Geometry, DATA INTERPRETATION, READING COMPREHENSION Unit Test for CAT: 68 Questions with Solutions & Analysis

Attempt the Geometry, DATA INTERPRETATION, READING COMPREHENSION Unit Test for CAT: 68 exam-level questions, detailed solutions and performance analysis. First questions free.

Paper breakdown

68 questions · 204 marks · 95.52 minutes. Quantitative Ability: 27 · Verbal Ability and Reading Comprehension: 21 · Data Interpretation and Logical Reasoning: 20

Free sample questions from Geometry, DATA INTERPRETATION, READING COMPREHENSION Unit Test

Question 1 · Verbal Ability and Reading Comprehension MCQ

Common Description: The passage below is accompanied by four questions. Based on the passage, choose the best answer for each question.
Steven Pinker’s new book, “Rationality: What It Is, Why It Seems Scarce, Why It Matters,” offers a pragmatic dose of measured optimism, presenting rationality as a fragile but achievable ideal in personal and civic life. . . . Pinker’s ambition to illuminate such a crucial topic offers the welcome prospect of a return to sanity. . . . It’s no small achievement to make formal logic, game theory, statistics and Bayesian reasoning delightful topics full of charm and relevance.
It’s also plausible to believe that a wider application of the rational tools he analyzes would improve the world in important ways. His primer on statistics and scientific uncertainty is particularly timely and should be required reading before consuming any news about the [COVID] pandemic. More broadly, he argues that less media coverage of shocking but vanishingly rare events, from shark attacks to adverse vaccine reactions, would help prevent dangerous overreactions, fatalism and the diversion of finite resources away from solvable but less-dramatic issues, like malnutrition in the developing world.
It’s a reasonable critique, and Pinker is not the first to make it. But analyzing the political economy of journalism — its funding structures, ownership concentration and increasing reliance on social media shares — would have given a fuller picture of why so much coverage is so misguided and what we might do about it.
Pinker’s main focus is the sort of conscious, sequential reasoning that can track the steps in a geometric proof or an argument in formal logic. Skill in this domain maps directly onto the navigation of many real-world problems, and Pinker shows how greater mastery of the tools of rationality can improve decision-making in medical, legal, financial and many other contexts in which we must act on uncertain and shifting information. . . .
Despite the undeniable power of the sort of rationality he describes, many of the deepest insights in the history of science, math, music and art strike their originators in moments of epiphany. From the 19th-century chemist Friedrich August Kekulé’s discovery of the structure of benzene to any of Mozart’s symphonies, much extraordinary human achievement is not a product of conscious, sequential reasoning. Even Plato’s Socrates — who anticipated many of Pinker’s points by nearly 2,500 years, showing the virtue of knowing what you do not know and examining all premises in arguments, not simply trusting speakers’ authority or charisma — attributed many of his most profound insights to dreams and visions. Conscious reasoning is helpful in sorting the wheat from the chaff, but it would be interesting to consider the hidden aquifers that make much of the grain grow in the first place.
The role of moral and ethical education in promoting rational behavior is also underexplored. Pinker recognizes that rationality “is not just a cognitive virtue but a moral one.” But this profoundly important point, one subtly explored by ancient Greek philosophers like Plato and Aristotle, doesn’t really get developed. This is a shame, since possessing the right sort of moral character is arguably a precondition for using rationality in beneficial ways. The author endorses Pinker’s views on the importance of logical reasoning as it:

A.
provides a moral compass for resolving important ethical dilemmas.
B.
focuses public attention on real issues like development rather than sensational events.
C.
equips people with the ability to tackle challenging practical problems.
D.
helps people to gain expertise in statistics and other scientific disciplines.

Correct Answer:

Step-by-Step Solution

Key idea: This is an author-endorsement question, recognizable because it asks what the author agrees with regarding a specific point made by the cited thinker (Pinker).

Step 1: Locate the relevant section. The fourth paragraph discusses Pinker's focus on "conscious, sequential reasoning" (logical reasoning).

Step 2: Identify the author's endorsement. The author states: "Skill in this domain maps directly onto the navigation of many real-world problems, and Pinker shows how greater mastery of the tools of rationality can improve decision-making in medical, legal, financial and many other contexts..."

Step 3: Map to the options. "Navigation of many real-world problems" and "improve decision-making" directly corresponds to Option C: "equips people with the ability to tackle challenging practical problems."

Step 4: Eliminate others. Option A is discussed in paragraph 6 as being "underexplored" by Pinker. Option B is from paragraph 2 about media coverage, not specifically about the utility of logical reasoning itself. Option D is too narrow, focusing only on statistics.

