for biodiversity. Studying the early Holocene, researchers have found that human presence was about as likely to increase biodiversity as reduce it. Of course, not all human-created landscapes have the same value. A paved subdivision with astroturfed lawns is very different to a village with diverse vegetable and flower gardens. But scientists continue to find evidence that the old idea of humans as antithetical to nature is also wrong-headed, and that rosy visions of thriving, human-free environments are more imaginary than real.
In our attempts to shape the world around us to our needs, humans have often created landscapes like meadows, gardens, and forests, which support hugely diverse species, and are more successful at preserving them, than parks created specifically for this.
In terms of preserving biodiversity, scientists are finding increasing evidence that human action is not always antithetical to nature, but often assists the preservation of meadows, landscapes, and flourishing of species.
Studying the early Holocene and human practices over millennia, researchers say that while agricultural meadows, gardens, and forests were not explicitly designed as nature reserves, they actually preserved a vast array of species, belying the idea that humans harm nature.
Contrary to the idea that humans always hurt nature and that it thrives in their absence, a lot of human action across history has been equally likely to increase biodiversity than reduce it, often creating varied ecological landscapes that support a vast array of species.
In our attempts to shape the world around us to our needs, humans have often created landscapes like meadows, gardens, and forests, which support hugely diverse species, and are more successful at preserving them, than parks created specifically for this.
The passage argues that human-shaped landscapes such as meadows and gardens often supported biodiversity as well as or better than reserves designed for that purpose, that human presence in the early Holocene was about as likely to increase biodiversity as reduce it, and that the idea of humans as purely harmful to nature and of pristine human-free environments is largely a myth.
Why Option A is
Incorrect: This captures only the comparative point about meadows outperforming reserves, missing the passage's larger conclusion that humans are not inherently harmful to nature.
Why Option B is
Incorrect: This overstates the claim, suggesting human action generally assists preservation, when the passage actually says human impact was equally likely to help or harm biodiversity.
Why Option C is
Incorrect: This captures the historical detail accurately but omits the nuance that human impact could go either way, and drops the closing point about human-free environments being more imaginary than real.
Why Option D is Correct: This captures the central contradiction, that human action across history has been just as likely to increase biodiversity as reduce it, often producing varied landscapes supporting diverse species, which directly reflects the passage's strongest claim.
Key Takeaway: A correct summary must include the passage's final and strongest claim, not just a supporting detail from earlier in the argument.
The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In 1903, left-wing feminist Elizabeth Magie invented The Landlord's Game, the original version of what became Monopoly. It was designed as a powerful teaching tool to illustrate the dangers of monopolies and how wealth could concentrate in the hands of a few. The game featured a circular path, properties, and a "Go to Jail" space. Magie created two rule sets: one "monopolist" version where players crushed opponents through accumulation, and another, more radical "Prosperity" version, where everyone shared in the wealth, promoting fairness and equity. Years later, unemployed Charles Darrow sold a simplified version to Parker Brothers. They paid Magie only $500 for her patent—without royalties—and credited Darrow as the sole inventor. For decades, his tale of inventing the game in his basement remained the official story, while Magie's name and her original, anti-capitalist message were left in the shadows.
It is ironical that a left-wing feminist lost credit for the Landlord's Game to an unemployed man, who plagiarised and sold one version of the twin game to Parker Brothers for a meagre sum, denying her royalties.
Celebrated icons of the gaming industry, Charles Darrow and Parker Brothers, snatched the feminist icon Elizabeth Magie's original design and transformed Monopoly into a worldwide phenomenon, while barely acknowledging her.
Only one version of Monopoly became famous because of Charles Darrow's relentless basement work, carefully refining Elizabeth Magie's original idea into an engaging and entertaining pastime that he successfully patented and sold, symbolizing what many regarded as the ultimate triumph of individual ingenuity.
Parker Brothers' capitalist intent led to them acquiring from Charles Darrow a simplified version of Elizabeth Magie's original game, transforming it into a widespread commercial success while providing her only minimal financial compensation and granting scant public recognition.
It is ironical that a left-wing feminist lost credit for the Landlord's Game to an unemployed man, who plagiarised and sold one version of the twin game to Parker Brothers for a meagre sum, denying her royalties.
The passage centers on the irony that Magie, the original feminist inventor, lost credit and fair compensation to Darrow, who sold a simplified version of her game to Parker Brothers for a small one-time sum without royalties, while his fabricated origin story replaced her name in history.
Why Option A is Correct: This captures the irony of credit being stolen, the meagre payment without royalties, and the erasure of Magie's name and message, matching the passage closely.
Why Option B is
Incorrect: This describes Darrow and Parker Brothers as "celebrated icons," which is not supported and shifts focus away from the irony of Magie's erasure to their success instead.
Why Option C is
Incorrect: This frames Darrow's basement work as careful refinement and individual ingenuity, which misrepresents the passage, since Darrow simply sold a simplified version and is not credited with the depth of innovation this option implies.
Why Option D is
Incorrect: This focuses heavily on Parker Brothers' capitalist intent and frames the transaction mechanically, missing the personal irony of Magie being a left-wing feminist whose anti-capitalist message was buried, which is central to the passage.
Key Takeaway: Choose the summary that captures both the factual sequence of events and the tone or irony the passage is building toward.
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. Sporting a copper-coloured pixie cut and a pair of pink feather antlers, Torres himself resembles a child's doodle.
2. His casual millennial delivery, peppered with "um"s and "ah"s, makes surreal concepts sound like items on a brunch menu.
3. Though he may have failed so far in his colour-scouting mission (he hasn't yet found a new one, he admits), this hour leaves you tickled pink.
4. Like his previous show, My Favourite Shapes, this is an hour of sit-down comedy aided by an overhead camera which relays Torres's theories — illustrated with crayon squiggles — on to a screen behind him.
5. His inquisitive mind produces interconnected ideas about Catholicism, the blandness of Pixar and what orange sounds like, while his insights train us to spot "highly purple behaviour".
Sentences 1, 2, 4 and 5 together describe Torres's appearance, his delivery style, the show's format with the overhead camera and crayon squiggles, and the content of his ideas. Sentence 3 introduces a conclusion about the show leaving the viewer "tickled pink" despite his failed colour-scouting mission, which is a closing judgment that does not fit with the descriptive sequence the other four sentences build.
Key Takeaway: In odd-one-out sentence sets, identify the common descriptive or narrative thread running through four sentences, and find the one sentence that introduces a conclusion or shift instead of continuing that thread.
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. Pfas are a class of about 15,000 compounds most frequently used to make products water-, stain- and grease-resistant.
2. New research suggests exposure to some common perfluoroalkyl and polyfluoroalkyl substances (Pfas) cause changes to gene activity and that these changes are linked to health problems including multiple cancers, neurological disorders and autoimmune disease.
3. These Pfas compounds are dubbed "forever chemicals" because they do not naturally break down in the environment.
4. The research may also point toward other diseases potentially caused by Pfas that have not yet been identified.
5. The findings are a major step toward determining the mechanism by which the chemicals cause disease and could help doctors identify, detect and treat health problems for those exposed to Pfas before the issues advance.
Sentences 1, 2, 4 and 5 form a coherent flow about Pfas being linked to gene activity changes and disease, what Pfas are used for, and the implications of the research for future disease detection. Sentence 3, while factually about Pfas, introduces the "forever chemicals" nickname and their non-biodegradability, which is a separate point not part of the main thread connecting the research findings to gene activity and disease detection.
Key Takeaway: Even a factually relevant sentence about the same topic can be the odd one out if it introduces a different angle that does not connect to the specific narrative thread the other sentences build.
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: The region's Western customers found it hard to believe that Dhaka muslin could possibly have been made by human hands — there were rumours that it was woven by mermaids, fairies and even ghosts.
Once upon the silty banks of the Meghna River, a miracle was spun — a fabric so light it was called "baft-hawa", or woven air. This was Dhaka Muslin — the world's most coveted cloth. (1). Handspun from a rare cotton called Phuti Karpas, found nowhere else on Earth, and woven with a 16-step sacred ritual — beginning with cleaning the cotton using the teeth of a river catfish! (2). Every spring, the maple-like leaves pushed up through the grey, silty soil to produce a single daffodil yellow flower twice a year, which gave way to a snowy floret of cotton fibres. (3). Spun at dawn on boats by sharp-eyed young women, its threads were so fine the elderly could barely see them. Motifs of wildflowers, river breeze, and soul were etched into each piece — some so light, a 91-metre bolt could pass through a ring, or a 60' length fit inside a snuffbox. It draped Greek goddesses, Roman nobles, Mughal emperors, and European aristocrats. Marie Antoinette, Empress Joséphine — even Jane Austen adored its floating grace. (4).
Option 4
Option 1
Option 3
Option 2
Option 1
The inserted sentence describes Western customers finding it hard to believe Dhaka muslin was made by human hands, with rumours of mermaids and fairies weaving it. This reaction of disbelief fits best right after the muslin is first introduced as "the world's most coveted cloth," before the passage moves into the specific technical details of how it was made, which is Option 1.
Why Option A is
Incorrect: Option 4 is at the very end, after the passage has already described the muslin's reach and admirers across history. Inserting a reaction of disbelief about human origin there breaks the narrative's closing flow rather than fitting naturally.
Why Option B is Correct: Option 1 comes right after the muslin is introduced as the world's most coveted cloth — the perfect spot for showing how almost mythical it appeared to the Western audience.
Why Option C is
Incorrect: Option 3 comes in the middle of the cotton plant description, where the passage is detailing the flower and floret formation process — an inappropriate place to insert a sentence about Western disbelief in human craftsmanship.
Why Option D is
Incorrect: Option 2 falls right after the description of the 16-step ritual, where the passage is still building the technical mystique of how it was made, not yet ready for the reaction of disbelief about human origin.
Key Takeaway: Match the tone and content of the missing sentence to the surrounding context — an awe-struck reaction fits best right after the subject is first introduced as extraordinary.
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. As books age, the cellulose and lignin in the paper begin to break down, releasing a mix of volatile organic compounds into the air.
2. Old books carry a scent that many people instantly recognize—and even love.
3. These compounds are benzaldehyde, which gives off an almond-like scent, vanillin, which smells like vanilla, ethyl hexanol (floral scent), toluene (sweet), and furfural (which has a slightly bready scent).
4. This familiar aroma isn't just dust or mildew; it's actually a result of slow chemical changes happening inside the paper and ink.
