CAT Mock Test- 4 (November 1) - CAT MCQ

"People's primary concern was to avoid living an unfortunate life. Therefore, they were more likely to order their thoughts, decisions, and behaviours to promote increased life satisfaction."

These lines from the first paragraph state that since people wanted to avoid living an unfortunate life, they were likely to alter their thought processes, decision making and behaviour such that it increased their life satisfaction. Hence, we can infer that a direct consequence would be people making changes to themselves, such that their life satisfaction is increased. Hence Option C can be inferred.

Option A is mentioned in the same paragraph but is not implied as a direct consequence.

Option B is not a direct consequence, as people refer to philosophies like them to find answers to difficult questions, not necessarily to avoid an unfortunate life.

Option D is also not a direct consequence mentioned, as it mentions people embracing Stoicism, which is too specific. Ordering of thought, decisions and behaviour is not the same as embracing the same from as school of philosophy. Hence, it is a distortion and can be eliminated.

"In times of plague and trauma, we moderns seek to protect ourselves with prayers, charms, sigils and spells as much as any medieval peasant." Option A can be inferred from this line.

"various occult sciences - astrology, alchemy, kabbalah, geomancy, dream interpretation - became an important basis for empire more than ever before. The ability to predict the future with divination, then change it with magic, was of obvious political, military and economic interest." Hence, we can infer that magic as a means to predict/portend the future and alter it for certain purposes has been presented in the passage. Option C can be inferred.

"Many considered it alchemy’s sister, both sciences being predicated on the harnessing of cosmic correspondences and natural sympathies to restore elemental equilibrium in the human body - the definition of health. Techniques for life-extension were also central to the alchemical quest." Option D echoes the above point and hence, can be eliminated.

Option B has not been stated in the passage. 

"Humans often appear to react irrationally in the face of disease, as the COVID-19 pandemic has shown. Many cling to religion or become superstitious. Others become fatalistic. In times of plague and trauma, we moderns seek to protect ourselves with prayers, charms, sigils and spells as much as any medieval peasant. That a surgical mask is hygienic doesn’t make it any less of a magical symbol."

The first paragraph is about how humans act irrationally in the face of disease, and how they have different coping mechanisms. The author also states that we protect ourselves with prayers and other occult practices like a medieval peasant would have done. But the last line of the passage has a deeper significance than just stating a fact. It acts as a justification for the previous lines. The author here justifies that a surgical mask could be construed as a magical symbol, in spite of it having scientific utility. The author has mentioned the mask here as a proof that modern humans do behave irrationally, just as they did in earlier times. Option C perfectly captures this.

The author explicitly mentions Option A, and this is not the purpose why he mentions the mask as explained above. Hence, A can be eliminated.

Option B is not the main contention of the author while mentioning the surgical masks, as it does not help his argument in any way. Hence B can be eliminated.

The author does not draw a parallel between occult and scientific practices but between the way humans reacted to disease in the pre-modern era and how they react to it in the modern era. Hence, D can be eliminated too.

Hence, Option C is the answer.

"Western Europe saw a parallel upsurge of occultism - much of it from Arabic sources - which we now call the Renaissance. The scientific revolution that followed continued the same trend: historians now admit that saints of science such as Johannes Kepler, Francis Bacon, Robert Boyle and Isaac Newton were likewise raving occultists."

So, while the Islamic world witnessed an occult-scientific revolution, the Western world also witnessed a surge in interest for the occult, thanks to Arabic sources. And this surge in interest/popularity continued during the subsequent scientific revolution as well. 

Only option D conveys the above inference. Option D is the answer 

The following is Mankiw's version of 'just deserts'- "Meritocracy holds that “a person who contributes more to society deserves a higher income that reflects those greater contributions.” Moreover, meritocracy measures each person’s contribution as the market value that she adds “to society’s production of goods and services.”

The following is the author's take on the same- "someone who contributes more to prosperity deserves a higher income that reflects their greater contribution."

Hence, the key difference between the two versions is how they quantify the contributions of an individual. While Mankiw focuses on the market value-added, the author is more concerned about the ability of the contribution to improve the prosperity of the society, i.e., the benefits to the society.

Comparing the options, option D is closest to the above interpretation and is the answer.

Option A is wrong. Mankiw does measure individual contribution based on market value-added.

Option B is a distortion. The author is concerned about prosperity in general and not just the social status of the individual. Both are not interchangeable.

Option C is wrong. The author only suggests a scale to quantify individual contributions and does not advance the exclusion of any particular kind of contribution.

"The solution to the inequalities produced by meritocracy’s success is to tax the winners rather than to bind Prometheus."

The last line of the passage is an argument in the favour of implementing pragmatic reforms rather than getting rid of meritocracy. Here, 'tax the winners' refers to the pragmatic reform advocated by the author and 'bind Prometheus' is the metaphor for doing away with meritocracy. (Prometheus is a greek mythological creature who signifies the striving and development of humans, and who was bound by Zeus. Hence the metaphor is apt in signifying that doing away with meritocracy, which has yielded many benefits, is like binding Prometheus. The mythological part is irrelevant to the answer, and one must only understand the context to identify the meaning of the metaphor). Hence, the answer is Option B.

The phrase does not advocate anything and just stands as a metaphor. Hence, A is eliminated.

Mentioning a Greek mythological character is neither the author's contention here, though the metaphor does mean this. C would have been the answer if the meaning of the phrase was asked, but will be eliminated here.

The phrase does not mean what is mentioned in Option D. Hence, it can be eliminated.

The situations category of individuals highlighted in Options A, B and D fulfil the criterion to be tagged as "Kierkegaard’s Socrates" - Individuals who are "happy to trade knowledge for freedom"; they are not truly invested in truth but are satisfied with their ignorance {Doubt is something we suffer insofar as we are invested in the truth but find it hard to verify our beliefs. The doubter feels alienated from reality and wants to get back in touch with it. The ironist, on the other hand, triumphs in this alienation. He does not love the truth; he loves the freedom that comes from not believing.}

Option C does not show a wilful ignoring of knowledge. The said companies might still sell those drinks for occasional consumption. 

The correct sequence of the sentences provided in the question is indeed 4, 2, 3, 1. Let's break down each sentence and understand how they logically flow together to form a coherent paragraph:

• The U.S. Supreme Court’s recent decision in Jones v. Mississippi makes it easier for judges to sentence children to life in prison with no chance of parole. This sentence introduces the central idea of the paragraph, which is the recent ruling by the U.S. Supreme Court regarding the sentencing of juveniles to life in prison without parole. It sets the context for the subsequent sentences.
• This new ruling claims that the early teen years cast the die for how someone is likely to behave for the rest of their lives. Sentence 2 elaborates on the implications of the recent ruling mentioned in sentence 4. It explains that the ruling is based on the belief that behavior during early teenage years determines future behavior. This sentence provides insight into the rationale behind the ruling.
• It ignores the very scientific evidence that influenced and guided previous court decisions. Following the introduction of the recent ruling and its rationale in sentences 4 and 2, sentence 3 provides a critical perspective. It suggests that the recent ruling disregards or overlooks scientific evidence that was previously considered in similar court decisions. This statement challenges the validity or completeness of the reasoning behind the recent ruling.
• This argument woefully ignores the neuroscience that explains why juveniles should not be treated as adults. Finally, sentence 1 concludes the paragraph by explicitly stating the main critique or argument against the recent ruling. It points out that the ruling fails to consider neuroscience evidence that supports the idea that juveniles should not be treated as adults in the criminal justice system. This sentence reinforces the perspective introduced in sentence 3 and provides a clear critique of the ruling.