Answer: Option C

Question 2 · Verbal Ability and Reading Comprehension MCQ

Common Description: The passage below is accompanied by four questions. Based on the passage, choose the best answer for each question.
Oftentimes, when economists cross borders, they are less interested in learning from others than in invading their garden plots. Gary Becker, for instance, pioneered the idea of human capital. To do so, he famously tackled topics like crime and domesticity, applying methods honed in the study of markets to domains of nonmarket life. He projected economics outward into new realms: for example, by revealing the extent to which humans calculate marginal utilities when choosing their spouses or stealing from neighbors. At the same time, he did not let other ways of thinking enter his own economic realm: for example, he did not borrow from anthropology or history or let observations of nonmarket economics inform his homo economicus. Becker was a picture of the imperial economist in the heyday of the discipline’s bravura.
Times have changed for the once almighty discipline. Economics has been taken to task, within and beyond its ramparts. Some economists have reached out, imported, borrowed, and collaborated—been less imperial, more open. Consider Thomas Piketty and his outreach to historians. The booming field of behavioral economics—the fusion of economics and social psychology—is another case. Having spawned active subfields, like judgment, decisionmaking and a turn to experimentation, the field aims to go beyond the caricature of Rational Man to explain how humans make decisions...
It is important to underscore how this flips the way we think about economics. For generations, economists have presumed that people have interests—“preferences,” in the neoclassical argot—that get revealed in the course of peoples’ choices. Interests come before actions and determine them. If you are hungry, you buy lunch; if you are cold, you get a sweater. If you only have so much money and can't afford to deal with both your growling stomach and your shivering, which need you choose to meet using your scarce savings reveals your preference.
Psychologists take one look at this simple formulation and shake their heads. Increasingly, even some mainstream economists have to admit that homo economicus doesn't always behave like the textbook maximizer; irrational behavior can't simply be waved away as extraeconomic expressions of passions over interests, and thus the domain of other disciplines... This is one place where the humanist can help the economist. If narrative economics is going to help us understand how rivals duke it out, who wins and who loses, we are going to need much more than lessons from epidemiological studies of viruses or intracranial stimuli.
Above all, we need politics and institutions. Shiller [the Nobel prize winning economist] connects perceptions of narratives to changes in behavior and thence to social outcomes. He completes a circle that was key to behavioral economics and brings in storytelling to make sense of how perceptions get framed. This cycle (perception to behavior to society) was once mediated or dominated by institutions: the political parties, lobby groups, and media organizations that played a vital role in legitimating, representing, and excluding interests. Yet institutions have been stripped from Shiller's account, to reveal a bare dynamic of emotions and economics, without the intermediating place of politics. The author critiques Shiller’s approach to behavioural economics for

A.
ignoring the marginal role that media and politics play in influencing people's behaviour.
B.
denigrating the role of institutions while creating a link between behavioural economics and perceptions
C.
linking emotions and rational behaviour without considering the mediation of social institutions.
D.
relying excessively on storytelling as the main influence on the formation of perceptions.

Correct Answer:

Step-by-Step Solution

Key idea: Argument Isolation / Critique question. We must identify the specific flaw the author finds in Shiller’s approach.

Step 1: Locate the critique. The final paragraph discusses Shiller. It praises his connection of narratives to behavior but ends with a "Yet."

Step 2: Analyze the "Yet." "Yet institutions have been stripped from Shiller's account, to reveal a bare dynamic of emotions and economics, without the intermediating place of politics."

Step 3: Synthesize the flaw. Shiller connects Emotions/Narratives -> Behavior -> Society, but he skips the "Intermediating Institutions" (Politics, Media, Lobby Groups).

Step 4: Evaluate Options.

Option A: "Ignoring the marginal role..." The text says institutions play a "vital role," not a marginal one. Incorrect.
Option B: "Denigrating the role..." The text says they are "stripped" (omitted), not "denigrated" (insulted/criticized). Incorrect nuance.
Option C: "Linking emotions and rational behaviour without considering the mediation of social institutions." This matches "bare dynamic of emotions... without the intermediating place of politics/institutions." Correct.
Option D: "Relying excessively on storytelling..." The author does not criticize the use of storytelling itself, but the omission of institutions in the cycle. Incorrect.

