CAT 50 Day Streak Day 30

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CAT 50 Day Streak – Day 30

30 questions across VARC, DILR & QA · +3 for a correct answer · −1 for a wrong MCQ · no negative marking for TITA

0/30Attempted

0Correct

0Wrong

0Score / 90

VARC · Verbal Ability & Reading Comprehension

Q1 – Q10 0/10

Passage · Q1 – Q4

Public goods are freely accessible to all members of a given public, each being able to benefit from it without paying for it. The reason standard theory puts forward for this anomaly is that public goods are by their technical character non-excludable. There is no way to exclude a person from access to such a good if it is produced at all. Examples cited include the defence of the realm, the rule of law, clean air or traffic control. If all can have it without contributing to its cost, nobody will contribute and the good will not be produced. This, in a nutshell, is the public goods dilemma, a form of market failure which requires taxation to overcome it. Its solution lies outside the economic calculus; it belongs to politics.

Access to a private good is controlled by its producer or owner by a variety of devices ranging from shop counters, safes, walls and fences to measures against theft, robbery, fraud, illicit copying and breach of contract. The cost of these devices and measures is the exclusion cost of the good. Every good is private or public according to whether exclusion cost is or is not incurred in making it available. A public good is distributed freely to all comers from a given public, avoiding the exclusion cost that would keep it private. This saving is the ‘productivity of publicness’.

Given sufficient imagination and clever technology, every good can be excluded at some cost. Arguably, some would be very awkward to exclude, but none is intrinsically ‘non-excludable’, i.e. doomed to be a public good. By the same token, every good, whether private or public, has many more or less imperfect substitutes that may also be private or public. Thus, contrary to received theory, a more general view tells us that while no good is intrinsically public, the higher is its exclusion cost and the more imperfect are its substitutes, the more efficient it is to provide it publicly.

There is one type of exclusion cost that is more important by far than all the rest in putting a good in the public category: it is social preference. It is intangible and is only revealed by the choices it inspires. A pure example is a children’s playground. Access to it is excludable at low cost by a fence and a ticket collector at the gate. However, society would suffer deep moral embarrassment if rich children could use the playground but poor ones could only watch them from the outside. Therefore real exclusion cost would be unbearably high, and children’s playgrounds are provided as public goods.

There are other, less pure but quantitatively far more important examples. One is free universal education. Most countries provide it to age 16, some to university degree level. In this case, technical-logistical exclusion cost would be quite low (indeed, in a broad sense negative as exclusion would permit student selection, and that would in turn lower production cost), but social ethics would not tolerate the exclusion of poor, dumb and sub-scholarship standard pupils. With education becoming a public good, affording free access, the share of public goods in the national product expands vastly.

Q1MCQ

The author suggests that the public character of a good may depend on

The moral discomfort society feels about exclusion
The degree to which substitutes exist for that good
The level of profit its private producers can generate

AOnly I and II

BOnly I and III

COnly II and III

DAll I, II, and III

Solution Answer: A

I is the “social preference” discussion: “deep moral embarrassment” makes exclusion too costly. II is “the more imperfect are its substitutes, the more efficient it is to provide it publicly”.

III: the passage never links public character to producers’ profits. It turns on exclusion cost and substitutes.

Q2MCQ

According to the author, the primary reason free universal education is provided as a public good is that:

Athe “productivity of publicness” is greater in education than in any other public good.

Bthe “production cost” would be unmanageably high if it were privatized.

Cthe intangible cost associated with the social unacceptability of exclusion is considered too high.

Dit is an “imperfect substitute” for private education, making its public provision more efficient.

Solution Answer: C

The technical exclusion cost of education “would be quite low”, but “social ethics would not tolerate the exclusion” of poor or weak pupils. That intangible social-preference cost is what makes it public.

A: no such comparison is made. B: the passage says exclusion would even lower production cost. D misuses the substitutes idea, which isn’t applied to education.

Q3MCQ

Which of the following scenarios best exemplifies the author’s concept of “social preference” as the most important exclusion cost?

AA city government builds a network of public roads, funded by taxes, because it is technically difficult to charge every individual driver for using each street.

BA pharmaceutical company, after public outrage, agrees to provide a life-saving antidote to all patients for free, even though it could easily restrict access to paying customers.

CA software firm makes its code open-source because the cost of prosecuting “illicit copying” is higher than the potential revenue from sales.

DA national park service maintains a “free entry” policy because the cost of building fences and staffing ticket booths in the remote wilderness is “unbearably high.”

Solution Answer: B

Like the playground, the antidote could easily be restricted at low technical cost. Public moral outrage, an intangible cost, makes exclusion unacceptable, so it is given freely.

A, C and D all turn on technical or financial exclusion costs (charging drivers, prosecuting copying, building fences), not social preference.

Q4MCQ

It can be inferred that the author believes in the “public goods dilemma” as described in the passage:

Ais a false paradox that stems from a flawed premise about technical non-excludability.

Bis an insoluble problem for goods like national defense, but not for goods like playgrounds.

Cis becoming more severe as the exclusion cost for most goods decreases.

Dis ultimately solved by the economic savings from the “productivity of publicness.”

Solution Answer: A

Standard theory builds the dilemma on goods being “by their technical character non-excludable”. The author rejects this: “none is intrinsically ‘non-excludable’”, “contrary to received theory”. So the dilemma rests on a flawed premise.

B: the author says every good can be excluded at some cost. C: no claim about severity rising. D: productivity of publicness is a saving, not a solution to the dilemma.

Q5TITA

The four sentences (labelled 1, 2, 3 and 4) given below, when properly sequenced, would yield a coherent paragraph. Decide on the proper sequencing of the order of the sentences and key in the sequence of the four numbers as your answer.

Conditions were ripe for the development of such a party.
Discharged from the hospital amid the social chaos that followed Germany’s defeat, Hitler took up political work in Munich in May–June 1919.
In 1920, he was put in charge of the party’s propaganda and left the army to devote himself to improving his position within the party, which in that year was renamed the Nationalsozialistische Deutsche Arbeiterpartei (Nazi).
As an army political agent, he joined the small German Workers’ Party in Munich in September 1919.

Solution Answer: 2431

The order runs like this:

2: Hitler begins political work in Munich in May–June 1919.
4: In September 1919 he joins the small German Workers’ Party.
3: In 1920 he heads “the party’s” propaganda, and the party is renamed Nazi.
1: “Such a party”, the newly named National Socialist party, found conditions ripe. This moves from the chronology to the wider context.

Q6MCQ

There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.

Sentence From a pre-20th-century perspective, this shift wouldn’t affect your experience because your experiences are part of the content of each moment.

Paragraph Imagine that everything that has happened to you today occurred an hour later. ___(1)___. Suppose that I enter a restaurant, look at a clock, see that it reads 2:47 pm, and then order a drink. ___(2)___. To those who lived before the 20th century, even if these events took place an hour later, everything would happen in the same way: the time would still read 2:47 pm on the clock, I would still order the drink, and still form the same sequence of memories. ___(3)___. From this perspective, the exact timing of events is irrelevant to their contents. ___(4)___. This led to the conclusion that time can stretch backwards indefinitely for if no moment is fundamentally different from any other and the moments themselves make no difference to the contents, there’s no way to mark a beginning.

AOption 1

BOption 2

COption 3

DOption 4

Solution Answer: A (Option 1)

“This shift” refers to everything happening “an hour later”, so the sentence must follow the opening line directly. It states the pre-20th-century claim, and the restaurant example then illustrates it.

At blanks 2–4 the paragraph has already given the example and restated the pre-20th-century view, so the sentence would be repetitive and “this shift” would be too far from its antecedent.

