CAT 2024 — Slot 2
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VARC topics covered: Education & Knowledge RC, History & Civilization RC, Nature And Environment RC, Odd One Out, Para Completion, Para Summary, Science And Technology RC
The history of any major technological or industrial advance is inevitably shadowed by a less predictable history of unintended consequences and secondary effects — what economists sometimes call "externalities." Sometimes those consequences are innocuous ones, or even beneficial. Gutenberg invents the printing press, and literacy rates rise, which causes a significant part of the reading public to require spectacles for the first time, which creates a surge of investment in lens-making across Europe, which leads to the invention of the telescope and the microscope.
Oftentimes the secondary effects seem to belong to an entirely different sphere of society. When Willis Carrier hit upon the idea of air-conditioning, the technology was primarily intended for industrial use: ensuring cool, dry air for factories that required low-humidity environments. But…it touched off one of the largest migrations in the history of the United States, enabling the rise of metropolitan areas like Phoenix and Las Vegas that barely existed when Carrier first started tinkering with the idea in the early 1900s.
Sometimes the unintended consequence comes about when consumers use an invention in a surprising way. Edison famously thought his phonograph, which he sometimes called "the talking machine," would primarily be used to take dictation….But then later innovators…discovered a much larger audience willing to pay for musical recordings made on descendants of Edison's original invention. In other cases, the original innovation comes into the world disguised as a plaything…the way the animatronic dolls of the mid-1700s inspired Jacquard to invent the first "programmable" loom and Charles Babbage to invent the first machine that fit the modern definition of a computer, setting the stage for the revolution in programmable technology that would transform the 21st century in countless ways.
We live under the gathering storm of modern history's most momentous unintended consequence….carbon-based climate change. Imagine the vast sweep of inventors whose ideas started the Industrial Revolution, all the entrepreneurs and scientists and hobbyists who had a hand in bringing it about. Line up a thousand of them and ask them all what they had been hoping to do with their work. Not one would say that their intent had been to deposit enough carbon in the atmosphere to create a greenhouse effect that trapped heat at the surface of the planet. And yet here we are.
Ethyl (leaded fuel) and Freon belonged to the same general class of secondary effect: innovations whose unintended consequences stem from some kind of waste by-product that they emit. But the potential health threats of Ethyl (unleaded fuel) were visible in the 1920s, unlike, say, the long-term effects of atmospheric carbon build up in the early days of the Industrial Revolution….
Indeed, it is reasonable to see CFCs (chlorofluorocarbons) as a forerunner of the kind of threat we will most likely face in the coming decades, as it becomes increasingly possible for individuals or small groups to create new scientific advances — through chemistry or biotechnology or materials science — setting off unintended consequences that reverberate on a global scale.
The author lists all of the following examples as "externalities" of major technical advances EXCEPT:
build-up of chlorofluorocarbons in the atmosphere
cooling and de-humidifying of factories through air-conditioning
application of the Jacquard loom to modern IT programming
extension of the phonograph to large-scale recording of music
cooling and de-humidifying of factories through air-conditioning
The passage identifies externalities as unintended consequences or secondary effects of technological advances. Air-conditioning's cooling and de-humidifying of factories was the original intended use of the technology, not an unintended consequence. All the other options describe secondary, unintended effects of inventions.
Which of the following best conveys the main point of the first paragraph?
The secondary effects of most major technological advances in the past, especially if they were unintended, have turned out to be beneficial.
The full impact of technological advances cannot be estimated in the short run as the ripple effects often extend far beyond the original intent.
It is important to judge an invention not by its immediate outcomes, but by the holistic impact of its secondary effects.
The entire impact of a technological advance should be evaluated by the boost its secondary effects gives to generating further technological advances.
The full impact of technological advances cannot be estimated in the short run as the ripple effects often extend far beyond the original intent.
The first paragraph uses the Gutenberg printing press example to show how one technological advance triggered a chain of consequences, literacy requiring spectacles, which drove investment in lenses, which led to telescopes and microscopes, that were far removed from the original intent and difficult to predict in advance.
Carrier, Babbage, and Edison are mentioned in the passage to illustrate the author's point that
the secondary effect of past inventions mostly resulted in the creation of new inventions.
these inventors could not have visualised the eventual impact of their inventions on society.
despite the original intention, the unintended consequences of their inventions were largely beneficial.
inventions typically end up being used for entirely different purposes than the intended ones.
these inventors could not have visualised the eventual impact of their inventions on society.
The passage mentions Carrier, Babbage and Edison specifically to show that their inventions had consequences and applications that went well beyond what they had originally envisioned, Carrier's air-conditioning transforming migration patterns, Edison's phonograph becoming a music medium, and Babbage's work inspired by toys setting the stage for programmable technology.
We can assume that the author would support all of the following views EXCEPT:
While technological advances in the past have had innocuous or beneficial outcomes, more recent advances have the potential to be more threatening globally.
The by-products of leaded fuel, rather than the fuel itself, were responsible for the build-up of carbon-related gases in the atmosphere.
It has become far easier for people today to bring out innovations with dire worldwide consequences than it was earlier.
The emissions caused by the large-scale use of leaded fuel ought to have been addressed earlier than they were.
While technological advances in the past have had innocuous or beneficial outcomes, more recent advances have the potential to be more threatening globally.
The passage explicitly states that the potential health threats of leaded fuel were visible as early as the 1920s, unlike the long-term effects of atmospheric carbon during the early Industrial Revolution. The passage's overall arc moves from innocuous or beneficial externalities in the past to increasingly threatening ones as technology becomes more accessible. The author does not claim that past advances had innocuous outcomes while more recent ones are threatening, as a blanket statement.
The job of a peer reviewer is thankless. Collectively, academics spend around 70 million hours every year evaluating each other's manuscripts on the behalf of scholarly journals — and they usually receive no monetary compensation and little if any recognition for their effort. Some do it as a way to keep abreast with developments in their field; some simply see it as a duty to the discipline. Either way, academic publishing would likely crumble without them.
In recent years, some scientists have begun posting their reviews online, mainly to claim credit for their work. Sites like Publons allow researchers to either share entire referee reports or simply list the journals for whom they've carried out a review…. The rise of Publons suggests that academics are increasingly placing value on the work of peer review and asking others, such as grant funders, to do the same. While that's vital in the publish-or-perish culture of academia, there's also immense value in the data underlying peer review. Sharing peer review data could help journals stamp out fraud, inefficiency, and systemic bias in academic publishing.…..Peer review data could also help root out bias. Last year, a study based on peer review data for nearly 24,000 submissions to the biomedical journal eLife found that women and non-Westerners were vastly underrepresented among peer reviewers. Only around one in every five reviewers was female, and less than two percent of reviewers were based in developing countries…. Openly publishing peer review data could perhaps also help journals address another problem in academic publishing: fraudulent peer reviews. For instance, a minority of authors have been known to use phony email addresses to pose as an outside expert and review their own manuscripts.…
Opponents of open peer review commonly argue that confidentiality is vital to the integrity of the review process; referees may be less critical of manuscripts if their reports are published, especially if they are revealing their identities by signing them. Some also hold concerns that open reviewing may deter referees from agreeing to judge manuscripts in the first place, or that they'll take longer to do so out of fear of scrutiny….
Even when the content of reviews and the identity of reviewers can't be shared publicly, perhaps journals could share the data with outside researchers for study. Or they could release other figures that wouldn't compromise the anonymity of reviews but that might answer important questions about how long the reviewing process takes, how many researchers editors have to reach out to on average to find one who will carry out the work, and the geographic distribution of peer reviewers.
Of course, opening up data underlying the reviewing process will not fix peer review entirely, and there may be instances in which there are valid reasons to keep the content of peer reviews hidden and the identity of the referees confidential. But the norm should shift from opacity in all cases to opacity only when necessary.
According to the passage, which of the following is the only reason NOT given in favour of making peer review data public?
It will deal with peer review fraud such as authors publishing bogus reviews of their work.
It would highlight the gender and race biases currently existing in the selection of reviewers.
It could address various inefficiencies and fraudulent practices that continue in academic publishing process.
It can tackle the problem of selecting appropriately qualified reviewers for academic writing.
It can tackle the problem of selecting appropriately qualified reviewers for academic writing.
The passage gives three reasons for making peer review data public, tackling fraud such as phony reviews, addressing gender and geographic bias among reviewers, and improving efficiency in the publishing process. The passage never mentions selecting appropriately qualified reviewers as a benefit of open peer review data.
All of the following are listed as reasons why academics choose to review other scholars' work EXCEPT:
It helps them keep current with cutting-edge ideas in their academic disciplines.
Some use this as an opportunity to publicise their own review work.
It is seen as a form of service to the academic community.
It is seen as an opportunity to expand their influence in the academic community.
It is seen as an opportunity to expand their influence in the academic community.
