CAT 2020 Slot 1 Question Paper

Option A: This element is discussed in the third paragraph: {"...The anarchists and their precursors were unique on the political Left in affirming that workers and peasants, grasping the chance that arose to bring an end to centuries of exploitation and tyranny, were inevitably betrayed by the new class of politicians, whose first priority was to re-establish a centralized state power..."} Prior to making this claim, the failure of the French revolution is highlighted, and this segment is subsequently tied to it. Hence, Option A is definitely a point that is made in the passage.

Option B: This has been outlined in the fourth paragraph: {"...For anarchists, the state itself is the enemy, and they have applied the same interpretation to the outcome of every revolution of the 19th and 20th centuries. This is not merely because every state keeps a watchful and sometimes punitive eye on its dissidents, but because every state protects the privileges of the powerful... "}

Option C: The last two paragraphs present the perception associated with individualist-anarchism: {"...desirable freedom of an individual or family to possess the resources needed for living, while not implying the right to own the resources needed by others. . . .There are, unsurprisingly, several traditions of individualist anarchism, one of them deriving from the ‘conscious egoism’ of the German writer Max Stirner (1806-56), and another from a remarkable series of 19th-century American figures who argued that in protecting our own autonomy and associating with others for common advantages, we are promoting the good of all..."} Thus, we can evidently understand the statement in C to be true.

Option D, however, is not inferable from the discussion undertaken in the passage. The author does not pinpoint "a mainstream mistrust of collectivism" as the source of the misconception of anarchism "espousing lawlessness and violence". No causal element is discussed, and thus, the statement in D is not an argument that is made in the passage. 

Hence, Option D is the correct answer.

The third paragraph renders us with sufficient information concerning the manner of betrayal. The failure of the Frech Revolution is being attributed to the deceit of the new class of politicians who not only prioritised the re-establishment and centralisation of authority but also unleashed a reign of terror on the ordinary masses. The author hints at this re-installation of power and the atrocities committed on the common people, who are the very drivers of the revolution, being the desertion of the principles of the French Revolution by the new ruling class of politicians. Any statement that aligns with this understanding would be a suitable answer. Option A brings in the concept of an impact on the market, which is not mentioned in the passage. Additionally, it does not signify betrayal of any kind. Same can be said about the statement in Option B. Option C is a strong candidate that symbolises betrayal. However, can we appropriately associate it with the underlying principles of the revolution?  The author's primary focus is on the cost borne by the ordinary populations, and despite the revolution being realised by their sacrifices, they are still the ones being subjected to inequity and torment. Although C highlights betrayal, it does not truly coincide with the idea being conveyed. The afore-mentioned element is appropriately captured by Option D: it highlights how the new class of politicians utilised the indignation of the masses/workers for the revolution - to topple authority and rise to power - and then stab them in the back by turning to oppress them.  

We can derive our understanding of these two groups from the following: {"...There are, unsurprisingly, several traditions of individualist anarchism, one of them deriving from the ‘conscious egoism’ of the German writer Max Stirner (1806-56), and another from a remarkable series of 19th-century American figures who argued that in protecting our own autonomy and associating with others for common advantages, we are promoting the good of all. These thinkers differed from free-market liberals in their absolute mistrust of American capitalism, and in their emphasis on mutualism..."} 

It is evident that both the groups prioritize individual autonomy; however, the latter does not put emphasis on mutualism. The mistrust of American capitalism is an additional facet marking the difference {but not captured in any of the options}. Option A aptly highlights the difference between the two factions, and is, hence, the correct answer.

Option B: Preferences concerning state configuration or involvement are not discussed in the passage, and hence, we can eliminate this option.

Option C: Perspectives regarding the nature of capitalism are not stated in the passage, and thus, we can effortlessly discard this option

Option D: None of the elements stated in this option can be understood from the passage, and therefore, can be rejected.

One has to be mindful of the question:  the set that is conceptually closest to the concerns of the passage.

The passage is clearly about 'Anarchism', and the discussion pertaining to 'Revolution' is a sub-element. This elementary understanding helps us pare down the viable choices to Options A and D. 'Betrayal' is again a minor sub-element used to supplement the core argument concerning the authority of the 'State' and the justified antagonism of the anarchists towards it. Additionally, towards the end of the discussion, individual autonomy and freedom serve as the focal points. Thus, of the two likely choices, Option A is more appropriate and conceptually closest to the concerns of the passage.

Option A: The various derivations of anarchism cannot be associated with Pierre-Joseph Proudhon. We don't have the requisite information to support this claim.

Option B: This has been presented as the motto of every anarchist faction: wherein the state is the enemy {opposition to the centralization of power}. While there are additional beliefs associated with different schools of anarchist thought, this antagonism towards centralised power forms their core. 

Option C: This belief has been shown to be limited to individualist anarchism {based on the information from the passage}; cannot be stated as a common attribute.

Option D: This assertion would be a gross understatement; the latter half of the option is especially a bit odd {no such idea has been implied}.

Hence, of the given choices, Option B is the correct answer.

Options A and C are economically sound decisions. Option B might be a viable choice, as well. However, Option D would not be an economically sound decision for a small purchase in the local market that is worth one-eighth of a bolt of cloth. Concerning small transactions, the author highlights the limitations that stemmed from a textile form of currency: "...Furthermore, a full bolt had a particular value. If consumers cut textiles into smaller pieces to buy or sell something worth less than a full bolt, that, too, greatly lessened the value of the textiles..." Thus, cutting the cloth would greatly diminish its value {and consequently cannot be considered as a sensible decision}. Hence, Option D is the correct answer.

The author evidently highlights the stability of utilizing textiles given its abundant supply{"...textile production was widespread and there were fewer problems with the supply of textiles..."}, the steady standard of measurement {"...the dimensions of a bolt of silk held remarkably steady from the third to the tenth century: 56 cm wide and 12 m long..."} and reliable quality {"...Furthermore, a full bolt had a particular value. If consumers ..."}. Transportation is not a dominant variable that is considered in the discussion pertaining to textiles. Hence, Option A is the correct choice.

Let us inspect the individual statements:

Option A: The author does not make any such claim. The following is stated in this regard: "...Grain, because it rotted easily, was not used nearly as much as coins and textiles...". Thus, the marginal usage of grains is presented relative to that of coins and textiles. Therefore, it cannot be understood that perishable currencies such as grains were not utilised for official work. 

Option B: The author simply states that "...Coins did have certain advantages: they were durable, recognisable and provided a convenient medium of exchange, especially for smaller transactions..." Presenting them as being more valuable than textile currency would be incorrect. 

Option C: This would be imprecise as the author portrays the following: "...Grain, because it rotted easily, was not used nearly as much as coins and textiles, but taxpayers were required to pay grain to the government as a share of their annual tax obligations, and official salaries were expressed in weights of grain..." The statement here deviates from this depiction.

Option D: This statement can be understood from the final paragraph, wherein the author states: "...In actuality, our own currency system today has some similarities even as it is changing in front of our eyes. . . . We have cash - coins for small transactions like paying for parking at a meter, and banknotes for other items; cheques...". Hence, Option D is a valid inference

Let us pay heed to the following excerpt: "...In actuality, our own currency system today has some similarities even as it is changing in front of our eyes. . . . We have cash - coins for small transactions like paying for parking at a meter, and banknotes for other items; cheques and debit/credit cards for other, often larger, types of payments. At the same time, we are shifting to electronic banking and making payments online. Some young people never use cash [and] do not know how to write a cheque..." 

Based on the above, we can infer Options A, B and C as the similarities that are implied. Concerning Option D, while the author does mention the present currency system as a changing/dynamic one, the same cannot be said about the Tang dynasty. The author does not highlight this element as a similarity.  Hence, Option D is the correct answer.

We need to find a statement, which if false, aligns with the discussion/serves as a supporting argument. Let us inspect the individual options:

Option A: In its current form, this statement is in tune with the author's assertion. However, if false, it is antagonistic to the claim being made in the passage. Hence, we can eliminate this option.

Options B and C: Regardless of whether these statements are true or false, they do very little to further the idea presented in the passage.

Option D: This statement implies that complete sentences do not need nouns and verbs. However, in the passage, the author says otherwise; thus, if this sentence is false, it perfectly aligns with the argument made in the passage. Therefore, Option D,  if false, could be seen as a supplementary argument.

The author intends to highlight that the rudimentary combination of a noun and a verb serves as the simplest form of expression; two basic yet immensely significant entities coupled together that is representative of a broader and perhaps, complex group of entities {a sentence}. Option A is closest to such a relationship:  vegetables and spices (two elements) combined to represent a larger group - 'dishes'. The same cannot be said about the rest of the options, since they evidently deviate from the core message being conveyed.

The author begins by highlighting the necessity for a set of codes (enabled by grammar) to organize communication and avoid confusion. He then proceeds to present supplementary arguments in this regard (elements associated with rhetoric and its specialists) and emphasizes how even proper, intentional simplification can be attained only through a firm grasp of the rudiments of grammar. Therefore, it can be observed that grammar is the focal point here and the correct choice should definitely align with this. Option B aptly captures the main concern raised in the passage. Options A and C fail to include the idea revolving around grammar and instead focuses on the additives. Option D is close; however, the mention of nouns and verbs is with the intention to supplement the idea highlighted in B. These simply serve as an illustration to emphasize the significance of grammar. Thus, between the two options B and D, Option B is the suitable choice.

Option A: This can be inferred from "...Must you write complete sentences each time, every time? Perish the thought. If your work consists only of fragments and floating clauses, the Grammar Police aren’t going to come and take you away..." The author presents an illustration to show how a simple combination of a noun and verb forms a complete expression.

Option B: Based on the limited information available from the passage, we can make this inference from "...no group of words can be a sentence, since a sentence is, by definition, a group of words containing a subject (noun) and a predicate (verb)..."

Option C: The author does not make any such claim. Grammar serves as a mechanism to organize communication and avoid confusion. However, the "primary purpose" of grammar is not to ensure that sentences remain "simple". Since we cannot infer this statement from the passage, it is the correct answer.

Option D: Although not explicitly mentioned, we can understand the sentiment behind using the term "Grammar Police". The author is using this as a metaphor for the strong adherents of the grammatic rules (who are perhaps swift to judge and criticize).

Therefore, we can infer all of the statements except for the one in Option C.

In this question, we need to identify the statement that coincides with the aspects discussed by the author. Putting ourselves in the author's shoes, we know that grammar (and the necessity to appropriately learn it) is the primary idea that needs to be conveyed. Writers with a substantial understanding of the elementary rules in grammar can appreciate the "comforting simplicity at its heart" (is what the author claims). Thus, the author will surely support any stance that concurs with the above.

Option A: This will definitely supplement the assertion made by the author. It will enable writers with the requisite understanding of the standard governing rules - a significant necessity highlighted by the author. He is bound to favor such an action. 

Option B: The author would endorse such drastic measures (does not match the tone). The pliability of grammatical rules is a noticeable comment made by the author (but only for those well-versed with it). Hence, we can eliminate this option.

Option C: The author does not portray any view that promotes the eschewal of rhetoric. This again deviates from the discussion in the passage and can be scrapped as the correct choice.

Option D: The focus is broad: on grammar (not just on punctuation and capitalization). 

It is evident that Option A is the only sensible statement that the author would support here.  

Let us inspect the individual statements:

Option A: This can be inferred from the second paragraph ["...It was precisely on the more recently colonized islands where Le Boeuf found that the tempos of the male vocal displays showed stronger differences to the ones from Isla Guadalupe, the founder colony..."]. The inception of the eventual dialects has been indirectly attributed to the dynamic changes that occurred as a result of the near extinction of the elephant seals. Hence, we can infer Option A from the passage.

Option B: This can be inferred from the fourth paragraph: ["...At the individual level, the pulse of the calls stayed the same: A male would maintain his vocal signature throughout his lifetime..."] The changing variables have little to no effect of the individual vocal signature of the elephant seals. Thus, Option B can be inferred from the passage.

Option C: No such claim is made in the passage.

Option D: This can be inferred from ["...could have been responsible for this increase, as in the early 1970s, 43 per cent of the males on Año Nuevo had come from southern rookeries that had a faster pulse rate..."] and the discussion at the beginning of the final paragraph: ["...As the population continued to expand and the islands kept on receiving immigrants from the original population, the calls in all locations would have eventually regressed to the average pulse rate of the founder colony...].

Therefore, of the given statements, we cannot infer Option C.

A noticeable clue to ensure that the male northern elephant seal dialects did not disappear is presented in the penultimate paragraph: ["...At the individual level, the pulse of the calls stayed the same: A male would maintain his vocal signature throughout his lifetime. But the average pulse rate was changing. Immigration could have been responsible for this increase, as in the early 1970s, 43 per cent of the males on Año Nuevo had come from southern rookeries that had a faster pulse rate... "]. The loss in the dialect is due to the influx of seals with a faster pulse rate and "as the population continued to expand and the islands kept on receiving immigrants from the original population, the calls in all locations would have eventually regressed to the average pulse rate of the founder colony." Thus, if the individual pulse rate of the immigrants varies or adapts to the existing population, this could preserve the dialect in a particular region. The statement in Option D reflects this specific idea and helps sustain the existing dialect in a given population. Options A and C do little to contribute to the cause of preventing the disappearance of the dialects.  Option B aligns with the discussion in the passage and is responsible for the regression of the dialects. Hence, Option D is the correct answer.

The following excerpt serves as an essential source for comparing the difference in the attributes of the elephant seals: ["...Yet there are other differences between the males from the late 1960s and their great-great-grandsons: Modern males exhibit more individual diversity, and their calls are more complex. While 50 years ago the drumming pattern was quite simple and the dialects denoted just a change in tempo, Casey explained, the calls recorded today have more complex structures, sometimes featuring doublets or triplets. . . ."]

In the late 1960s, the elephant seal calls were marked by having a simple drumming pattern which later transformed into calls with marked individual diversity and sophistication. Additionally, the dialects that were present in the 1960s were not evident during the study undertaken in the early 2010s, thereby indicating a decrease in the regional variations in the calls. These elements are aptly captured in Option B.

We can make a direct inference based on the following excerpt from the fourth paragraph: ["...This led Le Boeuf and his collaborator, Lewis Petrinovich, to deduce that the dialects were, perhaps, a result of isolation over time, after the breeding sites had been recolonised. For instance, the first settlers of Año Nuevo could have had, by chance, calls with low pulse rates. At other sites, where the scientists found faster pulse rates, the opposite would have happened—seals with faster rates would have happened to arrive first..."]

Based on the above information, the only reason behind the call pulse rate of male northern elephant seals in the southern rookeries being faster would be because the male northern elephant seals of Isla Guadalupe with faster call pulse rates might have been the original settlers of this region. Option A correctly highlights this reason and is, hence, the correct answer.

The passage is focused on delineating the mechanism of reading as perceived by feminists (and the criticism associated with it). The arrangement (1)-(5)-(4)-(2) forms a coherent paragraph and Statement (3) stands out like a sore thumb. While the passage describes the elements associated with a feminist perspective, (3) brings in a description of androcentric literature that does not align with the context. 

The paragraph highlights the following:

1. The validity of the ubiquitous perspective held by psychologists {of intelligence being a measurable, unalterable entity that is greatly influenced by heredity} is now being questioned by biologists.

2. The dubiety concerning the aspect of intelligence being hereditary {given the fact that "humans, who, unlike plants or animals, are not conceived and bred under controlled conditions."}

Thus, a statement capturing these elements is bound to be the answer. Option A aptly encompasses these two key points.

Option B: Calling the widely -held perspective as conventional wisdom would be inappropriate. Additionally, the statement here fails to capture point (2).

Option C: Although close, it misses out on the second half of the discussion.

Option D: This option might appear confusing, given that it touches upon both the key elements. However, it is unspecific and comes across as a bit odd {"ways in which what is inherited" doesn't make sense}. Between Options A and D, A is definitely the better choice.

Statement (1) introduces the predicament we face: the difficulty in understanding the indigenous significance of the nineteenth century San folktales by simply probing the narrative structure. The author indicates that there are additional variables that need to be considered. The aspect of evaluating elements beyond the "the narrative of verbatim" as highlighted in Statement (4). The outcome of such a methodology of exploring these narratives is then presented in the Statements (3) followed by (2). Statement (3) highlights dead-malevolent shamans while (2) talks about benign shamans. Hence, we obtain (1)-(4)-(3)-(2) as a coherent paragraph. 

The passage begins by highlighting Europe's continual attempt to adapt itself in a multipolar world by striving to be a dynamic entity- nationally and internationally{"changed its internal structure and invented new ways of thinking about the nature of international order"}.  Post this, the author portrays how certain stimuli in the modern world has lead Europe to review its political components {"set aside the political mechanisms through which it had conducted its affairs for three and a half centuries"} and to make changes in its economic structure {"established a common currency "}. Thus, the passage presents two key elements: (1) the fact that Europe has consistently tried to adapt to a changing world and (2) the manner in which Europe has attempted to achieve that in the existing multi-polar setup. Option D correctly highlights these points.

Option A: This misses out on the point (2). Furthermore, directly ascribing the unification of Europe to its attempt to rapidly change would be incorrect. 

Option B: This does not fully capture the essence of the passage and misses out on point 1.

Option C: Although the statement in this option captures point 1, it misses out on point 2.

Hence, Option D is the correct answer.

The passage is primarily about the misconception regarding the capacity of forensic phonetics stemming from its portrayal in movies and television. This aspect is correctly captured by the point stated in C. The author does not question the "scientific basis" of the evidence (based on voice recognition) that is presented in legal cases. Hence, Option A can be eliminated. The claim made in Option B cannot be understood from the passage. Same can be said about Option D. Hence, Option C is the correct answer.

Statement (3) introduces the core message: the continuing concerns in the region of South-East Asia. It highlights that since the 1990s there have been dynamic changes in the security environment of this region. Statment (1) adds details by listing the countries present in this region and the existing environment rife with tension/conflict {a transition from the 1990s into the present world}. Statement (2) continues on this line and cites an event in this regard: China becoming a pressing security concern for the regional countries. Statement (4) puts a cap on this discussion by indicating that the persisting regional instability attracts entities from outside the region. Hence, (3)-(1)-(2)-(4) forms a coherent arrangement.