Q7TITA

Five jumbled-up sentences (labelled 1, 2, 3, 4 and 5), related to a topic, are given below. Four of them can be put together to form a coherent paragraph. Identify the odd sentence and key in the number of that sentence as your answer.

As psychologists and behavioural economists have documented, people have various cognitive limitations.
This premise is also descriptive of decision-making, so long as these cognitive and other scarcity limitations are duly noted.
Mistakes are also rational, because of the resource scarcity that makes the economists’ rationality premise necessary.
But in economics, such changes operate by affecting the relative costs of pursuing different options.
People are prey to decision biases, irrationalities, and failures and make decision mistakes—many of which are calculated, planned, and expected.

Solution Answer: 4

The coherent paragraph is 1 → 5 → 3 → 2:

1: People have cognitive limitations.
5: So they make decision mistakes, many of them expected.
3: Such mistakes are “also rational” because of scarcity, which makes the rationality premise necessary.
2: “This premise” also describes decision-making once limitations are noted.

Sentence 4 talks of “such changes” affecting relative costs, but no changes are mentioned anywhere. It belongs to a different discussion.

Q8MCQ

The passage given below is followed by four alternate summaries. Choose the option that best captures the essence of the passage.

Passage A century later, we can observe another shift in how writers understood the relationship between mind and media. James Mill and his son John Stuart Mill laid out comprehensive psychological theories that built upon the 18th-century accounts of association. However, the most lasting impact of associationist theory came from Sigmund Freud’s development of free association as a practice of psychoanalytic therapy. In Associationism and the Literary Imagination, Cairns Craig observes that ‘Freud’s technique of “free association” invites the patient to do exactly what the associationist aesthetics of the empiricist tradition had always insisted readers do in experiencing a poem.’ He cites the late-18th-century essayist Archibald Alison, who described reading a poem as ‘letting our fancy be “busied in pursuit of all those trains of thought”’ of the poet.

AThe works of James and John Stuart Mill marked the end of associationist thought and paved the way for Freud’s distinct psychoanalytic approach.

BThe idea of association evolved from 18th-century theories of mind into Freud’s therapeutic practice, linking literary experience with psychological exploration.

CFreud’s development of free association rejected earlier empiricist views on the mind and created a wholly new understanding of human psychology.

DAssociationism influenced both literature and psychology, though its application in therapy differed fundamentally from its aesthetic origins.

Solution Answer: B

The passage traces association from 18th-century theory, through the Mills, to Freud’s free association. Craig’s point is that free association mirrors how readers experience a poem. B captures both the evolution and the literary–psychological link.

A: the Mills built on associationism; they didn’t end it. C: Freud extended the tradition rather than rejecting it. D: the passage stresses similarity (“exactly what”), not fundamental difference.

Q9TITA

Sentences given in the question below, when properly sequenced, form a coherent paragraph. The first and last sentences are A and F, and the four in between are labelled 1, 2, 3, and 4. Choose the most logical order of these four sentences to construct a coherent paragraph from sentences A to F.

Artificial Intelligence significantly influences the business management software market, transforming the way businesses operate and make decisions.
These systems analyze vast amounts of data, identify patterns, and provide actionable insights, enabling businesses to make data-driven decisions.
Furthermore, AI enhances customer service through chatbots and virtual assistants, providing instant responses and personalized experiences to customers.
AI-powered software automates routine tasks, freeing up valuable time for employees to focus on strategic tasks.
The integration of AI into business management software has led to the development of intelligent systems that enhance efficiency, productivity, and accuracy in various business operations.
The use of AI in business management software improves risk management by predicting potential risks and suggesting preventive measures.

Solution Answer: 4132

The order runs like this:

4: Integration of AI has produced “intelligent systems”.
1: “These systems” analyse data for decisions.
3: AI also automates routine tasks.
2: “Furthermore”, AI improves customer service. F then adds risk management.

Q10MCQ

There is a sentence that is missing in the paragraph below. Look at the paragraph and decide where (option 1, 2, 3, or 4) the following sentence would best fit.