The passage gives two reasons academics choose to review others' work, keeping abreast with developments in their field, and seeing it as a duty to the discipline. It also mentions that some post reviews online to claim credit for their work, which corresponds to option
B. The passage never mentions expanding influence in the academic community as a motivation for reviewing.
Based on the passage we can infer that the author would most probably support
more careful screening to ensure the recruitment of content-familiar peer reviewers.
preserving the anonymity of reviewers to protect them from criticism.
publicising peer review data rather than the publication of actual reviews.
greater transparency across the peer review process in academic publishing.
greater transparency across the peer review process in academic publishing.
The passage argues that the norm should shift from opacity in all cases to opacity only when necessary, advocating for greater transparency across the peer review process while acknowledging limited cases where confidentiality is warranted.
According to the passage, some are opposed to making peer reviews public for all the following reasons EXCEPT that it
makes reviewers reluctant to review manuscripts, especially if these are critical of the submitted work.
leaves the reviewers unexposed to unwarranted and unjustified criticism or comments from others.
deters reviewers from producing honest, if critical, reviews that are vital to the sound publishing process.
delays the manuscript evaluation process as reviewers would take longer to write their reviews.
leaves the reviewers unexposed to unwarranted and unjustified criticism or comments from others.
The passage lists three arguments opponents make against open peer review, that reviewers may be less critical if their reports are published, that open reviewing may deter reviewers from agreeing to participate, and that reviewers may take longer out of fear of scrutiny. The passage never says opponents argue that open review exposes reviewers to unjustified criticism, quite the opposite, the concern raised is about reviewers being more guarded because their own reviews become public.
[S]pices were a global commodity centuries before European voyages. There was a complex chain of relations, yet consumers had little knowledge of producers and vice versa. Desire for spices helped fuel European colonial empires to create political, military and commercial networks under a single power.
Historians know a fair amount about the supply of spices in Europe during the medieval period - the origins, methods of transportation, the prices - but less about demand. Why go to such extraordinary efforts to procure expensive products from exotic lands? Still, demand was great enough to inspire the voyages of Christopher Columbus and Vasco Da Gama, launching the first fateful wave of European colonialism. . . .
So, why were spices so highly prized in Europe in the centuries from about 1000 to 1500? One widely disseminated explanation for medieval demand for spices was that they covered the taste of spoiled meat. . . . Medieval purchasers consumed meat much fresher than what the average city-dweller in the developed world of today has at hand. However, refrigeration was not available, and some hot spices have been shown to serve as an anti-bacterial agent. Salting, smoking or drying meat were other means of preservation. Most spices used in cooking began as medical ingredients, and throughout the Middle Ages spices were used as both medicines and condiments. Above all, medieval recipes involve the combination of medical and culinary lore in order to balance food's humeral properties and prevent disease. Most spices were hot and dry and so appropriate in sauces to counteract the moist and wet properties supposedly possessed by most meat and fish. . . .
Where spices came from was known in a vague sense centuries before the voyages of Columbus. Just how vague may be judged by looking at medieval world maps . . . To the medieval European imagination, the East was exotic and alluring. Medieval maps often placed India close to the so-called Earthly Paradise, the Garden of Eden described in the Bible.
Geographical knowledge has a lot to do with the perceptions of spices' relative scarcity and the reasons for their high prices. An example of the varying notions of scarcity is the conflicting information about how pepper is harvested. As far back as the 7th century Europeans thought that pepper in India grew on trees "guarded" by serpents that would bite and poison anyone who attempted to gather the fruit. The only way to harvest pepper was to burn the trees, which would drive the snakes underground. Of course, this bit of lore would explain the shriveled black peppercorns, but not white, pink or other colors.
Spices never had the enduring allure or power of gold and silver or the commercial potential of new products such as tobacco, indigo or sugar. But the taste for spices did continue for a while beyond the Middle Ages. As late as the 17th century, the English and the Dutch were struggling for control of the Spice Islands: Dutch New Amsterdam, or New York, was exchanged by the British for one of the Moluccan Islands where nutmeg was grown.
It can be inferred that all of the following contributed to a decline in the allure of spices, EXCEPT:
the development of refrigeration techniques.
increase in the availability of spices.
changes in the system of medical treatment.
changes in European cuisine.
increase in the availability of spices.
The passage discusses how refrigeration eliminated the need for spices as anti-bacterial agents, how spices were primarily used in combination with medical knowledge about humoral properties, and how taste and culinary practices changed. An increase in availability of spices would lower their price but does not directly explain why the allure or medical and culinary need for spices declined.
In the context of the passage, the people who heard the story of pepper trees being guarded by snakes would be least likely to arrive at the conclusion that
this is why pepper is so hot.
pepper is costly for good reason.
it is not advisable to go to India to harvest the pepper themselves.
it is no surprise that the pepper supply is so limited.
this is why pepper is so hot.
The story about pepper trees guarded by snakes and burned to harvest peppercorns explains why pepper is scarce and costly and why going to India to harvest it yourself is dangerous, but it says nothing about why pepper is hot. The hotness of pepper is a physical property entirely unrelated to the legend of snakes and burning trees.
In the context of the passage, which one of the following conclusions CANNOT be reached?
The spice trade was a driver of colonial expansion.
India was colonised for its spices and gold.
Tobacco was more marketable than spices.
Colonialism was motivated by the demand for spices.
India was colonised for its spices and gold.
The passage discusses how the spice trade helped fuel European colonialism and inspired voyages of exploration. However, the passage never specifically states that India was colonised for its spices and gold. The passage mentions spices in relation to India but does not discuss gold in the context of Indian colonisation specifically, nor does it state that India was colonised for these two reasons together.
If a trader brought white peppercorns from India to medieval Europe, all of the following are unlikely to happen, EXCEPT:
medieval maps would be used as navigational aids.
Europeans would doubt the story of pepper harvesting.
the price of spices would decrease.
pepper would no longer be considered exotic.
Europeans would doubt the story of pepper harvesting.
The passage explains that medieval Europeans believed peppercorns were shriveled and black because the trees were burned to drive away snakes, which they thought was the only way to harvest pepper. White peppercorns would contradict this story since burning would not produce white peppercorns, so Europeans encountering white peppercorns would have reason to doubt the burning-snake narrative.
(. . .) There are three other common drivers for carnivore-human attacks, some of which are more preventable than others. Natural aggression-based conflicts - such as those involving females protecting their young or animals protecting a food source - can often be avoided as long as people stay away from those animals and their food.
Carnivores that recognise humans as a means to get food, are a different story. As they become more reliant on human food they might find at campsites or in rubbish bins, they become less avoidant of humans. Losing that instinctive fear response puts them into more situations where they could get into an altercation with a human, which often results in that bear being put down by humans. "A fed bear is a dead bear," says Servheen, referring to a common saying among biologists and conservationists. Predatory or predation-related attacks are quite rare, only accounting for 17% of attacks in North America since 1955. They occur when a carnivore views a human as prey and hunts it like it would any other animal it uses for food. (. . .)
Then there are animal attacks provoked by people taking pictures with them or feeding them in natural settings such as national parks which often end with animals being euthanised out of precaution. "Eventually, that animal becomes habituated to people, and [then] bad things happen to the animal. And the folks who initially wanted to make that connection don't necessarily realise that," says Christine Wilkinson, a postdoctoral researcher at UC Berkeley, California, who's been studying coyote-human conflicts.
After conducting countless postmortems on all types of carnivore-human attacks spanning 75 years, Penteriani's team believes 50% could have been avoided if humans reacted differently. A 2017 study co-authored by Penteriani found that engaging in risky behaviour around large carnivores increases the likelihood of an attack.
Two of the most common risky behaviours are parents leaving their children to play outside unattended and walking an unleashed dog, according to the study. Wilkinson says 66% of coyote attacks involve a dog. "[People] end up in a situation where their dog is being chased, or their dog chases a coyote, or maybe they're walking their dog near a den that's marked, and the coyote wants to escort them away," says Wilkinson.
Experts believe climate change also plays a part in the escalation of human-carnivore conflicts, but the correlation still needs to be ironed out. "As finite resources become scarcer, carnivores and people are coming into more frequent contact, which means that more conflict could occur," says Jen Miller, international programme specialist for the US Fish & Wildlife Service. For example, she says, there was an uptick in lion attacks in western India during a drought when lions and people were relying on the same water sources.
(. . .) The likelihood of human-carnivore conflicts appears to be higher in areas of low-income countries dominated by vast rural landscapes and farmland, according to Penteriani's research. "There are a lot of working landscapes in the Global South that are really heterogeneous, that are interspersed with carnivore habitats, forests and savannahs, which creates a lot more opportunity for these encounters, just statistically," says Wilkinson.
According to the passage, what is a significant factor that contributes to the habituation of carnivores to human presence?
The natural aggression exhibited by carnivores, exacerbated by human interference, particularly when they are safeguarding their offspring or food sources.
The increased scarcity of resources due to climate change, forcing carnivores to venture outside their natural habitats in search of sustenance.
The predatory perception of humans as potential prey within the carnivores' food chain.
The reduction in carnivores' instinctive fear response, resulting from their reliance upon human-provided food.
The natural aggression exhibited by carnivores, exacerbated by human interference, particularly when they are safeguarding their offspring or food sources.