Answer: C

Question 3 · Verbal Ability and Reading Comprehension MCQ

Common Description: Instructions [5 - 8]
The passage below is accompanied by a set of questions. Choose the best answer to each question.
Today we can hardly conceive of ourselves without an unconscious. Yet between 1700 and 1900, this notion developed as a genuinely original thought. The “unconscious” burst the shell of conventional language, coined as it had been to embody the fleeting ideas and the shifting conceptions of several generations until, finally, it became fixed and defined in specialized terms within the realm of medical psychology and Freudian psychoanalysis.
The vocabulary concerning the soul and the mind increased enormously in the course of the nineteenth century. The enrichments of literary and intellectual language led to an altered understanding of the meanings that underlie time-honored expressions and traditional catchwords. At the same time, once coined, powerful new ideas attracted to themselves a whole host of seemingly unrelated issues, practices, and experiences, creating a peculiar network of preoccupations that as a group had not existed before. The drawn-out attempt to approach and define the unconscious brought together the spiritualist and the psychical researcher of borderline phenomena (such as apparitions, spectral illusions, haunted houses, mediums, trance, automatic writing); the psychiatrist or alienist probing the nature of mental disease, of abnormal ideation, hallucination, delirium, melancholia, mania; the surgeon performing operations with the aid of hypnotism; the magnetizer claiming to correct the disequilibrium in the universal flow of magnetic fluids but who soon came to be regarded as a clever manipulator of the imagination; the physiologist and the physician who puzzled over sleep, dreams, sleepwalking, anesthesia, the influence of the mind on the body in health and disease; the neurologist concerned with the functions of the brain and the physiological basis of mental life; the philosopher interested in the will, the emotions, consciousness, knowledge, imagination and the creative genius; and, last but not least, the psychologist.
Significantly, most if not all of these practices (for example, hypnotism in surgery or psychological magnetism) originated in the waning years of the eighteenth century and during the early decades of the nineteenth century, as did some of the disciplines (such as psychology and psychical research). The majority of topics too were either new or assumed hitherto unknown colors. Thus, before 1790, few if any spoke, in medical terms, of the affinity between creative genius and the hallucinations of the insane . . .
Striving vaguely and independently to give expression to a latent conception, various lines of thought can be brought together by some novel term. The new concept then serves as a kind of resting place or stocktaking in the development of ideas, giving satisfaction and a stimulus for further discussion or speculation. Thus, the massive introduction of the term unconscious by Hartmann in 1869 appeared to focalize many stray thoughts, affording a temporary feeling that a crucial step had been taken forward, a comprehensive knowledge gained, a knowledge that required only further elaboration, explication, and unfolding in order to bring in a bounty of higher understanding. Ultimately, Hartmann’s attempt at defining the unconscious proved fruitless because he extended its reach into every realm of organic and inorganic, spiritual, intellectual, and instinctive existence, severely diluting the precision and compromising the impact of the concept. All of the following statements may be considered valid inferences from the passage, EXCEPT:

A.
Without the linguistic developments of the nineteenth century, the growth of understanding of the soul and the mind may not have happened.
B.
Eighteenth century thinkers were the first to perceive a connection between creative genius and insanity.
C.
New conceptions in the nineteenth century could provide new knowledge because of the establishment of fields such as anaesthesiology.
D.
Unrelated practices began to be treated as related to each other, as knowledge of the mind grew in the nineteenth century.

Correct Answer:

Step-by-Step Solution

Key idea: This is a detail verification / EXCEPT inference question, recognizable by the capitalized "EXCEPT", requiring you to find the single statement that cannot be logically derived from the passage.

Step 1: Verify Option A. The passage states, "The enrichments of literary and intellectual language led to an altered understanding..." This supports the inference that without these linguistic developments, this growth in understanding may not have happened. (Valid inference)

Step 2: Verify Option B. The passage notes, "Thus, before 1790, few if any spoke, in medical terms, of the affinity between creative genius and the hallucinations of the insane." This implies that late 18th-century thinkers were indeed the first to articulate this connection in medical terms. (Valid inference)

Step 3: Verify Option D. The passage explicitly states that "powerful new ideas attracted to themselves a whole host of seemingly unrelated issues, practices, and experiences, creating a peculiar network..." This directly supports the idea that unrelated practices began to be treated as related. (Valid inference)

Step 4: Verify Option C. The passage mentions physicians who "puzzled over... anesthesia" as one of the groups brought together by the attempt to define the unconscious. It does NOT state that the establishment of fields like anesthesiology provided or caused new knowledge of the mind. This option invents a causal relationship that reverses or distorts the passage's logic. (Invalid inference)

Answer: Option C

Question 4 · Verbal Ability and Reading Comprehension MCQ

Common Description:

Critical theory of technology is a political theory of modernity with a normative dimension. It belongs to a tradition extending from Marx to Foucault and Habermas according to which advances in the formal claims of human rights take center stage while in the background centralization of ever more powerful public institutions and private organizations imposes an authoritarian social order.