Sentence 2 introduces the topic, the distinctive scent of old books that people instantly recognise. Sentence 4 follows by clarifying that this aroma is not dust or mildew but a result of chemical changes in the paper and ink. Sentence 1 then explains the mechanism behind this, the breakdown of cellulose and lignin releasing volatile organic compounds. Sentence 3 closes by naming the specific compounds responsible and describing their individual scents. This gives the sequence 2413.
Key Takeaway: Build the sequence by first identifying the general introductory claim, then locating the sentence that clarifies or corrects a common misconception about it, before moving to the specific mechanism and details.
There are six spherical balls, B1, B2, B3, B4, B5, and B6, and four circular hoops H1, H2, H3, and H4.
Each ball was tested on each hoop once, by attempting to pass the ball through the hoop. If the diameter of a ball is not larger than the diameter of the hoop, the ball passes through the hoop and makes a “ping”. Any ball having a diameter larger than that of the hoop gets stuck on that hoop and does not make a ping.
The following additional information is known:
1. B1 and B6 each made a ping on H4, but B5 did not.
2. B4 made a ping on H3, but B1 did not.
3. All balls, except B3, made pings on H1.
4. None of the balls, except B2, made a ping on H2.
What was the total number of pings made by B1, B2, and B3?
From Condition (3), every ball except B3 pinged H1. Therefore, H1 is larger than B1, B2, B4, B5, and B6, while B3 is larger than H1.
Using Condition (4), only B2 pinged H2. Hence, B2 is larger than H2, whereas B1, B3, B4, B5, and B6 are all smaller than H2. This establishes that H2 is the smallest hoop.
From Condition (1), B1 and B6 pinged H4, but B5 did not. Therefore, B1 and B6 are larger than H4, while B5 is smaller than H4.
Similarly, from Condition (2), B4 pinged H3, whereas B1 did not. Hence, B4 is larger than H3, while B1 is smaller than H3.
Combining all these observations, we obtain the following order:
B2 < H2 < B4 < H3 < B1, B6 < H4 < B5 < H1 < B3
Accordingly, the only hoop order that satisfies all the conditions is:
H2 < H3 < H4 < H1
The deductions can be summarized in the table below.
The number of pings made by B1, B2, and B3 are 2, 4, and 0, respectively.
Hence, the required sum is: 2 + 4 + 0 = 6.
Which of the following statements about the relative sizes of the balls is NOT NECESSARILY true?
B2 < B1 < B5
B1 < B5 < B3
B4 < B5 < B3
B1 < B6 < B3
B1 < B6 < B3
From Condition (3), every ball except B3 pinged H1. Therefore, H1 is larger than B1, B2, B4, B5, and B6, while B3 is larger than H1.
Using Condition (4), only B2 pinged H2. Hence, B2 is larger than H2, whereas B1, B3, B4, B5, and B6 are all smaller than H2. This establishes that H2 is the smallest hoop.
From Condition (1), B1 and B6 pinged H4, but B5 did not. Therefore, B1 and B6 are larger than H4, while B5 is smaller than H4.
Similarly, from Condition (2), B4 pinged H3, whereas B1 did not. Hence, B4 is larger than H3, while B1 is smaller than H3.
Combining all these observations, we obtain the following order:
B2 < H2 < B4 < H3 < B1, B6 < H4 < B5 < H1 < B3
Accordingly, the only hoop order that satisfies all the conditions is:
H2 < H3 < H4 < H1
The deductions can be summarized in the table below.
The number of pings made by B1, B2, and B3 are 2, 4, and 0, respectively.
There is no information, on the basis of which B6 and B1 can be compared. Hence, B1 < B6 < B3 is not necessarily true.
Which of the following statements about the relative sizes of the hoops is true?
H2 < H4 < H3 < H1
H1 < H4 < H3 < H2
H2 < H3 < H4 < H1
H1 < H3 < H4 < H2
H2 < H3 < H4 < H1
From Condition (3), every ball except B3 pinged H1. Therefore, H1 is larger than B1, B2, B4, B5, and B6, while B3 is larger than H1.
Using Condition (4), only B2 pinged H2. Hence, B2 is larger than H2, whereas B1, B3, B4, B5, and B6 are all smaller than H2. This establishes that H2 is the smallest hoop.
From Condition (1), B1 and B6 pinged H4, but B5 did not. Therefore, B1 and B6 are larger than H4, while B5 is smaller than H4.
Similarly, from Condition (2), B4 pinged H3, whereas B1 did not. Hence, B4 is larger than H3, while B1 is smaller than H3.
Combining all these observations, we obtain the following order:
B2 < H2 < B4 < H3 < B1, B6 < H4 < B5 < H1 < B3
Accordingly, the only hoop order that satisfies all the conditions is:
H2 < H3 < H4 < H1
The deductions can be summarized in the table below.
The number of pings made by B1, B2, and B3 are 2, 4, and 0, respectively.
We can see that H2 < H3 < H4 < H1 is true.
What BEST can be said about the total number of pings from all the tests undertaken?
13 or 14
12 or 13
At least 9
12 or 13 or 14
12 or 13
From Condition (3), every ball except B3 pinged H1. Therefore, H1 is larger than B1, B2, B4, B5, and B6, while B3 is larger than H1.
Using Condition (4), only B2 pinged H2. Hence, B2 is larger than H2, whereas B1, B3, B4, B5, and B6 are all smaller than H2. This establishes that H2 is the smallest hoop.
From Condition (1), B1 and B6 pinged H4, but B5 did not. Therefore, B1 and B6 are larger than H4, while B5 is smaller than H4.
Similarly, from Condition (2), B4 pinged H3, whereas B1 did not. Hence, B4 is larger than H3, while B1 is smaller than H3.
Combining all these observations, we obtain the following order:
B2 < H2 < B4 < H3 < B1, B6 < H4 < B5 < H1 < B3
Accordingly, the only hoop order that satisfies all the conditions is:
H2 < H3 < H4 < H1
The deductions can be summarized in the table below.
The number of pings made by B1, B2, and B3 are 2, 4, and 0, respectively.
We can see from the table that the total number of pings can be 12 or 13.
The following charts depict details of research papers written by four authors, Arman, Brajen, Chintan, and Devon. The papers were of four types, single-author, two-author, three-author, and four-author, that is, written by one, two, three, or all four of these authors, respectively. No other authors were involved in writing these papers.
The following additional facts are known.
1. Each of the authors wrote at least one of each of the four types of papers.
2. The four authors wrote different numbers of single-author papers.
3. Both Chintan and Devon wrote more three-author papers than Brajen.
4. The number of single-author and two-author papers written by Brajen were the same.
What was the total number of two-author and threeauthor papers written by Brajen?
Step 1:
From the first bar graph, we know the total number of titles authored by each individual. We are also given the total number of single-author, two-author, three-author, and four-author papers.
Observe that 2 four-author papers contribute 2 × 4 = 8 author counts, since all four authors are involved in each paper. The same counting principle applies to the three-author and two-author papers.
Using Condition (1), each author contributed to at least one paper of every type. Therefore, none of the entries in the table can be 0.
Since Aman has authored 5 papers in total, and 2 of them are four-author papers, the remaining 3 papers must be distributed among the other three categories. As every category must have at least one paper, Aman must have 1 single-author, 1 two-author, and 1 three-author paper.
From Condition (2), every author has a distinct number of single-author papers. Since Aman already has 1, the remaining authors must have 2, 3, and 4 single-author papers.
Step 2:
Using Condition (4), Brajen cannot have 4 single-author and two-author papers combined, because the overall total for these categories is 8. He also cannot have 3, as that would leave him with 0 three-author papers, violating the given conditions.
Hence, Brajen must have 2 single-author papers and 2 two-author papers. Since his total is 8, he must also have 2 three-author papers.
Now, applying Condition (3), both Chintan and Devon must have more than 2 three-author papers, and the total number of three-author papers contributed by all authors is 9. The only feasible allocation is 3 each for Chintan and Devon.
The deductions obtained so far are summarized in the table below.
Therefore, the total number of two-author and three-author papers written by Brajen is:
2 + 2 = 4.
Which of the following statements is/are NECESSARILY true?
i. Chintan wrote exactly three two-author papers.
ii. Chintan wrote more single-author papers than Devon.
Neither i nor ii
Only i
Both i and ii
Only ii
Neither i nor ii
Step 1:
From the first bar graph, we know the total number of titles authored by each individual. We are also given the total number of single-author, two-author, three-author, and four-author papers.
Observe that 2 four-author papers contribute 2 × 4 = 8 author counts, since all four authors are involved in each paper. The same counting principle applies to the three-author and two-author papers.
Using Condition (1), each author contributed to at least one paper of every type. Therefore, none of the entries in the table can be 0.
Since Aman has authored 5 papers in total, and 2 of them are four-author papers, the remaining 3 papers must be distributed among the other three categories. As every category must have at least one paper, Aman must have 1 single-author, 1 two-author, and 1 three-author paper.
From Condition (2), every author has a distinct number of single-author papers. Since Aman already has 1, the remaining authors must have 2, 3, and 4 single-author papers.
Step 2:
Using Condition (4), Brajen cannot have 4 single-author and two-author papers combined, because the overall total for these categories is 8. He also cannot have 3, as that would leave him with 0 three-author papers, violating the given conditions.
Hence, Brajen must have 2 single-author papers and 2 two-author papers. Since his total is 8, he must also have 2 three-author papers.
Now, applying Condition (3), both Chintan and Devon must have more than 2 three-author papers, and the total number of three-author papers contributed by all authors is 9. The only feasible allocation is 3 each for Chintan and Devon.
The deductions obtained so far are summarized in the table below.
Let us check each statement:
i. The statement, Chintan wrote exactly three two-author papers may not necessarily be true.
ii. The statement, Chintan wrote more single-author papers than Devon may not necessarily be true.
Hence, neither i nor ii is definitely true.
Which of the following statements is/are NECESSARILY true?
i. Arman wrote three-author papers only with Chintan and Devon.
ii. Brajen wrote three-author papers only with Chintan and Devon.
Neither i or ii
Both i and ii
Only ii
Only i
Both i and ii
Step 1:
From the first bar graph, we know the total number of titles authored by each individual. We are also given the total number of single-author, two-author, three-author, and four-author papers.
Observe that 2 four-author papers contribute 2 × 4 = 8 author counts, since all four authors are involved in each paper. The same counting principle applies to the three-author and two-author papers.
Using Condition (1), each author contributed to at least one paper of every type. Therefore, none of the entries in the table can be 0.
Since Aman has authored 5 papers in total, and 2 of them are four-author papers, the remaining 3 papers must be distributed among the other three categories. As every category must have at least one paper, Aman must have 1 single-author, 1 two-author, and 1 three-author paper.
From Condition (2), every author has a distinct number of single-author papers. Since Aman already has 1, the remaining authors must have 2, 3, and 4 single-author papers.
Step 2:
Using Condition (4), Brajen cannot have 4 single-author and two-author papers combined, because the overall total for these categories is 8. He also cannot have 3, as that would leave him with 0 three-author papers, violating the given conditions.
Hence, Brajen must have 2 single-author papers and 2 two-author papers. Since his total is 8, he must also have 2 three-author papers.
Now, applying Condition (3), both Chintan and Devon must have more than 2 three-author papers, and the total number of three-author papers contributed by all authors is 9. The only feasible allocation is 3 each for Chintan and Devon.
The deductions obtained so far are summarized in the table below.
There are 3 three author papers and both Chintan and Devon wrote 3 three author papers whereas Aman wrote 1 and Brajen wrote 2. So the only possible combination will be {(Chintan, Devon, Aman), (Chintan, Devon, Brajen), (Chintan, Devon, Brajen)}. Hence, the statement, Arman wrote three-author papers only with Chintan and Devon, is true. Brajen wrote three-author papers only with Chintan and Devon is also true. Hence, both (i) and (ii) are true.
If Devon wrote more than one two-author papers, then how many two-author papers did Chintan write?
Step 1:
From the first bar graph, we know the total number of titles authored by each individual. We are also given the total number of single-author, two-author, three-author, and four-author papers.
Observe that 2 four-author papers contribute 2 × 4 = 8 author counts, since all four authors are involved in each paper. The same counting principle applies to the three-author and two-author papers.
Using Condition (1), each author contributed to at least one paper of every type. Therefore, none of the entries in the table can be 0.
Since Aman has authored 5 papers in total, and 2 of them are four-author papers, the remaining 3 papers must be distributed among the other three categories. As every category must have at least one paper, Aman must have 1 single-author, 1 two-author, and 1 three-author paper.
From Condition (2), every author has a distinct number of single-author papers. Since Aman already has 1, the remaining authors must have 2, 3, and 4 single-author papers.
Step 2:
Using Condition (4), Brajen cannot have 4 single-author and two-author papers combined, because the overall total for these categories is 8. He also cannot have 3, as that would leave him with 0 three-author papers, violating the given conditions.
Hence, Brajen must have 2 single-author papers and 2 two-author papers. Since his total is 8, he must also have 2 three-author papers.
Now, applying Condition (3), both Chintan and Devon must have more than 2 three-author papers, and the total number of three-author papers contributed by all authors is 9. The only feasible allocation is 3 each for Chintan and Devon.
The deductions obtained so far are summarized in the table below.
If Devon wrote more than one two-author papers, then the number of two-author papers written by Chintan is 3.
Ananya Raga, Bhaskar Tala, Charu Veena, and Devendra Sur are four musicians. Each of them started and completed their training as students under each of three Gurus — Pandit Meghnath, Ustad Samiran, and Acharya Raghunath between 2013 and 2024, including both the years. Each Guru trains any student for consecutive years only, for a span of 2, 3, or 4 years, with each Guru having a different span. During some of these years, a student may not have trained under these Gurus; however, they never trained under multiple Gurus in the same year.
In none of these years, any of these Gurus trained more than two of these students at the same time. When two students train under the same Guru at the same time, they are referred to as Gurubhai, irrespective of their gender.
The following additional facts are known.
1. Ustad Samiran never trained more than one of these students in the same year.
2. Acharya Raghunath did not train any of these students during 2015-2018, as well as during 2021-24.
3. Ananya and Devendra were never Gurubhai; neither were Bhaskar and Charu. All other pairs of musicians were Gurubhai for exactly 2 years.
4. In 2013, Ananya and Bhaskar started their trainings under Pandit Meghnath and under Ustad Samiran, respectively.
In which of the following years were Ananya and Bhaskar Gurubhai?
2020
2018
2021
2014
2020
Step 1:
From Condition (1), Ustad Samiran never trains more than one of the four students in the same year. Therefore, each student must have trained under Ustad Samiran for 3 years.
Using Condition (2), Acharya Raghunath does not train any student during 2015–2018 and 2021–2024, creating two gaps of four years each. Hence, Acharya Raghunath trains every student for exactly 2 years.
From Condition (4), Ananya and Bhaskar began their training under Pandit Meghnath and Ustad Samiran, respectively, in 2013. Since no student can train under more than one guru simultaneously, Charu and Devendra must have started under Acharya Raghunath in 2013.
It also follows that Ananya and Bhaskar trained under Acharya Raghunath during 2019–2020, completing their two-year training period. As the training durations under the three gurus are 2, 3, and 4 years, and we have already determined that Acharya Raghunath trains for 2 years while Pandit Meghnath trains for 3 years, Ustad Samiran must train each student for 4 years.
Since Ananya started her training in 2013, and a student cannot train under two gurus at the same time, Bhaskar can train under Pandit Meghnath only during 2021–2024.
Now, applying Condition (3), among the six possible student pairs (AB, AC, AD, BC, BD, and CD), the pairs AD and BC cannot be gurubhais. Each of the remaining four pairs must be gurubhais for exactly 2 years.
Accordingly, Ananya and Charu train together under Pandit Meghnath during 2015 and 2016. Similarly, Bhaskar and Devendra train together during 2021 and 2022.
Step 2:
Under Ustad Samiran, each student has a distinct 3-year training period with no overlap. Therefore, Ananya must train during 2022–2024, as any other three-year period would overlap with her training under the other two gurus.
The remaining schedules can now be determined uniquely. Devendra trains under Ustad Samiran from 2016–2018, while Charu trains under Ustad Samiran from 2019–2021.
The complete schedule is shown in the table below.
Hence, Ananya and Bhaskar were gurubhai in 2020.
In which year did Charu begin her training under Pandit Meghnath?
2017
2015
2016
2021
2015
Step 1:
From Condition (1), Ustad Samiran never trains more than one of the four students in the same year. Therefore, each student must have trained under Ustad Samiran for 3 years.
Using Condition (2), Acharya Raghunath does not train any student during 2015–2018 and 2021–2024, creating two gaps of four years each. Hence, Acharya Raghunath trains every student for exactly 2 years.
From Condition (4), Ananya and Bhaskar began their training under Pandit Meghnath and Ustad Samiran, respectively, in 2013. Since no student can train under more than one guru simultaneously, Charu and Devendra must have started under Acharya Raghunath in 2013.
It also follows that Ananya and Bhaskar trained under Acharya Raghunath during 2019–2020, completing their two-year training period. As the training durations under the three gurus are 2, 3, and 4 years, and we have already determined that Acharya Raghunath trains for 2 years while Pandit Meghnath trains for 3 years, Ustad Samiran must train each student for 4 years.
Since Ananya started her training in 2013, and a student cannot train under two gurus at the same time, Bhaskar can train under Pandit Meghnath only during 2021–2024.
Now, applying Condition (3), among the six possible student pairs (AB, AC, AD, BC, BD, and CD), the pairs AD and BC cannot be gurubhais. Each of the remaining four pairs must be gurubhais for exactly 2 years.
Accordingly, Ananya and Charu train together under Pandit Meghnath during 2015 and 2016. Similarly, Bhaskar and Devendra train together during 2021 and 2022.
Step 2:
Under Ustad Samiran, each student has a distinct 3-year training period with no overlap. Therefore, Ananya must train during 2022–2024, as any other three-year period would overlap with her training under the other two gurus.
The remaining schedules can now be determined uniquely. Devendra trains under Ustad Samiran from 2016–2018, while Charu trains under Ustad Samiran from 2019–2021.
The complete schedule is shown in the table below.
Charu began her training under Pandit Meghnath in 2015.
In which of the following years were Bhaskar and Devendra Gurubhai?
2015
2022
2018
2020
2022
Step 1:
From Condition (1), Ustad Samiran never trains more than one of the four students in the same year. Therefore, each student must have trained under Ustad Samiran for 3 years.
Using Condition (2), Acharya Raghunath does not train any student during 2015–2018 and 2021–2024, creating two gaps of four years each. Hence, Acharya Raghunath trains every student for exactly 2 years.
From Condition (4), Ananya and Bhaskar began their training under Pandit Meghnath and Ustad Samiran, respectively, in 2013. Since no student can train under more than one guru simultaneously, Charu and Devendra must have started under Acharya Raghunath in 2013.
It also follows that Ananya and Bhaskar trained under Acharya Raghunath during 2019–2020, completing their two-year training period. As the training durations under the three gurus are 2, 3, and 4 years, and we have already determined that Acharya Raghunath trains for 2 years while Pandit Meghnath trains for 3 years, Ustad Samiran must train each student for 4 years.
Since Ananya started her training in 2013, and a student cannot train under two gurus at the same time, Bhaskar can train under Pandit Meghnath only during 2021–2024.
Now, applying Condition (3), among the six possible student pairs (AB, AC, AD, BC, BD, and CD), the pairs AD and BC cannot be gurubhais. Each of the remaining four pairs must be gurubhais for exactly 2 years.
Accordingly, Ananya and Charu train together under Pandit Meghnath during 2015 and 2016. Similarly, Bhaskar and Devendra train together during 2021 and 2022.
Step 2:
Under Ustad Samiran, each student has a distinct 3-year training period with no overlap. Therefore, Ananya must train during 2022–2024, as any other three-year period would overlap with her training under the other two gurus.
The remaining schedules can now be determined uniquely. Devendra trains under Ustad Samiran from 2016–2018, while Charu trains under Ustad Samiran from 2019–2021.
The complete schedule is shown in the table below.
Bhaskar and Devendra were Gurubhai in 2022.
Which of the following statements is TRUE?
Ananya was training under Ustad Samiran in 2015.
Charu was training under Ustad Samiran in 2019.
Ananya was training under Ustad Samiran in 2018.
Charu was training under Ustad Samiran in 2018.
Charu was training under Ustad Samiran in 2019.
Step 1:
From Condition (1), Ustad Samiran never trains more than one of the four students in the same year. Therefore, each student must have trained under Ustad Samiran for 3 years.
Using Condition (2), Acharya Raghunath does not train any student during 2015–2018 and 2021–2024, creating two gaps of four years each. Hence, Acharya Raghunath trains every student for exactly 2 years.
From Condition (4), Ananya and Bhaskar began their training under Pandit Meghnath and Ustad Samiran, respectively, in 2013. Since no student can train under more than one guru simultaneously, Charu and Devendra must have started under Acharya Raghunath in 2013.
It also follows that Ananya and Bhaskar trained under Acharya Raghunath during 2019–2020, completing their two-year training period. As the training durations under the three gurus are 2, 3, and 4 years, and we have already determined that Acharya Raghunath trains for 2 years while Pandit Meghnath trains for 3 years, Ustad Samiran must train each student for 4 years.
Since Ananya started her training in 2013, and a student cannot train under two gurus at the same time, Bhaskar can train under Pandit Meghnath only during 2021–2024.
Now, applying Condition (3), among the six possible student pairs (AB, AC, AD, BC, BD, and CD), the pairs AD and BC cannot be gurubhais. Each of the remaining four pairs must be gurubhais for exactly 2 years.
Accordingly, Ananya and Charu train together under Pandit Meghnath during 2015 and 2016. Similarly, Bhaskar and Devendra train together during 2021 and 2022.
Step 2:
Under Ustad Samiran, each student has a distinct 3-year training period with no overlap. Therefore, Ananya must train during 2022–2024, as any other three-year period would overlap with her training under the other two gurus.
The remaining schedules can now be determined uniquely. Devendra trains under Ustad Samiran from 2016–2018, while Charu trains under Ustad Samiran from 2019–2021.
The complete schedule is shown in the table below.
Let us check each statement:
1. Ananya was training under Ustad Samiran in 2015, is not true.
2. Charu was training under Ustad Samiran in 2019, is true.
3. Ananya was training under Ustad Samiran in 2018, is not true
4. Charu was training under Ustad Samiran in 2018, is not true.
Between 2013-24, there were 4 years when only two of these four musicians were training under these three Gurus. These years were 2017, 2018, 2023 and 2024.
Step 1:
From Condition (1), Ustad Samiran never trains more than one of the four students in the same year. Therefore, each student must have trained under Ustad Samiran for 3 years.
Using Condition (2), Acharya Raghunath does not train any student during 2015–2018 and 2021–2024, creating two gaps of four years each. Hence, Acharya Raghunath trains every student for exactly 2 years.
From Condition (4), Ananya and Bhaskar began their training under Pandit Meghnath and Ustad Samiran, respectively, in 2013. Since no student can train under more than one guru simultaneously, Charu and Devendra must have started under Acharya Raghunath in 2013.
It also follows that Ananya and Bhaskar trained under Acharya Raghunath during 2019–2020, completing their two-year training period. As the training durations under the three gurus are 2, 3, and 4 years, and we have already determined that Acharya Raghunath trains for 2 years while Pandit Meghnath trains for 3 years, Ustad Samiran must train each student for 4 years.
Since Ananya started her training in 2013, and a student cannot train under two gurus at the same time, Bhaskar can train under Pandit Meghnath only during 2021–2024.
Now, applying Condition (3), among the six possible student pairs (AB, AC, AD, BC, BD, and CD), the pairs AD and BC cannot be gurubhais. Each of the remaining four pairs must be gurubhais for exactly 2 years.
Accordingly, Ananya and Charu train together under Pandit Meghnath during 2015 and 2016. Similarly, Bhaskar and Devendra train together during 2021 and 2022.
Step 2:
Under Ustad Samiran, each student has a distinct 3-year training period with no overlap. Therefore, Ananya must train during 2022–2024, as any other three-year period would overlap with her training under the other two gurus.
The remaining schedules can now be determined uniquely. Devendra trains under Ustad Samiran from 2016–2018, while Charu trains under Ustad Samiran from 2019–2021.
The complete schedule is shown in the table below.
Between 2013-24, there were 4 years when only two of these four musicians were training under these three Gurus. These years were 2017, 2018, 2023 and 2024.
The Sustainability Index (SI) of a country at a point in time is an integer between 1 and 100. This question is related to SI of six countries – A, B, C, D, E, and F – at three different points in time – 2016, 2020, and 2024. The plot represents the exact changes in their SI, with X-coordinate representing % increase in 2020 from 2016, i.e., (SI in 2020 minus SI in 2016) / (SI in 2016), and Y-coordinate representing % increase in 2024 from 2020. At any point in time, the country with highest SI is ranked 1, while the country with the lowest SI is ranked 6. The following additional facts are known.
1. In 2016, B, C, E, and A had ranks 1, 2, 3, and 4 respectively.
2. F had lower SI than any other country in 2016, 2020, and 2024.
3. In 2024, E was the only country with SI of 90.
4. The range of SI of the six countries was 60 in 2016 as well as in 2024.
What was the SI of E in 2016?
Step 1:
Only the values corresponding to B, C, E, and F are required to answer the questions.
In 2016, B had the highest SI, while F had the lowest. Also, the SI of B exceeded that of F by 60.
Let the SI of F in 2016 be x. Then, the SI of B in 2016 is x + 60.
Using the percentage changes shown in the graph, the SI of F becomes 2x in 2020 and 1.5x in 2024.
The SI of E in 2024 is 90. Applying the given percentage changes, the corresponding SI values of E are 75 in 2020 and 60 in 2016.
Since the range of SI values in 2024 is 60, the SI of F in 2024 (i.e., 1.5x) must be at least 30. Therefore, x ≥ 20, implying (x + 60) ≥ 80.
Now consider the percentage changes for B. Its SI decreases by 25% from 2016 to 2020, and again by 25% from 2020 to 2024. Thus, the overall multiplying factor from 2016 to 2024 is 9/16.
Since all SI values are integers, (x + 60) must be a multiple of 16. As it is already at least 80, the only possible values are 80 and 96.
- If (x + 60) = 96, then x = 36. This gives the SI of B in 2020 as 72, while the SI of F in 2020 also becomes 72, contradicting the fact that F had the lowest SI in 2020.
- Hence, this case is rejected.
Therefore, (x + 60) = 80, giving x = 20.
Thus, the SI values of B are:
- 2016: 80
- 2020: 60
- 2024: 45
Step 2:
In 2016, the SI of C lies between those of B and E. Therefore, its value must lie between 60 and 80.
From the graph, the SI of C becomes 4/5 of its 2016 value in 2020, and then 7/5 of its 2020 value in 2024. Hence, the overall multiplying factor from 2016 to 2024 is 28/25.
This implies that the SI of C in 2016 must be a multiple of 25. The only multiple of 25 between 60 and 80 is 75.
Accordingly, the SI of C is:
- 2016: 75
- 2020: 60
- 2024: 84
The derived values are summarized in the table below.
Hence, the SI of E in 2016 was 60.
What was the SI of F in 2020?
Step 1:
Only the values corresponding to B, C, E, and F are required to answer the questions.
In 2016, B had the highest SI, while F had the lowest. Also, the SI of B exceeded that of F by 60.
Let the SI of F in 2016 be x. Then, the SI of B in 2016 is x + 60.
Using the percentage changes shown in the graph, the SI of F becomes 2x in 2020 and 1.5x in 2024.
The SI of E in 2024 is 90. Applying the given percentage changes, the corresponding SI values of E are 75 in 2020 and 60 in 2016.
Since the range of SI values in 2024 is 60, the SI of F in 2024 (i.e., 1.5x) must be at least 30. Therefore, x ≥ 20, implying (x + 60) ≥ 80.
Now consider the percentage changes for B. Its SI decreases by 25% from 2016 to 2020, and again by 25% from 2020 to 2024. Thus, the overall multiplying factor from 2016 to 2024 is 9/16.
Since all SI values are integers, (x + 60) must be a multiple of 16. As it is already at least 80, the only possible values are 80 and 96.
- If (x + 60) = 96, then x = 36. This gives the SI of B in 2020 as 72, while the SI of F in 2020 also becomes 72, contradicting the fact that F had the lowest SI in 2020.
- Hence, this case is rejected.
Therefore, (x + 60) = 80, giving x = 20.
Thus, the SI values of B are:
- 2016: 80
- 2020: 60
- 2024: 45
Step 2:
In 2016, the SI of C lies between those of B and E. Therefore, its value must lie between 60 and 80.
From the graph, the SI of C becomes 4/5 of its 2016 value in 2020, and then 7/5 of its 2020 value in 2024. Hence, the overall multiplying factor from 2016 to 2024 is 28/25.
This implies that the SI of C in 2016 must be a multiple of 25. The only multiple of 25 between 60 and 80 is 75.
Accordingly, the SI of C is:
- 2016: 75
- 2020: 60
- 2024: 84
The derived values are summarized in the table below.
The SI of F in 2020 = 40.
What was the SI of C in 2024?
Step 1:
Only the values corresponding to B, C, E, and F are required to answer the questions.
In 2016, B had the highest SI, while F had the lowest. Also, the SI of B exceeded that of F by 60.
Let the SI of F in 2016 be x. Then, the SI of B in 2016 is x + 60.
Using the percentage changes shown in the graph, the SI of F becomes 2x in 2020 and 1.5x in 2024.
The SI of E in 2024 is 90. Applying the given percentage changes, the corresponding SI values of E are 75 in 2020 and 60 in 2016.
Since the range of SI values in 2024 is 60, the SI of F in 2024 (i.e., 1.5x) must be at least 30. Therefore, x ≥ 20, implying (x + 60) ≥ 80.
Now consider the percentage changes for B. Its SI decreases by 25% from 2016 to 2020, and again by 25% from 2020 to 2024. Thus, the overall multiplying factor from 2016 to 2024 is 9/16.
Since all SI values are integers, (x + 60) must be a multiple of 16. As it is already at least 80, the only possible values are 80 and 96.
- If (x + 60) = 96, then x = 36. This gives the SI of B in 2020 as 72, while the SI of F in 2020 also becomes 72, contradicting the fact that F had the lowest SI in 2020.
- Hence, this case is rejected.
Therefore, (x + 60) = 80, giving x = 20.
Thus, the SI values of B are:
- 2016: 80
- 2020: 60
- 2024: 45
Step 2:
In 2016, the SI of C lies between those of B and E. Therefore, its value must lie between 60 and 80.
From the graph, the SI of C becomes 4/5 of its 2016 value in 2020, and then 7/5 of its 2020 value in 2024. Hence, the overall multiplying factor from 2016 to 2024 is 28/25.
This implies that the SI of C in 2016 must be a multiple of 25. The only multiple of 25 between 60 and 80 is 75.
Accordingly, the SI of C is:
- 2016: 75
- 2020: 60
- 2024: 84
The derived values are summarized in the table below.
The SI of C in 2024 = 84
What was the SI of B in 2024?
54
45
60
80
Step 1:
Only the values corresponding to B, C, E, and F are required to answer the questions.
In 2016, B had the highest SI, while F had the lowest. Also, the SI of B exceeded that of F by 60.