The passage explicitly explains that carnivores that become reliant on human food lose their instinctive fear response, which leads to more situations where altercations with humans occur. This reduction in the fear response, driven by food habituation, is the mechanism through which carnivores become habituated to human presence.
Given the insights provided by Penteriani's research and Wilkinson's statement, which of the following conclusions can be drawn about the relationship between landscape heterogeneity and human-carnivore conflicts?
Low-income countries with vast, contiguous wilderness areas are less prone to human-carnivore conflicts because these areas lack the human presence necessary for such encounters.
Landscape heterogeneity, characterized by a mix of farmland and natural habitats, inherently reduces the chances of human-carnivore conflicts by providing more refuge for wildlife away from human activity.
Homogeneous landscapes with uniform agricultural practices are more likely to experience high rates of human-carnivore conflicts due to the predictability of resources.
The diversity and interspersion of working landscapes with carnivore habitats in rural areas increase the statistical probability of encounters between humans and carnivores.
Low-income countries with vast, contiguous wilderness areas are less prone to human-carnivore conflicts because these areas lack the human presence necessary for such encounters.
Wilkinson explicitly states that working landscapes in the Global South are interspersed with carnivore habitats, forests and savannahs, and that this creates more statistical opportunities for encounters between humans and carnivores.
Which of the following statements, if false, would be inconsistent with the concerns raised in the passage regarding the drivers of carnivore-human conflicts?
Climate change has had negligible effects on the frequency of carnivore-human interactions in affected regions.
Predatory attacks by carnivores are a common occurrence and have steadily increased over the past few decades.
Carnivores lose their instinctive fear of humans, when consistently exposed to human food sources
Human efforts to avoid risky behaviours around large carnivores have proven effective in reducing conflict incidents.
Climate change has had negligible effects on the frequency of carnivore-human interactions in affected regions.
The passage states that experts believe climate change plays a part in escalating human-carnivore conflicts, citing the example of increased lion attacks during drought in western India. If option A is true, meaning climate change has had negligible effects, this would directly contradict the passage's stated concern about climate change as a driver of conflict, making it inconsistent.
According to the passage, which of the following scenarios would MOST likely exacerbate the frequency of carnivore-human conflicts?
Implementing 'food waste' management strategies to prevent wild animals being attracted to human food sources.
Addressing the impact of climate change on the availability of resources for wildlife.
Attempting to photograph wild animals from within secured viewing areas in national parks and protected zones.
Unleashing dogs by pet owners in areas with known high concentrations of large carnivores.
Unleashing dogs by pet owners in areas with known high concentrations of large carnivores.
The passage explicitly states that walking an unleashed dog is one of the two most common risky behaviours around large carnivores, citing Wilkinson's finding that 66% of coyote attacks involve a dog. Unleashing dogs in areas with high concentrations of large carnivores would directly increase the likelihood of conflict.
There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.
Sentence: Science has officially crowned us superior to our early-rising brethren.
Paragraph: My fellow night owls, grab a strong cup of coffee and gather around: I have great news. (1). For a long time, our kind has been unfairly maligned. Stereotyped as lazy and undisciplined. Told we ought to be morning larks. Advised to go to bed early so we can wake before 5am and run a marathon before breakfast like all high-flyers seem to do. Now, however, we are having the last laugh. (2). It may be a tad more complicated than that. A study published last week, which you may have already seen while scrolling at 1am, suggests that staying up late could be good for brain power. (3). Is this study a thinly veiled PR exercise conducted by a caffeine-pill company? Nope, it's legit. (4). Research led by academics at Imperial College London studied data on more than 26,000 people and found that "self-declared 'night owls' generally tend to have higher cognitive scores".
Option 4
Option 3
Option 1
Option 2
Option 2
The inserted sentence announces that science has officially crowned night owls as superior. Option 2 falls right after the paragraph establishes that night owls are "having the last laugh," and right before the caveat that "it may be a tad more complicated than that." The inserted sentence fits perfectly here as the triumphant claim that the next sentence then immediately qualifies and complicates.
Five jumbled up sentences (labelled 1, 2, 3, 4 and 5), related to a topic, are given below. Four of them can be put together to form a coherent paragraph. Identify the odd sentence and key in the number of that sentence as your answer.
1. No known real researcher of human behaviour would say that gender is all nature or all nurture.
2. The evidence for a biological basis for gender certainly doesn't mean we should be complacent in the face of sexism.
3. Many people are uncomfortable with the idea that gender is not purely a social construct.
4. Despite this empirical truth, researchers who study the biological basis of gender often face political pushback.
5. There's a political preference for gender to be only a reflection of social factors and so entirely malleable.
Sentences 1, 3, 4 and 5 form a coherent argument, starting with the empirical fact that no serious researcher considers gender purely nature or nurture, then noting that many people are uncomfortable with the idea that it is not purely social, explaining that researchers face political pushback despite this empirical truth, and finally stating the political preference for gender to be entirely socially malleable as the source of that resistance. Sentence 2 shifts to a normative ethical point about not being complacent in the face of sexism, which is a separate argument about social consequence rather than a continuation of the empirical and political thread the other four sentences build.
There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.
Sentence: [T]he Europeans did not invent globalization.
Paragraph: The first phase of globalization occurred long before the introduction of either steam or electric power…Chinese consumers at all social levels consumed vast quantities of spices, fragrant woods and unusual plants. The peoples of Southeast Asia who lived in forests gave up their traditional livelihoods and completely reoriented their economies to supply Chinese consumers….(1). These exchanges of the year 1000 opened some of the routes through which goods and peoples continued to travel after Columbus traversed the mid-Atlantic. (2). Yet the world of 1000 differed from that of 1492 in important ways….the travellers who encountered one another in the year 1000 were much closer technologically. (3). They changed and augmented what was already there since 1000. (4). If globalization hadn't yet begun, Europeans wouldn't have been able to penetrate the markets in so many places as quickly as they did after 1492.
Option 1
Option 4
Option 2
Option 3
Option 3
The inserted sentence states that the Europeans did not invent globalisation. Option 3 falls right before the sentence "They changed and augmented what was already there since 1000," where "they" refers to the Europeans. The inserted sentence establishes precisely what the Europeans did not do, which the very next sentence then clarifies, that they built on what already existed, making Option 3 the natural place for this claim.
The passage given below is followed by four alternate summaries. Choose the option that best captures the essence of the passage.
Different from individuals, states conduct warfare operations using the DIME model— "diplomacy, information, military, and economics." Most states do everything they can to inflict pain and confusion on their enemies before deploying the military. In fact, attacks on vectors of information are a well-worn tactic of war and usually are the first target when the charge begins. It's common for telecom data and communications networks to be routinely monitored by governments, which is why the open data policies of the web are so concerning to many advocates of privacy and human rights. With the worldwide adoption of social media, more governments are getting involved in low-grade information warfare through the use of cyber troops. According to a study by the Oxford Internet Institute in 2020, cyber troops are "government or political party actors tasked with manipulating public opinion online." The Oxford research group was able to identify 81 countries with active cyber troop operations, utilizing many different strategies to spread false information, including spending millions on online advertising.
Following the DIME model, many governments have taken advantage of open data policies of the web to deploy cyber troops who manipulate domestic public opinion, using advertising and other strategies to spread false information.
Governments primarily use the DIME model to deploy cyber troops who practise low-grade information warfare, seeking to manipulate public opinion with the objective of inflicting pain and confusion on their enemies.
Using the DIME model, together with military operations, many governments simultaneously conduct information warfare with the help of cyber troops and routinely monitor telecom data and communications networks.
As part of conducting information warfare as per the DIME model, many governments routinely monitor telecom data and communications networks, and use cyber troops on social media to manipulate public opinion.
As part of conducting information warfare as per the DIME model, many governments routinely monitor telecom data and communications networks, and use cyber troops on social media to manipulate public opinion.
The passage explains the DIME model, notes that information is typically the first target in warfare, mentions governments monitoring telecom networks, introduces social media enabling cyber troops, and closes with the Oxford study's finding about 81 countries using cyber troops to spread false information. The correct summary must capture the information warfare thread through telecom monitoring and cyber troops without distorting the scope.
The passage given below is followed by four alternate summaries. Choose the option that best captures the essence of the passage.
John Cleese told Fox News Digital that comedians do not have the freedom to be funny in 2022. "There's always been limitations on what they're allowed to say," Cleese said. "I think it's particularly worrying at the moment because you can only create in an atmosphere of freedom, where you're not checking everything you say critically before you move on. What you have to be able to do is to build without knowing where you're going because you've never been there before. That's what creativity is — you have to be allowed to build. And a lot of comedians now are sitting there and when they think of something, they say something like, 'Can I get away with it? I don't think so. So and so got into trouble, and he said that, oh, she said that.' You see what I mean? And that's the death of creativity."
Comedians must not check what they think and say. They must go where no one has gone before.
Creativity and critical thinking cannot work together. Comedians must first be creative, and later be critical.
Comedians are being prevented from saying what they want and that is the death of this art form.