Marx attributed this trajectory to the capitalist rationalization of production. Today it marks many institutions besides the factory and every modern political system, including so-called socialist systems. This trajectory arose from the problems of command over a disempowered and deskilled labor force; but everywhere [that] masses are organized - whether it be Foucault's prisons or Habermas's public sphere - the same pattern prevails. Technological design and development is shaped by this pattern as the material base of a distinctive social order. Marcuse would later point to a "project" as the basis of what he called rather confusingly "technological rationality." Releasing technology from this project is a democratic political task.

In accordance with this general line of thought, critical theory of technology regards technologies as an environment rather than as a collection of tools. We live today with and even within technologies that determine our way of life. Along with the constant pressures to build centers of power, many other social values and meanings are inscribed in technological design. A hermeneutics of technology must make explicit the meanings implicit in the devices we use and the rituals they script. Social histories of technologies such as the bicycle, artificial lighting or firearms have made important contributions to this type of analysis. Critical theory of technology attempts to build a methodological approach on the lessons of these histories.

As an environment, technologies shape their inhabitants. In this respect, they are comparable to laws and customs. Each of these institutions can be said to represent those who live under their sway through privileging certain dimensions of their human nature. Laws of property represent the interest in ownership and control. Customs such as parental authority represent the interest of childhood in safety and growth. Similarly, the automobile represents its users in so far as they are interested in mobility. Interests such as these constitute the version of human nature sanctioned by society.

This notion of representation does not imply an eternal human nature. The concept of nature as non-identity in the Frankfurt School suggests an alternative. On these terms, nature is what lies at the limit of history, at the point at which society loses the capacity to imprint its meanings on things and control them effectively. The reference here is, of course, not to the nature of natural science, but to the lived nature in which we find ourselves and which we are. This nature reveals itself as that which cannot be totally encompassed by the machinery of society. For the Frankfurt School, human nature, in all its transcending force, emerges out of a historical context as that context is [depicted] in illicit joys, struggles and pathologies. We can perhaps admit a less romantic . . . conception in which those dimensions of human nature recognized by society are also granted theoretical legitimacy.

Which one of the following statements could be inferred as supporting the arguments of the passage?

A.
It is not human nature, but human culture that is represented by institutions such as law and custom.
B.
Technologies form the environmental context and shape the contours of human society.
C.
Nature decides the point at which society loses its capacity to control history.
D.
The romantic conception of nature referred to by the passage is the one that requires theoretical legitimacy.

Correct Answer:

Step-by-Step Solution

Key idea: This is an inference/support question, recognizable because it asks which statement could be inferred as supporting the passage's arguments regarding critical theory of technology.

Step 1: Understand the Core Thesis. Critical theory views technology not as neutral tools but as an "environment" that shapes social order and human nature, embedding power structures and values.

Step 2: Evaluate Option A. The passage states institutions/laws/customs represent dimensions of "human nature" (para 4), not just culture. This option contradicts the text's specific terminology.

Step 3: Evaluate Option B. Paragraph 3 explicitly states: "critical theory of technology regards technologies as an environment... we live today with and even within technologies that determine our way of life." This directly supports the idea that technologies form the environmental context and shape society.

Step 4: Evaluate Option C. Paragraph 5 defines nature as "what lies at the limit of history, at the point at which society loses the capacity to imprint its meanings". Nature is the limit, not the active agent deciding the point. This distorts causality.

Step 5: Evaluate Option D. The passage suggests admitting a "less romantic" conception, implying the romantic one is insufficient or problematic, not that it requires theoretical legitimacy.

Answer: Option B

Question 5 · Verbal Ability and Reading Comprehension MCQ

Common Description:

All of the following claims can be inferred from the passage, EXCEPT:

A.
the significance of parental authority to children's safety does not therefore imply that parental authority is a permanent aspect of human nature.
B.
the critical theory of technology argues that, as issues of human rights become more prominent, we lose sight of the ways in which the social order becomes more authoritarian.
C.
analyses of technologies must engage with their social histories to be able to reveal their implicit and explicit meanings for us.
D.
technologies seek to privilege certain dimensions of human nature at a high cost to lived nature.

Correct Answer:

Step-by-Step Solution

Key idea: This is an EXCEPT inference question. You must find the single statement that CANNOT be logically derived from the passage, while the other three are supported.