Statements (1), (2), (4) and (5) discuss the aspect of how talking/freedom of speech was a significant facet associated with the slaves. They emphasise how this element was central to slavery  { "slaves could never be completely muted"}. Contrarily, Statment (3) goes on a tangential route that associates the potency of "oaths, orations, and invocations" based on the origin of the slaves {discusses "...a connection to the all-powerful spirit world..." which is clearly out of place}. Thus, Statement (3) is the odd-one-out here.

Statement (1) serves as a general introduction to the topic: an inclination towards using poison {or associated agents}. The author mentions that the utility of this element has existed since the dawn of civilization. Statment (3) supplements this idea by mentioning that certain advancements served as a new avenue for the zealots of this subject. Statement (2) delineates on the safeguard that was put into place once the threat posed by this domain was realized. Statement (4) pinpoints the flaw in this safeguard. Hence, we notice that (1)-(3)-(2)-(4) forms a logical arrangement.

Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This indicates that the Maths score is bound to be the average of the best 3 out of the 4 exam scores obtained by this candidate. Based on this inference, we can proceed with identifying the math score that can be represented as an average of the rest of the scores. We can straightaway eliminate Deep and Esha as potential candidates, given that their Mathematics score is greater than the rest of the exam scores. 

For Alva: best 3 out of 4 - 80(English), 75(Hindi), 75(Science)

Avg. = 230/3 = 76.67 $$\ne\ $$ 70

For Carl: best 3 out of 4 - 80(Hindi), 90(Social Science), 100(Science)

Avg. = 270/3 = 90 which matches the given value 

$$\therefore\ $$ Carl most likely missed his Mathematics examination.

For Foni: best 3 out of 4 - 83(English), 83(Social Science), 88(Science)

Avg. = 254/3 = 84.67 $$\ne\ $$ 78

Hence, we observe that only Carl has missed his Mathematics examination. Hence, Option B is the correct answer.

Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This indicates that the Maths score is bound to be the average of the best 3 out of the 4 exam scores obtained by this candidate. Based on this inference, we can proceed with identifying the math score that can be represented as an average of the rest of the scores. We can straightaway eliminate Deep and Esha as potential candidates, given that their Mathematics score is greater than the rest of the exam scores. After estimating the average scores for the rest of the candidates, we observe that only Carl has missed his Mathematics examination.

For Carl: best 3 out of 4 - 80(Hindi), 90(Social Science), 100(Science)

Avg. = 270/3 = 90 which matches the given value

$$\therefore\ $$ Carl missed his Mathematics examination.

Further, based on Condition III, we can surmise that the student who missed Hindi and Science should have similar average scores in these two subjects. We notice that Alva has the same score of 75 in both Hindi and Science. The same can be said about Deep, who has a score of 90 in both these subjects. Thus, one out of Alva and Deep missed out on Hindi and Science examination, while the second individual missed out only on the Hindi examination. 

Since we know that Carl, Alva and Deep are unlikely to have missed out on the English exam, we can divert our attention to determining which individual out of Bithi, Esha and Foni failed to appear for this subject. However, we notice that Bithi's English score is greater than the rest of her scores, thereby helping us eliminate her as the potential candidate. 

For Esha:  best 3 out of 4 - 85(Hindi), 95(Mathematics), 60(Science)

Avg. = 240/3 = 80 which matches the given value

$$\therefore\ $$ Esha most likely missed her English examination.

For Foni: best 3 out of 4 - 78(Mathematics), 83(Social Science), 88(Science)

Avg. = 249/3 = 83 which matches the given value

$$\therefore\ $$ Foni most likely missed her English examination.

Based on Condition I, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science.

For English, we determined these individuals to be Esha and Foni. For Hindi, we determined these individuals to be Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Given that Carl, Alva and Deep cannot be a part of the group that missed Science or Social Science exam, we can proceed by carefully scrutinizing the rest of the group that includes Bithi, Esha and Foni. 

We notice that Bithi has a similar score in both Science and Social Science examination. Assuming that she did miss these exams, let us proceed to check if this was actually the case.  

For Bithi: Best 2 out 3 - 90(English), 80(Hindi)

Avg = 170/2 = 85 which matches the given value

$$\therefore\ $$ Bithi is likely to have missed her Science and Social Science examinations.

We additionally notice that Foni has a similar score in English and Social Science. On considering the best 2 out of 3 scores, the average value of the score for both the subject holds (equal to 83). Thus, we can conclude that Bithi and Foni missed their Social Science examination.

Thus, the students who missed just one exam were: Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi). 

Hence of the six students, we can correctly determine the missed subjects for four of them (except Alva and Deep):

Mathematics: Carl ; English: Esha & Foni ; Hindi: Alva & Deep; Science: Bithi & one out of Alva and Deep ; Social Science: Foni & Bithi

Hence, the correct answer to this question is Option D: Esha and Foni.

Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This indicates that the Maths score is bound to be the average of the best 3 out of the 4 exam scores obtained by this candidate. Based on this inference, we can proceed with identifying the math score that can be represented as an average of the rest of the scores. We can straightaway eliminate Deep and Esha as potential candidates, given that their Mathematics score is greater than the rest of the exam scores. After estimating the average scores for the rest of the candidates, we observe that only Carl has missed his Mathematics examination.

For Carl: best 3 out of 4 - 80(Hindi), 90(Social Science), 100(Science)

Avg. = 270/3 = 90 which matches the given value

$$\therefore\ $$ Carl missed his Mathematics examination.

Further, based on Condition III, we can surmise that the student who missed Hindi and Science should have similar average scores in these two subjects. We notice that Alva has the same score of 75 in both Hindi and Science. The same can be said about Deep, who has a score of 90 in both these subjects. Thus, one out of Alva and Deep missed out on Hindi and Science examination, while the second individual missed out only on the Hindi examination.

Since we know that Carl, Alva and Deep are unlikely to have missed out on the English exam, we can divert our attention to determining which individual out of Bithi, Esha and Foni failed to appear for this subject. However, we notice that Bithi's English score is greater than the rest of her scores, thereby helping us eliminate her as the potential candidate.

For Esha: best 3 out of 4 - 85(Hindi), 95(Mathematics), 60(Science)

Avg. = 240/3 = 80 which matches the given value

$$\therefore\ $$ Esha most likely missed her English examination.

For Foni: best 3 out of 4 - 78(Mathematics), 83(Social Science), 88(Science)

Avg. = 249/3 = 83 which matches the given value

$$\therefore\ $$ Foni most likely missed her English examination.

Based on Condition I, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science.

For English, we determined these individuals to be Esha and Foni. For Hindi, we determined these individuals to be Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Given that Carl, Alva and Deep cannot be a part of the group that missed Science or Social Science exam, we can proceed by carefully scrutinizing the rest of the group that includes Bithi, Esha and Foni.

We notice that Bithi has a similar score in both Science and Social Science examination. Assuming that she did miss these exams, let us proceed to check if this was actually the case.

For Bithi: Best 2 out 3 - 90(English), 80(Hindi)

Avg = 170/2 = 85 which matches the given value

$$\therefore\ $$ Bithi is likely to have missed her Science and Social Science examinations.

We additionally notice that Foni has a similar score in English and Social Science. On considering the best 2 out of 3 scores, the average value of the score for both the subject holds (equal to 83). Thus, we can conclude that Bithi and Foni missed their Social Science examination.

Thus, the students who missed just one exam were: Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi).

Hence of the six students, we can correctly determine the missed subjects for four of them (except Alva and Deep):

Mathematics: Carl ; English: Esha & Foni ; Hindi: Alva & Deep; Science: Bithi & one out of Alva and Deep ; Social Science: Foni & Bithi

Hence, the correct answer to this question is Option B: Alva and Deep.

Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This indicates that the Maths score is bound to be the average of the best 3 out of the 4 exam scores obtained by this candidate. Based on this inference, we can proceed with identifying the math score that can be represented as an average of the rest of the scores. We can straightaway eliminate Deep and Esha as potential candidates, given that their Mathematics score is greater than the rest of the exam scores. After estimating the average scores for the rest of the candidates, we observe that only Carl has missed his Mathematics examination.

For Carl: best 3 out of 4 - 80(Hindi), 90(Social Science), 100(Science)

Avg. = 270/3 = 90 which matches the given value

$$\therefore\ $$ Carl missed his Mathematics examination.

Further, based on Condition III, we can surmise that the student who missed Hindi and Science should have similar average scores in these two subjects. We notice that Alva has the same score of 75 in both Hindi and Science. The same can be said about Deep, who has a score of 90 in both these subjects. Thus, one out of Alva and Deep missed out on Hindi and Science examination, while the second individual missed out only on the Hindi examination.

Since we know that Carl, Alva and Deep are unlikely to have missed out on the English exam, we can divert our attention to determining which individual out of Bithi, Esha and Foni failed to appear for this subject. However, we notice that Bithi's English score is greater than the rest of her scores, thereby helping us eliminate her as the potential candidate.

For Esha: best 3 out of 4 - 85(Hindi), 95(Mathematics), 60(Science)

Avg. = 240/3 = 80 which matches the given value

$$\therefore\ $$ Esha most likely missed her English examination.