Freedom and creativity are essential for comedy. Fear about offending people hinders originality.
Freedom and creativity are essential for comedy. Fear about offending people hinders originality.
Cleese's argument is that creative work requires freedom from self-censorship, that the moment comedians start filtering their ideas by asking whether they can get away with something, creativity is killed. The passage is not about restrictions on what comedians are allowed to say in a legal sense, but about how fear of offending others stifles originality.
There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.
Sentence: Yet each day the flock produced eggs with calcareous shells though they apparently had not ingested any calcium from land which was entirely lacking in limestone.
Paragraph: Early in this century a young Breton schoolboy who preparing himself for a scientific career began to notice a strange fact about hens in his father's poultry yard. ___(1) ___. As they scratched the soil they constantly seemed to be pecking at specks of mica, a siliceous material dotting the ground. (2). No one could explain to Louis Kervran why the chickens selected the mica, or why each time a bird was killed for the family cooking pot no trace of the mica could be found in its gizzard. ___(3) ___. It took Kervran many years to establish that the chickens were transmuting one element into another. (4).
Option 1
Option 3
Option 2
Option 4
Option 3
The inserted sentence describes the mystery, the hens producing calcium-shelled eggs despite no calcium in the soil. This fits at Option 3, right after the paragraph has introduced that no one could explain why mica disappeared from the gizzard, and right before the statement that it took Kervran many years to establish the element transmutation explanation. The inserted sentence adds another unexplained mystery, the calcium paradox, which deepens the puzzle that Kervran eventually solved.
The passage given below is followed by four alternate summaries. Choose the option that best captures the essence of the passage.
Recent important scientific findings have emerged from crossing the boundaries of scientific fields. They stem from physicists collaborating with biologists, sociologists and others, to answer questions about our world. But physicists and their potential collaborators often find their cultures out of sync. For one, physicists often discard a lot of information while extracting broad patterns; for other scientists, information is not readily disposed. Further, many non-physicists are uncomfortable with mathematical models. Still, the desire to work on something new and different is real, and there are clear benefits from the collision of views.
Despite differences in their research styles, physicists' research collaborations with scholars from other disciplines have yielded important research findings.
Large data sets and mathematical models in physics research combined with the research methods of non-physicist collaborators have yielded important scientific findings.
The desire to diversify their research and answer important questions has led to several collaborations between physicists and other social scientists.
Physicists have successfully buried their differences on research methods applied in other fields in their desire to find answers to baffling scientific questions.
Despite differences in their research styles, physicists' research collaborations with scholars from other disciplines have yielded important research findings.
The passage states that important scientific findings have come from physicists crossing into other fields, acknowledges that there are cultural differences and methodological friction, but concludes that the desire to work on something new is real and the collision of views has clear benefits.
Five jumbled up sentences (labelled 1, 2, 3, 4 and 5), related to a topic, are given below. Four of them can be put together to form a coherent paragraph. Identify the odd sentence and key in the number of that sentence as your answer.
1. The UK is a world leader in developing cultivated meat and the approval of a cultivated pet food is an important milestone.
2. If we're to realise the full potential benefits of cultivated meat the government must invest in research and infrastructure.
3. The first UK applications for cultivated meat produced for humans remain under assessment with the Food Standards Agency.
4. The previous UK government had been looking at fast-tracking the approval of cultivated meat for human consumption.
5. It underscores the potential for new innovation to help reduce the negative impacts of intensive animal agriculture.
Sentences 1, 2, 3 and 5 form a coherent argument about cultivated meat in the UK, beginning with the UK being a world leader and the pet food approval being a milestone, explaining that the UK still has human consumption applications under assessment, noting the potential for innovation to reduce harm from intensive animal agriculture, and calling for government investment to realise that potential. Sentence 4, about the previous UK government fast-tracking cultivated meat approval for human consumption, introduces a separate historical policy point about a prior government's approach that does not logically continue or complete the forward-looking argument about current milestones, ongoing assessment, and future investment.
The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes(3 of them, each in the shape of rectangles – shaded in the diagram) of a gated area. Different points on the walkway are indicated by letters (A through P) with distances being OP = 150 m, ON = MN = 300 m, ML = 400 m, EL = 200 m, DE = 400 m.
The following additional information about the facilities in the area is known.
1. The only entry/exit point is at C.
2. There are many residences within the gated area; all of them are located on the path AH and ML with four of them being at A, H, M, and L.
3. The post office is located at P and the bank is located at B.
One resident whose house is located at L, needs to visit the post office as well as the bank. What is the minimum distance (in m) he has to walk starting from his residence and returning to his residence after visiting both the post office and the bank?
[Note: In actual paper CAT 2024, two options (3) and (4) i.e., 3000 were same.]
3200
2700
3000
3400
3200
One person enters the gated area and decides to walk as much as possible before leaving the area without walking along any path more than once and always walking next to one of the lakes. Note that he may cross a point multiple times. How much distance (in m) will he walk within the gated area?
3200
3000
2800
3800
3800
One resident takes a walk within the gated area starting from A and returning to A without going through any point (other than A) more than once. What is the maximum distance (in m) she can walk in this way?
Visitors coming for morning walks are allowed to enter as long as they do not pass by any of the residences and do not cross any point (except C) more than once.What is the maximum distance (in m) that such a visitor can walk within the gated area?
An online e-commerce firm receives daily integer product ratings from 1 through 5 given by buyers. The daily average is the average of the ratings given on that day. The cumulative average is the average of all ratings given on or before that day.
The rating system began on Day 1, and the cumulative averages were 3 and 3.1 at the end of Day 1 and Day 2, respectively. The distribution of ratings on Day 2 is given in the figure below.
The following information is known about ratings on Day 3.
1. 100 buyers gave product ratings on Day 3.
2. The modes of the product ratings were 4 and 5.
3. The numbers of buyers giving each product rating are non-zero multiples of 10.
4. The same number of buyers gave product ratings of 1 and 2, and that number is half the number of buyers who gave a rating of 3.
How many buyers gave ratings on Day 1?
What is the daily average rating of Day 3?
3.2
3.5
3.0
3.6
3.6
What is the median of all ratings given on Day 3?
Which of the following is true about the cumulative average ratings of Day 2 and Day 3?
The cumulative average of Day 3 decreased from Day 2.
The cumulative average of Day 3 increased by more than 8% from Day 2.
The cumulative average of Day 3 increased by a percentage between 5% and 8% from Day 2.
The cumulative average of Day 3 increased by less than 5% from Day 2.
The cumulative average of Day 3 increased by a percentage between 5% and 8% from Day 2.
The two plots below give the following information about six firms A, B, C, D, E, and F for 2019 and 2023.
PAT: The firm’s profits after taxes in Rs. crores,
ES: The firm’s employee strength, that is the number of employees in the firm, and
PRD: The percentage of the firm’s PAT that they spend on Research and Development (R&D).
In the plots, the horizontal and vertical coordinates of point representing each firm gives their ES and PAT values respectively. The PRD values of each firm are proportional to the areas around the points representing each firm. The areas are comparable between the two plots, i.e., equal areas in the two plots represent the same PRD values for the two years.
Assume that the annual rate of growth in PAT over the previous year (ARG) remained constant over the years for each of the six firms. Which among the firms A, B, C, and E had the highest ARG?
Firm B
Firm C
Firm A
Firm E
Firm E
Since the annual growth rate is assumed to remain constant, it is sufficient to compare the increase in each firm's PAT between 2019 and 2023.
- Firm A: 3000 → 3900, an increase of 900
- Firm B: 2800 → 3800, an increase of 1000
- Firm C: 2400 → 3000, an increase of 600
- Firm E: 2400 → 3500, an increase of 1100
Among these, Firm E records the highest increase in PAT.
Hence, Firm E has the highest annual growth rate.
The ratio of the amount of money spent by Firm C on R&D in 2019 to that in 2023 is closest to
9 : 4
5 : 9
9 : 5
5 : 6
9 : 5
This question requires a bit of approximation because the bubbles are not completely enclosed within the axes. In such cases, the best approach is to use the most visible bubbles for comparison.
For example, the bubble corresponding to Firm C has an approximate vertical diameter of 3 units in 2019, whereas in 2023 its diameter is about 2 units.
Therefore, the ratio of R&D expenditure in 2019 to that in 2024 is:
= (2400 × (1.5)²) / (3000 × (1)²)
= (8/10) × (9/4)
= 9 : 5
Which among the firms A, C, E, and F had the maximum PAT per employee in 2023?
Firm A
Firm F
Firm E
Firm C
Firm C
The PAT per employee for each firm in 2023 is calculated as follows:
- Firm A: 3900 / 1300 = 3
- Firm C: 3000 / 800 = 3.75
- Firm E: 3500 / 1400 = 2.5
- Firm F: 3200 / 1000 = 3.2
Among these, the highest PAT per employee is 3.75, achieved by Firm C.
Hence, the required answer is Firm C.
Which among the firms C, D, E, and F had the least amount of R&D spending per employee in 2023?