Let the SI of F in 2016 be x. Then, the SI of B in 2016 is x + 60.
Using the percentage changes shown in the graph, the SI of F becomes 2x in 2020 and 1.5x in 2024.
The SI of E in 2024 is 90. Applying the given percentage changes, the corresponding SI values of E are 75 in 2020 and 60 in 2016.
Since the range of SI values in 2024 is 60, the SI of F in 2024 (i.e., 1.5x) must be at least 30. Therefore, x ≥ 20, implying (x + 60) ≥ 80.
Now consider the percentage changes for B. Its SI decreases by 25% from 2016 to 2020, and again by 25% from 2020 to 2024. Thus, the overall multiplying factor from 2016 to 2024 is 9/16.
Since all SI values are integers, (x + 60) must be a multiple of 16. As it is already at least 80, the only possible values are 80 and 96.
- If (x + 60) = 96, then x = 36. This gives the SI of B in 2020 as 72, while the SI of F in 2020 also becomes 72, contradicting the fact that F had the lowest SI in 2020.
- Hence, this case is rejected.
Therefore, (x + 60) = 80, giving x = 20.
Thus, the SI values of B are:
- 2016: 80
- 2020: 60
- 2024: 45
Step 2:
In 2016, the SI of C lies between those of B and E. Therefore, its value must lie between 60 and 80.
From the graph, the SI of C becomes 4/5 of its 2016 value in 2020, and then 7/5 of its 2020 value in 2024. Hence, the overall multiplying factor from 2016 to 2024 is 28/25.
This implies that the SI of C in 2016 must be a multiple of 25. The only multiple of 25 between 60 and 80 is 75.
Accordingly, the SI of C is:
- 2016: 75
- 2020: 60
- 2024: 84
The derived values are summarized in the table below.
The SI of B in 2024 = 45
The two most populous cities and the non-urban region (NUR) of each of three states, Whimshire, Fogglia, and Humbleset, are assigned Pollution Measures (PMs). These nine PMs are all distinct multiples of 10, ranging from 10 to 90. The six cities in increasing order of their There is only one pair of an NUR and a city (considering all cities and all NURs) where the PM of the NUR is greater than that of the city. That NUR and the city both belong to Humbleset.
The PIs of all three states are distinct integers, with Humbleset and Fogglia having the highest and the lowest PI respectively.
PMs are: Blusterburg, Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo.
The Pollution Index (PI) of a state is a weighted average of the PMs of its NUR and cities, with a weight of 50% for the NUR, and 25% each for its two cities.
There is only one pair of an NUR and a city (considering all cities and all NURs) where the PM of the NUR is greater than that of the city. That NUR and the city both belong to Humbleset.
The PIs of all three states are distinct integers, with Humbleset and Fogglia having the highest and the lowest PI respectively.
What is the PI of Whimshire?
Step 1:
From the given information, the nine PI values are distinct multiples of 10, namely:
10, 20, 30, 40, 50, 60, 70, 80, and 90.
The cities are arranged in increasing order of PI as follows:
Blusterburg < Noodleton < Splutterville < Quackford < Mumpypore < Zingaloo
We are also told that there is only one city–NUR pair for which the NUR has a higher PI than the city. This is possible only if Blusterburg has a PI of 30, while the corresponding NUR has a PI of 40. Any other assignment would violate the given condition. Moreover, both of these belong to Humbleset.
Accordingly, the remaining cities take the PI values 50, 60, 70, 80, and 90 in increasing order, while the remaining two NURs receive the PI values 10 and 20.
The partial information is summarized in the table below.
Step 2:
The PI values of all three states are distinct integers, with Humbleset having the highest PI and Fogglia the lowest.
For Humbleset to have an integer PI, its remaining city must contribute a weighted PI of 12.5, 17.5, or 22.5.
However, choosing 12.5 or 17.5 would prevent Humbleset from having the highest PI. Therefore, the remaining city in Humbleset must be Zingaloo, whose weighted PI is 22.5.
Thus,
PI(Humbleset) = 7.5 + 20 + 22.5 = 50
Next, for the PI values of all the states to remain integers, the cities with weighted PIs 12.5 and 17.5 must belong to the same state. Their combined weighted PI is 30, while the remaining pair contributes 15 + 20 = 35.
Since Fogglia must have the lowest PI, the only feasible allocation is:
PI(Fogglia) = 12.5 + 17.5 + 5 = 35
PI(Whimshire) = 15 + 20 + 10 = 45
The completed table is shown below.
Hence, the PI of Whimshire is 45.
What is the PI of Fogglia?
Step 1:
From the given information, the nine PI values are distinct multiples of 10, namely:
10, 20, 30, 40, 50, 60, 70, 80, and 90.
The cities are arranged in increasing order of PI as follows:
Blusterburg < Noodleton < Splutterville < Quackford < Mumpypore < Zingaloo
We are also told that there is only one city–NUR pair for which the NUR has a higher PI than the city. This is possible only if Blusterburg has a PI of 30, while the corresponding NUR has a PI of 40. Any other assignment would violate the given condition. Moreover, both of these belong to Humbleset.
Accordingly, the remaining cities take the PI values 50, 60, 70, 80, and 90 in increasing order, while the remaining two NURs receive the PI values 10 and 20.
The partial information is summarized in the table below.
Step 2:
The PI values of all three states are distinct integers, with Humbleset having the highest PI and Fogglia the lowest.
For Humbleset to have an integer PI, its remaining city must contribute a weighted PI of 12.5, 17.5, or 22.5.
However, choosing 12.5 or 17.5 would prevent Humbleset from having the highest PI. Therefore, the remaining city in Humbleset must be Zingaloo, whose weighted PI is 22.5.
Thus,
PI(Humbleset) = 7.5 + 20 + 22.5 = 50
Next, for the PI values of all the states to remain integers, the cities with weighted PIs 12.5 and 17.5 must belong to the same state. Their combined weighted PI is 30, while the remaining pair contributes 15 + 20 = 35.
Since Fogglia must have the lowest PI, the only feasible allocation is:
PI(Fogglia) = 12.5 + 17.5 + 5 = 35
PI(Whimshire) = 15 + 20 + 10 = 45
The completed table is shown below.
The PI of Fogglia = 35
What is the PI of Humbleset?
Step 1:
From the given information, the nine PI values are distinct multiples of 10, namely:
10, 20, 30, 40, 50, 60, 70, 80, and 90.
The cities are arranged in increasing order of PI as follows:
Blusterburg < Noodleton < Splutterville < Quackford < Mumpypore < Zingaloo
We are also told that there is only one city–NUR pair for which the NUR has a higher PI than the city. This is possible only if Blusterburg has a PI of 30, while the corresponding NUR has a PI of 40. Any other assignment would violate the given condition. Moreover, both of these belong to Humbleset.
Accordingly, the remaining cities take the PI values 50, 60, 70, 80, and 90 in increasing order, while the remaining two NURs receive the PI values 10 and 20.
The partial information is summarized in the table below.
Step 2:
The PI values of all three states are distinct integers, with Humbleset having the highest PI and Fogglia the lowest.
For Humbleset to have an integer PI, its remaining city must contribute a weighted PI of 12.5, 17.5, or 22.5.
However, choosing 12.5 or 17.5 would prevent Humbleset from having the highest PI. Therefore, the remaining city in Humbleset must be Zingaloo, whose weighted PI is 22.5.
Thus,
PI(Humbleset) = 7.5 + 20 + 22.5 = 50
Next, for the PI values of all the states to remain integers, the cities with weighted PIs 12.5 and 17.5 must belong to the same state. Their combined weighted PI is 30, while the remaining pair contributes 15 + 20 = 35.
Since Fogglia must have the lowest PI, the only feasible allocation is:
PI(Fogglia) = 12.5 + 17.5 + 5 = 35
PI(Whimshire) = 15 + 20 + 10 = 45
The completed table is shown below.
The PI of Humbleset = 50
Which pair of cities definitely belong to the same state?
Splutterville, Quackford
Mumpypore, Zingaloo
Noodleton, Quackford
Blusterburg, Mumpypore
Noodleton, Quackford
Step 1:
From the given information, the nine PI values are distinct multiples of 10, namely:
10, 20, 30, 40, 50, 60, 70, 80, and 90.
The cities are arranged in increasing order of PI as follows:
Blusterburg < Noodleton < Splutterville < Quackford < Mumpypore < Zingaloo
We are also told that there is only one city–NUR pair for which the NUR has a higher PI than the city. This is possible only if Blusterburg has a PI of 30, while the corresponding NUR has a PI of 40. Any other assignment would violate the given condition. Moreover, both of these belong to Humbleset.
Accordingly, the remaining cities take the PI values 50, 60, 70, 80, and 90 in increasing order, while the remaining two NURs receive the PI values 10 and 20.
The partial information is summarized in the table below.
Step 2:
The PI values of all three states are distinct integers, with Humbleset having the highest PI and Fogglia the lowest.
For Humbleset to have an integer PI, its remaining city must contribute a weighted PI of 12.5, 17.5, or 22.5.
However, choosing 12.5 or 17.5 would prevent Humbleset from having the highest PI. Therefore, the remaining city in Humbleset must be Zingaloo, whose weighted PI is 22.5.
Thus,
PI(Humbleset) = 7.5 + 20 + 22.5 = 50
Next, for the PI values of all the states to remain integers, the cities with weighted PIs 12.5 and 17.5 must belong to the same state. Their combined weighted PI is 30, while the remaining pair contributes 15 + 20 = 35.
Since Fogglia must have the lowest PI, the only feasible allocation is:
PI(Fogglia) = 12.5 + 17.5 + 5 = 35
PI(Whimshire) = 15 + 20 + 10 = 45
The completed table is shown below.
The pair of cities that definitely belong to the same states Noodleton and Quackford.
For how many of the cities and NURs is it possible to identify their PM and the state they belong to?
Step 1:
From the given information, the nine PI values are distinct multiples of 10, namely:
10, 20, 30, 40, 50, 60, 70, 80, and 90.
The cities are arranged in increasing order of PI as follows:
Blusterburg < Noodleton < Splutterville < Quackford < Mumpypore < Zingaloo
We are also told that there is only one city–NUR pair for which the NUR has a higher PI than the city. This is possible only if Blusterburg has a PI of 30, while the corresponding NUR has a PI of 40. Any other assignment would violate the given condition. Moreover, both of these belong to Humbleset.
Accordingly, the remaining cities take the PI values 50, 60, 70, 80, and 90 in increasing order, while the remaining two NURs receive the PI values 10 and 20.
The partial information is summarized in the table below.