Sentence But is it used offensively now?

Paragraph Am I being oversensitive about something ostensibly friendly? ___(1)___. I tried to work it out through the traditional medium of Googling “Is buddy rude?” ___(2)___. Apparently, it might be derived from “butty”, a 19th-century coalminers’ term for a co-worker, or a childish version of “brother”, which seems innocuous. ___(3)___. People online wonder why it grates, at least, and an English teacher on YouTube calls it “not always friendly and hard to interpret”. ___(4)___. One Scottish Redditer claimed that, on the east coast, it’s habitually deployed in the phrase: “A’right buddy, you needin’ yer teeth smashed in?”, which I find persuasive.

AOption 1

BOption 2

COption 3

DOption 4

Solution Answer: C (Option 3)

The etymology ends with “which seems innocuous”. “But is it used offensively now?” contrasts that harmless origin with present usage. The next sentences answer it with people wondering why it grates and the YouTube teacher. So it fits at blank 3.

At blanks 1 and 2 the origin hasn’t been given, so “But… now” has nothing to contrast. At blank 4 the evidence about current usage has already started.

DILR · Data Interpretation & Logical Reasoning

Q11 – Q20 0/10

Set 1 · Q11 – Q15

During a particular month, each of the seven friends met one or more of the other six friends. For any pair of friends, say, Anita and Bhoomi, if Anita met Bhoomi, then Bhoomi also met Anita. It is also known that:

The number of friends that the seven friends met is 1, 2, 2, 3, 3, 4, and 5, not necessarily in any specific order.
Deepa met more friends than Chhaya.
Anita and Bhoomi did not meet each other, and among the friends that Bhoomi met, Anita did not meet exactly two friends.
Chhaya met Ekta and two other friends, but did not meet Bhoomi.
Gauri did not meet any friends who met Falguni.
Anita met all the friends that Falguni met, not considering Anita and Falguni.
Ekta met both Bhoomi and Falguni.

Q11MCQ

How many friends did Gauri meet?

Solution Answer: B

Chhaya and Deepa. Chhaya met 3, so Deepa met 4 or 5.
Ekta. Ekta met Falguni, so Anita met Ekta (Anita met everyone Falguni met). Ekta’s friends then include Anita, Bhoomi, Chhaya and Falguni, so Ekta met at least 4. Deepa and Ekta take 4 and 5.
Deepa ≠ 5. Deepa would then meet everyone, including Falguni and Gauri, so Gauri would have met someone who met Falguni ✗. So Deepa = 4 and Ekta = 5: Anita, Bhoomi, Chhaya, Deepa, Falguni.
Falguni and Bhoomi. If Falguni met Bhoomi, Anita would have to meet Bhoomi ✗. Bhoomi didn’t meet Anita or Chhaya, so Bhoomi’s possible friends are Deepa, Ekta and Gauri.
Bhoomi. Anita met Ekta, so the two of Bhoomi’s friends Anita didn’t meet must be Deepa and Gauri. Bhoomi = {Deepa, Ekta, Gauri}, and Anita met neither Deepa nor Gauri.
Falguni. Falguni’s other friends must also be Anita’s friends, so Falguni can’t meet Deepa or Gauri. Deepa needs 4: Bhoomi, Ekta, plus Chhaya and Gauri.
Gauri. Gauri met Bhoomi and Deepa, neither of whom met Falguni, so Gauri = 2.
Chhaya. Chhaya’s third friend is Anita or Falguni. If it were Falguni, Anita would also need to meet Chhaya, giving Chhaya 4 ✗. So Chhaya met Anita.
Anita and Falguni. The remaining degrees are 1 and 2. If they met, Anita would have 3 ✗. So Anita = {Chhaya, Ekta} and Falguni = {Ekta}.

Friend	Met	Count
Anita	Chhaya, Ekta	2
Bhoomi	Deepa, Ekta, Gauri	3
Chhaya	Anita, Deepa, Ekta	3
Deepa	Bhoomi, Chhaya, Ekta, Gauri	4
Ekta	Anita, Bhoomi, Chhaya, Deepa, Falguni	5
Falguni	Ekta	1
Gauri	Bhoomi, Deepa	2

Gauri met 2 friends.

Q12MCQ

Who has met only one friend?

AAnita

BDeepa

CFalguni

DGauri

Solution Answer: C

From the table in Q11, Falguni met only Ekta.

Q13TITA

How many common friends did both Deepa and Ekta meet?

Solution Answer: 2

From the table in Q11, Deepa met {Bhoomi, Chhaya, Ekta, Gauri} and Ekta met {Anita, Bhoomi, Chhaya, Deepa, Falguni}. They share Bhoomi and Chhaya, so 2.

Q14MCQ

Who among the following did Chhaya not meet?

AFalguni

BAnita

CDeepa

DEkta

Solution Answer: A

From the table in Q11, Chhaya met Anita, Deepa and Ekta. Of the options, she did not meet Falguni.

Q15MCQ

Which of the following statements is true?

ADeepa met the highest number of people.

BBoth Bhoomi and Chhaya met Gauri.

CAnita met more people than Gauri did.

DEkta and Falguni did not meet any common person.

Solution Answer: D

A: Ekta met the most (5), not Deepa ✗
B: Chhaya did not meet Gauri ✗
C: Anita and Gauri both met 2 ✗
D: Falguni’s only friend is Ekta herself, so they share no common friend ✓

Set 2 · Q16 – Q20

Five students enrolled for a mock series for MAHCET, which comprised five mock tests numbered M1, M2, M3, M4 and M5, conducted in that order. However, each student wrote exactly four mocks, and each mock was written by exactly four students. Further, it is known that, for any mock, the average score of the students who wrote that mock was not more than 125. The table below provides the marks scored by each student in the four mocks that they wrote. The four mocks are labelled mock 1, mock 2, mock 3 and mock 4, which represent the first, second, third and fourth mock written by the respective student, in that order.

	Mock 1	Mock 2	Mock 3	Mock 4
Manika	105	118	115	135
Yugal	140	132	123	128
Tanu	119	108	112	126
Harsh	137	123	124	107
Sakshi	132	143	121	126

Q16MCQ

Which of the mocks did Manika not take?

AM1

BM2

CM3

DM4

Solution Answer: D

Structure. Each student skipped exactly one mock, and each mock was skipped by exactly one student. Each mock total must be ≤ 4 × 125 = 500. All scores add to 2474, so the slack is only 26.
M1. M1 = 633 (sum of everyone’s first score) − first score of the student who skipped M1. To stay ≤ 500, the skipper’s first score must be ≥ 133: Yugal (140) or Harsh (137).
Yugal skips M1 ✗. M2 ≤ 500 then forces Sakshi to skip M2. Any assignment of the rest pushes M3 above 500.
Harsh skips M1. M2 ≤ 500 forces Sakshi to skip M2. M3 ≤ 500 then needs Manika and Tanu to skip M4 and M5, so Yugal skips M3.
M4 decides. If Manika skipped M5, M4 would be 368 + 135 = 503 ✗. So Tanu skips M5 and Manika skips M4.

Student	M1	M2	M3	M4	M5
Manika	105	118	115	—	135
Yugal	140	132	—	123	128
Tanu	119	108	112	126	—
Harsh	—	137	123	124	107
Sakshi	132	—	143	121	126
Total	496	495	493	494	496

Q17TITA

In how many tests was Tanu the least scorer?

Solution Answer: 2

From the table in Q16, the lowest scorers are: M1 Manika (105), M2 Tanu (108), M3 Tanu (112), M4 Sakshi (121) and M5 Harsh (107). Tanu was lowest in 2 tests.

Q18MCQ

Which two mocks had the same total marks obtained by the students who took them?