For Foni: best 3 out of 4 - 78(Mathematics), 83(Social Science), 88(Science)

Avg. = 249/3 = 83 which matches the given value

$$\therefore\ $$ Foni most likely missed her English examination.

Based on Condition I, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science.

For English, we determined these individuals to be Esha and Foni. For Hindi, we determined these individuals to be Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Given that Carl, Alva and Deep cannot be a part of the group that missed Science or Social Science exam, we can proceed by carefully scrutinizing the rest of the group that includes Bithi, Esha and Foni.

We notice that Bithi has a similar score in both Science and Social Science examination. Assuming that she did miss these exams, let us proceed to check if this was actually the case.

For Bithi: Best 2 out 3 - 90(English), 80(Hindi)

Avg = 170/2 = 85 which matches the given value

$$\therefore\ $$ Bithi is likely to have missed her Science and Social Science examinations.

We additionally notice that Foni has a similar score in English and Social Science. On considering the best 2 out of 3 scores, the average value of the score for both the subject holds (equal to 83). Thus, we can conclude that Bithi and Foni missed their Social Science examination.

Thus, the students who missed just one exam were: Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi).

Hence of the six students, we can correctly determine the missed subjects for four of them (except Alva and Deep):

Mathematics: Carl ; English: Esha & Foni ; Hindi: Alva & Deep; Science: Bithi & one out of Alva and Deep ; Social Science: Foni & Bithi

Hence, the correct answer to this question is Option A: Bithi & one out of Alva and Deep.

Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This indicates that the Maths score is bound to be the average of the best 3 out of the 4 exam scores obtained by this candidate. Based on this inference, we can proceed with identifying the math score that can be represented as an average of the rest of the scores. We can straightaway eliminate Deep and Esha as potential candidates, given that their Mathematics score is greater than the rest of the exam scores. After estimating the average scores for the rest of the candidates, we observe that only Carl has missed his Mathematics examination.

For Carl: best 3 out of 4 - 80(Hindi), 90(Social Science), 100(Science)

Avg. = 270/3 = 90 which matches the given value

$$\therefore\ $$ Carl missed his Mathematics examination.

Further, based on Condition III, we can surmise that the student who missed Hindi and Science should have similar average scores in these two subjects. We notice that Alva has the same score of 75 in both Hindi and Science. The same can be said about Deep, who has a score of 90 in both these subjects. Thus, one out of Alva and Deep missed out on Hindi and Science examination, while the second individual missed out only on the Hindi examination.

Since we know that Carl, Alva and Deep are unlikely to have missed out on the English exam, we can divert our attention to determining which individual out of Bithi, Esha and Foni failed to appear for this subject. However, we notice that Bithi's English score is greater than the rest of her scores, thereby helping us eliminate her as the potential candidate.

For Esha: best 3 out of 4 - 85(Hindi), 95(Mathematics), 60(Science)

Avg. = 240/3 = 80 which matches the given value

$$\therefore\ $$ Esha most likely missed her English examination.

For Foni: best 3 out of 4 - 78(Mathematics), 83(Social Science), 88(Science)

Avg. = 249/3 = 83 which matches the given value

$$\therefore\ $$ Foni most likely missed her English examination.

Based on Condition I, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science.

For English, we determined these individuals to be Esha and Foni. For Hindi, we determined these individuals to be Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Given that Carl, Alva and Deep cannot be a part of the group that missed Science or Social Science exam, we can proceed by carefully scrutinizing the rest of the group that includes Bithi, Esha and Foni.

We notice that Bithi has a similar score in both Science and Social Science examination. Assuming that she did miss these exams, let us proceed to check if this was actually the case.

For Bithi: Best 2 out 3 - 90(English), 80(Hindi)

Avg = 170/2 = 85 which matches the given value

$$\therefore\ $$ Bithi is likely to have missed her Science and Social Science examinations.

We additionally notice that Foni has a similar score in English and Social Science. On considering the best 2 out of 3 scores, the average value of the score for both the subject holds (equal to 83). Thus, we can conclude that Bithi and Foni missed their Social Science examination.

Thus, the students who missed just one exam were: Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi).

Hence of the six students, we can correctly determine the missed subjects for four of them (except Alva and Deep):

Mathematics: Carl ; English: Esha & Foni ; Hindi: Alva & Deep; Science: Bithi & one out of Alva and Deep ; Social Science: Foni & Bithi

Hence, the correct answer to this question is 3. {Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi)}

Based on Condition II, we understand that the student who missed the Mathematics examination did not miss any other examination. This indicates that the Maths score is bound to be the average of the best 3 out of the 4 exam scores obtained by this candidate. Based on this inference, we can proceed with identifying the math score that can be represented as an average of the rest of the scores. We can straightaway eliminate Deep and Esha as potential candidates, given that their Mathematics score is greater than the rest of the exam scores. After estimating the average scores for the rest of the candidates, we observe that only Carl has missed his Mathematics examination.

For Carl: best 3 out of 4 - 80(Hindi), 90(Social Science), 100(Science)

Avg. = 270/3 = 90 which matches the given value

$$\therefore\ $$ Carl missed his Mathematics examination.

Further, based on Condition III, we can surmise that the student who missed Hindi and Science should have similar average scores in these two subjects. We notice that Alva has the same score of 75 in both Hindi and Science. The same can be said about Deep, who has a score of 90 in both these subjects. Thus, one out of Alva and Deep missed out on Hindi and Science examination, while the second individual missed out only on the Hindi examination.

Since we know that Carl, Alva and Deep are unlikely to have missed out on the English exam, we can divert our attention to determining which individual out of Bithi, Esha and Foni failed to appear for this subject. However, we notice that Bithi's English score is greater than the rest of her scores, thereby helping us eliminate her as the potential candidate.

For Esha: best 3 out of 4 - 85(Hindi), 95(Mathematics), 60(Science)

Avg. = 240/3 = 80 which matches the given value

$$\therefore\ $$ Esha most likely missed her English examination.

For Foni: best 3 out of 4 - 78(Mathematics), 83(Social Science), 88(Science)

Avg. = 249/3 = 83 which matches the given value

$$\therefore\ $$ Foni most likely missed her English examination.

Based on Condition I, we know that exactly two candidates missed the examinations for English, Hindi, Science, and Social Science.

For English, we determined these individuals to be Esha and Foni. For Hindi, we determined these individuals to be Alva and Deep. For Science, we know one of the individuals is either Alva or Deep. Given that Carl, Alva and Deep cannot be a part of the group that missed Science or Social Science exam, we can proceed by carefully scrutinizing the rest of the group that includes Bithi, Esha and Foni.

We notice that Bithi has a similar score in both Science and Social Science examination. Assuming that she did miss these exams, let us proceed to check if this was actually the case.

For Bithi: Best 2 out 3 - 90(English), 80(Hindi)

Avg = 170/2 = 85 which matches the given value

$$\therefore\ $$ Bithi is likely to have missed her Science and Social Science examinations.

We additionally notice that Foni has a similar score in English and Social Science. On considering the best 2 out of 3 scores, the average value of the score for both the subject holds (equal to 83). Thus, we can conclude that Bithi and Foni missed their Social Science examination.

Thus, the students who missed just one exam were: Carl (Mathematics); Esha (English) and one out of Alva and Deep (Hindi).

Hence of the six students, we can correctly determine the missed subjects for four of them (except Alva and Deep):

Mathematics: Carl ; English: Esha & Foni ; Hindi: Alva & Deep; Science: Bithi & one out of Alva and Deep ; Social Science: Foni & Bithi

Except for Alva and Deep, we can conclusively comment of the missed subjects of the rest four. Hence, the correct answer is 4.

Based on the given information, we can form the following Venn-diagram for ease of understanding: 

The conditions help us to further bifurcate the individuals based on their expertise.

Mridangam: A and B (condition 1); one out of F and G (condition 3)

Tabla: one out of A and B (condition 1); F and G (condition 3); D (condition 2)

Ghatam: D (condition 2)

Based on condition 4, we infer that 'I' and 'J' are either experts in Ghatam or Mridangam. However, condition 5 adds that 'I' is not an expert in Mridangam. This helps us definitively zero-in on 'I' as an expert in Ghatam. Since 'I' is a Ghatam expert, J is an expert in Mridangam and 'H' is an expert in Tabla [based on conditions 4 and 5]. Thus, we can depict our understanding so far as follows:

Mridangam [total: 5] - A and B (condition 1); one out of F and G (condition 3); J (condition 4 and 5); one out of C and E {remaining experts}

Tabla [total: 6] - one out of A and B (condition 1); F and G (condition 3); D (condition 2); H (condition 4 and 5); one out of C and E {remaining experts} 

Ghatam [total: 2] - D (condition 2); I (condition 4 and 5)

Thus, we observe that H is definitely an expert in tabla but not in either mridangam or ghatam. Hence, Option D is the correct answer.

Based on the given information, we can form the following Venn-diagram for ease of understanding: 

The conditions help us to further bifurcate the individuals based on their expertise.