Step 2:
The PI values of all three states are distinct integers, with Humbleset having the highest PI and Fogglia the lowest.
For Humbleset to have an integer PI, its remaining city must contribute a weighted PI of 12.5, 17.5, or 22.5.
However, choosing 12.5 or 17.5 would prevent Humbleset from having the highest PI. Therefore, the remaining city in Humbleset must be Zingaloo, whose weighted PI is 22.5.
Thus,
PI(Humbleset) = 7.5 + 20 + 22.5 = 50
Next, for the PI values of all the states to remain integers, the cities with weighted PIs 12.5 and 17.5 must belong to the same state. Their combined weighted PI is 30, while the remaining pair contributes 15 + 20 = 35.
Since Fogglia must have the lowest PI, the only feasible allocation is:
PI(Fogglia) = 12.5 + 17.5 + 5 = 35
PI(Whimshire) = 15 + 20 + 10 = 45
The completed table is shown below.
We can identify the pIs of all cities and NURs and also identify the state they belong to.
If 9^(x²+2x−3) − 4(3^(x²+2x−2)) + 27 = 0 then the product of all possible values of x is
30
20
5
15
20
Step 1: Rewrite the equation using the same base. Given: 9^(x² + 2x − 3) − 4(3^(x² + 2x − 2)) + 27 = 0. Since 9 = 3², 9^(x² + 2x − 3) = 3^[2(x² + 2x − 3)] = 3^(2x² + 4x − 6). Also, 3^(2x² + 4x − 6) = (3^(x² + 2x − 3))². Let y = 3^(x² + 2x − 3). Then, 3^(x² + 2x − 2) = 3 × 3^(x² + 2x − 3) = 3y. Substitute into the equation: y² − 4(3y) + 27 = 0 → y² − 12y + 27 = 0.
Step 2: Solve the quadratic equation. y² − 12y + 27 = 0 → (y − 3)(y − 9) = 0. Therefore, y = 3 or y = 9.
Step 3: Find the corresponding values of x. Since y = 3^(x² + 2x − 3). Case 1: y = 3. 3^(x² + 2x − 3) = 3 → x² + 2x − 3 = 1 → x² + 2x − 4 = 0. Using the quadratic formula: x = (−2 ± √20)/2 = −1 ± √5. Case 2: y = 9. 3^(x² + 2x − 3) = 9 = 3² → x² + 2x − 3 = 2 → x² + 2x − 5 = 0. Using the quadratic formula: x = (−2 ± √24)/2 = −1 ± √6.
Step 4: Find the product of all possible values of x. For x² + 2x − 4 = 0, product of roots = −4. For x² + 2x − 5 = 0, product of roots = −5. Hence, the product of all four values is (−4) × (−5) = 20.
The average number of copies of a book sold per day by a shopkeeper is 60 in the initial seven days and 63 in the initial eight days, after the book launch. On the ninth day, she sells 11 copies less than the eighth day, and the average number of copies sold per day from second day to ninth day becomes 66. The number of copies sold on the first day of the book launch is
Step 1: Find the total number of books sold in the first 7 days. Average for the first 7 days = 60. Total books sold in the first 7 days = 7 × 60 = 420.
Step 2: Find the number of books sold on the 8th day. Average for the first 8 days = 63. Total books sold in the first 8 days = 8 × 63 = 504. Books sold on the 8th day = 504 − 420 = 84.
Step 3: Find the number of books sold on the 9th day. The 9th day sales are 11 less than the 8th day sales. Books sold on the 9th day = 84 − 11 = 73.
Step 4: Find the total number of books sold from the 2nd day to the 9th day. Average from the 2nd day to the 9th day = 66. Number of days = 8. Total books sold from the 2nd day to the 9th day = 8 × 66 = 528.
Step 5: Find the number of books sold on the 1st day. The total books sold from the 2nd day to the 9th day can also be written as: (Total books sold in the first 8 days − Books sold on the 1st day) + Books sold on the 9th day. So, (504 − First day sales) + 73 = 528 → 577 − First day sales = 528 → First day sales = 49.
The set of all real values of x for which (x² − |x + 9| + x) > 0, is
(−∞, −3) ∪ (3, ∞)
(−∞, −9) ∪ (3, ∞)
(−9, −3) ∪ (3, ∞)
(−∞, −9) ∪ (9, ∞)
(−∞, −3) ∪ (3, ∞)
Step 1: Split the expression based on the absolute value. Given, x² − |x + 9| + x > 0. The absolute value changes at x = −9. So, consider two cases.
Case 1: x ≥ −9. Here, |x + 9| = x + 9. Substituting: x² − (x + 9) + x > 0 → x² − 9 > 0 → (x − 3)(x + 3) > 0. This is positive when x < −3 or x > 3. Since this case requires x ≥ −9, the solution becomes: −9 ≤ x < −3 or x > 3, i.e., [−9, −3) ∪ (3, ∞).
Case 2: x < −9. Here, |x + 9| = −(x + 9). Substituting: x² − [−(x + 9)] + x > 0 → x² + x + 9 + x > 0 → x² + 2x + 9 > 0. Completing the square: (x + 1)² + 8 > 0. This expression is always positive for every real value of x. Since this case requires x < −9, all values x < −9 satisfy the inequality. So the solution is: (−∞, −9).
Step 3: Combine both cases. (−∞, −9) ∪ [−9, −3) ∪ (3, ∞). Since x = −9 satisfies the inequality, (−∞, −9) ∪ [−9, −3) = (−∞, −3). Therefore, the complete solution set is (−∞, −3) ∪ (3, ∞).
An item with a cost price of Rs. 1650 is sold at a certain discount on a fixed marked price to earn a profit of 20% on the cost price. If the discount was doubled, the profit would have been Rs. 110. The rate of discount, in percentage, at which the profit percentage would be equal to the rate of discount, is nearest to
16
18
14
12
14
Step 1: Find the selling price when the profit is 20%. Cost Price (CP) = Rs. 1650. Profit = 20% of 1650 = Rs. 330. Therefore, Selling Price (SP₁) = 1650 + 330 = Rs. 1980.
Step 2: Find the selling price when the discount is doubled. When the discount is doubled, the profit becomes Rs. 110. Therefore, Selling Price (SP₂) = 1650 + 110 = Rs. 1760.
Step 3: Find the marked price. Let the marked price be M and the original discount be d%. Then, SP₁ = M × (100 − d)/100 = 1980 and SP₂ = M × (100 − 2d)/100 = 1760. Subtracting: M × d/100 = 1980 − 1760 = 220 ... (1). From the first equation, M × (100 − d)/100 = 1980. Using equation (1), M = 1980 + 220 = Rs. 2200.
Step 4: Find the original discount percentage. Discount amount = 220. Marked Price = 2200. Discount percentage = (220/2200) × 100 = 10%.
Step 5: Let the required discount rate be x%. At this discount, Profit percentage = Discount percentage = x%. Selling Price = 1650 × (100 + x)/100. Also, Selling Price = 2200 × (100 − x)/100. Equating both: 1650(100 + x) = 2200(100 − x) → 165000 + 1650x = 220000 − 2200x → 3850x = 55000 → x = 55000/3850 = 100/7 ≈ 14.29%.
Step 6: The nearest percentage is 14%.
If m and n are integers such that (m + 2n)(2m + n) = 27, then the maximum possible value of 2m − 3n is
Step 1: Introduce new variables. Let a = m + 2n and b = 2m + n. Then, ab = 27. Since 27 has only a few integer factor pairs, we can test each one.
Step 2: Express m and n in terms of a and b. From m + 2n = a and 2m + n = b. Multiplying the first equation by 2: 2m + 4n = 2a. Subtracting the second equation: 3n = 2a − b → n = (2a − b)/3. Now, m = a − 2n = a − 2(2a − b)/3 = (2b − a)/3.
Step 3: Express the required quantity. 2m − 3n = 2 × (2b − a)/3 − 3 × (2a − b)/3 = (4b − 2a − 6a + 3b)/3 = (7b − 8a)/3.
Step 4: Check the integer factor pairs of 27. Possible factor pairs: (1, 27), (3, 9), (9, 3), (27, 1) and (−1, −27), (−3, −9), (−9, −3), (−27, −1). Evaluate only those pairs that give integer values of m and n.
For (a, b) = (3, 9): m = (18 − 3)/3 = 5, n = (6 − 9)/3 = −1. 2m − 3n = 10 + 3 = 13.
For (a, b) = (9, 3): m = (6 − 9)/3 = −1, n = (18 − 3)/3 = 5. 2m − 3n = −2 − 15 = −17.
For (a, b) = (−3, −9): m = (−18 + 3)/3 = −5, n = (−6 + 9)/3 = 1. 2m − 3n = −10 − 3 = −13.
For (a, b) = (−9, −3): m = (−6 + 9)/3 = 1, n = (−18 + 3)/3 = −5. 2m − 3n = 2 + 15 = 17.
The remaining factor pairs do not produce integer values of m and n.
Step 5: Find the maximum value. Among the valid values 13, −17, −13 and 17, the maximum is 17.
The sum of digits of the number (625)65 × (128)36 is
Step 1: Express each number as a power of a prime. 625 = 5⁴ and 128 = 2⁷. Therefore, (625)⁶⁵ × (128)³⁶ = (5⁴)⁶⁵ × (2⁷)³⁶ = 5²⁶⁰ × 2²⁵². Step 2: Simplify the expression. Write 5²⁶⁰ as 5²⁵² × 5⁸. So, 5²⁶⁰ × 2²⁵² = (5²⁵² × 2²⁵²) × 5⁸ = 10²⁵² × 5⁸. Step 3: Calculate 5⁸. 5⁸ = 390625. Hence, 10²⁵² × 5⁸ = 390625 × 10²⁵². This is the number 390625 followed by 252 zeros. Step 4: Find the sum of the digits. Sum of the digits of 390625 = 3 + 9 + 0 + 6 + 2 + 5 = 25. The trailing zeros do not affect the digit sum.
The equations 3x² − 5x + p = 0 and 2x² − 2x + q = 0 have one common root. The sum of the other roots of these equations is
8/3 − p + 3/2 q
2/3 − p + 3/2 q
8/3 + p + 1/3 q
2/3 − 2p + 2/3 q
8/3 − p + 3/2 q
Step 1: Let the common root be α. Let the other root of 3x² − 5x + p = 0 be β, and the other root of 2x² − 2x + q = 0 be γ. We have to find β + γ.
Step 2: Use the sum of roots. For 3x² − 5x + p = 0, sum of roots = 5/3. So, α + β = 5/3 → β = 5/3 − α. For 2x² − 2x + q = 0, sum of roots = 1. So, α + γ = 1 → γ = 1 − α. Therefore, β + γ = (5/3 − α) + (1 − α) = 8/3 − 2α.
Step 3: Find the common root α. Since α satisfies both equations: 3α² − 5α + p = 0 and 2α² − 2α + q = 0. Multiply the second equation by 3: 6α² − 6α + 3q = 0. Multiply the first equation by 2: 6α² − 10α + 2p = 0. Subtract: −4α + 2p − 3q = 0 → 4α = 2p − 3q → α = p/2 − 3q/4.
Step 4: Substitute α into β + γ. β + γ = 8/3 − 2α = 8/3 − 2(p/2 − 3q/4) = 8/3 − p + 3q/2.
If log₆₄ x² + log₈ √y + 3 log₅₁₂ (√yz) = 4, where x, y and z are positive real numbers, then the minimum possible value of (x + y + z) is
48
36
24
96
48
Step 1: Convert all logarithms to base 2. Since 64 = 2⁶, 8 = 2³, 512 = 2⁹: log₆₄ x² = (1/6) log₂ x² = (1/3) log₂ x. log₈ √y = (1/3) log₂ √y = (1/6) log₂ y. 3 log₅₁₂ (√yz) = 3 × (1/9) log₂ (√yz) = (1/3) × (1/2) log₂ (yz) = (1/6)(log₂ y + log₂ z). Substituting: (1/3) log₂ x + (1/6) log₂ y + (1/6) log₂ y + (1/6) log₂ z = 4. So, (1/3) log₂ x + (1/3) log₂ y + (1/6) log₂ z = 4. Multiply throughout by 6: 2 log₂ x + 2 log₂ y + log₂ z = 24. Using logarithm properties: log₂ (x²y²z) = 24. Hence, x²y²z = 2²⁴.
Step 2: Apply AM-GM Inequality. Apply AM-GM directly to x², y² and z: (x² + y² + z)/3 ≥ (x²y²z)^(1/3) = 2⁸ = 256. Equality holds when x² = y² = z = 256. Since x and y are positive: x = 16, y = 16, z = 16. Therefore, x + y + z = 16 + 16 + 16 = 48.
Rita and Sneha can row a boat at 5 km/h and 6 km/h in still water, respectively. In a river flowing with a constant velocity, Sneha takes 48 minutes more to row 14 km upstream than to row the same distance downstream. If Rita starts from a certain location in the river, and returns downstream to the same location, taking a total of 100 minutes, then the total distance, in km, Rita will cover is
Step 1: Find the speed of the river. Sneha's speed in still water = 6 km/h. Let the speed of the river be v km/h. Upstream speed = 6 − v. Downstream speed = 6 + v. Given that rowing 14 km upstream takes 48 minutes = 4/5 hour more than rowing 14 km downstream: 14/(6 − v) − 14/(6 + v) = 4/5.
Step 2: Solve for the speed of the river. Taking the LCM: 14[(6 + v) − (6 − v)]/[(6 − v)(6 + v)] = 4/5 → 28v/(36 − v²) = 4/5. Cross-multiplying: 140v = 144 − 4v² → v² + 35v − 36 = 0 → (v + 36)(v − 1) = 0. Since the speed of the river must be positive, v = 1 km/h.
Step 3: Find Rita's upstream and downstream speeds. Rita's speed in still water = 5 km/h. Upstream speed = 5 − 1 = 4 km/h. Downstream speed = 5 + 1 = 6 km/h.
Step 4: Let the one-way distance be d km. The total time for going upstream and returning downstream is 100 minutes = 5/3 hours. So, d/4 + d/6 = 5/3. Taking the LCM: 5d/12 = 5/3 → 5d = 20 → d = 4 km.
Step 5: Find the total distance covered. Rita rows 4 km upstream and 4 km downstream. Total distance = 4 + 4 = 8 km.
Suppose a, b, c are three distinct natural numbers, such that 3ac = 8(a + b). Then, the smallest possible value of 3a + 2b + c is
Step 1: Write the given equation. Given, 3ac = 8(a + b). Rearranging: 8b = 3ac − 8a → b = a(3c − 8)/8. Since b is a natural number, a(3c − 8) must be divisible by 8.
Step 2: Find the smallest possible values. We need to minimize 3a + 2b + c. Try the smallest natural values of c.
Case 1: c = 1. b = −5a/8, which is not a natural number. Not possible.
Case 2: c = 2. b = −a/4, which is not a natural number. Not possible.
Case 3: c = 3. b = a/8. For b to be a natural number, a must be a multiple of 8. Smallest such value is a = 8. Then b = 1. Numbers are distinct: 8, 1, 3. 3a + 2b + c = 3×8 + 2×1 + 3 = 24 + 2 + 3 = 29.
Case 4: c = 4. b = a/2. For b to be a natural number, a must be even. Take a = 2 (smallest even keeping numbers distinct). Then b = 1. Numbers are 2, 1 and 4, all distinct. 3a + 2b + c = 3×2 + 2×1 + 4 = 6 + 2 + 4 = 12.
Step 5: Check whether a smaller value is possible. For c = 1 and c = 2, no natural number solution exists. For c = 3, the minimum value obtained is 29. For c ≥ 5, the value of c itself increases, and the corresponding values of a and b remain positive, making the expression larger than 12. Hence, the smallest possible value is 12.
Let f(x) = x/(2x−1) and g(x) = x/(x−1). Then the domain of the function h(x) = f(g(x)) + g(f(x)) is all real numbers except
−1, 1/2, and 1
1/2, 1, and 3/2
−1/2, 1/2, and 1
1/2, and 1
−1, 1/2, and 1
Step 1: Find the domain of f(x) and g(x). f(x) = x/(2x − 1) is undefined when 2x − 1 = 0 → x = 1/2. Hence, x ≠ 1/2. g(x) = x/(x − 1) is undefined when x − 1 = 0 → x = 1. Hence, x ≠ 1.
Step 2: Find the domain of f(g(x)). For f(g(x)) to exist: (1) g(x) must be defined → x ≠ 1. (2) g(x) ≠ 1/2. Solving g(x) = 1/2: x/(x − 1) = 1/2 → 2x = x − 1 → x = −1. So, x ≠ −1. Therefore, f(g(x)) is defined only when x ≠ 1 and x ≠ −1.
Step 3: Find the domain of g(f(x)). For g(f(x)) to exist: (1) f(x) must be defined → x ≠ 1/2. (2) f(x) ≠ 1. Solving f(x) = 1: x/(2x − 1) = 1 → x = 2x − 1 → x = 1. So, x ≠ 1. Therefore, g(f(x)) is defined only when x ≠ 1/2 and x ≠ 1.
Step 4: Find the domain of h(x). Since h(x) = f(g(x)) + g(f(x)), both expressions must be defined simultaneously. Hence, x cannot be −1, 1/2, or 1.
A loan of Rs 1000 is fully repaid by two installments of Rs 530 and Rs 594, paid at the end of first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is
10
11
9
8
8
Step 1: Let the annual rate of interest be r%. The loan amount is Rs. 1000. At the end of the first year, the amount becomes 1000 × (1 + r/100). After paying the first installment of Rs. 530, the outstanding amount is 1000 × (1 + r/100) − 530.
Step 2: Form the equation after the second year. This outstanding amount earns interest for one more year. Hence, the amount before the second payment is [1000 × (1 + r/100) − 530] × (1 + r/100). Since the loan is fully repaid by paying Rs. 594: [1000 × (1 + r/100) − 530] × (1 + r/100) = 594.
Step 3: Simplify the equation. Let 1 + r/100 = x. Then, (1000x − 530)x = 594 → 1000x² − 530x − 594 = 0. Dividing by 2: 500x² − 265x − 297 = 0.
Step 4: Solve the quadratic equation. Try r = 8%: x = 1.08. Substituting: 1000 × (1.08)² − 530 × (1.08) = 1166.4 − 572.4 = 594. The equation is satisfied. Hence, r = 8%.
Two tangents drawn from a point P and a circle with center O at point Q and R. Point A and B lie on PQ and PR, respectively, such that AB is also a tangent to the same circle. If ∠AOB = 50°, then ∠APB, in degrees equals
Step 1: Identify the quadrilateral formed by the tangents. Since PQ, PR and AB are tangents to the circle, the radius is perpendicular to the tangent at the point of contact. Let the points of contact of the tangents PQ, PR and AB be X, Y and T respectively. Then, OX ⟂ PQ, OY ⟂ PR, OT ⟂ AB.
Step 2: Use the property of tangents from an external point. Since A is an external point, the two tangents from A are AX and AT. Therefore, OA bisects ∠XOT. Similarly, OB bisects ∠YOT. Given ∠AOB = 50°. Hence, ∠XOY = 2 × ∠AOB = 2 × 50° = 100°.
Step 3: Use the angle between two tangents. The angle between two tangents drawn from an external point is supplementary to the central angle subtended by the points of contact. Therefore, ∠APB = 180° − ∠XOY = 180° − 100° = 80°.
The number of divisors of (2⁶ × 3⁵ × 5³ × 7²), which are of the form (3r + 1), where r is a non-negative integer, is
36
56
24
42
42
Step 1: Write the general form of a divisor. The given number is 2⁶ × 3⁵ × 5³ × 7². A divisor is of the form 2ᵃ × 3ᵇ × 5ᶜ × 7ᵈ where 0 ≤ a ≤ 6, 0 ≤ b ≤ 5, 0 ≤ c ≤ 3, 0 ≤ d ≤ 2.
Step 2: Use the condition that the divisor is of the form 3r + 1. A number of the form 3r + 1 leaves remainder 1 when divided by 3. If b ≥ 1, then the divisor is divisible by 3. Hence, b = 0. Now the divisor becomes 2ᵃ × 5ᶜ × 7ᵈ.
Step 3: Find the remainder modulo 3. Modulo 3: 2 ≡ −1, 5 ≡ −1, 7 ≡ 1. Therefore, 2ᵃ × 5ᶜ × 7ᵈ ≡ (−1)ᵃ × (−1)ᶜ × 1ᵈ = (−1)^(a+c). For the remainder to be 1, a + c must be even.
Step 4: Count the valid values of a and c. Possible values of a: 0 to 6. Even values: 0, 2, 4, 6 → 4 choices. Odd values: 1, 3, 5 → 3 choices. Possible values of c: 0 to 3. Even values: 0, 2 → 2 choices. Odd values: 1, 3 → 2 choices. For a + c to be even: both even = 4 × 2 = 8 pairs; both odd = 3 × 2 = 6 pairs. Total valid pairs = 8 + 6 = 14.