AM1 and M5

BM2 and M3

CM3 and M4

DAll mocks had different totals

Solution Answer: A

From the table in Q16, the totals are 496, 495, 493, 494 and 496. M1 and M5 both total 496.

Q19MCQ

Who among the following did not appear in Mock M5?

AManika

BYugal

CTanu

DHarsh

Solution Answer: C

From the table in Q16, Tanu skipped M5.

Q20TITA

What are the total marks obtained by the students who took M3?

Solution Answer: 493

From the table in Q16, M3 = 115 + 112 + 123 + 143 = 493.

QA · Quantitative Ability

Q21 – Q30 0/10

Q21MCQ

Two boys start moving towards each other, one from point P and the other from point Q. The first boy left point P 6 hours later than the second boy left point Q, and it was found during their meeting that he had traveled 12 km less than the second boy. After their meeting, they continued at the same speeds, and the first boy reached point Q 8 hours later, while the second boy reached point P 9 hours later. Find the speed of the faster boy.

A4 km/h

B5 km/h

C6 km/h

D8 km/h

Solution Answer: C

Let the speeds be a (first boy) and b (second boy), and let the first boy walk t hours before the meeting.

After meeting, each covers the other’s distance. The first boy’s 8 hours cover the second’s distance: b(t + 6) = 8a. The second boy’s 9 hours cover the first’s distance: at = 9b.

From the second equation, t = 9b/a. Substituting gives 8a² = 9b² + 6ab. With k = a/b: 8k² − 6k − 9 = 0, so k = 3/2.

The distance difference is 8a − 9b = 12, i.e. 12b − 9b = 3b = 12. So b = 4 and a = 6. The faster boy goes 6 km/h.

Q22TITA

If x is a real number ≥1, what is the number of solutions for the equation (x⁹ + 15)^1/2 = x³ – 1?

Solution Answer: 0

Square both sides: x⁹ + 15 = x⁶ − 2x³ + 1. With u = x³ ≥ 1: u³ − u² + 2u + 14 = 0.

For u ≥ 1, u³ − u² ≥ 0 and 2u + 14 > 0, so the left side is always positive. There are 0 solutions.

Q23MCQ

In the given figure, AB is the diameter of the circle and BE is the bisector of ∠ABD. Find the length of EB.

Circle with diameter AB; line BC tangent at B with BC = 15; secant A–E–D–C with AE = 10 and DC = 9; BD drawn and BE = x bisects angle ABD

A6√5

B12

C6√6

D6√3

Solution Answer: A

Tangent-secant: BC² = CD × CA, so 225 = 9 × CA and CA = 25. Then AD = 16 and ED = 16 − 10 = 6.

AB is a diameter, so ∠ADB = 90°. The tangent is perpendicular to the radius, so ∠ABC = 90°. That gives AB = √(25² − 15²) = 20, and the altitude BD = (20 × 15)/25 = 12.

Check with the angle bisector in triangle ABD: AE/ED = AB/BD = 20/12 = 5/3, which matches 10 : 6 ✓.

EB = √(BD² + ED²) = √(144 + 36) = √180 = 6√5.

Q24MCQ

A village has 3 people for each horse, 4 sheep for each cow, and 3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in the village?

A41

B47

C59

D61

Solution Answer: B

With h horses and c cows: people = 3h, ducks = 9h, sheep = 4c. Total = 13h + 5c.

41 = 13(2) + 5(3) ✓
59 = 13(3) + 5(4) ✓
61 = 13(2) + 5(7) ✓
47: subtracting 0, 13, 26 or 39 leaves 47, 34, 21 or 8, none divisible by 5 ✗

The total can’t be 47.

Q25TITA

Six men can complete a piece of work in 12 days. Eight women can complete the same work in 18 days, while eighteen children can finish it in 10 days. If 4 men, 12 women, and 20 children work together for 2 days, and the remaining work is then to be completed by men alone on the next day, how many additional men will be required to finish the work on time?

