List of top Questions asked in CAT

To discover the relation between rules, paradigms, and normal science, consider first how the historian isolates the particular loci of commitment that have been described as accepted rules. Close historical investigation of a given specialty at a given time discloses a set of recurrent and quasi-standard illustrations of various theories in their conceptual, observational, and instrumental applications. These are the community’s paradigms, revealed in its textbooks, lectures, and laboratory exercises. By studying them and by practicing with them, the members of the corresponding community learn their trade. The historian, of course, will discover in addition a penumbral area occupied by achievements whose status is still in doubt, but the core of solved problems and techniques will usually be clear. Despite occasional ambiguities, the paradigms of a mature scientific community can be determined with relative ease.

That demands a second step and one of a somewhat different kind. When undertaking it, the historian must compare the community’s paradigms with each other and with its current research reports. In doing so, his object is to discover what isolable elements, explicit or implicit, the members of that community may have abstracted from their more global paradigms and deploy as rules in their research. Anyone who has attempted to describe or analyze the evolution of a particular scientific tradition will necessarily have sought accepted principles and rules of this sort. Almost certainly, he will have met with at least partial success. But, if his experience has been at all like my own, he will have found the search for rules both more difficult and less satisfying than the search for paradigms. Some of the generalizations he employs to describe the community’s shared beliefs will present more problems. Others, however, will seem a shade too strong. Phrased in just that way, or in any other way he can imagine, they would almost certainly have been rejected by some members of the group he studies. Nevertheless, if the coherence of the research tradition is to be understood in terms of rules, some specification of common ground in the corresponding area is needed. As a result, the search for a body of rules competent to constitute a given normal research tradition becomes a source of continual and deep frustration.

Recognizing that frustration, however, makes it possible to diagnose its source. Scientists can agree that a Newton, Lavoisier, Maxwell, or Einstein has produced an apparently permanent solution to a group of outstanding problems and still disagree, sometimes without being aware of it, about the particular abstract characteristics that make those solutions permanent. They can, that is, agree in their identification of a paradigm without agreeing on, or even attempting to produce, a full interpretation or rationalization of it. Lack of a standard interpretation or of an agreed reduction to rules will not prevent a paradigm from guiding research. Normal science can be determined in part by the direct inspection of paradigms, a process that is often aided by but does not depend upon the formulation of rules and assumptions. Indeed, the existence of a paradigm need not even imply that any full set of rules exists.

The difficulties historians face in establishing cause-and-effect relations in the history of human societies are broadly similar to the difficulties facing astronomers, climatologists, ecologists, evolutionary biologists, geologists, and palaeontologists. To varying degrees each of these fields is plagued by:

• The impossibility of performing replicated, controlled experimental interventions.
• The complexity arising from enormous numbers of variables.
• The resulting uniqueness of each system.
• The consequent impossibility of formulating universal laws.
• The difficulties of predicting emergent properties and future behaviour.

Prediction in history, as in other historical sciences, is most feasible on large spatial scales and over long times, when the unique features of millions of small-scale brief events become averaged out. Just as one could predict the sex ratio of the next \( 1,000 \) newborns but not the sexes of one's own two children, the historian can recognize factors that made inevitable the broad outcome of the collision between American and Eurasian societies after \( 13,000 \) years of separate developments, but not the outcome of the 1960 U.S. presidential election. The details of which candidate said what during a single televised debate in October 1960 could have given the electoral victory to Nixon instead of to Kennedy, but no details of who said what could have blocked the European conquest of Native Americans.

How can students of human history profit from the experience of scientists in other historical sciences? A methodology that has proved useful involves the comparative method and so-called natural experiments. While neither astronomers studying galaxy formation nor human historians can manipulate their systems in controlled laboratory experiments, they both can take advantage of natural experiments, by comparing systems differing in the presence or absence (or in the strong or weak effect) of some putative causative factor.

For example, epidemiologists, forbidden to feed large amounts of salt to people experimentally, have still been able to identify effects of high salt intake by comparing groups of humans who already differ greatly in their salt intake. Similarly, cultural anthropologists, unable to provide human groups experimentally with varying resource abundances for many centuries, study long-term effects of resource abundance on human societies by comparing recent Polynesian populations living on islands differing naturally in resource abundance.

The student of human history can draw on many more natural experiments than just comparisons among the five inhabited continents. Comparisons can also utilize large islands that have developed complex societies in a considerable degree of isolation (such as Japan, Madagascar, Native American Hispaniola, New Guinea, Hawaii, and many others), as well as societies on hundreds of smaller islands and regional societies within each of the continents.

Natural experiments in any field, whether in ecology or human history, are inherently open to potential methodological criticisms. Those include confounding effects of natural variation in additional variables besides the one of interest, as well as problems in inferring chains of causation from observed correlations between variables. Such methodological problems have been discussed in great detail for some of the historical sciences. In particular, epidemiology—the science of drawing inferences about human diseases by comparing groups of people (often by retrospective historical studies)—has for a long time successfully employed formalized procedures for dealing with problems similar to those facing historians of human societies.

In short, I acknowledge that it is much more difficult to understand human history than to understand problems in fields of science where history is unimportant and where fewer individual variables operate. Nevertheless, successful methodologies for analyzing historical problems have been worked out in several fields. As a result, the histories of dinosaurs, nebulae, and glaciers are generally acknowledged to belong to fields of science rather than to the humanities.

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: