Explanation:

To find which fraction has the minimum change in its value when 5 is added to both its numerator and denominator, we can evaluate each fraction after adding 5 and then calculate the difference between the original fraction and the modified fraction.

Step-by-Step Solution

Let’s denote each fraction as follows:

• $f_1 = \frac{2}{3}$
• $f_2 = \frac{3}{4}$
• $f_3 = \frac{4}{5}$
• $f_4 = \frac{5}{6}$

When 5 is added to both the numerator and the denominator, each fraction becomes:

• $f_1′ = \frac{2+5}{3+5} = \frac{7}{8}$
• $f_2′ = \frac{3+5}{4+5} = \frac{8}{9}$
• $f_3′ = \frac{4+5}{5+5} = \frac{9}{10}$
• $f_4′ = \frac{5+5}{6+5} = \frac{10}{11}$

Now, let’s calculate the absolute difference between each original fraction and its modified fraction:

1. For $f_1 = \frac{2}{3}$:

$$ \left| \frac{2}{3} – \frac{7}{8} \right| = \left| \frac{16 – 21}{24} \right| = \frac{5}{24} \approx 0.2083 $$

2. For $f_2 = \frac{3}{4}$:

$$ \left| \frac{3}{4} – \frac{8}{9} \right| = \left| \frac{27 – 32}{36} \right| = \frac{5}{36} \approx 0.1389 $$

3. For $f_3 = \frac{4}{5}$:

$$ \left| \frac{4}{5} – \frac{9}{10} \right| = \left| \frac{40 – 45}{50} \right| = \frac{5}{50} = 0.1 $$

4. For $f_4 = \frac{5}{6}$:

$$ \left| \frac{5}{6} – \frac{10}{11} \right| = \left| \frac{55 – 60}{66} \right| = \frac{5}{66} \approx 0.0758 $$

Conclusion

The fraction with the minimum change in value when 5 is added to both the numerator and the denominator is:

$$ \frac{5}{6} $$

Thus, the correct answer is: $ \text{(d) } \frac{5}{6} $

Explanation:

To determine the period with the maximum natural growth rate, we need to calculate the natural growth rate for each period. The natural growth rate is the difference between the birth rate and the death rate.

$$
\text{Natural Growth Rate} = \text{Birth Rate} – \text{Death Rate}
$$

We will calculate this for each period and identify the period with the highest growth rate.

Step-by-Step Calculations

1. 1911–1921:

$$ \text{Natural Growth Rate} = 48.1 – 35.5 = 12.6 $$

2. 1921–1931:

$$ \text{Natural Growth Rate} = 46.4 – 36.3 = 10.1 $$

3. 1931–1941:

$$ \text{Natural Growth Rate} = 45.2 – 31.2 = 14.0 $$

4. 1941–1951:

$$ \text{Natural Growth Rate} = 39.9 – 27.4 = 12.5 $$

5. 1951–1961:

$$ \text{Natural Growth Rate} = 41.7 – 22.8 = 18.9 $$

6. 1961–1971:

$$\text{Natural Growth Rate} = 41.1 – 18.9 = 22.2$$

7. 1971–1981:

$$ \text{Natural Growth Rate} = 37.1 – 14.8 = 22.3 $$

Conclusion

The period with the maximum natural growth rate is 1971–1981, with a growth rate of 22.3.

The correct answer is: (d) 1971–1981

Explanation:

To determine the value of $x$ in the above table, which represents the overall average marks in Hindi, we can use the concept of weighted averages.

Given Data

1. Average marks in English:

• For girls: $9$
• For boys: $8$
• Overall average in English: $8.8$

2. Average marks in Hindi:

• For girls: $8$
• For boys: $7$
• Overall average in Hindi: $x$ (this is what we need to find)

Step-by-Step Solution

Let:

• $G$ represent the number of girls,
• $B$ represent the number of boys.

Step 1: Set Up the Equation for English Marks

From the given data, the overall average marks in English are $8.8$. Therefore, we can write the equation for the total marks in English as follows:

$$
\frac{9G + 8B}{G + B} = 8.8
$$

Multiply both sides by $G + B$:

$$ 9G + 8B = 8.8(G + B) $$

Step 2: Simplify the Equation for the Ratio $ \frac{G}{B} $

Rearrange the equation:

$$
9G – 8.8G = 8.8B – 8B
$$

Simplifying further:

$$
0.2G = 0.8B
$$

Dividing both sides by $0.2$, we get:

$$
\frac{G}{B} = \frac{0.8}{0.2} = 4
$$

Thus, the ratio of the number of girls to the number of boys is:

$$
G : B = 4 : 1
$$

This means that for every 4 girls, there is 1 boy.

Step 3: Set Up the Equation for the Overall Average in Hindi

We are given the following information about the average marks in Hindi:

• Average marks in Hindi for girls = $8$
• Average marks in Hindi for boys = $7$

Now, we want to find the overall average marks in Hindi, denoted by $x$.

Using the weighted average formula for Hindi marks, we have:

$$
x = \frac{8G + 7B}{G + B}
$$

Step 4: Substitute the Ratio $ G : B = 4 : 1 $

Let $G = 4k$ and $B = k$ (where $k$ is a constant representing the number of boys).

Now substitute $G = 4k$ and $B = k$ into the equation for $x$:

$$
x = \frac{8(4k) + 7(k)}{4k + k}
$$

Simplify the expression:

$$
x = \frac{32k + 7k}{5k}
$$

Combine terms in the numerator:

$$
x = \frac{39k}{5k}
$$

Cancel $k$ from the numerator and denominator:

$$
x = \frac{39}{5} = 7.8
$$

Conclusion

The value of $x$ is: x = 7.8

The correct answer is: (a) 7.8

Explanation:

Let’s go through the problem step-by-step to find the correct answer.

Information Given

1. A is 16th from the left end.
2. V is 18th from the right end.
3. G is 11th from A towards the right.
4. G is 3rd from V towards the right end.

We need to determine the total number of boys in the row.

Step-by-Step Solution

1. Position of A from the left end:

• Since A is 16th from the left, the position of A is:
  $$
  \text{Position of A} = 16
  $$

2. Position of G relative to A:

• G is 11th to the right of A. Therefore, the position of G from the left end is:
  $$
  \text{Position of G from the left end} = 16 + 11 = 27
  $$

3. Position of G relative to V:

• G is 3rd from V towards the right end. This means that V is positioned 3 places to the left of G.
• Since G is at position 27 from the left end, the position of V from the left end is:
  $$
  \text{Position of V from the left end} = 27 – 3 = 24
  $$

4. Total Number of Boys:

• We know that V is 18th from the right end, and we’ve calculated V’s position from the left end to be 24. Using this information, we can find the total number of boys by adding V’s position from the left and right ends and subtracting 1 (since V is counted from both ends):
  $$
  \text{Total number of boys} = 24 + 18 – 1 = 41
  $$

Conclusion

The correct answer is: (b) 41

Explanation:

The passage critically examines the effects of economic liberalization in India, highlighting issues like “jobless growth, persistent poverty, and rising inequality.” It also points to four critical areas—agriculture, infrastructure, industrialization, and education—that are seen as constraints on India’s future economic prospects. Based on this information, let’s analyze each assumption:

Assumption 1: “India’s economy needs to be greatly integrated with the global economy so as to create a large number of jobs and to sustain its growth momentum.”

This assumption is not valid based on the passage. The passage does not suggest that further integration with the global economy is needed for job creation or sustained growth. Instead, it focuses on the internal limitations and challenges that India faces (such as issues in agriculture, infrastructure, and education) and does not imply that greater global integration is a solution to these problems. Therefore, there is no basis in the passage to assume that increased integration with the global economy would address these issues.

Assumption 2: “Economic liberalization would cause large economic growth which would reduce poverty and create sufficient employment in the long run.”

This assumption is also not valid according to the passage. In fact, the passage suggests that economic liberalization in India has led to “jobless growth, persistent poverty, and rising inequality,” indicating that liberalization has not been effective in reducing poverty or creating sufficient employment. The passage implies that the expectations of liberalization leading to inclusive growth and job creation have not materialized. Thus, this assumption contradicts the information provided in the passage.

Since neither of the assumptions is supported by the passage, the correct answer is: (d) Neither 1 nor 2.

Explanation:

The passage discusses the advantages of cultivating the pulse variety “Pusa Arhar 16” over paddy in northern India. Key points include its higher yield, suitability for mechanical harvesting, and the benefits of its straw, which can be ploughed back into the soil to decompose quickly. In contrast, paddy straw has a high silica content, making it difficult to decompose, which leads to the common practice of burning it.

Analyzing each statement:

Statement 1: “Farmers’ income will be higher with pulse cultivation than with paddy cultivation.”

While the passage mentions that the yield of Pusa Arhar 16 is higher than existing varieties, it does not provide any specific information comparing the income from pulse cultivation to that from paddy cultivation. Therefore, this statement cannot be inferred from the passage.

Statement 2: “Pulse cultivation causes less pollution as compared to paddy cultivation.”

This is a reasonable inference. The passage suggests that paddy straw is often burned due to its high silica content and difficulty in decomposition, which causes pollution. In contrast, Pusa Arhar straw can be easily ploughed back into the soil without burning, thus reducing pollution.

Statement 3: “Pulse straw can be used to improve soil quality.”

This is also a valid inference. The passage states that Arhar straw can be ploughed back into the soil and decomposes quickly, which implies that it can contribute to soil fertility and improve soil quality.

Statement 4: “In the context of northern Indian agriculture, paddy straw has no usefulness.”

This is incorrect. Although the passage highlights the difficulty of decomposing paddy straw and the pollution caused by burning it, it does not imply that paddy straw has absolutely no usefulness. Paddy straw could still have potential uses in other contexts, even if it is challenging to manage as a leftover in northern India.

Statement 5: “Mechanized agriculture is the main cause for stubble burning.”

This is not supported by the passage. Although mechanized harvesting leaves stubble behind, the primary reason for burning paddy straw mentioned in the passage is its high silica content and difficulty in decomposition, not mechanization itself.

Thus, the correct answer is: (c) 2 nd 3 only.

Explanation:

The passage discusses the issue of malnutrition in India, highlighting that it leads to severe health consequences for children, including stunted growth, impaired learning, and reduced lifetime earnings. Based on this information, practical implications to address malnutrition should include measures aimed at improving child health and nutrition from early stages, including maternal health, breastfeeding, safe drinking water, sanitation, and vaccination.

Analyzing each option:

Statement 1: “India’s Public Distribution System should be monitored by the Union Government.”

While monitoring the Public Distribution System (PDS) could potentially help, the passage does not discuss issues related to the PDS, nor does it suggest that this is a primary solution to malnutrition.

Statement 2: “Girls should be encouraged to delay marriage and first pregnancy.”

This is a relevant implication because early marriage and early pregnancy often lead to poorer health outcomes for both mother and child, increasing the risk of malnutrition in children. Delaying marriage and pregnancy allows mothers to be better prepared nutritionally and physically, benefiting child health.

Statement 3: “Mothers should be encouraged to breastfeed their children immediately after birth.”

Breastfeeding is crucial for newborns, providing essential nutrients and antibodies that protect against infections and malnutrition. Promoting breastfeeding aligns with the passage’s focus on improving child health.

Statement 4: “The supply of safe drinking water and proper sanitation facilities to all should be ensured.”

Safe drinking water and proper sanitation are fundamental for preventing waterborne diseases, which can worsen malnutrition in children. This is a rational implication given the focus on improving child health and reducing mortality.

Statement 5: “Authorities should ensure the vaccination as prescribed.”

Vaccination helps protect children from various diseases, thereby reducing health complications that could exacerbate malnutrition. This implication aligns well with the need to improve child health as described in the passage.

Thus, the correct answer is: (b) 2, 3, 4 and 5.

Explanation:

The passage highlights the historical event of the Irish Potato Famine in the 19th century, which was caused by a heavy reliance on a single variety of potato, the “Lumper.” This variety was genetically uniform and therefore vulnerable to the fungus Phytophthora infestans, which destroyed almost all of the potato crops in Ireland, leading to widespread famine and displacement.

The key message here is that reliance on a single, genetically uniform food source can be extremely risky, as it can lead to disastrous consequences if that source is affected by disease or other adverse conditions. The passage emphasizes the dangers of depending on a homogeneous food source, which supports option (d) as the best choice.

In contrast:

• Option (a) is incorrect because the passage does not discuss soil and climate suitability for foreign plants.
• Option (b) is incorrect because the passage does not argue against potatoes as a staple food or their ability to replace cereal crops.
• Option (c) is incorrect because, while the passage mentions the spread of a fungal infection, it does not suggest that preventing fungal infections is impossible; the main focus is on the risk of relying on a single crop variety.

The correct answer is (d) Relying on a homogeneous food source is not desirable.

Explanation:

The passage discusses Article 25 of the Indian Constitution, which grants individuals the freedom of conscience and the right to freely profess, practice, and propagate religion. However, this right is subject to conditions such as public order, morality, health, and other provisions in Part III of the Constitution, which includes fundamental rights like equality before the law. This implies that religious freedom is not absolute and can be limited by the State if it conflicts with other fundamental rights or public interest.

The phrase “subject to public order, morality, health, and to the other provisions of this Part” clearly indicates that the State has the authority to intervene in religious practices if they violate these conditions. This makes option (b) the most logical inference, as it aligns with the idea that religious freedom is conditional and open to State intervention.

In contrast:

• Option (a) is incorrect because the State can interfere under certain conditions.
• Option (c) is incorrect because religious freedom is indeed covered under fundamental rights, though with limitations.
• Option (d) is incorrect because the passage suggests that religious practices are not immune to State laws; they are subordinate to other fundamental rights.

The correct answer is (b) Religious freedom under the Constitution is open to State intervention.

Explanation:

To determine the correct answer, let’s analyze each option in light of the information provided in the passage.

Passage Summary

The passage discusses the health risks associated with poor solid waste management, particularly when combined with inadequate drainage and untreated sewage dumping. It explains how unscientific waste management practices, coupled with improper drainage systems, lead to choked drains filled with stagnant water, which in turn breeds mosquitoes and spreads water-borne diseases.

Analyzing Each Option

1. Option (a): “In India, the drainage networks are not separate for sewerage and storm water.”

• This option is partially related but does not fully capture the main message of the passage. While the passage describes how untreated sewage ends up in stormwater drains, which contributes to health risks, it does not specifically discuss the separation (or lack thereof) of sewerage and stormwater systems as the critical issue. The focus is more on the combined impact of poor waste management and drainage on health, rather than on the technical design of drainage networks.

2. Option (b): “Urban local bodies do not have enough resources and legislative authority to deal with the problems of waste management.”

• This option is not supported by the passage. The passage does not discuss the resources or authority of urban local bodies. Instead, it focuses on the health consequences of poor waste and drainage management, without mentioning administrative or legislative limitations.

3. Option (c): “Solid waste management should be integrated with the maintenance of drainage and sewerage networks.”

• This option best reflects the critical message of the passage. The passage emphasizes that unscientific solid waste management, combined with poor drainage and untreated sewage dumping, leads to choked drains and health risks. This suggests that an integrated approach to managing solid waste, drainage, and sewerage systems is necessary to prevent issues like stagnant water and the spread of water-borne diseases. This option aligns well with the main idea of the passage.

4. Option (d): “Bad management of solid waste and sewerage systems by our municipalities is the reason for drinking water shortages in our cities.”

• This option is not supported by the passage. The passage does not discuss drinking water shortages. Instead, it focuses on the health hazards related to poor waste and drainage management. Therefore, this option is not relevant to the passage’s main message.

Conclusion

The statement that best reflects the critical message of the passage is Option (c): “Solid waste management should be integrated with the maintenance of drainage and sewerage networks.” This option aligns with the passage’s emphasis on the need for a comprehensive approach to managing waste, drainage, and sewage to prevent health risks.

Thus, the correct answer is: (c) Solid waste management should be integrated with the maintenance of drainage and sewerage networks.

Explanation:

To determine the correct answer, let’s analyze each assumption based on the information provided in the passage.

Passage Summary

The passage explains that genome editing is distinct from genome modification. It involves identifying and precisely altering specific parts of a plant’s genome to make it more resistant to diseases or herbicides, or to increase yields. This editing process uses “molecular scissors” to dissect and repair the genome, a process similar to the natural evolutionary response of plants when they face environmental challenges. Genome editing allows researchers to accelerate this natural adaptation process by targeting specific genes in the laboratory.

Analyzing Each Assumption

1. Assumption 1: “Genome editing does not require the transfer of genes from one plant to another.”

• This assumption is valid. The passage describes genome editing as a process that involves modifying specific genes within the plant’s own genome to make it less vulnerable to disease or more resistant to herbicides. There is no mention of transferring genes from one plant to another. Instead, genome editing involves altering existing genes in a precise way, which implies that gene transfer between plants is not required.

2. Assumption 2: “Through genome editing, the chosen genes can be altered precisely in a manner akin to the natural process that helps plants to adapt to environmental factors.”

• This assumption is valid. The passage states that genome editing allows researchers to “speed up” the natural evolutionary process, which involves creating mutations that help plants adapt to diseases or other environmental stresses. By editing the genome precisely, scientists are replicating a process that would naturally occur but at an accelerated pace. Therefore, this assumption is supported by the passage.

Conclusion

Both Assumption 1 and Assumption 2 are valid based on the passage. The passage supports the idea that genome editing does not involve gene transfer between plants and that it enables precise changes similar to natural adaptation processes.

Therefore, the correct answer is: (c) Both 1 and 2

Explanation:

To determine the correct answer, let’s analyze each option in relation to the information provided in the passage.

Passage Summary

The passage emphasizes India’s potential for solar energy and the need to move away from fossil fuels. It points out that while India has made a strong commitment to reducing emissions by 33% by 2030, renewable energy technologies are still maturing and require further development to become cost-effective. In the meantime, the Indian government is promoting a gas-based economy alongside investing in renewables. However, business houses are cautious about investing heavily in renewables due to the current limitations and costs associated with the technology.

Analyzing Each Option

1. Option (a): “India’s commitment to reduce emissions by 33% is unlikely to be achieved.”

• This option is not directly supported by the passage. Although the passage mentions challenges associated with renewable energy investments, it does not imply that India’s emission reduction target is unachievable. Instead, the passage highlights India’s strong commitment and the steps being taken towards achieving this target, including a push towards a gas-based economy and investments in renewables.

2. Option (b): “India should import gas rather than invest in renewable resources.”

• This option is not supported by the passage. The passage does not suggest that India should prioritize importing gas over investing in renewable resources. Rather, it explains that India is promoting a gas-based economy as a transitional step while also investing in renewable energy. There is no indication that importing gas should replace renewable energy investments.

3. Option (c): “Getting renewable resources to market too soon may be costly.”

• This option best aligns with the passage. The passage mentions that renewable energy technologies are still going through “cost curves and learning curves” to reach optimal output, which implies that rushing them to market before they are fully mature may involve high costs. Furthermore, it notes that business houses are hesitant to invest heavily in renewables due to these cost and technology limitations, supporting the idea that entering the market prematurely could be expensive.

4. Option (d): “India should put in more efforts in the exploration of natural gas.”

• This option is **not directly supported by the passage**. While the passage mentions a shift towards a gas-based economy, it does not specifically suggest that India should increase efforts in natural gas exploration. The focus is more on balancing investments in gas and renewables rather than increasing exploration efforts for natural gas.

Conclusion

The statement that best reflects the logical and rational inference that can be made from the passage is Option (c): “Getting renewable resources to market too soon may be costly.” This option aligns with the passage’s mention of renewable resources still developing in terms of cost-efficiency and technology readiness, as well as the hesitation of business houses to invest heavily in them at this stage.

Thus, the correct answer is: (c) Getting renewable resources to market too soon may be costly.

Explanation:

To determine the correct answer, let’s analyze each option in light of the information provided in the passage.

Passage Summary

The passage discusses the shrinking of bank credit to the industrial sector in India, highlighting that this decline is a concern because credit growth is essential for reviving investment. The passage attributes this issue to the incomplete economic reforms since 1991, specifically pointing out the absence of a “resolution corporation” for banks. In a market economy, banks should be allowed to fail, but currently, India lacks a proper system to shut down weak banks. As a result, loss-making banks continue to require more capital without a mechanism for resolution.

Analyzing Each Option

1. Option (a): “Indian banking system is not able to help the country in its economic growth.”

• This option is too broad. While the passage indicates that the banking system faces challenges due to a lack of mechanisms to shut down weak banks, it does not imply that the entire Indian banking system is unable to contribute to economic growth. The focus is specifically on the need for a resolution mechanism for failing banks, not on the overall incapability of the banking system to support economic growth. Therefore, this option does not fully capture the main message of the passage.

2. Option (b): “Economic reforms that started in 1991 have not helped in improving the economy to expected levels.”

• This option is also too broad. The passage mentions that the 1991 reforms were incomplete because they did not establish a resolution corporation for banks. However, it does not imply that the entire set of economic reforms has failed to improve the economy. The passage is focused on a specific gap in the reform process (i.e., the lack of a resolution mechanism for banks), rather than a general failure of all the economic reforms.

3. Option (c): “India lacks the institutional mechanisms to deal with the failure of banks.”

• This option best reflects the inference that can be made from the passage. The passage specifically states that India lacks a “resolution corporation” for banks, which means there is no proper system to shut down failing banks. This lack of institutional mechanism to deal with bank failures is a key issue highlighted in the passage.

4. Option (d): “Encouraging the foreign investments in our industrial sector is a good alternative to this sector’s dependence on banks for credit.”

• This option is not supported by the passage. The passage does not suggest foreign investment as an alternative to bank credit. Instead, it focuses on the need for a resolution mechanism for banks to handle failing institutions. There is no discussion of foreign investment or alternatives to bank credit.

Conclusion

The statement that best reflects the inference that can be made from the passage is Option (c): “India lacks the institutional mechanisms to deal with the failure of banks.” This option aligns with the passage’s focus on the absence of a resolution corporation and the inability to shut down failing banks.

Thus, the correct answer is: (c) India lacks the institutional mechanisms to deal with the failure of banks.

Explanation:

To determine the correct answer, let’s analyze each option in relation to the information provided in the passage.

Passage Summary

The passage discusses the evolving role of biotechnology in agriculture, particularly in relation to genetically modified (GM) crops. It highlights that while current GM crops are mostly engineered for single traits, future GM crops will likely be developed with multiple traits, making biotechnology more complex and broad in scope. The passage suggests that biotechnology regulation should take a comprehensive approach, considering not only the generation of GM crops but also socio-economic impacts and other factors. Additionally, it emphasizes the importance of biotechnology in climate change mitigation and adaptation, and states that biotechnology regulation should not be isolated from trade policy and international treaties.

Analyzing Each Option

1. Option (a): “Precautionary principle is not given importance in current debate on developing GM crops.”

• This option is not supported by the passage. The passage does not discuss the precautionary principle or indicate that it is being neglected. Instead, it focuses on the need for a comprehensive approach to biotechnology regulation, including socio-economic impacts and integration with climate change strategies. Therefore, this option does not reflect the main message of the passage.

2. Option (b): “Biotechnology is not currently used in climate change mitigation and adaptation mechanisms.”

• This option is contradicted by the passage. The passage states that biotechnology is important for developing crop varieties that can help with climate change mitigation and adaptation, and it emphasizes that biotechnology should be included in climate change action plans. Therefore, the statement that biotechnology is “not currently used” in climate change mechanisms is incorrect.

3. Option (c): “Biotechnology’s role is not confined to the current priorities of developing GM crops.”

• This option best reflects the crux of the passage. The passage argues that biotechnology in agriculture should go beyond just developing GM crops for single traits. It calls for a broader view that includes multiple traits, socio-economic impacts, climate change mitigation, trade policy, and international obligations. This implies that the role of biotechnology is much broader than the current focus on GM crop development.

4. Option (d): “The negative impacts of biotechnology are not properly understood.”

• This option is not directly supported by the passage. While the passage mentions minimizing negative impacts, it does not imply that the negative impacts of biotechnology are poorly understood. The focus is on taking a comprehensive approach to regulation rather than indicating a lack of understanding about biotechnology’s negative effects.

Conclusion

The statement that best implies the crux of the passage is Option (c): “Biotechnology’s role is not confined to the current priorities of developing GM crops.” This option aligns with the passage’s emphasis on the broader role of biotechnology in agriculture, which includes multiple traits, socio-economic considerations, climate change, and international obligations.

Thus, the correct answer is: (c) Biotechnology’s role is not confined to the current priorities of developing GM crops.

Explanation:

To determine the correct answer, let’s analyze each assumption based on the information provided in the passage.

Passage Summary

The passage discusses the role of biotechnology in agriculture, specifically focusing on the regulation of genetically modified (GM) crops. It states that future GM crops will likely have multiple traits, which implies a growing complexity in biotechnology. The passage argues that the regulation of biotechnology should take a comprehensive approach, considering not just the generation of GM crops, but also socio-economic impacts. Additionally, it emphasizes the importance of using biotechnology as part of climate change mitigation and adaptation strategies, and points out that domestic regulation of biotechnology must align with trade policy and international obligations.

Analyzing Each Assumption

Assumption 1: “Biotechnology regulation is an evolving process.”

• This assumption is valid. The passage indicates that biotechnology, particularly in agriculture, is moving towards GM crops with multiple traits, which will increase complexity. This implies that regulation needs to adapt to these changes. The need for a comprehensive approach that considers various aspects, including socio-economic impacts, suggests that biotechnology regulation is an evolving process.

Assumption 2: “Participation of people is needed in policy decisions regarding biotechnology regulation.”

• This assumption is not directly supported by the passage. While the passage calls for a comprehensive look at biotechnology regulation that includes socio-economic impacts, it does not explicitly mention or imply that public participation is necessary in policy decisions. Therefore, this assumption is not valid based on the given passage.

Assumption 3**: “Biotechnology regulation should take into account socio-economic aspects in decision-making.”

• This assumption is valid. The passage explicitly states that biotechnology regulation should consider socio-economic impacts, in addition to focusing on the generation of GM crops. This assumption aligns with the passage’s suggestion for a broader approach to biotechnology regulation.

Assumption 4: “Wider involvement of political executive in biotechnology regulation improves its effectiveness in dealing with the country’s trade policies and international obligations.”

• This assumption is not directly supported by the passage. The passage mentions that biotechnology regulation should not be viewed in isolation from trade policy and international treaties, but it does not imply that the involvement of the political executive would improve the regulation’s effectiveness. There is no reference to political executive involvement in the passage.

Conclusion

Based on the passage, only Assumptions 1 and 3 are valid.

Therefore, the correct answer is: (b) 1 and 3 only

Explanation:

To determine the correct answer, let’s analyze each option in relation to the information provided in the passage.

Passage Summary

The passage emphasizes the importance of asset allocation in investment decisions. It discusses common misconceptions about investing in “risky” assets and points out that no asset is truly risk-free. The author argues that we cannot avoid risk entirely, as even seemingly safe investments (such as jobs, homes, banks, bonds, etc.) carry potential risks. Instead of avoiding risk completely, the author suggests that a balanced approach to asset allocation can help manage and mitigate risks without assuming that all extreme negative events will happen at once.

Analyzing Each Option

1. Option (a): “Distribute your wealth across different kinds of assets so that your risks would be minimized.”

• This option best aligns with the suggestion given by the author. The passage highlights the importance of asset allocation, which involves distributing investments across different types of assets to manage risk. By diversifying, an investor can reduce the impact of risks associated with any one asset. This option reflects the author’s emphasis on mitigating risk through diversified asset allocation rather than avoiding risky assets altogether.

2. Option (b): “Risk-taking behaviour should be a necessary component of your personality if you want to generate wealth.”

• This option is not the main suggestion of the passage. While the passage does encourage a realistic view of risk in investments, it does not suggest that an investor must have a “risk-taking personality” to generate wealth. Instead, it focuses on thoughtful asset allocation as a way to manage risk, rather than endorsing aggressive risk-taking.

3. Option (c): “While making investments, find a trustworthy asset management organization which would manage your wealth for you.”

• This option is not supported by the passage. The author does not discuss the need to rely on a third-party organization for managing investments. The focus is on individuals making informed decisions about asset allocation, rather than delegating those decisions to an asset management organization.

4. Option (d): “You should know that investing your money is a risky business.”

• This option is partially accurate but does not fully capture the suggestion of the passage. While the passage acknowledges that all assets carry some level of risk, it does not merely advise investors to be aware of risk. Instead, it suggests that investors use asset allocation as a strategy to manage and mitigate these risks. Therefore, this option does not fully reflect the author’s main recommendation.

Conclusion

The statement that best implies the suggestion given by the author is Option (a): “Distribute your wealth across different kinds of assets so that your risks would be minimized.” This option is consistent with the author’s emphasis on the importance of asset allocation as a way to manage investment risk.

Thus, the correct answer is: (a) Distribute your wealth across different kinds of assets so that your risks would be minimized.

Explanation:

To determine the correct answer, let’s analyze each option in relation to the information provided in the passage.

Passage Summary

The passage highlights a fundamental issue in public policy related to water management. It points out the irony that water, which comes from natural sources like rivers and wetlands, is often treated as if it is separate from these ecosystems. The passage suggests that public policy treats water as a resource to be exploited, rather than as an integral part of the natural ecosystem. Consequently, engineering efforts focus on increasing water supply, rather than enhancing the ecological systems that sustain water resources.

Analyzing Each Option

1. Option (a): “Rivers and other wetlands should be protected under Ramsar Convention.”

• This option is not directly supported by the passage. While protecting rivers and wetlands under the Ramsar Convention (an international treaty for the conservation of wetlands) could be beneficial, the passage does not mention or imply this specific approach. The focus of the passage is on the need to understand and treat water as part of the natural ecosystem, rather than on specific conservation frameworks.

2. Option (b): “Engineering water systems should be modernized and further augmented.”

• This option contradicts the main message of the passage. The passage critiques the current approach of engineering water systems primarily to increase supply, without considering the ecological systems that support water resources. It suggests that simply augmenting engineering efforts is not the right solution, as it does not address the need to strengthen the natural ecosystem. Therefore, this option is not aligned with the passage.

3. Option (c): “Wetlands need to be reinforced as more than just open sources of water.”

• This option best reflects the inference from the passage. The passage emphasizes the need for public policy to recognize that water is part of the natural ecosystem, and that wetlands and rivers should not be viewed solely as water sources to be exploited. Instead, there should be a focus on strengthening the ecological systems. This option captures the essence of the passage, which advocates for a more holistic view of water management that goes beyond treating wetlands as mere sources of water.

4. Option (d): “Water supply should not be free of cost so as to prevent its misuse or overuse.”

• This option is not supported by the passage. The passage does not discuss the cost of water or issues related to its misuse or overuse. Its primary focus is on the need to recognize and strengthen the natural ecosystems (such as wetlands and rivers) that provide water, rather than addressing water pricing or cost-related issues.

Conclusion

The statement that best reflects the logical and rational inference from the passage is Option (c): “Wetlands need to be reinforced as more than just open sources of water.” This option aligns with the passage’s emphasis on treating water as part of the natural ecosystem and enhancing ecological systems, rather than merely focusing on engineering solutions to increase water supply.

Thus, the correct answer is: (c) Wetlands need to be reinforced as more than just open sources of water.

Explanation:

To determine the correct answer, let’s analyze each assumption based on the information provided in the passage.

Passage Summary

The passage discusses the right to privacy in the context of the digital age. It argues that a right is meaningful only if it can be protected for everyone, in all situations. Using the example of free expression, the passage suggests that the role of the State is not only to refrain from infringing on rights, but also to actively protect individuals from private parties who might try to violate these rights. This implies that the State has a responsibility to enforce and protect citizens’ rights, including the right to privacy, against violations by private entities.

Analyzing Each Assumption

Assumption 1: “State should have some institutions to ensure its appropriate role in a digital society.”

This assumption is valid. The passage implies that the State has a responsibility to actively protect citizens’ rights, especially in the digital age where privacy concerns are prominent. For the State to fulfill this role effectively, it would logically need institutions or mechanisms to enforce these protections. Thus, the assumption that the State should have institutions to ensure its role in a digital society aligns with the passage.

Assumption 2: “State should ensure that private parties do not violate the citizens’ right to privacy.”

• This assumption is valid. The passage explicitly states that the State’s role includes actively ensuring that private parties cannot interfere with citizens’ rights (such as the right to free expression and, by extension, the right to privacy). This directly supports the assumption that the State should protect citizens from privacy violations by private entities.

Assumption 3: “Digital economy is not compatible with the idea of not violating the citizens’ privacy.”

• This assumption is not valid. The passage does not suggest that the digital economy is inherently incompatible with privacy protections. Instead, it emphasizes the State’s role in safeguarding privacy rights within the digital age. The assumption is too extreme, as the passage does not imply that privacy protection is impossible or incompatible within a digital economy.

Conclusion

Based on the passage, Assumptions 1 and 2 are valid, while Assumption 3 is not. Therefore, the correct answer is: $ \text{(a) 1 and 2} $

Explanation:

To determine the correct answer, let’s analyze each option in relation to the information given in the passage.

Passage Summary

The passage describes the trend of labor movement in India. Over the last decade, labor has been leaving agriculture and moving to sectors like construction and unregistered manufacturing, which do not offer significantly better job conditions. The service sector, despite being the most productive, is not creating enough jobs to meet the country’s employment needs. While e-commerce could potentially reduce the jobs gap, it can only close about half of it. To fill the remaining gap, sectors that drive domestic demand, such as health and education, are necessary.

Analyzing Each Option

1. Option (a): “Strong measures need to be taken to reduce the rural to urban migration of labour.”

• This option is not supported by the passage. The passage does not suggest that rural-to-urban migration should be curtailed. Instead, it focuses on the types of jobs available to laborers once they leave agriculture and points out the limited opportunities in sectors like construction and unregistered manufacturing. There is no indication that reducing migration is a solution to the employment challenge.

2. Option (b): “The working condition in construction and unregistered manufacturing needs to be improved.”

• This option is partially related but not the main implication of the passage. While the passage mentions that jobs in construction and unregistered manufacturing are “not markedly better,” it does not suggest improving working conditions in these sectors as a solution. The focus is on the need for more productive jobs in other sectors, particularly those driven by domestic demand.

3. Option (c): “Service sector has been reducing the problem of unemployment.”

• This option is incorrect. The passage explicitly states that the service sector, where labor is most productive, is not generating the additional jobs needed to address India’s employment needs. Therefore, this option contradicts the information in the passage.

4. Option (d): “Increased social sector spending is imperative for large-scale job creation.”

• This option best reflects the implication of the passage. The passage suggests that sectors driving domestic demand, such as health and education, are essential for filling the jobs gap. Health and education are part of the social sector, and increased spending in these areas would likely lead to job creation. This aligns well with the passage’s implication that the social sector is crucial for addressing India’s employment needs.

Conclusion

The statement that best reflects the implication of the passage is Option (d): “Increased social sector spending is imperative for large-scale job creation.” This option is consistent with the passage’s emphasis on the need for sectors like health and education to generate the remaining jobs.

Thus, the correct answer is: (d) Increased social sector spending is imperative for large-scale job creation.

Explanation:

To determine the correct answer, let’s analyze each option based on the information provided in the passage.

Passage Summary

The passage highlights the role of agriculture in poverty reduction in India. It notes that India’s poverty reduction was slow from 1988 to 2005, but accelerated dramatically from 2005 to 2012. The key factor behind this change was a significant shift in the relative price of agricultural products, driven by rising global prices. This shift led to a more than 50% increase in private investment in agriculture, which in turn boosted agricultural GDP growth, increased real farm wages, and led to a net agri-trade surplus. The overall impact was a substantial reduction in poverty. The passage suggests that agricultural growth, driven by price incentives and investment, played a crucial role in reducing poverty.

Analyzing Each Option

1. Option (a): “India should create large-scale off-farm rural employment to reduce poverty in the near future.”

• This option is not directly supported by the passage. While creating off-farm rural employment could potentially help reduce poverty, the passage does not discuss or emphasize off-farm employment as a solution. Instead, it focuses on the impact of growth within the agricultural sector itself on poverty reduction.

2. Option (b): “India should create a large number of farmer producer companies.”

• This option is not mentioned in the passage. While forming farmer producer companies could be beneficial for organizing small and marginal farmers, the passage does not suggest this as a critical measure for poverty reduction. The focus is on private investment in agriculture, higher agri-GDP, and real farm wage growth, not on the formation of specific organizational structures.

3. Option (c): “Private investment in agriculture should be given priority over public investment.”

• This option is partially supported but is not the main message of the passage. The passage highlights that private investment in agriculture increased due to favorable price conditions, which led to significant poverty reduction. However, it does not imply that private investment should be prioritized over public investment. It simply states that private investment responded positively to price incentives, without advocating for a shift in policy to prioritize private investment.

4. Option (d): “Inclusive agricultural growth is key to reduce poverty in the near future.”

• This option best reflects the critical message of the passage. The passage emphasizes how growth in the agricultural sector—particularly growth that benefits small and marginal farmers—led to substantial poverty reduction. Higher agri-GDP, increased farm wages, and a trade surplus all indicate that agricultural growth benefited a wide section of the population involved in agriculture, which aligns with the idea of “inclusive agricultural growth.”

Conclusion

The statement that best reflects the critical message of the passage is Option (d): “Inclusive agricultural growth is key to reduce poverty in the near future.” This option is consistent with the passage’s emphasis on agricultural growth, private investment, and the positive impact on poverty reduction.

Thus, the correct answer is: (d) Inclusive agricultural growth is key to reduce poverty in the near future.

Explanation:

Analyzing Each Assumption

Assumption 1: “Structural and rural transformation is impossible when farms are mainly small and marginal.”

• This assumption is not valid. The passage mentions that 85% of India’s farms are small and marginal, but it does not state that structural and rural transformation is *impossible* under these conditions. Rather, it suggests that transformation has been slower in India compared to other countries, but it does not imply that transformation cannot happen at all. Therefore, this assumption is too absolute and is not supported by the passage.

Assumption 2: “A good price incentive can trigger investments in agriculture.”

• This assumption is valid. The passage states that a favorable change in the relative price scenario (due to rising global prices) led to a boost in private investments in agriculture by more than 50%. This clearly supports the idea that a good price incentive can trigger investments in agriculture.

Assumption 3: “India needs to build value chains for high-value agri-products like livestock and horticulture.”

• This assumption is not directly supported by the passage. While building value chains for high-value agricultural products could potentially benefit India, the passage does not discuss this as a necessary strategy. The passage focuses on the impact of price incentives and investments in agriculture on poverty reduction, but it does not suggest that building value chains for specific products is required. Therefore, this assumption is not valid based on the given information.

Assumption 4: “Higher global prices of agricultural commodities are essential for India’s poverty reduction.”

• This assumption is not valid. While the passage indicates that higher global prices were a contributing factor to poverty reduction in India during the period from 2005 to 2012, it does not claim that such prices are *essential* for poverty reduction. Other factors could potentially contribute to poverty reduction as well, and relying solely on higher global prices is not presented as the only solution. Therefore, this assumption is too strong and not fully supported by the passage.

Conclusion

Based on the passage, only Assumption 2 is valid. Therefore, the correct answer is: $ \text{(b) 2 and 4} $

Explanation:

Analyzing Each Assumption

Assumption 1: “Rich cities only can ensure sustainable delivery of water.”

• This assumption is not valid. The passage does state that if cities were rich enough, they could deliver water for free. However, it does not imply that only rich cities can ensure sustainable water delivery. Instead, the passage suggests that sustainable delivery is possible with proper financing mechanisms, regardless of whether a city is rich. Therefore, the passage does not support the idea that only rich cities can achieve sustainable water delivery.

Assumption 2: “Sustainable delivery of water in cities means much more than supplying water to households.”

• This assumption is valid. The passage clearly states that sustainable water delivery includes not only the distribution of potable water but also investments in sewerage systems and sewage treatment plants. This indicates that sustainable water management involves a comprehensive approach, addressing both clean water supply and wastewater treatment, which goes beyond merely supplying water to households.

Conclusion

Based on the passage, only Assumption 2 is valid.

Therefore, the correct answer is: (b) 2 only

Explanation:

To determine the most logical and crucial message conveyed by the passage, let’s analyze each option in light of the information provided.

Passage Summary

The passage discusses the challenges faced by urban bodies in delivering water sustainably to city populations. It emphasizes that water delivery requires significant investment, not only in sourcing, treating, and distributing water, but also in managing wastewater through sewerage and treatment plants. The passage suggests that urban bodies need a reliable financing mechanism to cover these costs, as cities cannot afford to provide water for free due to limited resources.

Analyzing Each Option

1. Option (a): “Urban local bodies must recover costs through user charges.”

• This option is logical and aligns with the passage. The passage emphasizes the need for financing mechanisms to support the sustainable delivery of water services. Since cities are not rich enough to cover these costs entirely, one practical implication is that urban bodies would need to recover costs, potentially through user charges. This option addresses the core issue presented in the passage.

2. Option (b): “Urban local bodies are not efficient enough to meet the water requirements of our cities.”

• This option is not supported by the passage. The passage does not discuss the efficiency of urban local bodies; instead, it focuses on the financial challenges involved in delivering water sustainably. Therefore, this option does not capture the main message of the passage.

3. Option (c): “Water shortage in our cities is a perennial problem that cannot be solved.”

• This option is also incorrect. While the passage acknowledges the challenges in financing water infrastructure, it does not suggest that the water shortage is an unsolvable problem. Rather, it implies that sustainable delivery is possible if proper financing mechanisms are established.

4. Option (d): “In view of the water crisis in our cities, there is an urgent need to limit the population of cities by adopting an upper limit of population size.”

• This option is not mentioned or implied in the passage. The passage focuses on the need for financing mechanisms, not on population control as a solution to water issues.

Conclusion

The most logical and crucial message conveyed by the passage is Option (a): “Urban local bodies must recover costs through user charges.” This option accurately reflects the passage’s emphasis on the need for financing mechanisms to ensure sustainable water delivery.

Thus, the correct answer is: (a) Urban local bodies must recover costs through user charges.

Explanation:

To determine the correct answer, let’s analyze each assumption in light of the information provided in the passage.

Passage Summary

The passage argues that economic freedom, particularly in the form of lower tax rates, encourages people to invest in education because they are allowed to retain most of the higher income that results from educational attainment. Conversely, high tax rates on the income of educated individuals discourage further investment in education. The passage implies that economic freedom positively impacts individuals’ decisions to invest in education, which in turn enhances human capital.

Analyzing Each Assumption

Assumption 1: “Lower tax rates in a country invariably translate into greater investments in higher education.”

This assumption is partially valid but not entirely accurate. The passage suggests that lower tax rates encourage investment in education by allowing individuals to retain more of the benefits of their educational achievements. However, it does not claim that lower tax rates will invariably lead to greater investments in education, as other factors could also influence educational investment. The use of “invariably” makes this assumption too absolute and therefore invalid based on the passage.

Assumption 2: “Investment in the education of children ensures their economic freedom.”

This assumption is not valid according to the passage. While the passage suggests that economic freedom encourages investment in education, it does not claim that investing in education ensures economic freedom. The passage focuses on the impact of tax policy and economic incentives on educational investment rather than stating that education itself guarantees economic freedom.

3. Assumption 3: “Economic freedom has a positive impact on building up human capital.”

This assumption is valid. The passage clearly implies that economic freedom (through lower tax rates) encourages people to invest in education, which contributes to building up human capital. This aligns with the passage’s argument that economic freedom makes educational investment more attractive.

Conclusion

Based on the passage, only Assumption 3 is valid. Therefore, the correct answer is: (c) 3 only

Explanation:

To determine the correct answer, let’s analyze each statement and evaluate if they are sufficient to answer the question, “Are there more than 100 figures in that book?”

Given Statements

S1: There are not more than two figures on any page of a 51-page book.

This means that the maximum number of figures per page is 2.

S2: There is at least one figure on every page.

This implies that each page has at least 1 figure.

Analysis of Statements

Since there are 51 pages in the book, we can use the information from both statements to find the possible range for the total number of figures in the book.

1. Using S1:
Since there are at most 2 figures per page, the maximum possible number of figures in the book is:
$$
2 \times 51 = 102
$$

2. Using S2:
Since there is at least 1 figure on each page, the minimum possible number of figures in the book is:
$$
1 \times 51 = 51
$$

3. Combining S1 and S2:
Together, these statements imply that the number of figures in the book is between 51 and 102 (inclusive).

Therefore, the total number of figures could be anywhere from 51 to 102. However, this range does not allow us to conclusively answer the question, “Are there more than 100 figures in the book?” The answer could be either “Yes” (if there are exactly 102 figures) or “No” (if there are fewer than 100 figures).

Conclusion

• S1 alone gives us an upper limit (102) but does not provide a lower limit, so it is not sufficient by itself.
• S2 alone gives us a lower limit (51) but does not provide an upper limit, so it is also not sufficient by itself.
• S1 and S2 together provide a range (51 to 102), but this range does not allow us to definitively answer whether there are more than 100 figures in the book.

Thus, the correct answer is: $ \text{(c) S1 and S2 together are not sufficient to answer the Question.} $

Explanation:

To determine the correct answer, let’s analyze each assumption in light of the given passage.

Passage Summary

The passage suggests that foreign private investment (FDI) is volatile and cannot be relied upon for stable employment. It also indicates that FDI might lead to regional and income inequalities and that the diffusion of new technology from FDI is uncertain due to inadequate physical and human capital in India. The passage implicitly suggests caution in relying on FDI and hints at the need for strengthening domestic capacity.

Analyzing Each Assumption

1. Assumption 1: “Relying on foreign investment in the long run is not an economically sound policy.”

• This assumption is valid. The passage indicates that foreign private investment is volatile and creates unstable employment, suggesting that relying on it is risky and potentially unsound for long-term economic stability.

2. Assumption 2: “Policies must be undertaken to reduce volatility in foreign private investment.”

• This assumption is valid. The passage discusses the volatility of FDI and implies that it is a drawback, which suggests that measures to reduce such volatility would be beneficial.

3. Assumption 3: “Policies must be undertaken to strengthen domestic private investment.”

• This assumption is also valid. Since the passage highlights the volatility and unreliability of foreign investment, it implies that strengthening domestic investment could be a more stable alternative.

4. Assumption 4: “Public investment should be given priority over private investment.”

• This assumption is not directly supported by the passage. While the passage criticizes FDI, it does not explicitly or implicitly suggest prioritizing public investment over all private investment.

5. Assumption 5: “Substantial public investment in education and health should be undertaken.”

• This assumption is valid. The passage notes that India’s physical and human capital may be inadequate to support the diffusion of new technology brought in by FDI. This implies a need for investment in education and health to improve human capital.

Conclusion

The valid assumptions based on the passage are 1, 3, and 5. Therefore, the correct answer is: $ \text{(b) 1, 3 and 5} $

Explanation:

Let’s analyze each statement and see if it is sufficient to determine who is the youngest among $A$, $B$, $C$, and $D$.

Given Statements:

1. S1: $C$ is younger than $D$, but older than $A$ and $B$.

• This implies that the order of age is $D > C > A$ and $D > C > B$.
• However, from this statement alone, we cannot determine the exact relationship between $A$ and $B$ (i.e., which one is younger).

2. S2: $D$ is the oldest.

• This confirms that $D$ is the oldest among the four.
• However, this does not give us information about the relative ages of $A$, $B$, and $C$.

3. S3: $A$ is older than $B$.

• This tells us that $A > B$ in terms of age.
• However, by itself, this does not tell us the complete age order among all four individuals.

Analyzing Each Combination of Statements

Option (a): S1 alone is sufficient to answer the Question.

• From S1 alone, we know that $C$ is younger than $D$ but older than both $A$ and $B$.
• However, without knowing the relative ages of $A$ and $B$, we cannot determine which of them is the youngest.
• Conclusion: S1 alone is not sufficient to answer the question.

Option (b): S1 and S2 together are sufficient to answer the Question.

• Using S1, we know that $D > C > A$ and $D > C > B$.
• Using S2, we confirm that $D$ is the oldest.
• However, we still do not know whether $A$ is older than $B$ or vice versa.
• Conclusion: S1 and S2 together are not sufficient to answer the question.

Option (c): S2 and S3 together are sufficient to answer the Question.

• Using S2, we know that $D$ is the oldest.
• Using S3, we know that $A > B$.
• However, this combination does not include any information about $C$, so we cannot determine the youngest among all four individuals.
• Conclusion: S2 and S3 together are not sufficient to answer the question.

Option (d): S1 and S3 together are sufficient to answer the Question.

• Using S1, we know that $D > C > A$ and $D > C > B$.
• Using S3, we know that $A > B$.
• Combining these, we get the full age order: $D > C > A > B$.
• From this order, we can conclude that $B$ is the youngest.
• Conclusion: S1 and S3 together are sufficient to answer the question.

Final Answer
The correct answer is: (d) S1 and S3 together are sufficient to answer the Question.

Explanation:

Let’s analyze each statement and the conclusions to determine which conclusions logically follow from the statements, disregarding commonly known facts.

Given Statements:

1. Statement 1: All cats are dogs.

• This means that every cat is included in the group of dogs.

2. Statement 2: All cats are black.

• This means that every cat is included in the group of black animals.

Analyzing the Conclusions

Conclusion I: All dogs are black.

• From the statements, we know that all cats are black. However, we do not have any information about whether all dogs are black.
• There could be dogs that are not cats, and we have no information on whether those dogs are black.
• Therefore, this conclusion does not logically follow from the statements.
• Conclusion: Conclusion I does not follow.

Conclusion II: Some dogs are not black.

• We know from the statements that all cats are dogs and all cats are black. However, we do not have information on whether dogs that are not cats are black.
• Therefore, it is possible (but not certain) that some dogs may not be black.
• This conclusion, however, is not guaranteed by the statements; it is only a possibility, not a logical certainty.
• Conclusion: Conclusion II does not logically follow.

Final Answer

The correct answer is: (c) Neither Conclusion-I nor Conclusion-II

Explanation:

Let’s analyze each statement and the conclusions to determine which conclusions logically follow from the statements.

Given Statements:

1. Statement 1: All numbers are divisible by 2.

• This implies that every number under consideration is an even number (i.e., divisible by 2).

2. Statement 2: All numbers are divisible by 3.

• This implies that every number under consideration is also divisible by 3.

Analyzing the Conclusions

Conclusion I: All numbers are divisible by 6.

• If a number is divisible by both 2 and 3, it is also divisible by their least common multiple (LCM), which is 6.
• Since the statements say that all numbers are divisible by both 2 and 3, it logically follows that all numbers are divisible by 6.
• Conclusion: Conclusion I follows from the given statements.

Conclusion II: All numbers are divisible by 4.

• The fact that all numbers are divisible by 2 does not imply that they are also divisible by 4. For example, the number 6 is divisible by 2 and by 3 (as stated in the statements), but it is not divisible by 4.
• There is no information in the statements to suggest that every number is divisible by 4.
• Conclusion: Conclusion II does not follow from the given statements.

Final Answer

The correct answer is: (a) Only Conclusion-I

Explanation:

Let’s analyze each statement to see if it is sufficient to determine the two numbers.

Given Statements:

1. S1: The product of the two numbers is 21.

• Let the two numbers be $x$ and $y$. Then from S1, we have:
  $$
  x \cdot y = 21.
  $$
• However, this equation alone does not uniquely determine the values of $x$ and $y$ because there could be multiple pairs of numbers whose product is 21 (for example, $(3, 7)$ and $(-3, -7)$).
• Conclusion: S1 alone is not sufficient to answer the question.

2. S2: The sum of the two numbers is 10.

• From S2, we have:
  $$
  x + y = 10.
  $$
• This equation alone does not uniquely determine the values of $x$ and $y$ because multiple pairs of numbers could sum to 10 (for example, $(3, 7)$, $(6, 4)$, and $(5, 5)$).
• Conclusion: S2 alone is not sufficient to answer the question.

Analyzing Statements S1 and S2 Together

• Using both S1 and S2 together, we have the following system of equations:
  $$
  x \cdot y = 21 \quad \text{and} \quad x + y = 10.
  $$
• We can solve this system by expressing $y$ in terms of $x$ using the sum equation:
  $$
  y = 10 – x.
  $$
• Substituting $y = 10 – x$ into the product equation:
  $$
  x \cdot (10 – x) = 21.
  $$
• Expanding and rearranging terms, we get:
  $$
  x^2 – 10x + 21 = 0.
  $$
• This is a quadratic equation in $x$. We can solve it using the quadratic formula:
  $$
  x = \frac{10 \pm \sqrt{(10)^2 – 4 \cdot 1 \cdot 21}}{2 \cdot 1}.
  $$
  $$
  x = \frac{10 \pm \sqrt{100 – 84}}{2}.
  $$
  $$
  x = \frac{10 \pm \sqrt{16}}{2}.
  $$
  $$
  x = \frac{10 \pm 4}{2}.
  $$
• This gives two solutions for $x$:
  $$
  x = \frac{10 + 4}{2} = 7 \quad \text{and} \quad x = \frac{10 – 4}{2} = 3.
  $$
• Substituting these values back into $y = 10 – x$, we get the corresponding values of $y$:
• If $x = 7$, then $y = 3$.
• If $x = 3$, then $y = 7$.
• Therefore, the two numbers are $3$ and $7$.

Conclusion

Using both S1 and S2 together, we can uniquely determine the two numbers as $3$ and $7$.

Final Answer
The correct answer is: (c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.

Explanation:

Let’s analyze each statement and see if it is sufficient to determine the unique natural number $n$ between 10 and 20 that satisfies the conditions in the question.

Given Statements:

1. S1: $n$ is a prime number.

• This tells us that $n$ is a prime number, but alone it doesn’t specify the exact value of $n$ between 10 and 20.
• The prime numbers between 10 and 20 are $11$, $13$, $17$, and $19$.

2. S2: $n$ leaves a remainder of 1 when divided by 4.

• This implies that $n \equiv 1 \, (\text{mod} \, 4)$, meaning that when $n$ is divided by 4, the remainder is 1.

Analyzing Each Statement Individually

Step 1: Using S1 alone

• If we only know that $n$ is a prime number between 10 and 20, then possible values of $n$ are $11$, $13$, $17$, and $19$.
• However, we cannot determine a unique value for $n$ from this information alone.
• Conclusion: S1 alone is not sufficient to answer the question.

Step 2: Using S2 alone

• If we only know that $n \equiv 1 \, (\text{mod} \, 4)$, then we are looking for numbers between 10 and 20 that leave a remainder of 1 when divided by 4.
• The numbers in this range that satisfy $n \equiv 1 \, (\text{mod} \, 4)$ are $13$ and $17$.
• However, this information alone does not tell us if $n$ is prime, so we cannot uniquely determine $n$ based on S2 alone.
• Conclusion: S2 alone is not sufficient to answer the question.

Analyzing Statements S1 and S2 Together

• Using S1 and S2 together, we know that $n$ must be a prime number between 10 and 20 and must leave a remainder of 1 when divided by 4.
• From S1, we have the possible prime numbers $11$, $13$, $17$, and $19$.
• From S2, we are restricted to numbers that leave a remainder of 1 when divided by 4, which are $13$ and $17$.
• Thus, combining S1 and S2, we find that the only numbers that satisfy both conditions are $13$ and $17$.
• However, both $13$ and $17$ satisfy the conditions, and there is no way to determine a unique value of $n$ between them.

Conclusion

Since both $13$ and $17$ satisfy the conditions given in S1 and S2, we cannot determine a unique value for $n$ even when using both statements together.

Final Answer
The correct answer is: (d) S1 and S2 together are not sufficient to answer the Question.

Explanation:

Let’s analyze each statement and see if it is sufficient to determine which of the numbers $P$, $Q$, $R$, and $S$ is the largest.

Given Statements:
1. S1: $R$ is greater than $P$ as well as $Q$.

• This means $R > P$ and $R > Q$.
• However, this statement does not provide any information about the relationship between $R$ and $S$, so we cannot conclude if $R$ is the largest among all four.

2. S2: $S$ is not the largest.

• This implies that the largest number is either $P$, $Q$, or $R$, but does not give any direct comparison among $P$, $Q$, and $R$.

Analysis of Each Option:

Option (a): S1 alone is sufficient to answer the Question.

Analysis: S1 only tells us that $R$ is greater than $P$ and $Q$, but it does not tell us whether $R$ is greater than $S$. Therefore, S1 alone is not sufficient to determine the largest number.

• Conclusion: This option is incorrect.

Option (b): S2 alone is sufficient to answer the Question.

Analysis: S2 only tells us that $S$ is not the largest, but does not provide any information about the relationships between $P$, $Q$, and $R$. Therefore, S2 alone is also not sufficient to determine the largest number.

• Conclusion: This option is incorrect.

Option (c): S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the question.

Analysis:

• From S1, we know $R > P$ and $R > Q$.
• From S2, we know that $S$ is not the largest.
• Combining these statements, we can conclude that $R$ must be the largest, as it is greater than both $P$ and $Q$, and $S$ is explicitly stated not to be the largest.
• Therefore, using both S1 and S2 together, we can conclude that $R$ is the largest.
• Conclusion: This option is correct.

Option (d): S1 and S2 together are not sufficient to answer the Question.

Analysis: As shown above, combining S1 and S2 allows us to determine that $R$ is the largest. Thus, S1 and S2 together are indeed sufficient.

• Conclusion: This option is incorrect.

The correct answer is: (c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the question.

Explanation:

To solve this problem, we need to determine the value of $n$ for which the sum of the digits in the number $10^n + 1$ is 2.

Step-by-step Solution:

1. Understanding the Expression $10^n + 1$:
– The expression $10^n + 1$ represents a number that consists of a 1 followed by $n$ zeros, plus 1.
– For example:
– If $n = 1$, then $10^1 + 1 = 10 + 1 = 11$.
– If $n = 2$, then $10^2 + 1 = 100 + 1 = 101$.
– If $n = 3$, then $10^3 + 1 = 1000 + 1 = 1001$.

2. Analyzing the Sum of Digits:
– For any positive integer $n$, the number $10^n + 1$ will have exactly two non-zero digits: a 1 at the start and a 1 at the end.
– All digits in between are zeros.
– Therefore, the sum of the digits of $10^n + 1$ will always be $1 + 1 = 2$.

3. Checking for $n = 0$:
– If $n = 0$, then $10^0 + 1 = 1 + 1 = 2$.
– Here, the sum of the digits is 2.

4. Conclusion:
– The sum of the digits of $10^n + 1$ is 2 for $n = 0$ as well as for any positive integer $n$.
– This is because the expression always results in a number with two 1’s and zeros in between, regardless of the value of $n$ (as long as $n$ is a whole number).

Thus, the correct answer is (b) For any whole number $n$.

Explanation:

To solve this problem, we need to determine how many different sums can be formed using the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500, and Rs. 2,000 by selecting at least three denominations at a time.

Step-by-step Solution:

1. Total Denominations:
There are 5 different denominations available: Rs. 50, Rs. 100, Rs. 200, Rs. 500, and Rs. 2,000.

2. Combinations of Denominations:
We need to find combinations that involve at least three denominations. The possible combinations are:
– Combinations of 3 denominations out of 5.
– Combinations of 4 denominations out of 5.
– Combinations of all 5 denominations.

3. Calculating the Number of Combinations:
– The number of ways to choose 3 denominations from 5 is given by $\binom{5}{3}$.
– The number of ways to choose 4 denominations from 5 is given by $\binom{5}{4}$.
– The number of ways to choose all 5 denominations is given by $\binom{5}{5}$.

Using the formula for combinations, $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, we calculate:

\[
\binom{5}{3} = \frac{5!}{3! \times 2!} = \frac{5 \times 4}{2 \times 1} = 10
\]

\[
\binom{5}{4} = \frac{5!}{4! \times 1!} = \frac{5}{1} = 5
\]

\[
\binom{5}{5} = \frac{5!}{5! \times 0!} = 1
\]

4. Total Number of Combinations:
Adding these together gives the total number of different combinations of denominations:

\[
\binom{5}{3} + \binom{5}{4} + \binom{5}{5} = 10 + 5 + 1 = 16
\]

Thus, there are 16 different sums that can be formed using at least three denominations at a time.

The correct answer is (a) 16.

Explanation:

To solve this problem, we need to understand how shifting students between groups affects the average weight of the entire class.

Let’s analyze the problem step by step:

1. Understanding the Average Weight:
The average weight of the students in the class is the total weight of all the students divided by the total number of students in the class.

2. Effect of Shifting Students:
When one student from group A and two students from group B are moved to group C, there is no change in the total number of students in the class. Similarly, the total weight of the class remains unchanged because the same students are still in the class, only in different groups.

Since the total weight and the total number of students in the class remain constant, the average weight of the class also remains unchanged.

Thus, the correct answer is (c) It remains the same.

Explanation:

To solve this problem, we need to find the remainder when the product $51 \times 27 \times 35 \times 62 \times 75$ is divided by 100.

To simplify the calculation, we can use the property of modular arithmetic which states that:

\[
(a \times b) \mod m = [(a \mod m) \times (b \mod m)] \mod m
\]

Applying this property repeatedly, we can find the product modulo 100 step by step.

First, we find the remainder of each number modulo 100:

\[
51 \mod 100 = 51
\]
\[
27 \mod 100 = 27
\]
\[
35 \mod 100 = 35
\]
\[
62 \mod 100 = 62
\]
\[
75 \mod 100 = 75
\]

Now, let’s calculate the product of these numbers modulo 100:

\[
(51 \times 27 \times 35 \times 62 \times 75) \mod 100
\]

We can break this calculation into parts to make it easier. Let’s calculate two at a time:

1. Calculate $51 \times 27$ modulo 100:

\[
51 \times 27 = 1377
\]

\[
1377 \mod 100 = 77
\]

2. Now multiply the result by 35 and find modulo 100:

\[
77 \times 35 = 2695
\]

\[
2695 \mod 100 = 95
\]

3. Next, multiply by 62 and find modulo 100:

\[
95 \times 62 = 5890
\]

\[
5890 \mod 100 = 90
\]

4. Finally, multiply by 75 and find modulo 100:

\[
90 \times 75 = 6750
\]

\[
6750 \mod 100 = 50
\]

Thus, the remainder when $51 \times 27 \times 35 \times 62 \times 75$ is divided by 100 is 50.

The correct answer is (a) 50.

Explanation:

To solve this problem, we need to find the smallest four-digit number that leaves a remainder of 2 when divided by 3, 4, 5, and 6.

First, let’s find the least common multiple (LCM) of the numbers 3, 4, 5, and 6. The LCM of these numbers will give us a number that is divisible by all four without leaving a remainder.

The prime factorization of each number is as follows:

– $3 = 3$
– $4 = 2^2$
– $5 = 5$
– $6 = 2 \times 3$

To find the LCM, we take the highest power of each prime number:

$$
\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60
$$

Now, we need a number of the form $60k + 2$ that is a four-digit number (where $k$ is an integer). This is because when such a number is divided by 3, 4, 5, or 6, it leaves a remainder of 2.

The smallest four-digit number is 1000. To find the smallest value of $k$, we set up the inequality:

$$
60k + 2 \geq 1000
$$

Subtract 2 from both sides:

$$
60k \geq 998
$$

Divide by 60:

$$
k \geq \frac{998}{60} \approx 16.6333
$$

Since $k$ must be an integer, we round up to the next whole number, $k = 17$.

Now, calculate $60k + 2$ when $k = 17$:

$$
60 \times 17 + 2 = 1020 + 2 = 1022
$$

Thus, the least four-digit number that leaves a remainder of 2 when divided by 3, 4, 5, and 6 is 1022.

The correct answer is (b) 1022.

Explanation:

To solve the problem, we need to convert the recurring decimal $1.272727\ldots$ into a fraction.

Let’s denote the repeating decimal as $x$:

$$
x = 1.272727\ldots
$$

We can separate the non-repeating part and the repeating part. The repeating decimal $1.272727\ldots$ can be written as:

$$
x = 1.2 + 0.07272727\ldots
$$

Let’s focus on the repeating part, $0.07272727\ldots$. Denote this repeating decimal as $y$:

$$
y = 0.07272727\ldots
$$

To express $y$ as a fraction, multiply $y$ by 100 to shift the decimal point two places to the right:

$$
100y = 7.272727\ldots
$$

Now subtract the original $y$ from this equation:

$$
100y – y = 7.272727\ldots – 0.072727\ldots
$$

$$
99y = 7.2
$$

$$
y = \frac{7.2}{99}
$$

Simplify $y$ by multiplying the numerator and denominator by 10:

$$
y = \frac{72}{990}
$$

Now simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 9:

$$
y = \frac{72 \div 9}{990 \div 9} = \frac{8}{110} = \frac{4}{55}
$$

Now substitute back for $x$:

$$
x = 1.2 + \frac{4}{55}
$$

Convert $1.2$ to a fraction:

$$
1.2 = \frac{12}{10} = \frac{6}{5}
$$

Now add the two fractions:

$$
x = \frac{6}{5} + \frac{4}{55}
$$

To add these fractions, find a common denominator. The least common multiple of 5 and 55 is 55:

$$
x = \frac{6 \times 11}{5 \times 11} + \frac{4}{55} = \frac{66}{55} + \frac{4}{55}
$$

$$
x = \frac{66 + 4}{55} = \frac{70}{55}
$$

Simplify the fraction by dividing both the numerator and denominator by 5:

$$
x = \frac{70 \div 5}{55 \div 5} = \frac{14}{11}
$$

Thus, the recurring decimal $1.272727\ldots$ is equivalent to the fraction $\frac{14}{11}$.

The correct answer is (b) $\frac{14}{11}$.

Explanation:

To solve this problem, we need to find the greatest length $x$ such that both $3\frac{1}{2}$ meters and $8\frac{3}{4}$ meters are integral multiples of $x$.

Step-by-Step Solution:

1. Convert the mixed numbers to improper fractions:

– $3\frac{1}{2}$ meters can be written as:
$$
3\frac{1}{2} = \frac{7}{2} \text{ meters}
$$

– $8\frac{3}{4}$ meters can be written as:
$$
8\frac{3}{4} = \frac{35}{4} \text{ meters}
$$

2. Find the greatest length $x$ that is a common divisor of both fractions:

We need to find the greatest length $x$ such that $\frac{7}{2}$ and $\frac{35}{4}$ are integral multiples of $x$.

To do this, we first express both fractions with a common denominator.

– The common denominator for $\frac{7}{2}$ and $\frac{35}{4}$ is 4.

– Convert $\frac{7}{2}$ to have a denominator of 4:
$$
\frac{7}{2} = \frac{14}{4}
$$

– Now, both fractions are:
$$
\frac{14}{4} \text{ and } \frac{35}{4}
$$

3. Find the greatest common divisor (GCD) of the numerators:

– The numerators are 14 and 35.
– The GCD of 14 and 35 is 7.

4. Determine the greatest length $x$:

Since the common denominator is 4, the greatest length $x$ will be:
$$
x = \frac{7}{4} \text{ meters}
$$

– Converting $\frac{7}{4}$ to a mixed number:
$$
\frac{7}{4} = 1 \frac{3}{4} \text{ meters}
$$

Conclusion:

The greatest length $x$ is $1 \frac{3}{4} \text{ meters}$.

The correct answer is (d) $1 \frac{3}{4} \text{ m}$.

Explanation:

To solve this problem, we need to determine how many jumps the frog needs to make to climb out of a well that is 4.5 meters deep, given that the frog jumps up 30 cm and slides back down 15 cm after each jump.

Step-by-Step Solution:

1. Convert all measurements to the same unit:

– Depth of the well: $4.5 \text{ meters} = 450 \text{ cm}$
– Jump up: $30 \text{ cm}$
– Slide down: $15 \text{ cm}$
– Net gain per jump: $30 \text{ cm} – 15 \text{ cm} = 15 \text{ cm}$

2. Calculate the effective climbing distance per jump:

Each jump results in a net gain of $15 \text{ cm}$ except for the last jump when the frog gets out of the well.

3. Determine how many jumps are required to reach just below the top:

– To calculate the number of jumps to reach just below the top of the well, we find out how many jumps it takes to reach $450 \text{ cm}$ minus the final jump (where the frog does not slide back).

– The frog must reach $450 \text{ cm} – 30 \text{ cm} = 420 \text{ cm}$ with regular jumps before the final jump.

– The number of jumps required to climb $420 \text{ cm}$ with a net gain of $15 \text{ cm}$ per jump is:
$$
\frac{420 \text{ cm}}{15 \text{ cm/jump}} = 28 \text{ jumps}
$$

4. Final jump:

After reaching $420 \text{ cm}$, the frog makes one more jump of $30 \text{ cm}$ to reach the top of the well.

5. Total number of jumps:

– The total number of jumps is $28 \text{ jumps} + 1 \text{ jump} = 29 \text{ jumps}$.

Conclusion:

The frog needs 29 jumps to come out of the well.

The correct answer is (b) 29.

Explanation:

To solve this problem, we need to convert the weight of water from grams to cubic millimeters, given that 1 liter of water weighs 1 kilogram.

Step-by-Step Solution:

1. Understanding the given information:

– 1 liter of water = 1 kilogram
– 1 kilogram = 1000 grams
– Therefore, 1 liter of water = 1000 grams

2. Conversion between liters and cubic millimeters:

– 1 liter = 1000 cubic centimeters (cc)
– 1 cubic centimeter (cc) = 1 milliliter (ml)
– 1 milliliter of water weighs 1 gram (since 1 liter = 1000 grams)
– 1 cubic centimeter = \(10^3\) cubic millimeters (mm\(^3\))

Therefore:
\[
1 \, \text{liter} = 1000 \, \text{cubic centimeters} = 1000 \times 10^3 \, \text{cubic millimeters} = 10^6 \, \text{cubic millimeters}
\]

3. Determine the volume of water that weighs 0.1 grams:

– If 1 gram of water corresponds to 1 milliliter (or 1000 cubic millimeters), then 0.1 grams of water corresponds to:

\[
0.1 \, \text{grams} = 0.1 \, \text{milliliters}
\]

– Since 1 milliliter = 1000 cubic millimeters, 0.1 milliliters is:

\[
0.1 \, \text{milliliters} = 0.1 \times 1000 \, \text{cubic millimeters} = 100 \, \text{cubic millimeters}
\]

4. Conclusion:

The volume of water that weighs 0.1 grams is 100 cubic millimeters.

Therefore, The correct answer is (c) 100.

Explanation:

To solve this problem, we need to determine the weight of the empty vessel using the information provided about the weight of the vessel when it is fully filled and when it is one-third filled with water.

Given:
– The weight of the vessel when full of water is 40 kg.
– The weight of the vessel when one-third filled with water is 20 kg.

Let’s Define the Variables:
– Let $V$ be the weight of the empty vessel in kg.
– Let $W$ be the weight of the water that completely fills the vessel in kg.

Step-by-Step Solution:

1. Write the equations based on the given information:

– When the vessel is full of water:
$$
V + W = 40
$$

– When the vessel is one-third filled with water:
$$
V + \frac{W}{3} = 20
$$

2. Subtract the second equation from the first equation:

Subtracting the second equation from the first equation gives:
$$
(V + W) – \left(V + \frac{W}{3}\right) = 40 – 20
$$

Simplify the equation:
$$
W – \frac{W}{3} = 20
$$

Combine like terms:
$$
\frac{3W}{3} – \frac{W}{3} = 20
$$

$$
\frac{2W}{3} = 20
$$

3. Solve for $W$:

Multiply both sides by 3:
$$
2W = 60
$$

Divide by 2:
$$
W = 30
$$

4. Find the weight of the empty vessel $V$:

Substitute $W = 30$ back into the first equation:
$$
V + W = 40
$$

$$
V + 30 = 40
$$

Subtract 30 from both sides:
$$
V = 10
$$

Conclusion:

The weight of the empty vessel is 10 kg.

Therefore, the correct answer is (a) 10 kg.

Explanation:

To solve this problem, we need to find pairs of natural numbers $(x, y)$ such that the difference of their squares equals 63. In other words, we want to find all pairs of natural numbers $(x, y)$ satisfying:

\[
x^2 – y^2 = 63
\]

Step-by-Step Solution:

1. Use the Difference of Squares Formula:

The expression $x^2 – y^2$ can be factored using the difference of squares formula:

\[
x^2 – y^2 = (x – y)(x + y)
\]

So, we have:

\[
(x – y)(x + y) = 63
\]

2. Find Factor Pairs of 63:

To satisfy the equation, we need to find pairs of natural numbers $(x – y)$ and $(x + y)$ such that their product is 63. The factor pairs of 63 are:

\[
(1, 63), \, (3, 21), \, (7, 9)
\]

3. Solve for Each Factor Pair:

For each factor pair $(a, b)$, we solve the equations:

\[
x – y = a \quad \text{and} \quad x + y = b
\]

We can solve for $x$ and $y$ by adding and subtracting these equations:

\[
x = \frac{a + b}{2}, \quad y = \frac{b – a}{2}
\]

Let’s check each factor pair:

– Factor pair $(1, 63)$:

\[
x – y = 1, \quad x + y = 63
\]
\[
x = \frac{1 + 63}{2} = 32, \quad y = \frac{63 – 1}{2} = 31
\]
Pair: $(x, y) = (32, 31)$

– Factor pair $(3, 21)$:

\[
x – y = 3, \quad x + y = 21
\]
\[
x = \frac{3 + 21}{2} = 12, \quad y = \frac{21 – 3}{2} = 9
\]
Pair: $(x, y) = (12, 9)$

– Factor pair $(7, 9)$:

\[
x – y = 7, \quad x + y = 9
\]
\[
x = \frac{7 + 9}{2} = 8, \quad y = \frac{9 – 7}{2} = 1
\]
Pair: $(x, y) = (8, 1)$

4. Count the Number of Pairs:

From the factor pairs, we found three valid pairs of natural numbers: $(32, 31)$, $(12, 9)$, and $(8, 1)$.

Conclusion:

There are 3 pairs of natural numbers such that the difference of their squares is 63.

The correct answer is (a) 3.

Explanation:

To solve this problem, we need to find the ratio of the speed of the man in still water to the speed of the current based on the given information.

Given:
– The man takes half the time to row a certain distance downstream than upstream.

Let’s Define the Variables:
– Let $s$ be the speed of the man in still water (in km/hr).
– Let $c$ be the speed of the current (in km/hr).

Analyzing the Problem:

1. Speed of the man downstream:
The speed of the man when rowing downstream is $(s + c)$ km/hr.

2. Speed of the man upstream:
The speed of the man when rowing upstream is $(s – c)$ km/hr.

3. Time taken to row a certain distance:

Let the distance be $d$ km.

– Time taken to row downstream: $\frac{d}{s + c}$
– Time taken to row upstream: $\frac{d}{s – c}$

4. Given Condition:
It is given that the time taken to row downstream is half the time taken to row upstream.

So, we can write:
$$
\frac{d}{s + c} = \frac{1}{2} \times \frac{d}{s – c}
$$

5. Simplifying the Equation:

Canceling out $d$ from both sides and cross-multiplying gives:
$$
\frac{1}{s + c} = \frac{1}{2(s – c)}
$$

Multiplying both sides by $2(s – c)(s + c)$, we get:
$$
2(s – c) = s + c
$$

Expanding and simplifying:
$$
2s – 2c = s + c
$$

Rearrange the terms:
$$
2s – s = 2c + c
$$

$$
s = 3c
$$

6. Finding the Ratio:

The ratio of the speed in still water to the speed of the current is:
$$
\frac{s}{c} = \frac{3c}{c} = 3
$$

So, the ratio is $3:1$.

Conclusion:

The correct answer is (d) 3 : 1.

Explanation:

To solve this problem, we need to analyze the possible points of intersection between a square and a circle.

Analysis of the Statements:

1. Minimum Number of Points of Intersection:

– A square and a circle can intersect in different ways, but the minimum number of points of intersection depends on the relative position and size of the circle with respect to the square.

– If the circle touches the square tangentially (i.e., the circle touches one side of the square at exactly one point), the number of intersection points is 1.

– If the circle is entirely inside the square or completely outside without touching, there will be 0 points of intersection.

– Therefore, the minimum number of points of intersection of a square and a circle can be 0.

2. Maximum Number of Points of Intersection:

– The maximum number of intersection points occurs when the circle intersects each side of the square at two distinct points.

– Since a square has four sides, and if the circle intersects each side at two points, the maximum number of points of intersection will be $4 \times 2 = 8$.

– Thus, the maximum number of points of intersection between a square and a circle is 8.

Evaluating the Statements:

1. Statement 1: “The minimum number of points of intersection of a square and a circle is 2.”

This statement is incorrect because the minimum number of points of intersection can be 0 or 1.

2. Statement 2: “The maximum number of points of intersection of a square and circle is 8.”

This statement is correct, as explained above.

The correct answer is (b) 2 only.

Explanation:

To solve this problem, we need to calculate the average speed of the car for the entire journey. The average speed is defined as the total distance traveled divided by the total time taken.

1. Define Variables:

Let the distance between places X and Y be $d$ km.

2. Calculate the time taken for each part of the journey:

From X to Y at speed $V$:

Time taken = $\frac{d}{V}$ hours

From Y to X at speed $2V$:

Time taken = $\frac{d}{2V}$ hours

From X to Y again at speed $3V$:

Time taken = $\frac{d}{3V}$ hours

From Y to X again at speed $4V$:

Time taken = $\frac{d}{4V}$ hours

3. Total Distance Traveled:

The car travels from X to Y and back to X twice, so the total distance is:
$$
\text{Total Distance} = 4d
$$

4. Total Time Taken:

The total time taken for the entire journey is the sum of the times for each segment:
$$
\text{Total Time} = \frac{d}{V} + \frac{d}{2V} + \frac{d}{3V} + \frac{d}{4V}
$$

To combine these fractions, find the least common multiple (LCM) of the denominators:

$$
\text{LCM of } V, 2V, 3V, 4V = 12V
$$

Convert each fraction to have a common denominator:
$$
\frac{d}{V} = \frac{12d}{12V}, \quad \frac{d}{2V} = \frac{6d}{12V}, \quad \frac{d}{3V} = \frac{4d}{12V}, \quad \frac{d}{4V} = \frac{3d}{12V}
$$

Thus, the total time is:
$$
\text{Total Time} = \frac{12d}{12V} + \frac{6d}{12V} + \frac{4d}{12V} + \frac{3d}{12V} = \frac{25d}{12V}
$$

5. Calculate the Average Speed:

The average speed is the total distance divided by the total time:
$$
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{4d}{\frac{25d}{12V}}
$$

Simplify the expression:
$$
\text{Average Speed} = \frac{4d \times 12V}{25d} = \frac{48V}{25}
$$

6. Determine the Range of the Average Speed:

We know $V > 0$, and we want to find where $\frac{48V}{25}$ lies compared to $V, 2V, 3V,$ and $4V$.

– $\frac{48V}{25}$ is approximately $1.92V$

Thus, $1.92V$ lies between $V$ and $2V$.

The correct answer is (b) lies between $v$ and $2v$ km/hr.

Explanation:

To solve this problem, we need to determine the rate at which each person, X and Y, can work individually and then find out how long it will take them to complete 40% of the work together.

1. Determine the work rate of X:

Person X can complete 20% of the work in 8 days.
Let’s find the work rate of X in terms of work done per day.

$$
\text{Work rate of X} = \frac{20\%}{8 \text{ days}} = \frac{20}{100} \times \frac{1}{8} = \frac{1}{40} \text{ of the work per day}
$$

2. Determine the work rate of Y:

Person Y can complete 25% of the work in 6 days.
Let’s find the work rate of Y in terms of work done per day.

$$
\text{Work rate of Y} = \frac{25\%}{6 \text{ days}} = \frac{25}{100} \times \frac{1}{6} = \frac{1}{24} \text{ of the work per day}
$$

3. Combined work rate of X and Y:

To find out how much work they can complete together in one day, we add their individual work rates:

$$
\text{Combined work rate} = \frac{1}{40} + \frac{1}{24}
$$

To add these fractions, we find a common denominator:

$$
\text{Least common multiple of } 40 \text{ and } 24 = 120
$$

Converting each fraction:

$$
\frac{1}{40} = \frac{3}{120}, \quad \frac{1}{24} = \frac{5}{120}
$$

So, the combined work rate is:

$$
\text{Combined work rate} = \frac{3}{120} + \frac{5}{120} = \frac{8}{120} = \frac{1}{15}
$$

This means X and Y together can complete $\frac{1}{15}$ of the work per day.

4. Calculate the time to complete 40% of the work:

To find how many days they need to complete 40% of the work together:

$$
\text{Days required} = \frac{40\%}{\frac{1}{15}} = \frac{40}{100} \times 15 = 6
$$

Thus, X and Y together will complete 40% of the work in 6 days.

The correct answer is (a) 6.

Explanation:

To determine the largest number among the given options, we evaluate each expression using the property of exponents:

$$
\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
$$

Let’s evaluate each expression:

1. Option (a): $\left(\frac{1}{2}\right)^{-6}$

$$
\left(\frac{1}{2}\right)^{-6} = \left(\frac{2}{1}\right)^6 = 2^6 = 64
$$

2. Option (b): $\left(\frac{1}{4}\right)^{-3}$

$$
\left(\frac{1}{4}\right)^{-3} = \left(\frac{4}{1}\right)^3 = 4^3 = 64
$$

3. Option (c): $\left(\frac{1}{3}\right)^{-4}$

$$
\left(\frac{1}{3}\right)^{-4} = \left(\frac{3}{1}\right)^4 = 3^4 = 81
$$

4. Option (d): $\left(\frac{1}{6}\right)^{-2}$

$$
\left(\frac{1}{6}\right)^{-2} = \left(\frac{6}{1}\right)^2 = 6^2 = 36
$$

After evaluating all options, we find:
– Option (a) = 64
– Option (b) = 64
– Option (c) = 81
– Option (d) = 36

The largest number is $81$, which corresponds to option (c).

Thus, the correct answer is (c) $\left(\frac{1}{3}\right)^{-4}$.

Explanation:

To solve the given problem, we need to determine the pattern between the groups of letters and use it to find the missing term represented by “@”.

Given sequences:
– ACPQ : BESU : MNGI : @

First, let’s analyze the pattern between ACPQ and BESU:

\[
\begin{aligned}
&\text{A} \rightarrow \text{B} \, (\text{+1 position}) \\
&\text{C} \rightarrow \text{E} \, (\text{+2 positions}) \\
&\text{P} \rightarrow \text{S} \, (\text{+3 positions}) \\
&\text{Q} \rightarrow \text{U} \, (\text{+4 positions}) \\
\end{aligned}
\]

Now, let’s analyze the pattern between BESU and MNGI:

\[
\begin{aligned}
&\text{B} \rightarrow \text{M} \, (\text{+11 positions}) \\
&\text{E} \rightarrow \text{N} \, (\text{+9 positions}) \\
&\text{S} \rightarrow \text{G} \, (\text{-12 positions}) \\
&\text{U} \rightarrow \text{I} \, (\text{-12 positions}) \\
\end{aligned}
\]

To find the missing term represented by “@”, apply the same pattern to MNGI:

\[
\begin{aligned}
&\text{M} \rightarrow \text{N} \, (\text{+1 position}) \\
&\text{N} \rightarrow \text{P} \, (\text{+2 positions}) \\
&\text{G} \rightarrow \text{J} \, (\text{+3 positions}) \\
&\text{I} \rightarrow \text{M} \, (\text{+4 positions}) \\
\end{aligned}
\]

Thus, the missing term “@” is $\textbf{NPJM}.$

Therefore, the correct answer is $\textbf{(d) NPJM}.$

Explanation:

Let the original price of the article be $P$.

Option 1: Successive discounts of 10% and 20%, then pay a service tax of 10%

1. Apply the first discount of 10%:

$$
\text{Price after 10% discount} = P \times (1 – 0.10) = 0.9P
$$

2. Apply the second discount of 20%:

$$
\text{Price after additional 20% discount} = 0.9P \times (1 – 0.20) = 0.9P \times 0.8 = 0.72P
$$

3. Apply the service tax of 10%:

$$
\text{Final price} = 0.72P \times (1 + 0.10) = 0.72P \times 1.10 = 0.792P
$$

Option 2: Successive discounts of 20% and 10%, then pay a service tax of 10%

1. Apply the first discount of 20%:

$$
\text{Price after 20% discount} = P \times (1 – 0.20) = 0.8P
$$

2. Apply the second discount of 10%:

$$
\text{Price after additional 10% discount} = 0.8P \times (1 – 0.10) = 0.8P \times 0.9 = 0.72P
$$

3. Apply the service tax of 10%:

$$
\text{Final price} = 0.72P \times (1 + 0.10) = 0.72P \times 1.10 = 0.792P
$$

Option 3: Pay a service tax of 10% first, then successive discounts of 20% and 10%

1. Apply the service tax of 10% first:

$$
\text{Price after service tax} = P \times (1 + 0.10) = 1.10P
$$

2. Apply the first discount of 20%:

$$
\text{Price after 20% discount} = 1.10P \times (1 – 0.20) = 1.10P \times 0.8 = 0.88P
$$

3. Apply the second discount of 10%:

$$
\text{Final price} = 0.88P \times (1 – 0.10) = 0.88P \times 0.9 = 0.792P
$$

Conclusion

All three options result in the same final price of $0.792P$.

The correct answer is (d) All the options are equally good for the customer.

Explanation:

1. Let the original price of rice per kg be $x$ rupees.

2. After a 25\% hike, the new price per kg becomes:
$$
x + 0.25x = 1.25x
$$

3. The person could buy a certain amount of rice for Rs. 1,200 before the price hike:

The quantity of rice that could be bought before the hike:
$$
\frac{1200}{x} \text{ kg}
$$

4. The quantity of rice that can be bought after the price hike:

$$
\frac{1200}{1.25x} \text{ kg}
$$

5. According to the problem, the person is able to buy 6 kg less rice after the price hike:

$$
\frac{1200}{x} – \frac{1200}{1.25x} = 6
$$

6. Simplify the equation:

First, find a common denominator:
$$
\frac{1200 \times 1.25 – 1200}{1.25x} = 6
$$

Simplifying further:
$$
\frac{1200(1.25 – 1)}{1.25x} = 6
$$

$$
\frac{1200 \times 0.25}{1.25x} = 6
$$

$$
\frac{300}{1.25x} = 6
$$

$$
300 = 6 \times 1.25x
$$

7. Solve for $x$:

$$
x = \frac{300}{7.5} = 40
$$

Conclusion

The original price of rice per kg was Rs. 40.

Thus, the correct answer is $\textbf{(b) Rs. 40}.$

Explanation:

1. Initial Information:

– The average score after 50 innings is 46.4.
– This means the total score after 50 innings is:

$$
\text{Total score after 50 innings} = 50 \times 46.4 = 2320
$$

2. Information After 60 Innings:

– The average score after 60 innings is $46.4 + 2.6 = 49$.
– This means the total score after 60 innings is:

$$
\text{Total score after 60 innings} = 60 \times 49 = 2940
$$

3. Calculate the Total Score in the Last 10 Innings:

– The total score in the last 10 innings is the difference between the total scores after 60 and 50 innings:

$$
\text{Total score in the last 10 innings} = 2940 – 2320 = 620
$$

4. Calculate the Average Score in the Last 10 Innings:

– The average score in the last 10 innings is:

$$
\text{Average score in the last 10 innings} = \frac{620}{10} = 62
$$

Conclusion

The correct answer is $\textbf{(c) 62}.$

Explanation:

Given the word “DELHI,” we need to create 5-letter words that start with D and end with I.

1. Fixing the First and Last Letters:

Since the first letter must be D and the last letter must be I, the letters in positions 2, 3, and 4 need to be arranged from the remaining letters: E, L, and H.

2. Arranging the Middle Three Letters:

The number of ways to arrange the letters E, L, and H is given by the permutation of 3 distinct items.

The number of permutations of 3 distinct letters is given by $3!$:

\[
3! = 3 \times 2 \times 1 = 6
\]

3. Calculating the Total Number of Words:

Since there is only one way to choose D as the first letter and I as the last letter, the total number of different 5-letter words is equal to the number of arrangements of the middle three letters.

Therefore, the total number of different 5-letter words is: 6

Conclusion

The correct answer is $\textbf{(d) 6}.$

Explanation:

1. Initial Setup:

The bottle initially contains 20 liters of liquid A.

2. First Replacement:

– 4 liters of liquid A is taken out.
– This leaves $20 – 4 = 16$ liters of liquid A in the bottle.
– 4 liters of liquid B is added, so the bottle now has 16 liters of liquid A and 4 liters of liquid B.

3. Second Replacement:

– 4 liters of the mixture is taken out. Since the mixture consists of 16 liters of A and 4 liters of B, the concentration of A and B in the mixture are:
$$
\text{Concentration of A} = \frac{16}{20}, \quad \text{Concentration of B} = \frac{4}{20}
$$
– Therefore, the 4 liters of mixture taken out will consist of:
$$
\text{Liquid A taken out} = 4 \times \frac{16}{20} = 3.2 \text{ liters}
$$
$$
\text{Liquid B taken out} = 4 \times \frac{4}{20} = 0.8 \text{ liters}
$$
– After removing 4 liters of mixture, the remaining quantities are:
$$
\text{Remaining Liquid A} = 16 – 3.2 = 12.8 \text{ liters}
$$
$$
\text{Remaining Liquid B} = 4 – 0.8 = 3.2 \text{ liters}
$$
– 4 liters of liquid B is then added, so the final quantities in the mixture are:
$$
\text{Final Liquid A} = 12.8 \text{ liters}, \quad \text{Final Liquid B} = 3.2 + 4 = 7.2 \text{ liters}
$$

4. Calculating the Ratio:

The final ratio of liquid A to liquid B is:
$$
\text{Ratio of A to B} = \frac{12.8}{7.2} = \frac{128}{72} = \frac{16}{9}
$$

Conclusion

The correct answer is $\textbf{(c) 16 : 9}.$

Explanation:

Given that 12th January is a Sunday, we need to determine the day of the week for 12th July and 15th July in both leap and non-leap years.

Non-Leap Year:

Days from 12th January to 12th July:

$19 + 28 + 31 + 30 + 31 + 30 + 12 = 181$ days

181 days modulo 7:

$181 \div 7 = 25$ remainder $6$

Since 12th January is a Sunday, 181 days later is 6 days after Sunday, which is a $\textbf{Saturday}$.

Days from 12th January to 15th July:

$181 + 3 = 184$ days

184 days modulo 7:

$184 \div 7 = 26$ remainder $2$

Since 12th January is a Sunday, 184 days later is 2 days after Sunday, which is a $\textbf{Tuesday}.$

Leap Year:

Days from 12th January to 12th July:

$19 + 29 + 31 + 30 + 31 + 30 + 12 = 182$ days

182 days modulo 7:

$182 \div 7 = 26$ remainder $0$

Since 12th January is a Sunday, 182 days later is 0 days after Sunday, which is a $\textbf{Sunday}.$

Days from 12th January to 15th July:

$182 + 3 = 185$ days

185 days modulo 7:

$185 \div 7 = 26$ remainder $3$

Since 12th January is a Sunday, 185 days later is 3 days after Sunday, which is a $\textbf{Wednesday}.$

Conclusion:

The correct answer is $\textbf{(c) 12th July is Sunday if the year is a leap year.}$

Explanation:

Given the four-digit numbers $A3BC$ and $DE2F$, where each letter represents a different digit greater than 3, we have the equation:

\[
A3BC + DE2F = 15902
\]

Step 1: Expand the numbers

The numbers can be expanded as:

\[
A3BC = 1000A + 300 + 10B + C
\]

$$ DE2F = 1000D + 100E + 20 + F $$

Substitute these into the equation:

\[
(1000A + 300 + 10B + C) + (1000D + 100E + 20 + F) = 15902
\]

Combine like terms:

\[
1000A + 1000D + 10B + C + 100E + F + 320 = 15902
\]

Subtract 320 from both sides:

\[
1000A + 1000D + 10B + C + 100E + F = 15582
\]

Step 2: Analyze the possible values for $A$ and $D$

Since $1000A + 1000D = 1000(A + D)$, and the result should match the thousands place value of 15582:

\[
A + D = 15
\]

Both $A$ and $D$ are digits greater than 3, so the possible pairs $(A, D)$ that sum to 15 are:

\[
(6, 9) \quad \text{and} \quad (9, 6)
\]

Step 3: Determine the difference between $A$ and $D$

In both possible cases:

– If $A = 6$ and $D = 9$, the difference $|A – D|$ is $|6 – 9| = 3$.
– If $A = 9$ and $D = 6$, the difference $|A – D|$ is $|9 – 6| = 3$.

Conclusion

The difference between the values of $A$ and $D$ is: $\boxed{3} $

Thus, the correct answer is (c) 3.

Explanation:

To solve this problem, we need to identify the pattern or operation applied to each number in the sequence $14, 18, 20, 24, 30, 32, \ldots$ that results in a sequence related to prime numbers.

Step-by-Step Solution

1. Identify the sequence:

– The given sequence is: $14, 18, 20, 24, 30, 32, \ldots$

2. Find a possible operation related to prime numbers:

– Notice the differences between each pair of consecutive numbers:

\[
18 – 14 = 4, \quad 20 – 18 = 2, \quad 24 – 20 = 4, \quad 30 – 24 = 6, \quad 32 – 30 = 2
\]

– This sequence of differences $(4, 2, 4, 6, 2)$ doesn’t immediately suggest a connection to prime numbers.

– However, let’s check if the numbers themselves are close to being prime or can be transformed to prime numbers through some consistent operation.

3. Apply an operation related to prime numbers:

– Look for a pattern by subtracting 1 from each number in the sequence:

\[
14 – 1 = 13, \quad 18 – 1 = 17, \quad 20 – 1 = 19, \quad 24 – 1 = 23, \quad 30 – 1 = 29, \quad 32 – 1 = 31
\]

– The numbers obtained after subtracting 1 are: $13, 17, 19, 23, 29, 31$

– These are all consecutive prime numbers.

4. Determine the next number in the sequence:

– The next prime number after $31$ is $37$.

– To find the next number in the sequence, add 1 to this next prime number:

\[
37 + 1 = 38
\]

Conclusion

– The next number in the sequence is $38$.

Therefore, the correct answer is (c) 38.

Explanation:

To find the minimum length that can be measured using two sticks of lengths 7.5 feet and 3.25 feet, we need to determine the greatest common divisor (GCD) of the two lengths. The GCD represents the smallest unit of length that can be exactly measured using the given lengths.

Step-by-Step Solution

1. Convert the lengths to a common unit (feet):

– The lengths of the sticks are already in feet:
\[
\text{Length of Stick 1} = 7.5 \, \text{feet}
\]

\[
\text{Length of Stick 2} = 3.25 \, \text{feet}
\]

2. Express the lengths as fractions:

– Convert the decimal lengths to fractions for easier calculation of the GCD:
\[
7.5 = \frac{75}{10} = \frac{15}{2} \, \text{feet}
\]
\[
3.25 = \frac{325}{100} = \frac{13}{4} \, \text{feet}
\]

3. Find the GCD of the fractions:

– To find the GCD of $\frac{15}{2}$ and $\frac{13}{4}$, first find the GCD of the numerators (15 and 13) and the least common multiple (LCM) of the denominators (2 and 4).

GCD of numerators (15 and 13):
– Since 15 and 13 are coprime (no common factors other than 1), the GCD is 1.

LCM of denominators (2 and 4):
– The LCM of 2 and 4 is 4.

– Combine the GCD of the numerators and the LCM of the denominators:

\[
\text{GCD} = \frac{\text{GCD of numerators}}{\text{LCM of denominators}} = \frac{1}{4} \, \text{feet}
\]

4. Conclusion:

– The minimum length that can be measured using the two sticks is $ \frac{1}{4} $ feet, which is 0.25 feet.

Therefore, the correct answer is (b) 0.25 foot.

Explanation:

To solve this problem, we need to determine which two consecutive page numbers were torn out from a booklet such that the sum of the remaining page numbers is 195.

Step-by-Step Solution

1. Understand the total sum of the booklet’s pages:

– Let $n$ be the total number of pages in the booklet.
– Since the pages are numbered starting from 1, the sum of all the page numbers from 1 to $n$ can be calculated using the formula for the sum of an arithmetic series:

\[
\text{Sum} = \frac{n(n + 1)}{2}
\]

2. Find the total sum of pages if no pages were torn out:

We know the sum of the remaining pages is 195. Let $p$ and $p+1$ be the page numbers of the torn pages. Then, the sum of the pages can be represented as:

\[
\frac{n(n + 1)}{2} – (p + (p + 1)) = 195
\]

Simplifying this equation, we get:

\[
\frac{n(n + 1)}{2} – (2p + 1) = 195
\]

Rearrange to solve for $ \frac{n(n + 1)}{2} $:

\[
\frac{n(n + 1)}{2} = 195 + 2p + 1 = 196 + 2p
\]

3. Find possible values for $n$:

To find $n$, we need to test values that make $\frac{n(n + 1)}{2}$ close to 196.

Calculate $n$ when $n(n + 1)/2$ is close to 196:

For $n = 19$:

\[
\frac{19 \times 20}{2} = 190 \quad (\text{too low})
\]

For $n = 20$:

\[
\frac{20 \times 21}{2} = 210 \quad (\text{too high})
\]

4. Determine the torn page numbers $p$ and $p+1$:

Since $n = 20$ gives $\frac{20 \times 21}{2} = 210$, and we have $210 – (p + p + 1) = 195$, simplify this to find $p$:

\[
210 – (2p + 1) = 195
\]

\[
210 – 2p – 1 = 195
\]

\[
209 – 2p = 195
\]

\[
2p = 14
\]

\[
p = 7
\]

– Therefore, the torn pages are 7 and 8.

Conclusion

Therefore, the correct answer is (b) 7, 8.

Explanation:

To solve the problem, we need to find the digit represented by the symbol $ \text{Ä} $ that satisfies the given equation:

\[
\text{Ä} + 1\text{Ä} + 5\text{Ä} + \text{ÄÄ} + \text{Ä}1 = 1\text{ÄÄ}
\]

Step-by-Step Solution

1. Express the numbers in terms of $ \text{Ä} $:

– Let $ \text{Ä} = x $, where $ x $ is a digit from 0 to 9.
– Then, convert each number involving $ \text{Ä} $ to its numerical equivalent:
– $ \text{Ä} = x $
– $ 1\text{Ä} = 10 + x $
– $ 5\text{Ä} = 50 + x $
– $ \text{ÄÄ} = 10x + x = 11x $
– $ \text{Ä}1 = 10x + 1 $
– $ 1\text{ÄÄ} = 100 + 11x $

2. Set up the equation:

– Substituting the values into the given equation:
\[
x + (10 + x) + (50 + x) + (11x) + (10x + 1) = 100 + 11x
\]

3. Simplify the equation:

– Combine all the like terms:
\[
x + 10 + x + 50 + x + 11x + 10x + 1 = 100 + 11x
\]

– Combine $ x $ terms and constant terms:
\[
24x + 61 = 100 + 11x
\]

4. Solve for $ x $:

– Subtract $ 11x $ from both sides:
\[
24x – 11x + 61 = 100
\]
\[
13x + 61 = 100
\]
– Subtract 61 from both sides:
\[
13x = 39
\]
– Divide by 13:
\[
x = 3
\]

Conclusion:

– The symbol $ \text{Ä} $ stands for the digit 3.

Therefore, the correct answer is (b) 3.

Explanation:

To find out how much the person spent to buy the car, we can use the information given about the selling price and the loss percentage.

Step-by-Step Solution

1. Define the variables:

– Let the cost price (the amount spent to buy the car) be $C$ rupees.
– The selling price (the amount for which the car was sold) is given as Rs. 3,00,000.
– The loss percentage incurred on the sale is 20%.

2. Understand the relationship between cost price, selling price, and loss:

– A loss of 20% means the selling price is 80% of the cost price because:
\[
\text{Selling Price} = \text{Cost Price} – \text{Loss}
\]
– The loss can also be expressed in terms of the cost price:
\[
\text{Loss} = \text{Cost Price} \times \frac{20}{100}
\]
– Therefore, the selling price can be written as:
\[
\text{Selling Price} = \text{Cost Price} \times \left(1 – \frac{20}{100}\right)
\]

3. Set up the equation:

– Given that the selling price is Rs. 3,00,000 and using the formula above:
\[
3,00,000 = C \times \left(1 – \frac{20}{100}\right)
\]
– Simplify the equation:
\[
3,00,000 = C \times 0.80
\]

4. Solve for the cost price $C$:

\[
C = \frac{3,00,000}{0.80}
\]
\[
C = 3,75,000
\]

Conclusion

– The cost price of the car, or the amount the person spent to buy the car, is Rs. 3,75,000.

Therefore, the correct answer is (d) Rs. 3,75,000.

Explanation:

To solve this problem, we need to use the information given to set up an equation that relates the ages of the teacher and the students.

Step-by-Step Solution

1. Define the variables:

– Let the age of each student be $x$ years.
– The age of the teacher is $x + 20$ years (since the difference between the age of the teacher and each student is 20 years).

2. Set up the equation for the average age:

– The average age of the teacher and three students is given as 20 years.
– The total age of the teacher and the three students is:

\[
(x + 20) + 3x
\]

– This simplifies to:

\[
x + 20 + 3x = 4x + 20
\]

3. Use the average to form an equation:

– The average age is given by dividing the total age by the number of people (4 in this case):

\[
\text{Average age} = \frac{4x + 20}{4}
\]

– Set this equal to 20:

\[
\frac{4x + 20}{4} = 20
\]

4. Solve for $x$:

– Multiply both sides of the equation by 4 to clear the fraction:
\[
4x + 20 = 80
\]

– Subtract 20 from both sides:
\[
4x = 60
\]
– Divide both sides by 4:
\[
x = 15
\]

5. Find the age of the teacher:

– The age of the teacher is $x + 20$:

\[
x + 20 = 15 + 20 = 35
\]

Conclusion

– The age of the teacher is 35 years.

Therefore, the correct answer is (c) 35 years.

Explanation:

To solve this problem, we need to understand the relationships between the volumes, masses, and the densities of the two metallic cubes $P$ and $Q$.

Given Information

1. Let the side length of cube $P$ be $a$. The volume $x$ of cube $P$ is:
\[
x = a^3
\]

2. The side length of cube $Q$ is two times that of cube $P$. So, the side length of $Q$ is $2a$. The volume $y$ of cube $Q$ is:
\[
y = (2a)^3 = 8a^3
\]

3. The mass $m$ of cube $P$ and the mass $n$ of cube $Q$ are given, with $n = 2m$.

4. We are given:
– $u = \frac{m}{x}$
– $v = \frac{n}{y}$

Step-by-Step Solution

1. Calculate $u = \frac{m}{x}$:
\[
u = \frac{m}{a^3}
\]

2. Calculate $v = \frac{n}{y}$:

– Since $n = 2m$ and $y = 8a^3$:
\[
v = \frac{n}{y} = \frac{2m}{8a^3} = \frac{m}{4a^3}
\]

3. Determine the relationship between $u$ and $v$:

– We have:
\[
u = \frac{m}{a^3} \quad \text{and} \quad v = \frac{m}{4a^3}
\]
– Dividing $u$ by $v$:
\[
\frac{u}{v} = \frac{\frac{m}{a^3}}{\frac{m}{4a^3}} = \frac{4m/a^3}{m/a^3} = 4
\]
– Therefore:
\[
u = 4v
\]

Conclusion:

– The correct relationship is $u = 4v$.

Therefore, the correct answer is (a) $u = 4v$.

Explanation:

To solve this problem, we need to find the integers between 1 and 100 that have the digit 4 and are not divisible by 4.

Step-by-Step Solution

1. Identify numbers between 1 and 100 that contain the digit 4:

– The numbers between 1 and 100 that contain the digit 4 are:

4, 14, 24, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54, 64, 74, 84, 94

2. Count the total numbers that have the digit 4:

– Total numbers: 19.

3. Identify numbers that are divisible by 4:

– A number is divisible by 4 if the last two digits form a number divisible by 4.
– The numbers from the above list that are divisible by 4 are:

4, 24, 40, 44, 48, 64, 84

– Total numbers divisible by 4: 7.

4. Calculate the numbers that have the digit 4 but are not divisible by 4:

– Subtract the numbers divisible by 4 from the total numbers with digit 4:

19 – 7 = 12

Conclusion:

– There are 12 numbers between 1 and 100 that have the digit 4 and are not divisible by 4.

Therefore, the correct answer is (c) 12.

Explanation:

To determine the man’s direction from the starting point after he completes his walk, we need to carefully analyze his movements step-by-step with respect to his house’s orientation.

Step-by-Step Solution

1. Understand the initial conditions and movements:

– The man’s house faces East.
– The backside of the house is therefore to the West.

2. Trace the man’s path:

– First movement: The man walks 25 meters down the backside of his house. Since the house faces East, walking down the backside means he moves West.
– Second movement: He then turns to the right and walks 50 meters. Facing West and turning right means he now faces North.
– Third movement: The man turns to the left and walks 25 meters. Facing North and turning left means he now faces West again.

3. Analyze the final position:

– After the first movement, the man is 25 meters West of the starting point.
– After the second movement, he is 50 meters North and 25 meters West of the starting point.
– After the third movement, he moves 25 meters further West, resulting in a total of 50 meters West.

4. Determine his final position:

– He is now 50 meters West and 50 meters North of the starting point.

5. Identify the direction from the starting point:

– To determine the direction from the starting point, visualize the position relative to the starting point.
– The man is 50 meters West and 50 meters North of the starting point, which places him in the North-West direction relative to the starting point.

Conclusion

– The man’s direction from the starting point is North-West.

Therefore, the correct answer is (d) North-West.

Explanation:

To solve this problem, we need to count the number of times an odd number in the sequence is immediately followed by another odd number.

Step-by-Step Solution

1. Identify the odd numbers in the sequence:

– Odd numbers are those that are not divisible by 2. In this sequence, the odd numbers are 5, 1, 7, 3, 9, 5, 7, 3, 1, 5, 3, 5, 3, 9, 5, and 3.

2. Analyze the sequence to find consecutive odd numbers:

– The sequence given is:
\[
5, 1, 4, 7, 3, 9, 8, 5, 7, 2, 6, 3, 1, 5, 8, 6, 3, 8, 5, 2, 2, 4, 3, 4, 9, 6
\]

3. Count the pairs of consecutive odd numbers:

– Let’s go through the sequence step-by-step:
– 5 is followed by 1 (both are odd).
– 1 is followed by 4 (not odd).
– 7 is followed by 3 (both are odd).
– 3 is followed by 9 (both are odd).
– 9 is followed by 8 (not odd).
– 5 is followed by 7 (both are odd).
– 7 is followed by 2 (not odd).
– 3 is followed by 1 (both are odd).
– 1 is followed by 5 (both are odd).
– 5 is followed by 8 (not odd).
– 3 is followed by 8 (not odd).
– 5 is followed by 2 (not odd).
– 3 is followed by 4 (not odd).
– 9 is followed by 6 (not odd).

4. Calculate the total number of consecutive odd pairs:

– From the above analysis, the pairs of consecutive odd numbers are:
– (5, 1)
– (7, 3)
– (3, 9)
– (5, 7)
– (3, 1)
– (1, 5)

– Therefore, there are 6 pairs of consecutive odd numbers.

Conclusion

– The total number of odd numbers followed by another odd number in the sequence is 6.

Therefore, the correct answer is (b) 6.

Explanation:

To solve this problem, we need to determine how many five-digit prime numbers can be formed using all the digits 1, 2, 3, 4, and 5 without repetition.

Step-by-Step Solution

1. Understanding Prime Numbers:

– A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

2. Analyzing the Permutations of the Digits:

– We are given the digits 1, 2, 3, 4, and 5.
– To form a five-digit number using all these digits without repetition, we can use permutations of these digits.
– The total number of permutations of these five digits is $5! = 120$.

3. Checking for Divisibility by 3:

– For a number to be prime, it cannot be divisible by any number other than 1 and itself.
– A number is divisible by 3 if the sum of its digits is divisible by 3.
– Sum of the digits 1, 2, 3, 4, and 5 is:
\[
1 + 2 + 3 + 4 + 5 = 15
\]

– Since 15 is divisible by 3, any permutation of these digits will result in a number that is divisible by 3.

4. Conclusion:

– Since every permutation of the digits 1, 2, 3, 4, and 5 results in a number that is divisible by 3, there are no prime numbers that can be formed using all five digits.

Therefore, the correct answer is (a) Zero.

Explanation:

To solve this problem, we need to determine the largest value among the natural numbers $p, q, r,$ and $s$ given the following relationships:

\[
p – 2016 = q + 2017 = r – 2018 = s + 2019
\]

Let’s denote the common value for all these expressions by $k$. Then, we can write the following equations:

\[
p – 2016 = k
\]
\[
q + 2017 = k
\]
\[
r – 2018 = k
\]
\[
s + 2019 = k
\]

Step-by-Step Solution

1. Express each variable in terms of $k$:

– From $p – 2016 = k$, solve for $p$:
\[
p = k + 2016
\]

– From $q + 2017 = k$, solve for $q$:
\[
q = k – 2017
\]

– From $r – 2018 = k$, solve for $r$:
\[
r = k + 2018
\]

– From $s + 2019 = k$, solve for $s$:
\[
s = k – 2019
\]

2. Determine the relationships between $p, q, r,$ and $s$:

– We have $p = k + 2016$.
– We have $q = k – 2017$.
– We have $r = k + 2018$.
– We have $s = k – 2019$.

3. Compare the values of $p, q, r,$ and $s$:

– Clearly, $r = k + 2018$ is the largest because 2018 is the highest positive number added to $k$.
– $p = k + 2016$ is slightly smaller than $r$.
– $q = k – 2017$ and $s = k – 2019$ are smaller than both $p$ and $r$.

4. Conclusion:

– The largest value among $p, q, r,$ and $s$ is $r$.

Therefore, the correct answer is (c) $r$.

Explanation:

To solve this problem, we need to analyze the divisibility of the sum $XYZ + YZX + ZXY$ under the condition that $(X + Y + Z)$ is not a multiple of 3. Let’s break down the solution step-by-step.

Step-by-Step Solution

1. Understanding the three-digit number representation:

– A three-digit number $XYZ$ can be represented as $100X + 10Y + Z$.
– Similarly, $YZX = 100Y + 10Z + X$ and $ZXY = 100Z + 10X + Y$.

2. Formulate the expression for $XYZ + YZX + ZXY$:
\[
XYZ + YZX + ZXY = (100X + 10Y + Z) + (100Y + 10Z + X) + (100Z + 10X + Y)
\]
– Combine like terms:
\[
XYZ + YZX + ZXY = (100X + X + 10X) + (10Y + 100Y + Y) + (Z + 10Z + 100Z)
\]
\[
XYZ + YZX + ZXY = 111X + 111Y + 111Z
\]
\[
XYZ + YZX + ZXY = 111(X + Y + Z)
\]

3. Analyze divisibility of the expression $111(X + Y + Z)$:

– Notice that $111 = 3 \times 37$.
– Thus, $111(X + Y + Z) = 3 \times 37 \times (X + Y + Z)$.

4. Check the divisibility conditions:

– Divisibility by 3: The expression $111(X + Y + Z)$ is always divisible by 3 since $111$ itself is divisible by 3.
– Divisibility by 9: If $(X + Y + Z)$ is not a multiple of 3, then $(X + Y + Z)$ is not a multiple of 9, meaning $111(X + Y + Z)$ is not necessarily divisible by 9.
– Divisibility by 37: The expression $111(X + Y + Z)$ is always divisible by 37 because $111$ is divisible by 37.
– Divisibility by $X + Y + Z$: Since the expression is $111(X + Y + Z)$, it is clearly divisible by $X + Y + Z$ regardless of whether $X + Y + Z$ is a multiple of 3.

5. Conclusion:

– The expression $XYZ + YZX + ZXY = 111(X + Y + Z)$ is not divisible by 9 when $X + Y + Z$ is not a multiple of 3.

Therefore, the correct answer is (b) 9.