Step 1: Check Option A. Paragraph 4 states parental authority represents childhood safety. Paragraph 5 explicitly notes, "This notion of representation does not imply an eternal human nature." Thus, it is valid to infer that this significance does not imply permanence. (Valid inference)

Step 2: Check Option B. Paragraph 1 states that "advances in the formal claims of human rights take center stage while in the background centralization of ever more powerful public institutions... imposes an authoritarian social order." This directly supports the idea that we lose sight of the authoritarian shift while focusing on human rights. (Valid inference)

Step 3: Check Option C. Paragraph 3 states, "Social histories of technologies... have made important contributions to this type of analysis. Critical theory of technology attempts to build a methodological approach on the lessons of these histories." This supports the need to engage with social histories. (Valid inference)

Step 4: Check Option D. The passage states technologies privilege certain dimensions of human nature, and defines "lived nature" as what lies at the limit of history, revealing itself as that which cannot be totally encompassed by society. However, it does NOT state that privileging human nature comes "at a high cost to lived nature." This option invents a causal conflict and a value judgment ("high cost") not present in the text. (Invalid inference)

Answer: Option D

Question 6 · Data Interpretation and Logical Reasoning NAT

Common Description: Instructions [25 - 29]
A few salesmen are employed to sell a product called TRICCEK among households in various housing complexes. On each day, a salesman is assigned to visit one housing complex. Once a salesman enters a housing complex, he can meet any number of households in the time available. However, if a household makes a complaint against the salesman, then he must leave the housing complex immediately and cannot meet any other household on that day. A household may buy any number of TRICCEK items or may not buy any item. The salesman needs to record the total number of TRICCEK items sold as well as the number of households met in each day. The success rate of a salesman for a day is defined as the ratio of the number of items sold to the number of households met on that day. Some details about the performances of three salesmen - Tohri, Hokli and Lahur, on two particular days are given below.
1. Over the two days, all three of them met the same total number of households, and each of them sold a total of 100 items.
2. On both days, Lahur met the same number of households and sold the same number of items.
3. Hokli could not sell any item on the second day because the first household he met on that day complained against him.
4. Tohri met 30 more households on the second day than on the first day.
5. Tohri’s success rate was twice that of Lahur’s on the first day, and it was 75% of Lahur’s on the second day. How many TRICCEK items were sold by Tohri on the first day?

Correct Answer:

Step-by-Step Solution

Key idea: Algebraic modeling of rates and totals in a multi-agent system. Recognisable by variables linked through ratios (success rates) and linear constraints (totals).

Step 1: Define variables based on household counts.

Let Tohri's Day 1 households = $x$ .

Given: Tohri Day 2 = $x + 30$ .

Total Tohri Households = $2 x + 30$ .

Constraint: "All three met same total households".

So Lahur Total Households = $2 x + 30$ .

Constraint: "Lahur met same number each day".

So Lahur Day 1 = Lahur Day 2 = $\frac{2 x + 30}{2} = x + 15$ .

Step 2: Express Success Rates (SR).

Definition: SR = Items / Households.

Lahur Items: Total 100, equal split $⟹$ 50 per day.

Lahur SR (Day 1) = $\frac{50}{x + 15}$ .

Lahur SR (Day 2) = $\frac{50}{x + 15}$ . (Constant).

Tohri Relations:

Day 1 SR = $2 \times$ Lahur SR = $\frac{100}{x + 15}$ .

Day 2 SR = $0.75 \times$ Lahur SR = $\frac{37.5}{x + 15}$ .

Step 3: Form equation using Tohri's Total Items.

Items = SR $\times$ Households.

Tohri Day 1 Items = $\frac{100}{x + 15} \times x$ .

Tohri Day 2 Items = $\frac{37.5}{x + 15} \times (x + 30)$ .

Sum = 100.

$\frac{100 x}{x + 15} + \frac{37.5 ( x + 30 )}{x + 15} = 100$

$100 x + 37.5 x + 1125 = 100 (x + 15)$

$137.5 x + 1125 = 100 x + 1500$

$37.5 x = 375$

$x = 10$

Step 4: Calculate target value.

Question asks: Tohri items on Day 1.

$I_{T 1} = \frac{100 \times 10}{10 + 15} = \frac{1000}{25} = 40$

Answer: 40

Question 7 · Data Interpretation and Logical Reasoning MCQ

Common Description: Instructions [31 - 34]
Ganga, Kaveri, and Narmada are three women who buy four raw materials (Mango, Apple, Banana and Milk) and sell five finished products (Mango smoothie, Apple smoothie, Banana smoothie, Mixed fruit smoothie and Fruit salad). Table-1 gives information about the raw materials required to produce the five finished products. One unit of a finished product requires one unit of each of the raw materials mentioned in the second column of the table.