- The missing sentence, "This stark reality has driven the growth of movements advocating for a drastic change in our lifestyle and economic priorities," fits best in Option A (1). It introduces societal reactions to the mounting environmental concerns outlined in the paragraph.
- This sentence highlights the impact of environmental issues on society, prompting movements for significant changes. It connects the urgency of environmental problems with broader societal actions.
- The sentence emphasizes the need for drastic changes in lifestyle and economic practices to address environmental challenges effectively. Its placement at the beginning of the paragraph provides context for understanding subsequent discussions about collective action and individual participation.
- Overall, it enhances the coherence and completeness of the paragraph by illustrating societal responses to environmental crises.

Option C encapsulates the primary message of the passage, emphasizing the new preference for adaptability and learning in the face of technological advancements, without distorting the passage’s nuances.

Option B accurately summarizes the passage, highlighting the ecological importance of coral reefs and the wider implications of their destruction.

Let us draw Venn diagram.

From statement (1),
(b + c) - (b + e) = 8
⇒ c = 8 + e ... (1)
From statement (2),
(a + b) - (d + e) = 7
⇒ a + b = 7 + d + e ... (2)
From statement (3),
(a + b + e + f) - (b + c) = 21
⇒ a + e + f = 21 + c
⇒ a + e + f = 21 + 8 + e [From equation (1), c = 8 + e]
⇒ a + f = 29 ... (3)
From statement (4) ,
(a + b + e + f) + (d + e) = 66
⇒ a + b + d + 2e + f = 66 ... (4)
From statement (5),
b + e = 21 ... (5)
⇒ c = 21 - 4 = 17 ... (6)
From equation (1),
e = c - 8 = 17 - 8 = 9 ... (7)
From equations (5) and (7),
b = 21 - e = 21 - 9 = 12 ... (8)
From equations (3) and (4),
b + d + 2e = 37
⇒ d = 37 - 12 - 2 x 9 = 7 ... (9)
From equations (2), (7), (8) and (9),
a + b = 7 + d + e
⇒ a = 7 + 7 + 9 - 12 = 11 ... (10)
From equations (3) and (10),
a + f = 29
⇒ f = 29 - 11 = 18

Number of friends who only watched movies = 7

Let us draw a Venn diagram

From statement (1):
⇒ (b + c) - (b + e) = 8
⇒ c = 8 + e ... (1)
From statement (2):
⇒ (a + b) - (d + e) = 7
⇒ a + b = 7 + d + e ... (2)
From statement (3):
⇒ (a + b + e + f) - (b + c) = 21
⇒ a + e + f = 21 + c
⇒ a + e + f = 21 + 8 + e (From equation (1): c = 8 + e)
⇒ a + f = 29 ... (3)
From statement (4):
⇒ (a + b + e + f) + (d + e) = 66
⇒ a + b + d + 2e + f = 66 ... (4)
From statement (5):
⇒ b + e = 21 ... (5)
⇒ c = 21 - 4 = 17 ... (6)
From equation (1):
e = c - 8 = 17 - 8 = 9 ... (7)
From equations (5) and (7):
⇒ b = 21 - e = 21 - 9 = 12 ... (8)
From equations (3) and (4):
⇒ b + d + 2e = 37
⇒ d = 37 - 12 - 2 x 9 = 7 ... (9)
From equations (2), (7), (8) and (9):
⇒ a + b = 7 + d + e
⇒ a = 7 + 7 + 9 - 12 = 11 ... (10)
From equations (3) and (10):
⇒ a + f = 29
⇒ f = 29 - 11 = 18

Difference between the number of friends who went to Goa and the number of friends who stayed at home = (11 + 12 + 17) - (7 + 9 + 18) = 40 - 34 = 6

Let us draw a Venn diagram

From statement (1):
⇒ (b + c) - (b + e) = 8
⇒ c = 8 + e ... (1)
From statement (2):
⇒ (a + b) - (d + e) = 7
⇒ a + b = 7 + d + e ... (2)
From statement (3):
⇒ (a + b + e + f) - (b + c) = 21
⇒ a + e + f = 21 + c
⇒ a + e + f = 21 + 8 + e (From equation (1): c = 8 + e)
⇒ a + f = 29 ... (3)
From statement (4):
⇒ (a + b + e + f) + (d + e) = 66
⇒ a + b + d + 2e + f = 66 ... (4)
From statement (5):
⇒ b + e = 21 ... (5)
⇒ c = 21 - 4 = 17 ... (6)
From equation (1):
e = c - 8 = 17 - 8 = 9 ... (7)
From equations (5) and (7):
⇒ b = 21 - e = 21 - 9 = 12 ... (8)
From equations (3) and (4):
⇒ b + d + 2e = 37
⇒ d = 37 - 12 - 2 x 9 = 7 ... (9)
From equations (2), (7), (8) and (9):
⇒ a + b = 7 + d + e
⇒ a = 7 + 7 + 9 - 12 = 11 ... (10)
From equations (3) and (10):
⇒ a + f = 29
⇒ f = 29 - 11 = 18

Number of friends who played charades among the ones who stayed at home = 18 + 9 = 27

Number of hours Mr. Xavier spent coding any mobile app = 92
Average time per day == 5.11 hours

Total time left for all the mobile apps combined = 184
Time to be spent per day = 184/18 = 10.22 hours

The following table presents the 10% range of prices for each car and the cars that can be a part of the racing team, if that car was made the leader's car:

If the cars given in the options must be the only cars in the racing team, one of them must be the leader's car.
Option 1: If Car-1 is the leader's car, Car-2 cannot be in the team. If Car-2 is the leader's car, Car-4 and Car-6 cannot be in the team. If Car-4 is the leader's car, Car-2 cannot be in the team. If Car-6 is the leader's car, Car-2 cannot be in the team. Hence, this is not possible.
Option 2: If Car-3 is the leader's car, Car-12 cannot be in the team. If Car-6 is the leader's car, Car-7 cannot be in the team. If Car-7 is the leader's car, Car-6 and Car-12 cannot be in the team.
If Car-12 is the leader's car, Car-3 and Car-7 cannot be in the team.
Option 3: If Car-3 is the leader, Car-12 cannot be in the team. If Car-9 is the leader's car, this team can be formed. Hence, this is possible with Car-9 as the leader of the team.
Option 4: If Car-6 is the leader's car, Car-7 cannot be in the team. If Car-7 is the leader's car, Car-6. Car-9, Car-10 cannot be in the team. If Car-9 is the leader's car, Car-7 and Car-9 cannot be in the team. If Car-10 is the leader's car, Car-7 cannot be in the team. Hence, this is also not possible. Therefore, only the team given in option3 is possible.

Let us first convert frequency bar graph into a table so that it's easier to read the data.

It is given that Hanne scored a total of 23 and exactly 3 students featured in the extended merit list, have the highest score in exactly one parameter.
We can see that Hanne is one of those three students who featured in extended merit list. Hence, we can say that he must have scored the highest score in one parameter.

Case 1: When Hanne scored the highest in CA:Let us first find out the maximum possible score that Hanne can get in each section.Hanne scored the highest in CA. Hence, Hanne's score in CA = 9Since Hanne scored the highest in CA, he cannot have a score of 9 in AA. Hence, the best possible score which Hanne can have in AA = 7Similarly, Hanne cannot have a score of 8 in OTBT. Hence, the best possible score which Hanne can have in OTBT = 7We can see that that Hanne can have a total score of 23 only when he scores 9, 7, 7 in CA, AA and OTBT, respectively. Hence, we have a unique solution for this case.

Case 2: When Hanne scored the highest in AALet us first find out the maximum possible score that Hanne can get in each section.Hanne scored the highest in AA. Hence, Hanne's score in AA = 9Since Hanne scored the highest in AA, he can't have a score of 9 in CA. Hence, the best possible score which Hanne can have in CA = 7Similarly, Hanne can't have a score of 8 in OTBT. Hence, the best possible score which Hanne can have in OTBT = 7We can see that Hanne can have a total score of 23 only when he scores 9, 7, 7 in CA, AA and OTBT, respectively. Hence, we have a unique solution for this case.

Case 3: When Hanne scored the highest in OTBTLet us first find out the maximum possible score that Hanne can get in each section.Hanne scored the highest in OTBT. Hence, Hanne's score in OTBT = 8Since Hanne has scored the highest in CA, he can't have a score of 9 in CA. Hence, the best possible score which Hanne can have in CA = 7Similarly, Hanne can't have a score of 9 in AA. Hence, the best possible score which Hanne can have in AA = 7We can see that that Hanne can have a maximum total score of 22. He can have a total score of 23 in this case. So, this case can be rejected.

 

We can see that Hanne can score 23 in two different ways. We have to deal with these 2 possible cases separately and figure out the maximum individual score that Ianne and Janne can score.
Two students Ianne and Janne, categorised in extended merit list, are tied with the same total score. To be categorised in extended merit list, one has to score anything from 19 to 24.
It is also given that exactly 3 students featured in extended merit list, have the highest score in exactly one parameter. Hence, we can say that Hanne, Ianne and Janne scored the highest marks of all parameters.

Case 1: When Hanne scored 9, 7, 7 in CA, AA and OTBT, respectively. After removing Hanne's score, frequency table will look like this:

Let us assume that Ianne scored the highest in AA and Janne scored the highest in OTBT. {Naming differently won't matter as we are calculating the maximum possible score for both.}
Ianne can score a maximum of 7 in CA. Similarly, he/she can score a maximum of 7 in OTBT. Hence, the maximum total which Ianne can score = 7 + 9 + 7 = 23
Janne can score a maximum of 7 in CA. Similarly, he/she can score a maximum of 6 in AA. Hence, the maximum total which Janne can score = 7 + 6 + 8 = 21
However, both can't score 7 at a time in CA. To maximise the individual score, let us assume that Ianne scored 6 in CA and Janne scored 7 in CA. Even in that case, the maximum individual score that Ianne and Janne can have = 21
We can represent this in a tabular format.

Case 2: When Hanne scored 7, 9, 7 in CA, AA and OTBT, respectively. After removing Hanne's score, frequency table will look like this:

Let us assume that Ianne scored the highest in CA and Janne scored the highest in OTBT. {Naming differently won't matter as we are calculating the maximum possible score for both.}
Ianne can score a maximum of 7 in AA. Similarly, he/she can score a maximum of 7 in OTBT. Hence, a maximum total which Ianne can score = 9 + 7 + 7 = 23
Janne can score a maximum of 6 in CA. Similarly, he/she can score a maximum of 7 in AA. Hence, a maximum total which Janne can score = 6 + 7 + 8 = 21
However, both can't score 7 at a time in AA. To maximise the individual score, let us assume that Ianne scored 6 in AA and Janne scored 7 in AA. Even in that case, the maximum individual score that Ianne and Janne can have = 21
We can represent this in a tabular format.

Therefore, we can say that Ianne and Janne can score a maximum total score of 21 individually.

The area of room T is 16 m²..
One side of room T is 6 m in length. Therefore, the area of room S is 4 m².
It has been given that five rooms have same area and rooms P and R have same area.
Rooms P, Q and R have same area.
Area of room P + area of room Q = 32 m²
Rooms P and Q share a wall.
Therefore, the lengths of rooms P and Q must be the same.
Let the length of room P be l.
Length of room U = length of room V = 2l
Let the width of room P be b.
lb = 16 --- (1)

Let the breadth of room U be 'x'.
2lx = 16 --- (2)
⇒ x = b/2
The same can be said to be true with respect to room V as well.
Now, we know that
⇒ b = 4 and l = 4
The length and breadth of all the rooms are as follows:

 

Amar has only 2 neighbours.
Amar can live in room P, V, S or T.
The areas of room assigned to Edward and Ganesh are same, but the perimeters are different. Therefore, one of them must be living in room U, V or R, and the other must be living in P or Q. The perimeter of the room in which Amar lives is equal to the perimeter of the room in which Ganesh lives. The perimeter of the room in which Ganesh lives can be 20 or 16. The perimeters of room P, V, S and T are 16, 20, 8 and 16, respectively. Therefore, Amar cannot live in room S. It has been given that Amar does not live in Room P. Therefore, Amar can live in room V or T.

Case 1: Amar lives in room V.
The areas of rooms in which Edward and Ganesh live are same, but the perimeters are different. Edward and Ganesh are not neighbours. Amar and Edward are not neighbours. Therefore, Edward must live in room P and Ganesh must live in room R.

Faisal is not a neighbour of Amar or Edward. Faisal does not live in the smallest room. Therefore, Faisal must live in room T. Faisal and Ganesh are neighbours of Deepak. Therefore, Deepak must live in room S. The perimeter of the room in which Bindu lives is less than the perimeter of the room in which Charan lives. Therefore, Charan must live in room U and Bindu must live in room Q. The arrangement will be as follows:

Case 2: Amar lives in room T.
The areas of rooms in which Edward and Ganesh live are same, but the perimeters are different. Edward and Ganesh are not neighbours. Amar and Edward are not neighbours. Faisal is not a neighbour of Amar or Edward and he does not live in the smallest room. Ganesh does not live in room Q. Therefore, Ganesh must live in room P, Faisal must live in room Q and Edward must live in room V.

Ganesh and Faisal are neighbours of Deepak. The perimeter of the room in which Bindu lives is less than the perimeter of the room in which Charan lives. Therefore, the arrangement will be as follows:

The positions of all the seven persons are different in the two cases. Therefore, we cannot determine the room of any of the persons they live in. Hence, option 4 is the right answer.

It is given that there were 6 attempts in the game and all the five players got different number of dollars. The maximum number of attempts a single player could have rolled the slot machine has to be 2 (if one player rolled the slot machine in 3 attempts, then two players would end up with the same number of dollars).
Jack was the only player to get 2 attempts consecutively and he did not get the 1st attempt. The number of maximum possible attempts by a player is 2, which means Jack got $10 in the first attempt, and in the second attempt he paid a penalty amount of $5, but from (iv), the player who rolled the slot machine in the last attempt won the game.
Jack couldn't be at 6th.
Possible positions for Jack are: 2, 3, or 3, 4, or 4, 5, but there is change in sequence (from (ii)) at the beginning of the 4th attempt and the 5th attempt. So, Jack couldn't go at the 4th and 5th attempt.
Sequence[attempt number(name, amount)]: 1, 2(Jack, $10), 3(Jack, -$5), 4, 5, 6(xx, $10)
This means, the 1st, 4th and 5th attempts didn't hit correctly and they would pay the penalty.
Sequence[attempt number(name, amount)]: 1(xx, -$5), 2(Jack, $10), 3(Jack, -$5), 4(xx, -$5), 5(xx, -$5), 6(xx, $10)
There has to be another person who rolled the slot machine in 2 attempts. This person also could not have rolled the slot machine in the 5th and 6th attempts.
Hence, he must have rolled the slot machine in the 1st and 5th attempts.
From (ii), this person has to be Luke because only by getting the 5th attempt could Luke move to front of Jack.
Sequence[attempt number(name, amount)]: 1(Luke, -$5), 2(Jack, $10), 3(Jack, -$5), 4(xx, -$5), 5(Luke, -$5), 6(xx, $10)
Will must have rolled the slot machine in one attempt without hitting the jackpot and Franklin must have rolled the slot machine in one attempt and hit the jackpot.
Sequence[attempt number(name, amount)]: 1(Luke, -$5), 2(Jack, $10), 3(Jack, -$5), 4(Will, -$5), 5(Luke, -$5), 6(Franklin, $10)

From the above table, it is clear that Franklin got the highest amount, i.e. $10, so he stood first in the game.

It is given that there were 6 attempts in the game and all the five players got different number of dollars. The maximum number of attempts a single player could have rolled the slot machine has to be 2 (if one player rolled the slot machine in 3 attempts, then two players would end up with the same number of dollars).
Jack was the only player to get 2 attempts consecutively and he did not get the 1st attempt. The number of maximum possible attempts by a player is 2, which means Jack got $10 in the first attempt, and in the second attempt he paid a penalty amount of $5, but from (iv), the player who rolled the slot machine in the last attempt won the game.
Jack couldn't be at 6th.
Possible positions for Jack are: 2, 3, or 3, 4, or 4, 5, but there is change in sequence (from (ii)) at the beginning of the 4th attempt and the 5th attempt. So, Jack couldn't go at the 4th and 5th attempt.
Sequence[attempt number(name, amount)]: 1, 2(Jack, $10), 3(Jack, -$5), 4, 5, 6(xx, $10)
This means, the 1st, 4th and 5th attempts didn't hit correctly and they would pay the penalty.
Sequence[attempt number(name, amount)]: 1(xx, -$5), 2(Jack, $10), 3(Jack, -$5), 4(xx, -$5), 5(xx, -$5), 6(xx, $10)
There has to be another person who rolled the slot machine in 2 attempts. This person also could not have rolled the slot machine in the 5th and 6th attempts.
Hence, he must have rolled the slot machine in the 1st and 5th attempts.
From (ii), this person has to be Luke because only by getting the 5th attempt could Luke move to front of Jack.
Sequence[attempt number(name, amount)]: 1(Luke, -$5), 2(Jack, $10), 3(Jack, -$5), 4(xx, -$5), 5(Luke, -$5), 6(xx, $10)
Will must have rolled the slot machine in one attempt without hitting the jackpot and Franklin must have rolled the slot machine in one attempt and hit the jackpot.
Sequence[attempt number(name, amount)]: 1(Luke, -$5), 2(Jack, $10), 3(Jack, -$5), 4(Will, -$5), 5(Luke, -$5), 6(Franklin, $10)

From the above table, after the final analysis of game, Benjamin's score is 0 and Will's score is -5. So the difference between the amounts (in $) with Benjamin and Will is 5.

It is given that there were 6 attempts in the game and all the five players got different number of dollars. The maximum number of attempts a single player could have rolled the slot machine has to be 2 (if one player rolled the slot machine in 3 attempts, then two players would end up with the same number of dollars).
Jack was the only player to get 2 attempts consecutively and he did not get the 1st attempt. The number of maximum possible attempts by a player is 2, which means Jack got $10 in the first attempt, and in the second attempt he paid a penalty amount of $5, but from (iv), the player who rolled the slot machine in the last attempt won the game.
Jack couldn't be at 6th.
Possible positions for Jack are: 2, 3, or 3, 4, or 4, 5, but there is change in sequence (from (ii)) at the beginning of the 4th attempt and the 5th attempt. So, Jack couldn't go at the 4th and 5th attempt.
Sequence[attempt number(name, amount)]: 1, 2(Jack, $10), 3(Jack, -$5), 4, 5, 6(xx, $10)
This means, the 1st, 4th and 5th attempts didn't hit correctly and they would pay the penalty.
Sequence[attempt number(name, amount)]: 1(xx, -$5), 2(Jack, $10), 3(Jack, -$5), 4(xx, -$5), 5(xx, -$5), 6(xx, $10)
There has to be another person who rolled the slot machine in 2 attempts. This person also could not have rolled the slot machine in the 5th and 6th attempts.
Hence, he must have rolled the slot machine in the 1st and 5th attempts.
From (ii), this person has to be Luke because only by getting the 5th attempt could Luke move to front of Jack.
Sequence[attempt number(name, amount)]: 1(Luke, -$5), 2(Jack, $10), 3(Jack, -$5), 4(xx, -$5), 5(Luke, -$5), 6(xx, $10)
Will must have rolled the slot machine in one attempt without hitting the jackpot and Franklin must have rolled the slot machine in one attempt and hit the jackpot.
Sequence[attempt number(name, amount)]: 1(Luke, -$5), 2(Jack, $10), 3(Jack, -$5), 4(Will, -$5), 5(Luke, -$5), 6(Franklin, $10)

Will and Luke respectively scored -10 and -5 debt at the end of the game, so both were in debt.

