List of top Questions asked in CAT

Members: \( K, L, M, N, P, Q, R, S, U, W \) are the only ten members in a department.

There is a proposal to form a team from within the members of the department, subject to the following conditions:

1. A team must include exactly one among \( P, R, S \).
2. A team must include either \( M \) or \( Q \), but not both.
3. If a team includes \( K \), then it must also include \( L \), and vice versa. \[ K \in \text{Team} \iff L \in \text{Team} \]
4. If a team includes one among \( S, U, W \), then it must also include the other two. \[ (S \in T) \lor (U \in T) \lor (W \in T) \Rightarrow S, U, W \in T \]
5. \( L \) and \( N \) cannot be members of the same team.
6. \( L \) and \( U \) cannot be members of the same team.

The size of a team is defined as the number of members in the team.

In a Class X Board examination, ten papers are distributed over five Groups: PCB, Mathematics, Social Science, Vernacular, and English. Each paper is evaluated out of \(100\).

The final score of a student is calculated as follows:

1. First, the Group Scores are obtained by averaging marks in the papers within the group.
2. The Final Score is the simple average of the five Group Scores.

The data for the top ten students are given below. (Dipan’s score in English Paper II is missing.)

Note: (B) or (G) indicates whether the student is a boy or girl.

Mathematicians are assigned a number called Erdős number (named after the famous mathematician, Paul Erdős). Only Paul Erdős himself has an Erdős number of \(0\). Any mathematician who has written a research paper with Erdős has an Erdős number of \(1\). For other mathematicians, the calculation of his/her Erdős number is as follows:

Suppose a mathematician \(X\) has co-authored papers with several other mathematicians. From among them, mathematician \(Y\) has the smallest Erdős number. Let the Erdős number of \(Y\) be \(y\). Then \(X\) has an Erdős number of \(y + 1\). Hence, any mathematician with no co-authorship chain connected to Erdős has an Erdős number of \(\infty\).

In a seven-day long mini-conference organized in memory of Paul Erdős, a close group of eight mathematicians, call them \(A, B, C, D, E, F, G,\) and \(H\), discussed some research problems. At the beginning of the conference:

• \(A\) was the only participant with an Erdős number of \(\infty\).
• Nobody had an Erdős number less than that of \(F\).

Event Timeline

Day 3 Event:

• \(F\) co-authored a paper jointly with \(A\) and \(C\).
• This reduced the average Erdős number of the group of eight mathematicians to \(3\).
• The Erdős numbers of \(B, D, E, G, H\) remained unchanged.
• No other co-authorship among any three members would have reduced the average Erdős number of the group to as low as \(3\).

At the end of Day 3:

• Five members of the group had identical Erdős numbers.
• The other three had Erdős numbers distinct from each other.

Day 5 Event:

• \(E\) co-authored a paper with \(F\).
• This reduced the group’s average Erdős number by \(0.5\).
• The Erdős numbers of the other six members were unchanged.

Note: No other paper was written during the conference.

Two traders, Chetan and Michael, were involved in the buying and selling of MCS shares over five trading days. At the beginning of the first day, the MCS share was priced at Rs 100, while at the end of the fifth day it was priced at Rs 110. At the end of each day, the MCS share price either went up by Rs 10, or else, it came down by Rs 10. Both Chetan and Michael took buying and selling decisions at the end of each trading day. The beginning price of MCS share on a given day was the same as the ending price of the previous day. Chetan and Michael started with the same number of shares and amount of cash, and had enough of both.
Below are some additional facts about how Chetan and Michael traded over the five trading days:

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs 1200 and Rs 2400 respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs 5400. 

Each question has a set of four sequentially ordered statements. Each statement can be classified as one of the following:
• Facts (F): Deal with pieces of information that one has heard, seen, or read, and which are open to discovery or verification. The answer option indicates such a statement with an ‘F’.
• Inferences (I): Conclusions drawn about the unknown, on the basis of the known. The answer option indicates such a statement with an ‘I’.
• Judgements (J): Opinions that imply approval or disapproval of persons, objects, situations, and occurrences in the past, present, or future. The answer option indicates such a statement with a ‘J’. 
Select the answer option that best describes the set of four statements.

My aim is to present a conception of justice which generalizes and carries to a higher level of abstraction the familiar theory of the social contract. In order to do this we are not to think of the original contract as one to enter a particular society or to set up a particular form of government. Rather, the idea is that the principles of justice for the basic structure of society are the object of the original agreement. They are the principles that free and rational persons concerned to further their own interests would accept in an initial position of equality. These principles are to regulate all further agreements; they specify the kinds of social cooperation that can be entered into and the forms of government that can be established. This way of regarding the principles of justice, I shall call justice as fairness. Thus, we are to imagine that those who engage in social cooperation choose together, in one joint act, the principles which are to assign basic rights and duties and to determine the division of social benefits. Just as each person must decide by rational reflection what constitutes his good, that is, the system of ends which it is rational for him to pursue, so a group of persons must decide once and for all what is to count among them as just and unjust. The choice which rational men would make in this hypothetical situation of equal liberty determines the principles of justice.
In justice as fairness, the original position is not an actual historical state of affairs. It is understood as a purely hypothetical situation characterized so as to lead to a certain conception of justice. Among the essential features of this situation is that no one knows his place in society, his class position or social status, nor does anyone know his fortune in the distribution of natural assets and abilities, his intelligence, strength, and the like. I shall even assume that the parties do not know their conceptions of the good or their special psychological propensities. The principles of justice are chosen behind a veil of ignorance. This ensures that no one is advantaged or disadvantaged in the choice of principles by the outcome of natural chance or the contingency of social circumstances. Since all are similarly situated and no one is able to design principles to favor his particular condition, the principles of justice are the result of a fair agreement or bargain.
Justice as fairness begins with one of the most general of all choices which persons might make together, namely, with the choice of the first principles of a conception of justice which is to regulate all subsequent criticism and reform of institutions. Then, having chosen a conception of justice, we can suppose that they are to choose a constitution and a legislature to enact laws, and so on, all in accordance with the principles of justice initially agreed upon. Our social situation is just if it is such that by this sequence of hypothetical agreements, we would have contracted into the general system of rules which defines it. Moreover, assuming that the original position does determine a set of principles, it will then be true that whenever social institutions satisfy these principles, those engaged in them can say to one another that they are cooperating on terms to which they would agree if they were free and equal persons whose relations with respect to one another were fair. They could all view their arrangements as meeting the stipulations which they would acknowledge in an initial situation that embodies widely accepted and reasonable constraints on the choice of principles. The general recognition of this fact would provide the basis for a public acceptance of the corresponding principles of justice. No society can, of course, be a scheme of cooperation which men enter voluntarily in a literal sense; each person finds himself placed at birth in some particular position in some particular society, and the nature of this position materially affects his life prospects. Yet a society satisfying the principles of justice as fairness comes as close as a society can to being a voluntary scheme, for it meets the principles which free and equal persons would assent to under circumstances that are fair

Fifteen years after communism was officially pronounced dead, its spectre seems once again to be haunting Europe. Last month, the Council of Europe’s parliamentary assembly voted to condemn the “crimes of totalitarian communist regimes,” linking them with Nazism and complaining that communist parties are still “legal and active in some countries.” Now Goran Lindblad, the conservative Swedish MP behind the resolution, wants to go further. Demands that European Ministers launch a continent-wide anti-communist campaign — including school textbook revisions, official memorial days, and museums — only narrowly missed the necessary two-thirds majority. Mr. Lindblad pledged to bring the wider plans back to the Council of Europe in the coming months.
He has chosen a good year for his ideological offensive: this is the 50th anniversary of Nikita Khrushchev’s denunciation of Josef Stalin and the subsequent Hungarian uprising, which will doubtless be the cue for further excoriation of the communist record. Paradoxically, given that there is no communist government left in Europe outside Moldova, the attacks have if anything, become more extreme as time has gone on. A clue as to why that might be can be found in the rambling report by Mr. Lindblad that led to the Council of Europe declaration. Blaming class struggle and public ownership, he explained “different elements of communist ideology such as equality or social justice still seduce many” and “a sort of nostalgia for communism is still alive.” Perhaps the real problem for Mr. Lindblad and his right-wing allies in Eastern Europe is that communism is not dead enough — and they will only be content when they have driven a stake through its heart.
The fashionable attempt to equate communism and Nazism is in reality a moral and historical nonsense. Despite the cruelties of the Stalin terror, there was no Soviet Treblinka or Sorbibor, no extermination camps built to murder millions. Nor did the Soviet Union launch the most devastating war in history at a cost of more than 50 million lives — in fact it played the decisive role in the defeat of the German war machine. Mr. Lindblad and the Council of Europe adopt as fact the wildest estimates of those “killed by communist regimes” (mostly in famines) from the fiercely contested Black Book of Communism, which also underplays the number of deaths attributable to Hitler. But, in any case, none of this explains why anyone might be nostalgic in former communist states, now enjoying the delights of capitalist restoration. The dominant account gives no sense of how communist regimes renewed themselves after 1956 or why Western leaders feared they might overtake the capitalist world well into the 1960s. For all its brutalities and failures, communism in the Soviet Union, Eastern Europe, and elsewhere delivered rapid industrialization, mass education, job security, and huge advances in social and gender equality. Its existence helped to drive up welfare standards in the West, and provided a powerful counterweight to Western global domination. 
It would be easier to take the Council of Europe’s condemnation of communist state crimes seriously if it had also seen fit to denounce the far bloodier record of European colonialism —which only finally came to an end in the 1970s. This was a system of racist despotism, which dominated the globe in Stalin’s time. And while there is precious little connection between the ideas of fascism and communism, there is an intimate link between colonialism and Nazism. The terms lebensraum and konzentrationslager were both first used by the German colonial regime in south-west Africa (now Namibia), which committed genocide against the Herero and Nama peoples and bequeathed its ideas and personnel directly to the Nazi party. Around 10 million Congolese died as a result of Belgian forced labour and mass murder in the early twentieth century; tens of millions perished in avoidable or enforced famines in British-ruled India; up to a million Algerians died in their war for independence, while controversy now rages in France about a new law requiring teachers to put a positive spin on colonial history. Comparable atrocities were carried out by all European colonialists, but not a word of condemnation from the Council of Europe. Presumably, European lives count for more.
No major twentieth century political tradition is without blood on its hands, but battles over history are more about the future than the past. Part of the current enthusiasm in official Western circles for dancing on the grave of communism is no doubt about relations with today’s Russia and China. But it also reflects a determination to prove there is no alternative to the new global capitalist order — and that any attempt to find one is bound to lead to suffering. With the new imperialism now being resisted in the Muslim world and Latin America, growing international demands for social justice and ever greater doubts about whether the environmental crisis can be solved within the existing economic system, the pressure for alternatives will increase.