Mridangam: A and B (condition 1); one out of F and G (condition 3)

Tabla: one out of A and B (condition 1); F and G (condition 3); D (condition 2)

Ghatam: D (condition 2)

Based on condition 4, we infer that 'I' and 'J' are either experts in Ghatam or Mridangam. However, condition 5 adds that 'I' is not an expert in Mridangam. This helps us definitively zero-in on 'I' as an expert in Ghatam. Since 'I' is a Ghatam expert, J is an expert in Mridangam and 'H' is an expert in Tabla [based on conditions 4 and 5]. Thus, we can depict our understanding so far as follows:

Mridangam [total: 5] - A and B (condition 1); one out of F and G (condition 3); J (condition 4 and 5); one out of C and E {remaining experts}

Tabla [total: 6] - one out of A and B (condition 1); F and G (condition 3); D (condition 2); H (condition 4 and 5); one out of C and E {remaining experts} 

Ghatam [total: 2] - D (condition 2); I (condition 4 and 5)

Thus, we observe that J is definitely an expert in mridangam but not in either tabla or ghatam. Hence, Option B is the correct answer.

Based on the given information, we can form the following Venn-diagram for ease of understanding: 

The conditions help us to further bifurcate the individuals based on their expertise.

Mridangam: A and B (condition 1); one out of F and G (condition 3)

Tabla: one out of A and B (condition 1); F and G (condition 3); D (condition 2)

Ghatam: D (condition 2)

Based on condition 4, we infer that 'I' and 'J' are either experts in Ghatam or Mridangam. However, condition 5 adds that 'I' is not an expert in Mridangam. This helps us definitively zero-in on 'I' as an expert in Ghatam. Since 'I' is a Ghatam expert, J is an expert in Mridangam and 'H' is an expert in Tabla [based on conditions 4 and 5]. Thus, we can depict our understanding so far as follows:

Mridangam [total: 5] - A and B (condition 1); one out of F and G (condition 3); J (condition 4 and 5); one out of C and E {remaining experts}

Tabla [total: 6] - one out of A and B (condition 1); F and G (condition 3); D (condition 2); H (condition 4 and 5); one out of C and E {remaining experts} 

Ghatam [total: 2] - D (condition 2); I (condition 4 and 5)

We observe that the pair C and E cannot have any musician who is an expert in both tabla and mridangam but not in ghatam. Hence, Option B is the correct answer.

Based on the given information, we can form the following Venn-diagram for ease of understanding: 

The conditions help us to further bifurcate the individuals based on their expertise.

Mridangam: A and B (condition 1); one out of F and G (condition 3)

Tabla: one out of A and B (condition 1); F and G (condition 3); D (condition 2)

Ghatam: D (condition 2)

Based on condition 4, we infer that 'I' and 'J' are either experts in Ghatam or Mridangam. However, condition 5 adds that 'I' is not an expert in Mridangam. This helps us definitively zero-in on 'I' as an expert in Ghatam. Since 'I' is a Ghatam expert, J is an expert in Mridangam and 'H' is an expert in Tabla [based on conditions 4 and 5]. Thus, we can depict our understanding so far as follows:

Mridangam [total: 5] - A and B (condition 1); one out of F and G (condition 3); J (condition 4 and 5); one out of C and E {remaining experts}

Tabla [total: 6] - one out of A and B (condition 1); F and G (condition 3); D (condition 2); H (condition 4 and 5); one out of C and E {remaining experts} 

Ghatam [total: 2] - D (condition 2); I (condition 4 and 5)

If C is an expert in Mridangam, E has to be an expert in Tabla. Additionally, if F is not an expert in Mridangam, he has to be an expert only in Tabla while G will be an expert in both Tabla and Mridangam.

Tabla [total: 6] - one out of A and B (condition 1); F and G (condition 3); D (condition 2); H (condition 4 and 5);  E {based on the condition given in the question}

Of these experts, A, B and G are experts of Mridangam and D is an expert of Ghatam as well. Thus, excluding these, we have  F, H and E who are the experts solely in Tabla. Thus, Option A is the correct answer.

Based on the given conditions, Damodaran misses out on the bonus if he gets a rating of 1 in any of the five parameters. He additionally needs to obtain a rating of 5 in at least one of the parameters. Thus, the maximum value of the range of ratings that he can acquire would be 1+3(given)+5+5+5. However, based on condition 2, he can have similar ratings in only two of the parameters. Thus, the maximum value of the final rating would be (1+3+5+5+4) /5 = 18/5 = 3.6. Hence, Option C is the correct answer. 

Since Eman got a bonus, he must have obtained a rating of 5 in at least one of the parameters. To minimize his final rating we need to consider the following range of values: 5(mandatory)+2(given)+2+3+3 = 15. The least value of his final rating is, therefore, 15/5 = 3. Hence, Option D is the correct answer. 

Our objective here is to minimize the final ratings in order to find the minimum value of the monthly payment. We cannot have a rating of 1 in any of the parameters since all the drives got the bonus and we need to have at least one parameter with a rating of 5. With this understanding, we obtain the following:

Arun:

Final rating = (5+4+2+2+3)/5 = 16/5 = 3.2; Fixed payment = Rs.1000

Variable payment = 3.2 * Rs. 250 = Rs.800 ; Total = Rs.(1000+800) = Rs. 1800 

Barun:

Final rating = (5+3+2+2+3)/5 = 15/5 = 3; Fixed payment = Rs.1200

Variable payment = 3 * Rs. 200 = Rs.600 ; Total = Rs.(1200+600) = Rs. 1800

Chandan: {rating of 5 in exactly two parameters based on condition 1}

Final rating = (5+5+2+2+3)/5 = 17/5 = 3.4; Fixed payment = Rs.1400

Variable payment = 3.4 * Rs. 100 = Rs.340 ; Total = Rs.(1400+340) = Rs. 1740

Damodaran:

Final rating = (5+3+2+2+3)/5 = 15/5 = 3; Fixed payment = Rs.1300

Variable payment = 3 * Rs. 150 = Rs.450 ; Total = Rs.(1300+450) = Rs. 1750

Eman:

Final rating = (5+3+2+2+3)/5 = 15/5 = 3; Fixed payment = Rs.1100

Variable payment = 3 * Rs. 200 = Rs.600 ; Total = Rs.(1100+600) = Rs. 1700

Hence, we observe that the minimum value of the monthly payment is Rs. 1700. Option D is the correct answer.

Our objective here is to maximize the final ratings in order to find the maximum possible value of the monthly payment. We cannot have a rating of 1 in any of the parameters since all the drives got the bonus and we need to have at least one parameter with a rating of 5. With this understanding, we obtain the following:

Arun: {can have only one rating of 5 based on condition 1}

Final rating = (5+4+4+3+3)/5 = 19/5 = 3.8; Fixed payment = Rs.1000

Variable payment = 3.8 * Rs. 250 = Rs.950 ; Total = Rs.(1000+950) = Rs. 1950

Barun: {can have only one rating of 5 based on condition 1}

Final rating = (5+4+4+3+3)/5 = 19/5 = 3.8; Fixed payment = Rs.1200

Variable payment = 3.8 * Rs. 200 = Rs.760 ; Total = Rs.(1200+760) = Rs. 1960

Chandan:

Final rating = (5+5+2+4+4)/5 = 20/5 = 4; Fixed payment = Rs.1400

Variable payment = 4 * Rs. 100 = Rs.400 ; Total = Rs.(1400+400) = Rs. 1800

Damodaran:

Final rating = (5+5+4+4+3)/5 = 21/5 = 4.2; Fixed payment = Rs.1300

Variable payment = 4.2 * Rs. 150 = Rs.630 ; Total = Rs.(1300+630) = Rs. 1930

Eman:

Final rating = (5+5+2+4+4)/5 = 20/5 = 4; Fixed payment = Rs.1100

Variable payment = 4 * Rs. 200 = Rs.800 ; Total = Rs.(1100+800) = Rs. 1900

Hence, we observe that the maximum possible value of the monthly payment is Rs. 1960. Option A is the correct answer.

Of the 1000 subjects, only 500 have been considered for the treatment. This constitutes our sample set. Thus the four drugs- A, B, C and D have been administered to this set of 500 individuals, while the rest 500 have been given the placebo. Based on the given information, we can then draw the following 4-set Venn diagram:

We can solve for the number of patients who were administered the drugs A, B and D excluding C by putting in the values for set A. The required value = 250 - (25+20+30+40+20+50+35) = 30. Based on condition (c), we know that 100 patients were treated with exactly three types of medicines. Thus, we can fill the slot for the number of patients who were administered only B, C and D excluding A by 100 - (40+20+30) = 10. 

 Similarly, based on condition (f), we know that the candidates who were administered only dug B are 75 - (25+20+10) = 20. Post this, we can easily calculate the number of people administered with only drugs B and C by 210 - (30+20+40+50+10+20+20) = 20. We can fill in the above values to obtain the following diagram:

The sum of all the values should add up to 500. On solving for 'x' [which represents the number of people who were administered drugs B and D only], we obtain x = 150. The final representation would appear as follows:

Based on the above, the number of patients who were treated with medicine type B is equal to 340.