Let the function be f(x) = mx+c, as it is given that the function, is linear, here 'm' is the slope of the function

f(f(x)) = m²x+c+mc

similarly f(f(f(x))) = f(m²x + c + m³x+c+mc+m²c

and f(f(f(f(x)))) = m⁴x + c + mc + m²c + m³c

f(f(f(f(1)))) = f(f(1)), Thus, m⁴+c+mc+m²c+m³c = m²+c+mc

On rearranging we get m⁴+m²c+m³c = m²

Since the function is not constant we can say that m ≠ 0

Dividing both side by m² and shifting all term to one side we get

m² + c + mc − 1 = 0

(m²−1)+(c+mc) = 0

(m+1)(m−1)+c(1+m)=0

(m+1)(m+c−1) = 0

Since the function does not make 45⁰ with x-axis the value of m can not be 1 or -1

Thus m+c-1 = 0 or m+c = 1....(I)

f(3) = 3m+c and f(0) = c

f(3)+2f(0) = 3m+c+2c 

= 3m+c+2c = 3(m+c) = 3 (inputting m+c = 1 from (I))

If x ≥ 0 it implies that x+2 is also always positive.

The minimum value the LHS will be

thus x = 0 as we are given that x ≥ 0
The RHS has a cos term and the maximum value of RHS is 4 when the 

It is possible when cos 

or x² + 2x = 4n

Thus there are many solutions for the equation but LHS achieves the value only once when x = 0. There is one solution possible.

Let the x units of soy-milk solution is used and y units of an almond-based solution is used

Amount of water in soy-milk = 0.125x units

Amount of water in almond-milk = 0.4y

After the mixing, the concentration of water is 0.3

Hence 0.125x+0.4y = 0.3(x+y)

0.125x+0.4y = 0.3x+0.3y

0.1y = 0.175x

y = 1.75x

y = 47​x

We can see that at x= 2,6,10,14,18 the function becomes 0

After every root, the function changes its value. 

for x > 18; the equation is always positive hence no solution 

for 14<x<18, the equation is negative and 15 & 17 are 2 odd numbers satisfying it

for 10<x<14; the equation is always positive

for 6 < x < 10 ; the equation is negative and 7 & 9 are 2 odd numbers satisfying it

for 2<x<6, the function will be always positive hence no solution

For 0<x<2 the function will be negative. 1 is the only odd number in this range

Total 5 such numbers are there.

Let us start with calculating the value of c − 1/2b​

Now we need to calculate the value of (1/a​−1/b​)

Putting (I) and (II) in original equation

we get 

Let the amount invested in the scheme I  xx

The return from scheme I at the end of year 2 (1.1)²x = 1.21

Let the amount invested in scheme II be yy

Since it is compounded half-yearly, the rate of interest per half-year will be 20%/2 ​= 10%

The return from scheme II will be y(1.1)⁴ = 1.4641y

Hence 1.21x+1.4641y=136246 ...(I)

Also, x+y = 100000 ...(II)

1.4641(II)-(I)  gives

0.2541x=10164

x=40,000

For compound interest, the interest gained per year is in G.P with a common ratio of (1+r%) where r is the rate of interest

Let the rate be R%

Then, 1+R%) = 1728/1440 ​= 1.2

R% = 0.2 or 20%

We can select any three points as vertices of the triangle, such that all 3 points cannot belong to a particular straight line
We can select any 3 points out of 15 points in a plane in ¹⁵C₃ = 455 ways
We can select 3 points from the 1st straight line in ³C₃ = 1 way
We can select 3 points from the 2nd straight line in ⁴C₃ = 4 ways
Thus total number of triangles that can be formed is = 455 - (1 + 4) = 450