Firm D
Firm C
Firm E
Firm F
Firm D
The PAT per employee for Firms F, C, and E was calculated in the previous question.
Now, for Firm D:
PAT per employee = 2400 / 800 = 3
Among Firms F, C, and D, the bubble sizes are identical. Therefore, the only quantity that needs to be compared is the PAT per employee.
- Firm F = 3.2
- Firm C = 3.75
- Firm D = 3
Thus, Firm D has the lowest PAT per employee among these three.
Now compare Firm D with Firm E by accounting for the proportionality between bubble area and R&D expenditure.
For Firm E:
- PAT per employee = 2.5
- Bubble radius = 1.5 units
R&D expenditure per employee:
= (5/2) × π × (3/2)²
= π × 25/8
For Firm D:
- PAT per employee = 3
- Bubble radius = 1 unit
R&D expenditure per employee:
= 3 × π × (1)²
= 3π
Since 25π/8 > 3π, Firm E has the higher R&D expenditure per employee.
Hence, Firm D had the lowest R&D spending per employee in 2023.
Eight gymnastics players numbered 1 through 8 underwent a training camp where they were coached by three coaches - Xena, Yuki, and Zara. Each coach trained at least two players.Yuki trained only even numbered players, while Zara trained only odd numbered players. After the camp, the coaches evaluated the players and gave integer ratings to the respective players trained by them on a scale of 1 to 7, with 1 being the lowest rating and 7 the highest.
The following additional information is known.
1. Xena trained more players than Yuki.
2. Player-1 and Player-4 were trained by the same coach, while the coaches who trained Player-2, Player-3 and Player-5 were all different.
3. Player-5 and Player-7 were trained by the same coach and got the same rating. All other players got a unique rating.
4. The average of the ratings of all the players was 4.
5. Player-2 got the highest rating.
6. The average of the ratings of the players trained by Yuki was twice that of the player strained by Xena and two more than that of the players trained by Zara.
7. Player-4’s rating was double of Player-8's and less than Player-5’s.
What best can be concluded about the number of players coached by Zara?
Exactly 3
Either 2 or 3
Exactly 2
Either 2 or 3 or 4
Exactly 2
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
What was the rating of Player-7?
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
What was the rating of Player-6?
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
For how many players the ratings can be determined with certainty?
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
Who all were the players trained by Xena?
Player-1, Player-3, Player-4
Player-1, Player-3, Player-4, Player-6
Player-1, Player-3, Player-4, Player-8
Player-1, Player-4, Player-6, Player-8
Player-1, Player-3, Player-4, Player-8
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are placed in ten slots of the following grid based on the conditions below.
1. Numbers in any row appear in an increasing order from left to right.
2. Numbers in any column appear in a decreasing order from top to bottom.
3. 1 is placed either in the same row or in the same column as 10.
4. Neither 2 nor 3 is placed in the same row or in the same column as 10.
5. Neither 7 nor 8 is placed in the same row or in the same column as 9.
6. 4 and 6 are placed in the same row.
What is the row number which has the least sum of numbers placed in that row?
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
Which of the following statements MUST be true?
I. 10 is placed in a slot in Row 1.
II. 1 is placed in a slot in Row 4.
Only I
Both I and II
Only II
Neither I nor II
Both I and II
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
Which of the following statements MUST be true?
I. 2 is placed in a slot in Column 2.
II. 3 is placed in a slot in Column 3.
Both I and II
Neither I nor II
Only II
Only I
Neither I nor II
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
For how many slots in the grid, placement of numbers CANNOT be determined with certainty?
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
What is the sum of the numbers placed in Column 4?
Step 1:
Since 10 is the largest number, it must occupy the cell in Row 1, Column 4.
Also, 2 and 3 cannot lie in the same row or column as 10.
Now consider the placement of 9, the next largest number. It can be placed either in Row 1, Column 3 or Row 2, Column 4.
Step 2:
Assume 9 is placed in Row 2, Column 4.
In that case, 8 and 7 cannot share the same row or column with 9, forcing them into fixed positions.
However, this arrangement leads to a contradiction.
If 4 and 6 are placed in Row 2, there is no valid position left for 2 and 3.
On the other hand, if 4 and 6 are placed in Row 3, the entries in Column 3 will no longer be in decreasing order from top to bottom.
Hence, this case is not feasible.
Step 3:
Therefore, 9 must be placed in Row 1, Column 3.
Consequently, 8 and 7 cannot be in the same row or column as 9, which fixes their positions.
If 4 and 6 are placed in Row 2, there is no valid location available for 2 and 3.
Hence, 4 and 6 must be placed in Row 1.
The final arrangement is:
- Row 1: 4, 6, 9, 10
- Row 2: 2/3, 5, 8
- Row 3: 3/2, 7
- Row 4: 1
− Bina incurs 19% loss when she sells a product at Rs. 4860 to Shyam, who in turn sells this product to Hari. If Bina would have sold this product to Shyam at the purchase price of Hari, she would have obtained 17% profit. Then, the profit, in rupees, made by Shyam is
Given:
Bina sells a product to Shyam for Rs. 4860 at a loss of 19%.
If Bina had sold the product at the price at which Hari purchased it, she would have earned a profit of 17%.
Find Shyam's profit.
Step 1: Find Bina's cost price
Since Rs. 4860 is 81% of the cost price,
Cost Price
= 4860/0.81
= Rs. 6000
Step 2: Find Hari's purchase price
If Bina had earned a profit of 17%,
Selling Price
= 117% of 6000
= 1.17 × 6000
= Rs. 7020
Thus, Hari purchased the product for Rs. 7020.
Step 3: Find Shyam's profit
Shyam purchased the product for Rs. 4860 and sold it for Rs. 7020.
Profit
= 7020 − 4860
= Rs. 2160
Answer: Rs. 2160
− The coordinates of the three vertices of a triangle are: (1, 2), (7, 2), and (1, 10). Then the radius of the incircle of the triangle is
Given: The vertices of the triangle are
A(1, 2), B(7, 2), and C(1, 10).
Find the radius of the incircle.
Step 1: Find the lengths of the sides
AB = 7 − 1 = 6
AC = 10 − 2 = 8
Using the distance formula,
BC = √[(7 − 1)² + (2 − 10)²]
= √(36 + 64)
= 10
Thus, the triangle is a right-angled triangle with sides 6, 8 and 10.
Step 2: Find the area
Area
= (1/2) × 6 × 8
= 24 sq. units
Step 3: Find the semiperimeter
Semiperimeter
= (6 + 8 + 10)/2
= 12
Step 4: Find the inradius
For any triangle,
Inradius = Area / Semiperimeter
= 24/12
= 2
Answer: 2
− A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is
Given:
A fruit seller has mangoes, bananas and apples.
At the beginning:
● Mangoes constitute 40% of the total stock.
● At least one fruit of each type is present.
During the day, he sells:
● Half of the mangoes.
● 96 bananas.
● 40% of the apples.
At the end of the day, exactly 50% of the total fruits have been sold.
Find the smallest possible initial stock.
Step 1: Let the initial total number of fruits be T.
Since mangoes are 40% of the stock,
Mangoes = 2T/5
Let the number of apples be
A.
Then,
Bananas = T − 2T/5 − A
= 3T/5 − A
Step 2: Write the total fruits sold
Half of the mangoes sold
= (1/2) × (2T/5)
= T/5
Bananas sold = 96
Apples sold = 40% of A
= 2A/5
Since exactly half the fruits are sold,
T/5 + 96 + 2A/5 = T/2
Multiply throughout by 10,
2T + 960 + 4A = 5T
4A = 3T − 960
A = (3T − 960)/4
Step 3: Use divisibility conditions
Since 40% of apples are sold, A must be divisible by 5.
So,
(3T − 960)/4
must be a multiple of 5.
Hence,
3T − 960
must be divisible by 20.
Since 960 is divisible by 20,
3T must be divisible by 20.
As 3 and 20 are coprime,
T must be divisible by 20.
Also, mangoes are 2T/5, so T must be divisible by 5, which is already satisfied.
Step 4: Check the smallest possible value
Let
T = 20k.
Then,
A = (60k − 960)/4
= 15k − 240.
Bananas
= 3T/5 − A
= 12k − (15k − 240)
= 240 − 3k.
Since at least 96 bananas must be available,
240 − 3k ≥ 96
k ≤ 48.
Also, apples must be at least 1,
15k − 240 ≥ 1
k ≥ 17.
The smallest possible value is
k = 17.
Hence,
T = 20 × 17
= 340
Verification
Mangoes = 136
Bananas = 189
Apples = 15
Sold:
● Mangoes = 68
● Bananas = 96
● Apples = 6
Total sold
= 68 + 96 + 6
= 170
which is exactly half of 340.
Answer: 340
− If a, b and c are positive real numbers such that a > 10 ≥ b ≥ c and log₈(a + b)/log₂c + log₂₇(a − b)/log₃c = 2/3, then the greatest possible integer value of a is
Given:
a, b and c are positive real numbers such that
a > 10 ≥ b ≥ c
and
log₈(a + b)/log₂c + log₂₇(a − b)/log₃c = 2/3
Find the greatest possible integer value of a.