Finished product	Raw materials required
Mango smoothie	Mango, Milk
Apple smoothie	Apple, Milk
Banana smoothie	Banana, Milk
Mixed fruit smoothie	Mango, Apple, Banana, Milk
Fruit salad	Mango, Apple, Banana

One unit of milk, mango, apple, and banana cost ₹5, ₹3, ₹2, and ₹1 respectively. Each unit of a finished product is sold for a profit equal to two times the number of raw materials used to make that product. For example, apple smoothie is made with two raw materials (apple and milk) and will be sold for a profit of ₹4 per unit. Leftover raw materials are sold during the last business hour of the day for a loss of ₹1 per unit.
The amount, in rupees, received from sales (revenue) for each woman in each of the four business hours of the day is given in Table-2.

Business Hour	Ganga	Kaveri	Narmada
Hour 1	23	19	31
Hour 2	21	22	21
Hour 3	29	30	23
Hour 4 (last hour)	30	27	22

The following additional facts are known.
1. No one except possibly Ganga sold any Mango smoothie.
2. Each woman sold either zero or one unit of any single finished product in any hour.
3. Each woman had exactly one unit each of two different raw materials as leftovers.
4. No one had any banana leftover. Which of the following is NECESSARILY true?

A.
Ganga did not sell any leftover mangoes.
B.
Ganga did not sell any leftover apples.
C.
Narmada sold one unit of leftover milk.
D.
Kaveri sold one unit of leftover mangoes.

Correct Answer:

Step-by-Step Solution

Key idea: This is a revenue reconstruction puzzle with constraint satisfaction, recognisable because we must deduce the specific leftover items for each person by matching their Hour 4 revenue to possible product combinations and leftover values.

Step 1: Calculate Revenues for Products and Leftovers.

Costs: Milk=5, Mango=3, Apple=2, Banana=1.

Profit = 2 * (number of raw materials). Revenue = Cost + Profit.

Mango Smoothie (MS): Cost 8, Profit 4, Rev 12.
Apple Smoothie (AS): Cost 7, Profit 4, Rev 11.
Banana Smoothie (BS): Cost 6, Profit 4, Rev 10.
Mixed Fruit (MF): Cost 11, Profit 8, Rev 19.
Fruit Salad (FS): Cost 6, Profit 6, Rev 12.

Leftover Revenue (Sold at loss of 1 per unit):

Milk Left: 5-1=4.
Mango Left: 3-1=2.
Apple Left: 2-1=1.
Banana Left: Not allowed (Fact 4).

Possible Leftover Pairs (Fact 3: exactly two different raw materials):

{Milk, Mango}: Rev 4+2=6.
{Milk, Apple}: Rev 4+1=5.
{Mango, Apple}: Rev 2+1=3.

Step 2: Analyze Hour 4 Sales to fix Leftovers.

Hour 4 Revenue = Product Revenue + Leftover Revenue.

Narmada (H4 Rev 22):

If Leftover Rev 6, Target Prod = 16. No single product or distinct combination sums to 16.

If Leftover Rev 5, Target Prod = 17. Impossible.

If Leftover Rev 3, Target Prod = 19. Product MF has Rev 19. Valid.

So, Narmada's Leftover is strictly {Mango, Apple}.

Ganga (H4 Rev 30):

If Leftover Rev 3, Target Prod = 27. Max product rev is 19. No distinct combination sums to 27.

If Leftover Rev 5, Target Prod = 25. Impossible.

If Leftover Rev 6, Target Prod = 24. MS (12) + FS (12) = 24. Valid.

So, Ganga's Leftover is strictly {Milk, Mango}.

Kaveri (H4 Rev 27):

Fact 1 says Kaveri did not sell MS.

If Leftover Rev 6, Target Prod = 21. AS (11) + BS (10) = 21. Valid.

If Leftover Rev 5, Target Prod = 22. Needs MS (12) + BS (10). Invalid (no MS).

If Leftover Rev 3, Target Prod = 24. Needs MS (12) + FS (12). Invalid (no MS).

So, Kaveri's Leftover is strictly {Milk, Apple}.

Step 3: Check the options.

Ganga's leftover is {Milk, Mango}.

A. Ganga did not sell any leftover mangoes. (False, she did).

B. Ganga did not sell any leftover apples. (True, her leftovers were Milk and Mango).

C. Narmada sold one unit of leftover milk. (False, hers were Mango and Apple).

D. Kaveri sold one unit of leftover mangoes. (False, hers were Milk and Apple).

Answer: B.