Of the 1000 subjects, only 500 have been considered for the treatment. This constitutes our sample set. Thus the four drugs- A, B, C and D have been administered to this set of 500 individuals, while the rest 500 have been given the placebo. Based on the given information, we can then draw the following 4-set Venn diagram:

We can solve for the number of patients who were administered the drugs A, B and D excluding C by putting in the values for set A. The required value = 250 - (25+20+30+40+20+50+35) = 30. Based on condition (c), we know that 100 patients were treated with exactly three types of medicines. Thus, we can fill the slot for the number of patients who were administered only B, C and D excluding A by 100 - (40+20+30) = 10. 

 Similarly, based on condition (f), we know that the candidates who were administered only dug B are 75 - (25+20+10) = 20. Post this, we can easily calculate the number of people administered with only drugs B and C by 210 - (30+20+40+50+10+20+20) = 20. We can fill in the above values to obtain the following diagram:

The sum of all the values should add up to 500. On solving for 'x' [which represents the number of people who were administered drugs B and D only], we obtain x = 150. The final representation would appear as follows:

The number of patients who were treated with medicine types B, C and D, but not type A was: 10.

Of the 1000 subjects, only 500 have been considered for the treatment. This constitutes our sample set. Thus the four drugs- A, B, C and D have been administered to this set of 500 individuals, while the rest 500 have been given the placebo. Based on the given information, we can then draw the following 4-set Venn diagram:

We can solve for the number of patients who were administered the drugs A, B and D excluding C by putting in the values for set A. The required value = 250 - (25+20+30+40+20+50+35) = 30. Based on condition (c), we know that 100 patients were treated with exactly three types of medicines. Thus, we can fill the slot for the number of patients who were administered only B, C and D excluding A by 100 - (40+20+30) = 10. 

 Similarly, based on condition (f), we know that the candidates who were administered only dug B are 75 - (25+20+10) = 20. Post this, we can easily calculate the number of people administered with only drugs B and C by 210 - (30+20+40+50+10+20+20) = 20. We can fill in the above values to obtain the following diagram:

The sum of all the values should add up to 500. On solving for 'x' [which represents the number of people who were administered drugs B and D only], we obtain x = 150. The final representation would appear as follows:

The number of people who were administered drugs B and D only were 150.

Of the 1000 subjects, only 500 have been considered for the treatment. This constitutes our sample set. Thus the four drugs- A, B, C and D have been administered to this set of 500 individuals, while the rest 500 have been given the placebo. Based on the given information, we can then draw the following 4-set Venn diagram:

We can solve for the number of patients who were administered the drugs A, B and D excluding C by putting in the values for set A. The required value = 250 - (25+20+30+40+20+50+35) = 30. Based on condition (c), we know that 100 patients were treated with exactly three types of medicines. Thus, we can fill the slot for the number of patients who were administered only B, C and D excluding A by 100 - (40+20+30) = 10. 

 Similarly, based on condition (f), we know that the candidates who were administered only dug B are 75 - (25+20+10) = 20. Post this, we can easily calculate the number of people administered with only drugs B and C by 210 - (30+20+40+50+10+20+20) = 20. We can fill in the above values to obtain the following diagram:

The sum of all the values should add up to 500. On solving for 'x' [which represents the number of people who were administered drugs B and D only], we obtain x = 150. The final representation would appear as follows:

The number of patients who were treated with medicine type D was 325.

From IV: A, B, C, D have one 3, 7, 3, 4-year contract respectively and all other contracts are one-year contracts.

From I, Z has at least one contract every year, the only possible combination is 7+3 or 7+4 year contract and that 7-year contract must be from B.

From III, Vendor W had contracts only in 2012 and from VII, Institutes B and D each had exactly one contract in 2012 => W has got contracts from A and C.
From II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that and from VI, VII: C and D didn't have any contract in 2011 and 2010 respectively => A should have X as a 3-year contract from 2010-2012. Now, for 2013-2015 X can't have B for the same. So, X must have got contracts from either C or D in that period.

Case 1:

X has C as a 3-year contract from 2013-2015 but in this case, D can't have any contract in 2012 so, this case is not valid.

Case 2:

X has D for a 4-year contract from 2012-2015 and C must have Z for a three-year contract in the period 2017-2019 such that Z has at least one contract every year.

It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes => A hasn't got any contract from 2013-2019 as it has X, W in the period 2010-2012 and similarly, C shouldn't have any contracts in the years 2010, 2013, 2014, 2015, 2016.

From III, Vendor Y had contracts in 2010 and 2019 and in 2010 D and C hasn't got any contract and A has already got 2 different contracts from two different vendors => Y has a contract from B in 2010 => B hasn't got any contracts in 2017, 2018, 2019.

For Y the only possible contract will be from D => D has got no contracts in the years 2011, 2016, 2017, 2018.

Now, the table looks like:

'N' represents no contract.

Out of the given options, only 2015 has two contracts and rest have only one contract in that particular year.

From IV: A, B, C, D have one 3, 7, 3, 4-year contract respectively and all other contracts are one-year contracts.

From I, Z has at least one contract every year, the only possible combination is 7+3 or 7+4 year contract and that 7-year contract must be from B.

From III, Vendor W had contracts only in 2012 and from VII, Institutes B and D each had exactly one contract in 2012 => W has got contracts from A and C.
From II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that and from VI, VII: C and D didn't have any contract in 2011 and 2010 respectively => A should have X as a 3-year contract from 2010-2012. Now, for 2013-2015 X can't have B for the same. So, X must have got contracts from either C or D in that period.

Case 1:

X has C as a 3-year contract from 2013-2015 but in this case, D can't have any contract in 2012 so, this case is not valid.

Case 2:

X has D for a 4-year contract from 2012-2015 and C must have Z for a three-year contract in the period 2017-2019 such that Z has at least one contract every year.

It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes => A hasn't got any contract from 2013-2019 as it has X, W in the period 2010-2012 and similarly, C shouldn't have any contracts in the years 2010, 2013, 2014, 2015, 2016.

From III, Vendor Y had contracts in 2010 and 2019 and in 2010 D and C hasn't got any contract and A has already got 2 different contracts from two different vendors => Y has a contract from B in 2010 => B hasn't got any contracts in 2017, 2018, 2019.

For Y the only possible contract will be from D => D has got no contracts in the years 2011, 2016, 2017, 2018.

Now, the table looks like:

'N' represents no contract.

Option D is true. 

From IV: A, B, C, D have one 3, 7, 3, 4-year contract respectively and all other contracts are one-year contracts.

From I, Z has at least one contract every year, the only possible combination is 7+3 or 7+4 year contract and that 7-year contract must be from B.

From III, Vendor W had contracts only in 2012 and from VII, Institutes B and D each had exactly one contract in 2012 => W has got contracts from A and C.
From II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that and from VI, VII: C and D didn't have any contract in 2011 and 2010 respectively => A should have X as a 3-year contract from 2010-2012. Now, for 2013-2015 X can't have B for the same. So, X must have got contracts from either C or D in that period.

Case 1:

X has C as a 3-year contract from 2013-2015 but in this case, D can't have any contract in 2012 so, this case is not valid.

Case 2:

X has D for a 4-year contract from 2012-2015 and C must have Z for a three-year contract in the period 2017-2019 such that Z has at least one contract every year.

It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes => A hasn't got any contract from 2013-2019 as it has X, W in the period 2010-2012 and similarly, C shouldn't have any contracts in the years 2010, 2013, 2014, 2015, 2016.

From III, Vendor Y had contracts in 2010 and 2019 and in 2010 D and C hasn't got any contract and A has already got 2 different contracts from two different vendors => Y has a contract from B in 2010 => B hasn't got any contracts in 2017, 2018, 2019.

For Y the only possible contract will be from D => D has got no contracts in the years 2011, 2016, 2017, 2018.

Now, the table looks like:

'N' represents no contract.

Only during 2016, 2017 and 2018, there was only one contract. 

From IV: A, B, C, D have one 3, 7, 3, 4-year contract respectively and all other contracts are one-year contracts.

From I, Z has at least one contract every year, the only possible combination is 7+3 or 7+4 year contract and that 7-year contract must be from B.

From III, Vendor W had contracts only in 2012 and from VII, Institutes B and D each had exactly one contract in 2012 => W has got contracts from A and C.
From II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that and from VI, VII: C and D didn't have any contract in 2011 and 2010 respectively => A should have X as a 3-year contract from 2010-2012. Now, for 2013-2015 X can't have B for the same. So, X must have got contracts from either C or D in that period.

Case 1:

X has C as a 3-year contract from 2013-2015 but in this case, D can't have any contract in 2012 so, this case is not valid.

Case 2:

X has D for a 4-year contract from 2012-2015 and C must have Z for a three-year contract in the period 2017-2019 such that Z has at least one contract every year.

It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes => A hasn't got any contract from 2013-2019 as it has X, W in the period 2010-2012 and similarly, C shouldn't have any contracts in the years 2010, 2013, 2014, 2015, 2016.

From III, Vendor Y had contracts in 2010 and 2019 and in 2010 D and C hasn't got any contract and A has already got 2 different contracts from two different vendors => Y has a contract from B in 2010 => B hasn't got any contracts in 2017, 2018, 2019.

For Y the only possible contract will be from D => D has got no contracts in the years 2011, 2016, 2017, 2018.

Now, the table looks like:

'N' represents no contract.

The Number of contracts in 2010 is three. 

From IV: A, B, C, D have one 3, 7, 3, 4-year contract respectively and all other contracts are one-year contracts.

From I, Z has at least one contract every year, the only possible combination is 7+3 or 7+4 year contract and that 7-year contract must be from B.

From III, Vendor W had contracts only in 2012 and from VII, Institutes B and D each had exactly one contract in 2012 => W has got contracts from A and C.
From II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that and from VI, VII: C and D didn't have any contract in 2011 and 2010 respectively => A should have X as a 3-year contract from 2010-2012. Now, for 2013-2015 X can't have B for the same. So, X must have got contracts from either C or D in that period.

Case 1:

X has C as a 3-year contract from 2013-2015 but in this case, D can't have any contract in 2012 so, this case is not valid.

Case 2:

X has D for a 4-year contract from 2012-2015 and C must have Z for a three-year contract in the period 2017-2019 such that Z has at least one contract every year.

It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes => A hasn't got any contract from 2013-2019 as it has X, W in the period 2010-2012 and similarly, C shouldn't have any contracts in the years 2010, 2013, 2014, 2015, 2016.

From III, Vendor Y had contracts in 2010 and 2019 and in 2010 D and C hasn't got any contract and A has already got 2 different contracts from two different vendors => Y has a contract from B in 2010 => B hasn't got any contracts in 2017, 2018, 2019.

For Y the only possible contract will be from D => D has got no contracts in the years 2011, 2016, 2017, 2018.

Now, the table looks like:

'N' represents no contract.

B and A have multiple contracts in a single year. 

From IV: A, B, C, D have one 3, 7, 3, 4-year contract respectively and all other contracts are one-year contracts.

From I, Z has at least one contract every year, the only possible combination is 7+3 or 7+4 year contract and that 7-year contract must be from B.

From III, Vendor W had contracts only in 2012 and from VII, Institutes B and D each had exactly one contract in 2012 => W has got contracts from A and C.
From II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that and from VI, VII: C and D didn't have any contract in 2011 and 2010 respectively => A should have X as a 3-year contract from 2010-2012. Now, for 2013-2015 X can't have B for the same. So, X must have got contracts from either C or D in that period.

Case 1:

X has C as a 3-year contract from 2013-2015 but in this case, D can't have any contract in 2012 so, this case is not valid.

Case 2:

X has D for a 4-year contract from 2012-2015 and C must have Z for a three-year contract in the period 2017-2019 such that Z has at least one contract every year.

It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes => A hasn't got any contract from 2013-2019 as it has X, W in the period 2010-2012 and similarly, C shouldn't have any contracts in the years 2010, 2013, 2014, 2015, 2016.

From III, Vendor Y had contracts in 2010 and 2019 and in 2010 D and C hasn't got any contract and A has already got 2 different contracts from two different vendors => Y has a contract from B in 2010 => B hasn't got any contracts in 2017, 2018, 2019.

For Y the only possible contract will be from D => D has got no contracts in the years 2011, 2016, 2017, 2018.

Now, the table looks like:

'N' represents no contract.

A, B, W, X had more than one contracts in a single year

$$2^{Y^2({\log_{3}{5})}}=5^{Y^2(\log_3 2)}$$

Given, $$5^{Y^2\left(\log_32\right)}=5^{\left(\log_23\right)}$$

=> $$Y^2\left(\log_32\right)=\left(\log_23\right)=>Y^2=\left(\log_23\right)^2$$

=>$$Y=\left(-\log_23\right)^{\ }or\ \left(\log_23\right)$$

since Y is a negative number, Y=$$\left(-\log_23\right)=\left(\log_2\frac{1}{3}\right)$$

Let the initial number of chocolates be 64x.

First child gets 32x+1 and 32x-1 are left.

2nd child gets 16x+1/2 and 16x-3/2 are left

3rd child gets 8x+1/4 and 8x-7/4 are left

4th child gets 4x+1/8 and 4x-15/8 are left

5th child gets 2x+1/16 and 2x-31/16 are left.

Given, 2x-31/16=0=> 2x=31/16 => x=31/32.

.'. Initially the Gentleman has 64x i.e. 64*31/32 =62 chocolates.

Let the number be 'abc'. Then, $$2<a\times\ b\times\ c<7$$. The product can be 3,4,5,6.

We can obtain each of these as products with the combination 1,1, x where x = 3,4,5,6. Each number can be arranged in 3 ways, and we have 4 such numbers: hence, a total of 12 numbers fulfilling the criteria.

We can factories 4 as 2*2 and the combination 2,2,1 can be used to form 3 more distinct numbers. 

We can factorize 6 as 2*3 and the combination 1,2,3 can be used to form 6 additional distinct numbers. 

Thus a total of 12 + 3 + 6 = 21 such numbers can be formed.

The graphs of $$2^{x}+2^{-x} and 2-(x-2)^{2}$$ never intersect. So, number of solutions=0.

Alternate method:

We notice that the minimum value of the term in the LHS will be greater than or equal to 2 {at x=0; LHS = 2}. However, the term in the RHS is less than or equal to 2 {at x=2; RHS = 2}. The values of x at which both the sides become 2 are distinct; hence, there are zero real-valued solutions to the above equation.

$$(x^{2}-7x+11)^{(x^{2}-13x+42)}=1$$

if $$(x^{2}-13x+42)$$=0 or $$(x^{2}-7x+11)$$=1 or $$(x^{2}-7x+11)$$=-1 and $$(x^{2}-13x+42)$$ is even number

For x=6,7 the value $$(x^{2}-13x+42)$$=0

$$(x^{2}-7x+11)$$=1 for x=5,2.

$$(x^{2}-7x+11)$$=-1 for x=3,4 and for X=3 or 4, $$(x^{2}-13x+42)$$ is even number.

.'. {2,3,4,5,6,7} is the solution set of x.

.'. x can take six values. 

Let the base radius be 3r.

Height of upper cone is 9 so, by symmetry radius of upper cone will be r.

Volume of frustum=$$\frac{\pi}{3}\left(9r^2\cdot27-r^2.9\right)$$

Volume of upper cone = $$\frac{\pi}{3}.r^2.9$$

Difference= $$\frac{\pi}{3}\cdot9\cdot r^2\cdot25=225$$ => $$\frac{\pi}{3}\cdot r^2=1$$

Volume of larger cone = $$\frac{\pi}{3}\cdot9r^2\cdot27=243$$

The area of the region contained by the lines $$\mid x\mid-y\leq1,y\geq0$$ and $$y\leq1$$ is the white region.

Total area = Area of rectangle + 2 * Area of triangle = $$2+\left(\frac{1}{2}\times\ 2\times\ 1\right)\ =3$$

Hence, 3 is the correct answer.

Let ABCD be the rectangle with length 2l and breadth 2b respectively.

Area of the circle i.e. area of painted region = $$\pi\ b^2$$.

Given, 4lb-$$\pi\ b^2$$=(2/3)$$\pi\ b^2$$.

=> 4lb=(5/3)$$\pi\ b^2$$.

=>l=$$\frac{5\pi}{12}b$$.

Given, 4lb=135 => 4*$$\frac{5\pi}{12}b^2$$=135 => b= $$\frac{9}{\sqrt{\ \pi\ }}$$

=> l=$$\frac{15}{4}\sqrt{\ \pi\ }$$
Perimeter of rectangle =4(l+b)=4($$\frac{15}{4}\sqrt{\ \pi\ }$$+$$\frac{9}{\sqrt{\ \pi\ }}$$ )=$$3\sqrt{\pi}(5+\frac{12}{\pi})$$.

Hence option A is correct. 

Let the length of radius be 'r'.

From the above diagram,

$$x^2+r^2=6^2\ $$....(i)

$$\left(10-x\right)^2+r^2=8^2\ $$----(ii)

Subtracting (i) from (ii), we get: 

x=3.6 => $$r^2=36-\left(3.6\right)^2$$ ==> $$r^2=36-\left(3.6\right)^2\ =23.04$$.

Area of circle = $$\pi\ r^2=23.04\pi\ $$

Area of rhombus= 1/2*d1*d2=1/2*12*16=96.

.'. Ratio of areas = 23.04$$\pi\ $$/96=$$\frac{6\pi}{25}$$

$$x_0=max(x_1,x_2,....,x_{12})$$

$$x_0$$ will be minimum if x1,x2...x12 are close to each other

100/12=8.33

.'. max$$(x_1,x_2,....,x_{12})$$ will be minimum if $$(x_1,x_2,....,x_{12})$$=(9,9,9,9,8,8,8,8,8,8,8,8,)

.'. Option B is correct.

Let the speed of Car 2 be 'x' kmph and the time taken by the two cars to meet be 't' hours.

In 't' hours, Car 1 travels $$\left(60\ \times\ t\right)\ km$$ while Car 2 travels $$\left(x\ \times\ t\right)\ km$$

It is given that the time taken by Car 1 to travel $$\left(x\ \times\ t\right)\ km$$ is 45 minutes or (3/4) hours. $$\therefore\ \frac{\left(x\ \times\ t\right)}{60}\ =\ \frac{3}{4}\ $$ or $$t=\frac{180}{4x}$$....(i)

Similarly, the time taken by Car 2 to travel $$\left(60\ \times\ t\right)\ km$$ is 20 minutes or (1/3) hours. $$\therefore\ \frac{\left(60\times\ t\right)}{x}=\frac{1}{3}$$ or $$\therefore\ t=\frac{x}{180}$$....(ii)

Equating the values in (i) and (ii), and solving for x:

$$\therefore\ \frac{180}{4x}=\frac{x}{180}\ \ \longrightarrow\ \ \ x\ =90\ kmph$$

Hence, Option B is the correct answer.

Let the price of desktop and laptop be x,y respectively.

Given,

x+y=50000...(i)

1.2x+0.9y=50000(1.02)=51000...(ii)

(ii)-0.9(i) gives

0.3x=6000=> x=20000.

Initially the amount of Dye and Water are 16,24 respectively.

To make the ratio of Dye to Water to 2:5 the amount of water should be 40l for 16l of Dye=> 16l of water is added.

Now, the Dye and Water arr 16,40 respectively.

After removing 1/4th of solution the amount of Dye and Water will be 12,30l respectively.

To have Dye and Water in the ratio of 2:3, for 30l of water we need 20l of Dye => 8l of Dye should be added.

Hence , 8 is correct answer. 

$$x=2^{12\left(7+4\sqrt{\ 3}\right)}$$.

$$x^{\frac{7}{2}}=2^{42\left(7+4\sqrt{\ 3}\right)}$$

$$x^{2\sqrt{\ 3}}=2^{24\sqrt{\ 3}\left(7+4\sqrt{\ 3}\right)}$$

$$\frac{x^{\frac{7}{2}}}{x^{2{\sqrt{3}}}}$$ = $$2^{\left(7+4\sqrt{\ 3}\right)\left(42-24\sqrt{\ 3}\right)}=2^{\left(7+4\sqrt{\ 3}\right)\left(7-4\sqrt{\ 3}\right)6}$$ =$$2^6$$.

Hence C is correct answer.

Let the volume of Metals A,B,C we 3x, 4x, 7x

Ratio weights of given volume be 5y,2y,6y

.'. 15xy+8xy+42xy=130 => 65xy=130 => xy=2.

.'.`The weight, in kg. of the metal C is 42xy=84.

Let $$a=x+\frac{1}{x}$$
So, the given equation is $$a^2-3a+2=0$$
So, $$a$$ can be either 2 or 1.

If $$a=1$$, $$x+\frac{1}{x}=1$$ and it has no real roots. 
If $$a=2$$, $$x+\frac{1}{x}=2$$ and it has exactly one real root which is $$x=1$$

So, the total number of distinct real roots of the given equation is 1

$$\frac{\log\ 5}{2\log2}\ =\frac{\log\ y}{2\log2}\cdot\frac{\log\ 5}{2\log6}$$

$$\log\ 36\ =\ \log\ y;\ \therefore\ y\ =36$$

The difference in the time take to traverse the same distance $$'d'$$ at two different speeds is 35 minutes. Equating this: $$\frac{d}{8}-\frac{d}{15}\ =\ \frac{35}{60}$$

On solving, we obtain $$d = 10 kms$$. Let $$x kmph$$ be the speed at which Amal needs to travel to reach the office in 50 minutes; then

$$\frac{10}{x}=\frac{50}{60}\ or\ x\ =\ 12\ kmph$$.Hence, Option B is the correct answer.

Let the total distance be 'D' km and the speed of the train be 's' kmph. The time taken to cover D at speed d is 't' hours. Based on the information: on equating the distance, we get $$s\ \times\ t\ =\ \frac{s}{3}\times\ \left(t+\frac{1}{2}\right)$$ 
On solving we acquire the value of $$\ t\ =\frac{1}{4}$$ or 15 mins. We understand that during the return journey, the first 5 minutes are spent traveling at speed 's' {distance traveled in terms of s = $$\ \frac{s}{12}$$}. Remaining distance in terms of 's' = $$\ \frac{s}{4}-\frac{s}{12}\ =\frac{s}{6}$$

The rest 4 minutes of stoppage added to this initial 5 minutes amounts to a total of 9 minutes. The train has to complete the rest of the journey in $$15 - 9 = 6 mins$$ or {1/10 hours}. Thus, let 'x' kmph be the new value of speed. Based on the above, we get $$\frac{s}{\frac{6}{x}}\ =\frac{1}{10}\ or\ x\ =\frac{10s}{6}$$

Since the increase in speed needs to be calculated: $$\frac{\left(\frac{10s}{6}\ -s\right)}{s}\times\ 100\ =\frac{200}{3}\approx\ 67\%$$ increase.

Hence, Option C is the correct answer.

The four digit even numbers will be of form:

1100, 1122, 1144 ... 1188, 2200, 2222, 2244 ... 9900, 9922, 9944, 9966, 9988

Their sum 'S' will be (1100+1100+22+1100+44+1100+66+1100+88)+(2200+2200+22+2200+44+...)....+(9900+9900+22+9900+44+9900+66+9900+88)

=> S=1100*5+(22+44+66+88)+2200*5+(22+44+66+88)....+9900*5+(22+44+66+88)

=> S=5*1100(1+2+3+...9)+9(22+44+66+88)

=>S=5*1100*9*10/2 + 9*11*20

Total number of numbers are 9*5=45

.'. Mean will be S/45 = 5*1100+44=5544.

Option D

Let the length of the train be $$ l kms$$ and speed be $$ s kmph$$. Base on the two scenarios presented, we obtain:

$$\frac{l}{s-2}=\frac{90}{3600}$$....(i) and $$\frac{l}{s-4}=\frac{100}{3600}$$...(ii)

On dividing (ii) by (i) and simplifying we acquire the value of $$s$$ as $$22 kmph$$. Substituting this value in (i), we have $$l=\frac{90}{3600}\times\ 20\ kms$$ {keeping it in km and hours for convenience}

Since we need to find $$\frac{l}{s}$$, let this be equal to $$x$$. Then, $$x\ =\ 90\times\frac{20}{22}\ =81.81\ \approx\ 82\ \sec onds\ $$

Hence, Option B is the correct choice.

Since $$c<9$$, we can have the following viable combinations for $$b\times\ c\ =96$$ (given our objective is to minimize the sum):

$$48\times\ 2$$ ; $$32\times3$$ ; $$24\times\ 4$$ ; $$16\times6$$ ; $$12\times8$$ 

Similarly, we can factorize $$a\times\ b\ = 432$$ into its factors. On close observation, we notice that $$18\times24\ and\ 24\ \times\ 4\ $$ corresponding to $$a\times b\ and\ b\times\ c\ $$ respectively together render us with the least value of the sum of $$a+b\ +\ c\ \ =\ 18+24+4\ =46$$

Hence, Option D is the correct answer.

Let 'x' be the strength of group G. Based on the information, $$0.65x$$ constitutes of literate people {the rest $$0.35x$$ = illiterate}
Of this $$0.65x$$, 75% are old people =(0.75)0.65x old literates. The total number of old people in group G is $$0.72x$$ {72% of the total}. Thus, the total number of old people who are illiterate = $$0.72x-0.4875x\ =\ 0.2325x$$. This is $$\frac{0.2325x}{0.35x}\times\ 100\ \approx\ \ 66\%$$ of the total number of illiterates. Hence, Option D is the correct answer.

Let their individual Amounts be equal after 't' years. Let their initial investments amount to $$A_V$$ and $$A_J$$ ; 

$$A_V\ =10,000\left(1+\frac{5t}{100}\right)$$ and $$A_J\ =8,000\left(1+\frac{10\left(t-2\right)}{100}\right)$$

Equating both: $$10,000\left(1+\frac{5t}{100}\right)\ =8,000\left(1+\frac{10\left(t-2\right)}{100}\right)$$

On simplifying both sides, we get: $$15t\ =\ 180\ ;\ t\ =\ 12$$

Let 'r' be the root of the function. It follows that f(r) = 0. We can represent this as $$f\left(r\right)=f\left\{5-\left(5-r\right)\right\}$$

Based on the relation: $$f\left(5-x\right)=f\left(5+x\right)$$; $$f\left(r\right)=f\left\{5-\left(5-r\right)\right\}=f\left\{5+\left(5-r\right)\right\}$$

$$\therefore\ f\left(r\right)=f\left(10-r\right)$$

Thus, every root 'r' is associated with another root '(10-r)' [these form a pair]. For even distinct roots, in this case four, let us assume the roots to be as follows: $$r_1,\ \left(10-r_1\right),\ r_2,\ \left(10-r_2\right)$$

The sum of these roots = $$r_1\ +\left(10-r_1\right)+\ r_2+\ \left(10-r_2\right)\ =\ 20$$

Hence, Option D is the correct answer.

Given

A+(B+C)/2=5 => 2A+B+C=10....(i)

(A+C)/2 +B=7 => A+2B+C=14.....(ii)

(i)-(ii)=> B-A=4 => B=4+A.

Given, A, B, C are positive integers

If A=1=>B=5 => C=3

If A=2=>B=6 => C=0 but this is invalid as C is positive.

Similarly if A>2 C will be negative and cases are not valid.

Hence, A+B=6.