Step 1: Simplify the logarithmic expression
Using the change of base formula,
log₈(a + b)
= log₂(a + b)/3
and
log₂₇(a − b)
= log₃(a − b)/3
Hence,
(1/3) logc(a + b) + (1/3) logc(a − b) = 2/3
Multiplying by 3,
logc(a + b) + logc(a − b) = 2
Using the product rule of logarithms,
logc[(a + b)(a − b)] = 2
Therefore,
a² − b² = c²
Step 2: Use the given constraints
Since
b ≤ 10
and
c ≤ 10,
we have
b² ≤ 100
and
c² ≤ 100
Thus,
a² = b² + c² ≤ 200
Hence,
a ≤ √200 ≈ 14.14
Therefore, the greatest possible integer value of a can be at most 14.
Step 3: Check whether a = 14 is possible
Let
a = 14
and
b = 10.
Then,
c² = a² − b²
= 196 − 100
= 96
So,
c = √96 = 4√6 ≈ 9.80
This satisfies
10 ≥ b ≥ c
since
10 ≥ 9.80.
Hence, all the given conditions are satisfied.
Therefore, the greatest possible integer value of a is
14
Answer: 14
A function f maps the set of natural numbers to whole numbers, such that f(xy) = f(x)f(y) + f(x) + f(y) for all x, y and f(p) = 1 for every prime number p. Then, the value of f(160000) is
4095
8191
2047
1023
4095
Given:
A function f satisfies
f(xy) = f(x)f(y) + f(x) + f(y)
for all natural numbers x and y.
Also,
f(p) = 1 for every prime number p.
Find the value of
f(160000).
Step 1: Define a new function
Let
g(x) = f(x) + 1
Then,
g(xy)
= f(xy) + 1
= f(x)f(y) + f(x) + f(y) + 1
= (f(x) + 1)(f(y) + 1)
= g(x)g(y)
Thus,
g(xy) = g(x)g(y)
Step 2: Find g for prime numbers
Since
f(p) = 1,
we have
g(p) = 2
for every prime p.
Step 3: Prime factorize 160000
160000
= 160 × 1000
= (2⁵ × 5) × (2³ × 5³)
= 2⁸ × 5⁴
Since g is multiplicative,
g(160000)
= g(2)⁸ × g(5)⁴
= 2⁸ × 2⁴
= 2¹²
= 4096
Step 4: Find f(160000)
f(160000)
= g(160000) − 1
= 4096 − 1
= 4095
Answer:
A. 4095
The roots α, β of the equation 3x² + λx − 1 = 0, satisfy 1/α² + 1/β² = 15. The value of (α³ + β³)², is
16
4
1
9
4
Given:
The roots α and β of
3x² + λx − 1 = 0
satisfy
1/α² + 1/β² = 15.
Find the value of
(α³ + β³)².
Step 1: Find the sum and product of the roots
For the quadratic,
α + β = −λ/3
αβ = −1/3
Step 2: Use the given condition
1/α² + 1/β²
= (α² + β²)/(αβ)²
= 15
Since
(αβ)² = (−1/3)² = 1/9,
we get
α² + β² = 15 × 1/9
= 5/3
Step 3: Find α + β
Using
α² + β² = (α + β)² − 2αβ,
5/3 = (α + β)² − 2(−1/3)
5/3 = (α + β)² + 2/3
(α + β)² = 1
Thus,
α + β = ±1
Step 4: Find α³ + β³
Using
α³ + β³ = (α + β)³ − 3αβ(α + β),
α³ + β³
= (α + β)³ + (α + β)
= (α + β)[(α + β)² + 1]
Since
(α + β)² = 1,
α³ + β³
= 2(α + β)
Step 5: Find the required value
(α³ + β³)²
= [2(α + β)]²
= 4(α + β)²
= 4 × 1
= 4
Answer:
B. 4
When Rajesh's age was same as the present age of Garima, the ratio of their ages was 3 : 2. When Garima's age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become
3 : 2
4 : 3
5 : 4
2 : 1
5 : 4
Given:
When Rajesh's age was the same as Garima's present age, the ratio of their ages was 3 : 2.
Find the ratio of their ages when Garima's age becomes equal to Rajesh's present age.
Step 1: Let the present ages be
Rajesh = R years
Garima = G years
where R > G.
Step 2: Use the first condition
When Rajesh's age was G years, he was
R − G
years younger than now.
At that time,
Garima's age = G − (R − G)
= 2G − R
Given,
G : (2G − R) = 3 : 2
Cross-multiplying,
2G = 3(2G − R)
2G = 6G − 3R
3R = 4G
Hence,
R : G = 4 : 3
Step 3: Let
R = 4x
and
G = 3x
Step 4: Find the required ratio
Garima reaches Rajesh's present age after
4x − 3x = x
years.
At that time,
Rajesh's age = 4x + x = 5x
Garima's age = 3x + x = 4x
Therefore, the required ratio is
5 : 4
Answer:
C. 5 : 4
Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is
7 + 4√3 : 1
4 + 2√3 : 1
4 + √3 : 1
2 + √3 : 1
7 + 4√3 : 1
Given:
Three equal circles of radius r touch each other externally.
Two more circles, X and Y, are drawn such that each touches all the three circles.
Radius of X is greater than that of Y.
Find the ratio of the radii of X and Y.
Step 1: Locate the centres
The centres of the three equal circles form an equilateral triangle of side
2r.
The centre of both circles X and Y is the common centre (circumcentre) of this equilateral triangle.
The distance from this centre to each vertex is the circumradius of the equilateral triangle.
Circumradius
= (2r)/√3
= 2r/√3
Step 2: Find the radius of the larger circle X
Since X encloses the three circles, the distance between its centre and the centre of each small circle is
R − r,
where R is the radius of X.
Hence,
R − r = 2r/√3
R = r + 2r/√3
= r(1 + 2/√3)
Step 3: Find the radius of the smaller circle Y
Since Y lies inside the gap, the distance between its centre and the centre of each small circle is
r + y,
where y is the radius of Y.
Hence,
r + y = 2r/√3
y = 2r/√3 − r
= r(2/√3 − 1)
Step 4: Find the ratio
R : y
= (1 + 2/√3) : (2/√3 − 1)
Multiply both terms by √3,
= (√3 + 2) : (2 − √3)
Rationalizing,
= (√3 + 2)² : (4 − 3)
= (3 + 4 + 4√3) : 1
= (7 + 4√3) : 1
Answer:
A. (7 + 4√3) : 1
ABCD is a trapezium in which AB is parallel to CD. The sides AD and BC when extended, intersect at point E. If AB = 2 cm, CD = 1 cm, and perimeter of ABCD is 6 cm, then the perimeter, in cm, of △AEB is
8
10
9
7
8
Given:
ABCD is a trapezium with
AB ∥ CD
AB = 2 cm
CD = 1 cm
Perimeter of ABCD = 6 cm.
The extensions of AD and BC meet at E.
Find the perimeter of △AEB.
Step 1: Use similarity of triangles
Since
AB ∥ CD,
triangles AEB and DEC are similar.
Hence,
AB/CD = AE/DE = BE/CE
Substituting the given values,
AE/DE = BE/CE = 2/1
Therefore,
AE = 2DE
and
BE = 2CE
Step 2: Express AD and BC
Since D lies on AE,
AD = AE − DE
= 2DE − DE
= DE
Thus,
AD = DE.
Similarly,
BC = BE − CE
= 2CE − CE
= CE
Thus,
BC = CE.
Hence,
AE = 2AD
and
BE = 2BC.
Step 3: Use the perimeter of the trapezium
Perimeter of ABCD
= AB + BC + CD + AD
6 = 2 + BC + 1 + AD
AD + BC = 3
Step 4: Find the perimeter of △AEB
Perimeter
= AE + BE + AB
= 2AD + 2BC + 2
= 2(AD + BC) + 2
= 2 × 3 + 2
= 8 cm
Answer:
A. 8
A company has 40 employees whose names are listed in a certain order. In the year 2022, the average bonus of the first 30 employees was Rs. 40000, of the last 30 employees was Rs. 60000, and of the first 10 and last 10 employees together was Rs. 50000. Next year, the average bonus of the first 10 employees increased by 100%, of the last 10 employees increased by 200% and of the remaining employees was unchanged. Then, the average bonus, in rupees, of all the 40 employees together in the year 2023 was
95000
90000
80000
85000
95000
Given:
There are 40 employees.
In 2022,
● Average bonus of the first 30 employees = Rs. 40000
● Average bonus of the last 30 employees = Rs. 60000
● Average bonus of the first 10 and last 10 employees together = Rs. 50000
In 2023,
● Bonus of the first 10 employees doubles.
● Bonus of the last 10 employees triples.
● Bonus of the remaining 20 employees remains unchanged.
Find the average bonus of all 40 employees in 2023.