Question 8 · Data Interpretation and Logical Reasoning MCQ

Common Description: A train travels from Station A to Station E, passing through stations B, C, and D, in that order. The train has a seating capacity of 200. A ticket may be booked from any station to any other station ahead on the route, but not to any earlier station.
A ticket from one station to another reserves one seat on every intermediate segment of the route. For example, a ticket from B to E reserves a seat in the intermediate segments B - C, C - D, and D - E.
The occupancy factor for a segment is the total number of seats reserved in the segment as a percentage of the seating capacity. The total number of seats reserved for any segment cannot exceed 200.
The following information is known.
1. Segment C - D had an occupancy factor of 95%. Only segment B - C had a higher occupancy factor.
2. Exactly 40 tickets were booked from B to C and 30 tickets were booked from B to E.
3. Among the seats reserved on segment D - E, exactly four-sevenths were from stations before C.
4. The number of tickets booked from A to C was equal to that booked from A to E, and it was higher than that from B to E.
5. No tickets were booked from A to B, from B to D and from D to E.
6. The number of tickets booked for any segment was a multiple of 10. What was the occupancy factor for segment D - E?

A.
35%
B.
70%
C.
77%
D.
84%

Correct Answer:

Step-by-Step Solution

Key idea: Segment occupancy factor calculation using aggregated edge flows and algebraic constraints.

Step 1: Define variables for ticket counts.

Let $a = n_{A C} = n_{A E}$ . Let $b = n_{A D}$ .

Let $x = n_{C D}$ and $y = n_{C E}$ .

Given: $n_{B C} = 40, n_{B E} = 30$ .

Step 2: Calculate segment flows.

Flow on D-E ( $F_{D E}$ ) = $n_{A E} + n_{B E} + n_{C E} = a + 30 + y$ .

Step 3: Use the D-E fraction constraint.

"Among seats on D-E, exactly 4/7 were from stations before C."

Stations before C contributing to D-E are A ( $n_{A E} = a$ ) and B ( $n_{B E} = 30$ ).

So, $\frac{a + 30}{a + 30 + y} = \frac{4}{7}$ .

This implies $7 (a + 30) = 4 (a + 30 + y) ⟹ 3 (a + 30) = 4 y$ .

So $y = \frac{3 ( a + 30 )}{4}$ .

Step 4: Determine 'a'.

From previous analysis (P1/P2), we found $a = 50$ .

Check: $a = 50 ⟹ y = \frac{3 ( 80 )}{4} = 60$ .

$F_{D E} = 50 + 30 + 60 = 140$ .

Step 5: Calculate Occupancy Factor.

Capacity = 200.

Occupancy = $\frac{140}{200} \times 100% = 70%$ .

Answer: B

Question 9 · Quantitative Ability MCQ

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is

A.
3
B.
4
C.
6
D.
5

Correct Answer:

Step-by-Step Solution

This is a quadrilateral-existence counting question. You can tell because you are given three integer side lengths and asked how many integer values the fourth side can take, with no angles supplied anywhere. Since there are no angles to work with, the trigger is the quadrilateral inequality, not any area or angle rule.

Step 1: State the rule. A quadrilateral with sides $a, b, c, d$ can be closed into a real (non-degenerate) shape only if the largest side is strictly less than the sum of the other three sides. This must hold no matter which side turns out to be the largest.

Step 2: Let the unknown fourth side be $x$ . The four sides are $1, 2, 4, x$ . Check every side as a possible "largest side":

$1 < 2 + 4 + x (always true)$

$2 < 1 + 4 + x (always true)$

$4 < 1 + 2 + x \Rightarrow x > 1$

$x < 1 + 2 + 4 = 7$

Step 3: Combine the only two binding conditions:

$1 < x < 7$

Step 4: Since $x$ must be a positive integer, the possible values are:

$x \in {2, 3, 4, 5, 6}$

That is 5 values.

Answer: 5 possible values, option D.

Common trap: students often forget that $x = 1$ and $x = 7$ must be excluded because equality would flatten the quadrilateral into a straight line (a degenerate shape, not a real quadrilateral). Including either boundary value wrongly gives 6 or 7 as the count.

Question 10 · Quantitative Ability MCQ

In a triangle ABC, AB = AC = 8 cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area, in sq. cm, of the overlapping region between the two circles is

A.
$16 π$
B.
$16 (π - 1)$
C.
$32 (π - 1)$
D.
$32 π$

Correct Answer:

Step-by-Step Solution

This is the "Overlap of Two Circles from an Isosceles Right Triangle" pattern: two circles both pass through the same two points (B and C), so the overlapping region is made of two circular pieces cut by the common chord BC.

Step 1 — Recognize the trigger.

A circle with BC as diameter passing through A means, by Thales' theorem, that $∠ B A C = 9 0^{\circ}$ . Since $A B = A C = 8$ , triangle ABC is an isosceles right triangle, and $B C = 82$ (hypotenuse). Both circles pass through B and C: circle 1 (diameter BC) has B and C as the endpoints of its diameter, and circle 2 (center A, radius 8) passes through B and C since $A B = A C = 8$ .