Our propensity to look out for regularities, and to impose laws upon nature, leads to the psychological phenomenon of dogmatic thinking or, more generally, dogmatic behaviour: we expect regularities everywhere and attempt to find them even where there are none; events which do not yield to these attempts we are inclined to treat as a kind of ‘background noise’; and we stick to our expectations even when they are inadequate and we ought to accept defeat. This dogmatism is to some extent necessary. It is demanded by a situation which can only be dealt with by forcing our conjectures upon the world. Moreover, this dogmatism allows us to approach a good theory in stages, by way of approximations: if we accept defeat too easily, we may prevent ourselves from finding that we were very nearly right.
It is clear that this dogmatic attitude, which makes us stick to our first impressions, is indicative of a strong belief; while a critical attitude, which is ready to modify its tenets, which admits doubt and demands tests, is indicative of a weaker belief. Now according to Hume’s theory, and to the popular theory, the strength of a belief should be a product of repetition; thus it should always grow with experience, and always be greater in less primitive persons. But dogmatic thinking, an uncontrolled wish to impose regularities, a manifest pleasure in rites and in repetition as such, is characteristic of primitives and children; and increasing experience and maturity sometimes create an attitude of caution and criticism rather than of dogmatism.
Mylogical criticism of Hume’s psychological theory, and the considerations connected with it, may seem a little removed from the field of the philosophy of science. But the distinction between dogmatic and critical thinking, or the dogmatic and the critical attitude, brings us right back to our central problem. For the dogmatic attitude is clearly related to the tendency to verify our laws and schemata by seeking to apply them and to confirm them, even to the point of neglecting refutations, whereas the critical attitude is one of readiness to change them —to test them; to refute them; to falsify them, if possible. This suggests that we may identify the critical attitude with the scientific attitude, and the dogmatic attitude with the one which we have described as pseudo-scientific. It further suggests that genetically speaking the pseudo-scientific attitude is more primitive than, and prior to, the scientific attitude: that it is a pre-scientific attitude. And this primitivity or priority also has its logical aspect. For the critical attitude is not so much opposed to the dogmatic attitude as super-imposed upon it: criticism must be directed against existing and influential beliefs in need of critical revision — in other words, dogmatic beliefs. A critical attitude needs for its raw material, as it were, theories or beliefs which are held more or less dogmatically.
Thus, science must begin with myths, and with the criticism of myths; neither with the collection of observations, nor with the invention of experiments, but with the critical discussion of myths, and of magical techniques and practices. The scientific tradition is distinguished from the pre-scientific tradition in having two layers. Like the latter, it passes on its theories; but it also passes on a critical attitude towards them. The theories are passed on, not as dogmas, but rather with the challenge to discuss them and improve upon them. The critical attitude, the tradition of free discussion of theories with the aim of discovering their weak spots so that they may be improved upon, is the attitude of reasonableness, of rationality. From the point of view here developed, all laws, all theories, remain essentially tentative, or conjectural, or hypothetical, even when we feel unable to doubt them any longer. Before a theory has been refuted we can never know in what way it may have to be modified.

If $\log_x(a \cdot \log_y) = (b \cdot \log_z) = ab$, then which of the following pairs of values for $(a,b)$ is not possible?