Step 1: Find the total bonuses in 2022
First 30 employees
= 30 × 40000
= Rs. 12,00,000
Last 30 employees
= 30 × 60000
= Rs. 18,00,000
First 10 and last 10 together
= 20 × 50000
= Rs. 10,00,000
Step 2: Find the total bonus of all 40 employees in 2022
Adding the totals of the first 30 and last 30 counts the middle 20 employees twice.
Hence,
Total bonus of all 40 employees
= (12,00,000 + 18,00,000 + 10,00,000)/2
= Rs. 20,00,000
Step 3: Find the bonuses of different groups
Let
First 10 = F
Middle 20 = M
Last 10 = L
Then,
F + M = 12,00,000
M + L = 18,00,000
F + L = 10,00,000
Adding the first two equations,
F + 2M + L = 30,00,000
Since
F + L = 10,00,000,
2M = 20,00,000
M = Rs. 10,00,000
Therefore,
F = 12,00,000 − 10,00,000
= Rs. 2,00,000
L = 18,00,000 − 10,00,000
= Rs. 8,00,000
Step 4: Find the total bonus in 2023
First 10 bonus doubles
= 2 × 2,00,000
= Rs. 4,00,000
Middle 20 remains
= Rs. 10,00,000
Last 10 bonus triples
= 3 × 8,00,000
= Rs. 24,00,000
Total bonus
= 4,00,000 + 10,00,000 + 24,00,000
= Rs. 38,00,000
Step 5: Find the average bonus
Average
= 38,00,000/40
= Rs. 95,000
Answer:
A. 95,000
− Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been
Given:
● Amal and Vimal together complete the work in 150 days.
● Vimal and Sunil together complete the work in 100 days.
Amal works for 75 days.
Then Vimal works for 135 days.
Finally, Sunil works for 45 days and completes the task.
Find the number of days required if:
● Amal works every day.
● Vimal works on every second day.
● Sunil works on every third day.
Step 1: Let the daily work rates be
Amal = A
Vimal = V
Sunil = S
Then,
A + V = 1/150
V + S = 1/100
Step 2: Use the second work schedule
The total work done is
75A + 135V + 45S = 1
Using
S = 1/100 − V,
75A + 135V + 45(1/100 − V) = 1
75A + 90V = 11/20
Divide throughout by 15,
5A + 6V = 11/300
Step 3: Solve for the individual rates
From
A + V = 1/150,
A = 1/150 − V
Substitute into
5A + 6V = 11/300,
5(1/150 − V) + 6V = 11/300
1/30 + V = 11/300
V = 1/300
Therefore,
A = 1/150 − 1/300
= 1/300
Also,
S = 1/100 − 1/300
= 1/150
Step 4: Determine the work pattern
Every 6 days,
● Amal works all 6 days.
● Vimal works on days 2, 4 and 6 (3 days).
● Sunil works on days 3 and 6 (2 days).
Work done in 6 days
= 6A + 3V + 2S
= 6 × 1/300 + 3 × 1/300 + 2 × 1/150
= 6/300 + 3/300 + 4/300
= 13/300
Step 5: Find the number of complete cycles
After 23 cycles (138 days),
Work done
= 23 × 13/300
= 299/300
Remaining work
= 1/300
Step 6: Find the last day
On the 139th day,
Amal works.
Since Amal's daily work is
1/300,
the remaining work is completed.
Hence, the total number of days required is
139
Answer: 139
All the values of x satisfying the inequality 1/(x + 5) ≤ 1/(2x − 3) are
x < −5 or 3/2 < x ≤ 8
−5 < x < 3/2 or x > 3/2
x < −5 or x > 3/2
−5 < x < 3/2 or 3/2 < x ≤ 8
x < −5 or 3/2 < x ≤ 8
Given:
1/(x + 5) ≤ 1/(2x − 3)
Find all values of x.
Step 1: Note the restrictions
The denominators cannot be zero.
So,
x ≠ −5
and
x ≠ 3/2
Step 2: Bring all terms to one side
1/(x + 5) − 1/(2x − 3) ≤ 0
Taking the LCM,
[(2x − 3) − (x + 5)] / [(x + 5)(2x − 3)] ≤ 0
(x − 8) / [(x + 5)(2x − 3)] ≤ 0
Step 3: Find the critical points
The critical points are
x = −5, 3/2, 8
These divide the number line into four intervals:
● x < −5
● −5 < x < 3/2
● 3/2 < x < 8
● x > 8
Step 4: Check the sign in each interval
For x < −5
Choose x = −6.
Expression = (−)/(+)
= Negative
Satisfied.
For −5 < x < 3/2
Choose x = 0.
Expression = (−)/(−)
= Positive
Not satisfied.
For 3/2 < x < 8
Choose x = 2.
Expression = (−)/(+)
= Negative
Satisfied.
For x > 8
Choose x = 9.
Expression = (+)/(+)
= Positive
Not satisfied.
Step 5: Check the boundary point
At x = 8,
the numerator becomes zero while the denominator is non-zero.
Hence,
the inequality is satisfied.
The points
x = −5
and
x = 3/2
are not included since the expression is undefined.
Final Answer
The solution set is
x < −5
or
3/2 < x ≤ 8
Answer:
A. x < −5 or 3/2 < x ≤ 8
Anil invests Rs 22000 for 6 years in a scheme with 4% interest per annum, compounded half-yearly. Separately, Sunil invests a certain amount in the same scheme for 5 years, and then reinvests the entire amount he receives at the end of 5 years, for one year at 10% simple interest. If the amounts received by both at the end of 6 years are equal, then the initial investment, in rupees, made by Sunil is
20860
20640
20480
20808
20808
Given:
Anil invests Rs. 22000 for 6 years at 4% per annum, compounded half-yearly.
Sunil invests a certain amount in the same scheme for 5 years and then reinvests the maturity amount for 1 year at 10% simple interest.
The final amounts received by both are equal.
Find Sunil's initial investment.
Step 1: Find Anil's maturity amount
Half-yearly interest rate
= 4%/2
= 2%
Number of half-years
= 6 × 2
= 12
Amount received by Anil
= 22000 × (1.02)¹²
Step 2: Find Sunil's maturity amount
Let Sunil's initial investment be x.
After 5 years at the same scheme,
Amount
= x × (1.02)¹⁰
This amount is reinvested for 1 year at 10% simple interest.
Final amount
= x × (1.02)¹⁰ × 1.10
Step 3: Equate the two amounts
22000 × (1.02)¹²
= x × (1.02)¹⁰ × 1.10
Dividing both sides by (1.02)¹⁰,
22000 × (1.02)²
= 1.10x
Since
(1.02)² = 1.0404,
22000 × 1.0404 = 1.10x
22888.8 = 1.10x
x = 22888.8/1.10
= 20808
Final Answer
Sunil's initial investment was
Rs. 20808
Answer:
D. 20808
A bus starts at 9 am and follows a fixed route every day. One day, it traveled at a constant speed of 60 km per hour and reached its destination 3.5 hours later than its scheduled arrival time. Next day, it traveled two-thirds of its route in one-third of its total scheduled travel time, and the remaining part of the route at 40 km per hour to reach just on time. The scheduled arrival time of the bus is
7 : 30 pm
7 : 00 pm
9 : 00 pm
10 : 30 pm
7 : 30 pm
Given:
The bus starts every day at 9:00 am.
One day, it travels at 60 km/h and reaches 3.5 hours late.
The next day,
● It covers two-thirds of the route in one-third of the scheduled travel time.
● It covers the remaining one-third of the route at 40 km/h.
● It reaches exactly on time.
Find the scheduled arrival time.
Step 1: Let the total distance be D km.
At 60 km/h, the bus takes
D/60 hours.
If the scheduled travel time is T hours, then
D/60 = T + 3.5
Step 2: Form the second equation
The first two-thirds of the journey is completed in
T/3 hours.
Hence, the speed during this part is
(2D/3)/(T/3)
= 2D/T
The remaining one-third of the journey is covered at 40 km/h.
Time taken for this part
= (D/3)/40
= D/120
Since the total journey is completed in the scheduled time,
T/3 + D/120 = T
D/120 = 2T/3
D = 80T
Step 3: Substitute into the first equation
D/60 = T + 3.5
80T/60 = T + 3.5
4T/3 = T + 3.5
T/3 = 3.5
T = 10.5 hours
Step 4: Find the scheduled arrival time
The bus starts at 9:00 am.
Adding 10.5 hours,
Scheduled arrival time
= 7:30 pm
Answer:
A. 7:30 pm
If m and n are natural numbers such that n > 1, and m^n = 2²⁵ × 3⁴⁰, then m − n equals
209932
209937
209942
209947
209947
Given:
m and n are natural numbers with
n > 1
and
mⁿ = 2²⁵ × 3⁴⁰.
Find the value of
m − n.
Step 1: Find the possible value of n
Since m is an integer, the exponents of every prime in mⁿ must be divisible by n.
Therefore,
n must divide both 25 and 40.
The common divisors of 25 and 40 are
1 and 5.
Since
n > 1,
we get
n = 5.