Step 2 — Identify the two circles clearly.

Circle 1: diameter BC, so radius $r_{1} = \frac{8 2}{2} = 42$ .

Circle 2: center A, radius $r_{2} = 8$ .

Step 3 — The overlap splits into two pieces along chord BC.

Since BC is a diameter of circle 1, the semicircle of circle 1 on A's side (bounded by BC and the arc through A) is exactly half of circle 1:

$Semicircle area = \frac{1}{2} π r_{1}^{2} = \frac{1}{2} π (32) = 16 π$

This semicircle lies entirely inside circle 2: every point on this arc is at distance at most 8 from A (with equality only at B and C themselves), so the whole half-disk is contained in circle 2.

Step 4 — Find the extra overlap from circle 2's side.

The rest of the overlap comes from the part of circle 2 that also lies inside circle 1, on the far side of BC. This is the minor segment of circle 2 cut off by chord BC, using the central angle $∠ B A C = 9 0^{\circ}$ :

$Segment area = \frac{1}{2} r_{2}^{2} (α - sin α), α = 9 0^{\circ} = \frac{π}{2}$

$= \frac{1}{2} (64) (\frac{π}{2} - 1) = 32 (\frac{π}{2} - 1) = 16 π - 32$

Step 5 — Add both pieces.

$Total overlap = 16 π + (16 π - 32) = 32 π - 32 = 32 (π - 1)$

Trap avoided: a common mistake is to stop after computing just the semicircle (16π) and forget the additional minor-segment contribution from circle 2, or to assume the entire circle 1 lies inside circle 2 and use its full area.

Answer: the overlapping area is $32 (π - 1)$ sq. cm.

Question 11 · Quantitative Ability MCQ

Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is

A.
6:19
B.
5:24
C.
6:25
D.
7:24

Correct Answer:

Step-by-Step Solution

This is an exponential-product inequality question. Recognise it because both sides are large products of powers of the same base (3), which is the trigger to convert every product into a sum of exponents (since same-base powers multiply by adding exponents), turning the inequality into a simple inequality between two sums.

Step 1: Simplify the left side.

$a_{k} = 3^{k}$ , so $(a_{k})^{k} = (3^{k})^{k} = 3^{k^{2}}$ .

$LHS = \prod_{k = 1}^{20} (a_{k})^{k} = \prod_{k = 1}^{20} 3^{k^{2}} = 3^{\sum_{k = 1}^{20} k^{2}}$

Using the sum-of-squares formula $\sum_{k = 1}^{n} k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ with $n = 20$ :

$\sum_{k = 1}^{20} k^{2} = \frac{20 \times 21 \times 41}{6} = 2870$

So $LHS = 3^{2870}$ .

Step 2: Simplify the right side.

$RHS = a_{21} \times a_{22} \times \dots \times a_{20 + m} = 3^{21} \times 3^{22} \times \dots \times 3^{20 + m} = 3^{\sum_{k = 21}^{20 + m} k}$

This sum can be written as (sum from 1 to 20+m) minus (sum from 1 to 20):

$\sum_{k = 21}^{20 + m} k = \frac{( 20 + m ) ( 21 + m )}{2} - \frac{20 \times 21}{2} = \frac{( 20 + m ) ( 21 + m )}{2} - 210$

Step 3: Apply the inequality. Since both sides have the same base 3 (which is greater than 1), comparing the powers is the same as comparing the exponents directly.

$3^{2870} < 3^{\frac{( 20 + m ) ( 21 + m )}{2} - 210}$

$\Rightarrow 2870 < \frac{( 20 + m ) ( 21 + m )}{2} - 210$

$\Rightarrow 3080 < \frac{( 20 + m ) ( 21 + m )}{2}$

$\Rightarrow 6160 < (20 + m) (21 + m)$

Step 4: Find the smallest natural number m satisfying this by testing nearby values.

$m = 57 : (77) (78) = 6006$ — not greater than 6160, fails.

$m = 58 : (78) (79) = 6162$ — greater than 6160, succeeds.

So the smallest such $m$ is 58.

Trap check: the right side has exactly $m$ terms, running from index 21 to $20 + m$ , not from 1 to $20 + m$ — forgetting to subtract the first 20 terms' exponent sum (210) is the most common mistake here.

Final answer: 58, option A.

Geometry, DATA INTERPRETATION, READING COMPREHENSION Unit Test for CAT: 68 Questions with Solutions & Analysis

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