Solution Answer: 32

Daily work rates: 1 man = 1/72, 1 woman = 1/144, 1 child = 1/180.

The team does 4/72 + 12/144 + 20/180 = 1/18 + 1/12 + 1/9 = 9/36 = 1/4 per day. In 2 days that is 1/2, leaving 1/2.

Men needed in one day = (1/2) ÷ (1/72) = 36. Additional men = 36 − 4 = 32.

Q26MCQ

If all the roots of the equation x⁸ + 8x⁷ + … + ax⁴ + … + 1 = 0 are integers, find the value of a.

A56

B70

C72

D80

Solution Answer: B

The product of the roots is 1, and every root is an integer, so each root is ±1. The sum of the roots is −8, so all eight roots are −1.

The polynomial is (x + 1)⁸, so a = ⁸C₄ = 70.

Q27MCQ

A shopkeeper marked up the price of an article by p%. When an article was not sold even after giving two successive discounts of 10% and 11.11%, he decided to offer one more discount, which is equal to 1/12 of the price after the two discounts. In this transaction, he incurred a loss of 12%. Find the marked-up price as a percentage of the selling price of the article.

A122.22%

B133.33%

C120%

D136.36%

Solution Answer: D

Marked price M after the two discounts: M × 0.9 × 8/9 = 0.8M.

After the third discount: SP = 0.8M × 11/12 = 11M/15.

M as a percentage of SP = 15/11 = 136.36%. (The 12% loss isn’t needed for this ratio.)

Q28MCQ

A Gulab Jamun contains sugar syrup equal to 30% of its total volume. How much syrup would be present in 90 Gulab Jamuns, each having the shape of a cylinder with two hemispherical ends, given that the total length of each Gulab Jamun is 5 cm, and its diameter is 2.8 cm?

A616 cm³

B576 cm³

C639 cm³

D676 cm³

Solution Answer: D

r = 1.4 cm, and the cylinder part is 5 − 2.8 = 2.2 cm long.

Volume = πr²h + 43πr³ = 13.552 + 11.499 ≈ 25.05 cm³ (with π = 22/7).

Syrup = 0.3 × 90 × 25.05 ≈ 676.4, i.e. 676 cm³.

Q29TITA

Find the number of real solutions of the equation: log_x+1 (2x² + 7x + 5) + log_2x+5 (x + 1)² – 4 = 0

Solution Answer: 1

Factor: 2x² + 7x + 5 = (2x + 5)(x + 1). With t = log_x+1(2x + 5), the equation becomes (1 + t) + 2/t − 4 = 0, i.e. t² − 3t + 2 = 0, so t = 1 or 2.

t = 1: 2x + 5 = x + 1 gives x = −4. The base would be negative ✗.
t = 2: 2x + 5 = (x + 1)² gives x² = 4, so x = ±2. x = −2 makes the base −1 ✗. x = 2 gives bases 3 and 9 ✓.

There is 1 real solution.

Q30MCQ

Smriti invested a certain part of a sum at 33.33% simple interest for two years and the remaining part at 16.67% p.a. compound interest for two years. If the amount becomes 18/13 times the principal, find the ratio of sums invested at S.I. and C.I. by Smriti.

A3:10

B1:6

C1:12

D5:13

Solution Answer: C

Let the S.I. part be x and the C.I. part be y.

S.I. at 1/3 per year for 2 years: amount = x(1 + 2/3) = 5x/3. C.I. at 1/6 per year for 2 years: amount = y(7/6)² = 49y/36.

The total amount is 18/13 of the principal: 5x/3 + 49y/36 = 18(x + y)/13.

Multiply through by 468: 780x + 637y = 648x + 648y, so 132x = 11y and x : y = 1 : 12.

Check: x = 1 and y = 12 give 5/3 + 49/3 = 18 on a principal of 13, which is 18/13 ✓.

Day 30 complete! 🔥

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