Step 2: Find m
m
= (2²⁵ × 3⁴⁰)^(1/5)
= 2⁵ × 3⁸
Now,
2⁵ = 32
3⁸ = 6561
Therefore,
m = 32 × 6561
= 209952
Step 3: Find m − n
m − n
= 209952 − 5
= 209947
Answer:
D. 209947
When 3³³³ is divided by 11, the remainder is
5
10
1
6
5
Given:
Find the remainder when
3³³³
is divided by 11.
Step 1: Find the repeating pattern
Since
3⁵ = 243
and
243 ÷ 11 leaves a remainder of 1,
we have
3⁵ ≡ 1 (mod 11)
Thus, the powers of 3 repeat every 5 terms modulo 11.
Step 2: Reduce the exponent
333 ÷ 5 leaves a remainder of 3.
Hence,
3³³³ ≡ 3³ (mod 11)
Step 3: Find the remainder
3³ = 27
27 ÷ 11 leaves a remainder of 5.
Therefore,
3³³³ leaves a remainder of 5 when divided by 11.
Answer:
A. 5
− If x and y are real numbers such that 4x² + 4y² − 4xy − 6y + 3 = 0, then the value of (4x + 5y) is
Given:
4x² + 4y² − 4xy − 6y + 3 = 0
Find the value of
4x + 5y.
Step 1: Rearrange the expression
Write the equation as
4(x² − xy + y²) − 6y + 3 = 0
Now,
x² − xy + y²
= (x − y/2)² + 3y²/4
Substituting,
4(x − y/2)² + 3y² − 6y + 3 = 0
Step 2: Complete the square
Since
3y² − 6y + 3 = 3(y − 1)²,
the equation becomes
4(x − y/2)² + 3(y − 1)² = 0
Step 3: Find x and y
Both terms are perfect squares and are non-negative.
Their sum is zero only if each is zero.
So,
x − y/2 = 0
and
y − 1 = 0
Hence,
y = 1
and
x = 1/2
Step 4: Find the required value
4x + 5y
= 4 × 1/2 + 5 × 1
= 2 + 5
= 7
Answer: 7
− If (x + 6√2)^(1/2) − (x − 6√2)^(1/2) = 2√2, then x equals
Given:
√(x + 6√2) − √(x − 6√2) = 2√2
Find x.
Step 1: Square both sides
[√(x + 6√2) − √(x − 6√2)]² = (2√2)²
x + 6√2 + x − 6√2 − 2√[(x + 6√2)(x − 6√2)] = 8
2x − 2√(x² − 72) = 8
Divide by 2,
x − √(x² − 72) = 4
Step 2: Isolate the square root
√(x² − 72) = x − 4
Since the left side is non-negative,
x ≥ 4.
Step 3: Square again
x² − 72 = (x − 4)²
x² − 72 = x² − 8x + 16
8x = 88
x = 11
Step 4: Verify the solution
Substituting x = 11,
√(11 + 6√2) − √(11 − 6√2)
= √9 − √1
= 3 − 1
= 2
= 2√2/√2
Since
11 + 6√2 = (3 + √2)²
and
11 − 6√2 = (3 − √2)²,
we have
√(11 + 6√2) = 3 + √2
√(11 − 6√2) = 3 − √2
Therefore,
(3 + √2) − (3 − √2)
= 2√2,
which satisfies the given equation.
Hence,
x = 11
Answer: 11
− P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is
Given:
● Direct paths between P and Q = 3
● Direct paths between Q and S = 4
● Direct paths between P and R = 4
● No direct path between P and S
● Let the number of direct paths between Q and R = x
● Let the number of direct paths between R and S = y
Also,
● Number of ways to travel from P to S is 62.
● Number of ways to travel from Q to R is 27.
Find x.
Step 1: Form the equation for travel from P to S
Travel is possible in three ways.
Via Q
Ways = 3 × 4 = 12
Via R
Ways = 4 × y = 4y
Via Q and then R
Ways = 3 × x × y = 3xy
Hence,
12 + 4y + 3xy = 62
or,
4y + 3xy = 50
y(4 + 3x) = 50
Step 2: Form the equation for travel from Q to R
Travel is possible in three ways.
Directly
Ways = x
Via P
Ways = 3 × 4 = 12
Via S
Ways = 4 × y = 4y
Hence,
x + 12 + 4y = 27
or,
x + 4y = 15
Step 3: Solve the equations
From the second equation,
4y = 15 − x
Substitute into the first equation,
((15 − x)/4)(4 + 3x) = 50
Multiply by 4,
(15 − x)(4 + 3x) = 200
Expand,
60 + 45x − 4x − 3x² = 200
3x² − 41x + 140 = 0
Factorizing,
(3x − 20)(x − 7) = 0
Thus,
x = 20/3 or x = 7
Since the number of paths must be an integer,
x = 7
Final Answer
The number of direct paths between Q and R is
7
If x and y satisfy the equations |x| + x + y = 15 and x + |y| − y = 20, then (x − y) equals
20
15
5
10
15
Given:
|x| + x + y = 15
x + |y| − y = 20
Find the value of
x − y.
Step 1: Analyze the first equation
We know,
● If x ≥ 0, then |x| = x.
● If x < 0, then |x| = −x.
Case 1: x ≥ 0
Then,
2x + y = 15
Case 2: x < 0
Then,
y = 15
Step 2: Analyze the second equation
We know,
● If y ≥ 0, then |y| = y.
● If y < 0, then |y| = −y.
Case 1: y ≥ 0
Then,
x = 20
Case 2: y < 0
Then,
x − 2y = 20
Step 3: Check possible cases
Case 1: x ≥ 0 and y ≥ 0
From the equations,
x = 20
2x + y = 15
40 + y = 15
y = −25
Contradiction since y ≥ 0.
Not possible.
Case 2: x ≥ 0 and y < 0
From the equations,
2x + y = 15
x − 2y = 20
Multiply the first equation by 2,
4x + 2y = 30
Add the second equation,
5x = 50
x = 10
Substituting,
20 + y = 15
y = −5
This satisfies the conditions.
Case 3: x < 0 and y ≥ 0
From the equations,
y = 15
x = 20
Contradiction since x < 0.
Not possible.
Case 4: x < 0 and y < 0
From the equations,
y = 15
Contradiction since y < 0.
Not possible.
Step 4: Find the required value
x − y
= 10 − (−5)
= 15
Answer:
B. 15
A vessel contained a certain amount of a solution of acid and water. When 2 litres of water was added to it, the new solution had 50% acid concentration. When 15 litres of acid was further added to this new solution, the final solution had 80% acid concentration. The ratio of water and acid in the original solution was
5 : 3
3 : 5
5 : 4
4 : 5
3 : 5
Given:
A vessel contains a solution of acid and water.
● After adding 2 litres of water, the solution becomes 50% acid.
● Then, 15 litres of acid is added, making the solution 80% acid.
Find the ratio of water to acid in the original solution.
Step 1: Let the original quantities be
Acid = A litres
Water = W litres
Step 2: Use the first condition
After adding 2 litres of water,
Acid = A
Water = W + 2
Since the solution is 50% acid,
A = W + 2
Step 3: Use the second condition
After adding 15 litres of acid,
Acid = A + 15
Water = W + 2
The acid concentration becomes 80%.
Therefore,
(A + 15)/(A + W + 17) = 4/5
Cross-multiplying,
5(A + 15) = 4(A + W + 17)
5A + 75 = 4A + 4W + 68
A = 4W − 7
Step 4: Solve the equations
From Step 2,
A = W + 2
Substitute into
A = 4W − 7
W + 2 = 4W − 7
3W = 9
W = 3
Hence,
A = 5
Step 5: Find the required ratio
Water : Acid
= 3 : 5
Answer:
B. 3 : 5
The sum of the infinite series (1/5)(1/5 − 1/7) + (1/5)²((1/5)² − (1/7)²) + (1/5)³((1/5)³ − (1/7)³) + …… is equal to
7/816
5/408
7/408
5/816
5/408
Given:
Find the sum of the infinite series
(1/5)(1/5 − 1/7) + (1/5)²[(1/5)² − (1/7)²] + (1/5)³[(1/5)³ − (1/7)³] + ...
Step 1: Write the general term
The nth term is
(1/5)ⁿ[(1/5)ⁿ − (1/7)ⁿ]
= (1/25)ⁿ − (1/35)ⁿ
Hence,
S = Σ[(1/25)ⁿ − (1/35)ⁿ], n = 1 to ∞
Step 2: Split into two geometric series
S
= Σ(1/25)ⁿ − Σ(1/35)ⁿ
Step 3: Use the sum of an infinite geometric series
For a geometric series,
Sum = a/(1 − r)
For the first series,
a = r = 1/25
Sum
= (1/25)/(1 − 1/25)
= 1/24
For the second series,
a = r = 1/35
Sum
= (1/35)/(1 − 1/35)
= 1/34
Step 4: Find the required sum
S
= 1/24 − 1/34
= (34 − 24)/816
= 10/816
= 5/408
Answer:
B. 5/408
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