Step 5: Choose the value of d. Since 7 ≡ 1 (mod 3), the value of d does not affect the remainder. Possible values of d are 0, 1 and 2, giving 3 choices. Total number of divisors = 14 × 3 = 42.
Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is
6:19
5:24
6:25
7:24
5:24
Step 1: Assign coordinates to the hexagon. Let the side length of the regular hexagon be 2. Coordinates: A = (2, 0), B = (1, √3), C = (−1, √3), D = (−2, 0), E = (−1, −√3), F = (1, −√3). Since P and Q are midpoints: P = (3/2, √3/2), Q = (−3/2, √3/2).
Step 2: Find the lengths of the parallel sides of trapezium PBCQ. PQ = distance between P and Q = 3. BC = 2.
Step 3: Find the height of the trapezium. The y-coordinate of BC is √3. The y-coordinate of PQ is √3/2. Hence, height = √3 − √3/2 = √3/2.
Step 4: Find the area of trapezium PBCQ. Area = (1/2) × (sum of parallel sides) × height = (1/2) × (3 + 2) × (√3/2) = 5√3/4.
Step 5: Find the area of the hexagon. A regular hexagon consists of six equilateral triangles of side 2. Area of one equilateral triangle = (√3/4) × 2² = √3. Therefore, area of the hexagon = 6√3.
Step 6: Find the required ratio. Area of trapezium : Area of hexagon = (5√3/4) : 6√3 = 5 : 24.
If a, b, c and d are integers such that their sum is 46, then the minimum possible value of (a − b)² + (a − c)² + (a − d)² is
Step 1: Write the given condition. We have a + b + c + d = 46. We need to minimize (a − b)² + (a − c)² + (a − d)². Since the expression contains only differences with a, the values of b, c and d should be as close to a as possible.
Step 2: Use the sum condition. Let b = a + x, c = a + y, d = a + z. Then, a + (a + x) + (a + y) + (a + z) = 46 → 4a + (x + y + z) = 46. Since 46 ≡ 2 (mod 4) and 4a is always a multiple of 4, x + y + z ≡ 2 (mod 4).
Step 3: Rewrite the expression. The required expression becomes x² + y² + z². So, we need to minimize x² + y² + z² subject to x + y + z ≡ 2 (mod 4).
Step 4: Find the minimum possible value. The smallest possible values satisfying x + y + z = 2 are 1, 1, 0 (or any permutation). Then x² + y² + z² = 1² + 1² + 0² = 2. This is achievable. For example, a = 11, b = 12, c = 12, d = 11. Their sum = 11 + 12 + 12 + 11 = 46 and (a − b)² + (a − c)² + (a − d)² = (−1)² + (−1)² + 0² = 2. No smaller value is possible because the sum of three integer squares is non-negative, and a value of 1 cannot satisfy the required congruence condition.
The ratio of expenditures of Lakshmi and Meenakshi is 2 : 3, and the ratio of income of Lakshmi to expenditure of Meenakshi is 6 : 7. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is 4 : 9, then the ratio of their incomes is
3:5
5:6
2:1
7:8
3:5
Step 1: Assume the expenditures. Let the expenditures of Lakshmi and Meenakshi be 2x and 3x, respectively.
Step 2: Find Lakshmi's income. Given, Income of Lakshmi : Expenditure of Meenakshi = 6 : 7. So, Lakshmi's income = (6/7) × 3x = 18x/7.
Step 3: Find Lakshmi's savings. Lakshmi's savings = Income − Expenditure = 18x/7 − 2x = 18x/7 − 14x/7 = 4x/7.
Step 4: Find Meenakshi's savings. Let Meenakshi's income be M. Then, Meenakshi's savings = M − 3x. Given, Lakshmi's savings : Meenakshi's savings = 4 : 9. So, (4x/7) : (M − 3x) = 4 : 9. Cancelling 4: x/7 : (M − 3x) = 1 : 9. Therefore, M − 3x = 9x/7 → M = 3x + 9x/7 = 21x/7 + 9x/7 = 30x/7.
Step 5: Find the required ratio. Lakshmi's income : Meenakshi's income = 18x/7 : 30x/7 = 18 : 30 = 3 : 5.
Let aₙ be the nᵗʰ term of a decreasing infinite geometric progression. If a₁ + a₂ + a₃ = 52 and a₁a₂ + a₂a₃ + a₃a₁ = 624, then the sum of this infinite geometric progression is
57
54
60
63
54
Step 1: Let the first term and common ratio be a and r. Then a₁ = a, a₂ = ar, a₃ = ar². Since the GP is decreasing and infinite, 0 < r < 1.
Step 2: Form the given equations. From the first condition: a + ar + ar² = 52 → a(1 + r + r²) = 52. Therefore, a = 52/(1 + r + r²). Also, a₁a₂ + a₂a₃ + a₃a₁ = 624 → a²r + a²r² + a²r³ = 624 → a²(r + r² + r³) = 624.
Step 3: Substitute the value of a. [52²/(1 + r + r²)²] × (r + r² + r³) = 624. Since r + r² + r³ = r(1 + r + r²): 52²r/(1 + r + r²) = 624. Since 52² = 2704: 2704r/(1 + r + r²) = 624. Dividing both sides by 208: 13r/(1 + r + r²) = 3.
Step 4: Solve for r. 13r = 3(1 + r + r²) → 13r = 3 + 3r + 3r² → 3r² − 10r + 3 = 0 → (3r − 1)(r − 3) = 0. So r = 1/3 or r = 3. Since the GP is decreasing, r = 1/3.
Step 5: Find the first term. a = 52/(1 + 1/3 + 1/9) = 52/(13/9) = 36.
Step 6: Find the sum of the infinite GP. Sum = a/(1 − r) = 36/(1 − 1/3) = 36/(2/3) = 54.
A mixture of coffee and cocoa, 16% of which is coffee, costs Rs 240 per kg. Another mixture of coffee and cocoa, of which 36% is coffee, costs Rs 320 per kg. If a new mixture of coffee and cocoa costs Rs 376 per kg, then the quantity, in kg, of coffee in 10 kg of this new mixture is
5
4
2.5
6
5
Step 1: Let the price of pure coffee be C Rs/kg and pure cocoa be K Rs/kg. From the first mixture: 0.16C + 0.84K = 240 → 16C + 84K = 24000 → 4C + 21K = 6000. From the second mixture: 0.36C + 0.64K = 320 → 36C + 64K = 32000 → 9C + 16K = 8000.
Step 2: Find the prices of pure coffee and cocoa. Multiply first equation by 9: 36C + 189K = 54000. Multiply second equation by 4: 36C + 64K = 32000. Subtract: 125K = 22000 → K = 176. Substitute into 9C + 16K = 8000: 9C + 2816 = 8000 → 9C = 5184 → C = 576.
Step 3: Let the percentage of coffee in the new mixture be x. Since the new mixture costs Rs. 376 per kg: 576x + 176(1 − x) = 376 → 576x + 176 − 176x = 376 → 400x = 200 → x = 0.5. Thus, the new mixture contains 50% coffee.
Step 4: Find the quantity of coffee in 10 kg. Coffee = 50% of 10 kg = 5 kg.
In △ABC, points D and E are on the sides BC and AC, respectively. BE and AD intersect at point T such that AD:AT = 4:3, and BE:BT = 5:4. Point F lies on AC such that DF is parallel to BE. Then, BD:CD is
15:4
11:4
7:4
9:4
11:4
Step 1: Use the given ratios on the cevians. Since AD : AT = 4 : 3, we have AT : TD = 3 : 1. By the Mass Points Theorem, Mass at D : Mass at A = 3 : 1. Let the mass at A be x. Then, Mass at D = 3x. Similarly, BE : BT = 5 : 4, so BT : TE = 4 : 1. By the Mass Points Theorem, Mass at E : Mass at B = 4 : 1. Let the mass at B be y. Then, Mass at E = 4y.
Step 2: Express the masses at D and E. Since D lies on BC: Mass at D = Mass at B + Mass at C → 3x = y + Mass at C → Mass at C = 3x − y. Since E lies on AC: Mass at E = Mass at A + Mass at C → 4y = x + Mass at
C. Substituting Mass at C = 3x − y: 4y = x + 3x − y → 5y = 4x → y = 4x/5.
Step 3: Find the mass at
C. Mass at C = 3x − 4x/5 = 11x/5. Thus, Mass at B : Mass at C = 4x/5 : 11x/5 = 4 : 11.
Step 4: Find BD : CD. A point on a line divides the segment in the inverse ratio of the masses at the endpoints. Therefore, BD : CD = Mass at C : Mass at B = 11 : 4.
Ankita is twice as efficient as Bipin, while Bipin is twice as efficient as Chandan. All three of them start together on a job, and Bipin leaves the job after 20 days. If the job got completed in 60 days, the number of days needed by Chandan to complete the job alone, is
Step 1: Assume Chandan's work rate. Let Chandan's work rate be 1 unit/day. Then, Bipin's work rate = 2 units/day and Ankita's work rate = 4 units/day. Together they work at 4 + 2 + 1 = 7 units/day.
Step 2: Work completed in the first 20 days. All three work together for 20 days. Work done = 20 × 7 = 140 units.
Step 3: Work completed in the remaining 40 days. After 20 days, Bipin leaves. Only Ankita and Chandan continue. Combined work rate = 4 + 1 = 5 units/day. They work for another 40 days. Work done = 40 × 5 = 200 units.
Step 4: Find the total work. Total work = 140 + 200 = 340 units. Since Chandan alone does 1 unit/day, he will complete 340 units in 340 days.
A certain amount of money was divided among Pinu, Meena, Rinu and Seema. Pinu received 20% of the total amount and Meena received 40% of the remaining amount. If Seema received 20% less than Pinu, the ratio of the amounts received by Pinu and Rinu is
2:1
1:2
5:8
8:5
5:8
Step 1: Assume the total amount. Let the total amount be 100. Pinu receives 20% of 100 = 20. Remaining amount = 100 − 20 = 80.
Step 2: Find Meena's share. Meena receives 40% of the remaining 80 = (40/100) × 80 = 32. Amount left for Rinu and Seema = 80 − 32 = 48.
Step 3: Find Seema's share. Seema receives 20% less than Pinu. Pinu's share = 20. So, Seema's share = 20 − 20% of 20 = 20 − 4 = 16.
Step 4: Find Rinu's share. Rinu's share = 48 − 16 = 32.
Step 5: Find the required ratio. Pinu : Rinu = 20 : 32 = 5 : 8.
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