UPPSC LT Grade Assistant Teacher Maths Official Paper Held On: 06 Dec 2025 Shift 1 (Previous Year Paper)

The correct answer is Service sector.

Key Points

• The Service sector has the largest contribution to the Gross Domestic Product (GDP) of India in recent years. It accounts for more than 50% of the GDP.
• This sector includes industries like Information Technology (IT), telecommunications, banking, healthcare, education, hospitality, and real estate.
• The growth of the IT and Software industry in cities such as Bengaluru, Hyderabad, and Pune has significantly boosted the contribution of the service sector.
• India is a global leader in IT and Business Process Outsourcing (BPO) services, which has further propelled its service sector growth.
• The service sector also significantly contributes to employment generation, especially in urban areas, and is a key driver of India's economic development.
• It is supported by increasing urbanization, rising disposable incomes, and a growing middle-class population that demands more services.
• The service sector has played a critical role in transforming India into a knowledge-based economy, with a focus on innovation and technology-driven solutions.

Additional Information

• Large-scale industries
• These industries include steel, automobile, textiles, and petrochemicals.
• They play a significant role in the industrial and manufacturing growth of the country.
• However, their contribution to the GDP is smaller compared to the service sector.
• Small and cottage industries
• These industries involve handicrafts, handlooms, and artisanal products.
• They are vital for rural employment and preserving India's cultural heritage.
• Despite their importance, their GDP contribution is relatively limited compared to the service sector.
• Agricultural sector
• The agricultural sector was historically the largest contributor to India's GDP but has now been surpassed by the service sector.
• It still remains the primary source of livelihood for a large portion of the population, particularly in rural areas.
• This sector includes activities like crop production, fishing, and forestry.

The Correct answer is More than 20,000 Hz.

Key Points

• Ultrasonic waves are defined as sound waves with a frequency higher than the upper limit of human hearing, which is approximately 20,000 Hz.
• These waves are extensively used in medical imaging, such as ultrasound scans, and also in industrial applications like sonography and cleaning processes.
• They are utilized in non-destructive testing to evaluate the integrity of materials without causing damage.
• Ultrasonic waves are also employed in animal communication. For instance, bats use these high-frequency sound waves for echolocation.
• The production of ultrasonic waves often involves the use of piezoelectric crystals, which vibrate rapidly when subjected to electrical energy.

Additional Information

• 20 Hz to 20,000 Hz
• This range represents the audible frequency range for humans, commonly referred to as the hearing range.
• Humans can detect sounds within this range, but anything beyond this (like ultrasonic waves) is inaudible.
• Less than 20 Hz
• Sound waves with frequencies below 20 Hz are termed infrasonic waves.
• These are used in seismology to study earthquakes and volcanic eruptions.
• Some animals, like elephants, communicate using infrasonic frequencies.
• More than 50,000 Hz
• While this is a correct description of ultrasonic waves, the standard definition begins at 20,000 Hz, making "More than 20,000 Hz" the more precise answer.
• Applications of frequencies above 50,000 Hz include advanced medical imaging and scientific research.

The Correct answer is A-iv, B-iii, C-i, D-ii.

Key Points

• Maasai are a tribal group found in East Africa, mainly in regions of Kenya and Tanzania.
• The Semai are an indigenous tribe located in Malaysia, particularly in the central regions of the country.
• The Bushmen, also known as San people, are indigenous hunter-gatherers living in the Kalahari Desert, which spans parts of Botswana, Namibia, and South Africa.
• The Maori are the indigenous people of New Zealand, known for their rich cultural heritage and traditions.

Additional Information

• Maasai
• The Maasai tribe is known for its colorful attire and nomadic lifestyle.
• They primarily rely on cattle herding as their source of livelihood.
• Semai
• The Semai tribe is recognized for its emphasis on peaceful coexistence and avoidance of conflict.
• They practice shifting cultivation and are skilled in basket weaving.
• Bushmen
• The Bushmen are known for their profound knowledge of the desert ecosystem.
• They use traditional methods for hunting and gathering food.
• Maori
• The Maori are famous for their distinctive tattoo art called ta moko.
• Their traditional war dance, haka, is internationally recognized.

The Correct answer is Mandovi River.

Key Points

• The Dudhsagar Falls is a spectacular waterfall located in the Indian state of Goa.
• It is situated on the Mandovi River, which originates in the Western Ghats and flows westward into the Arabian Sea.
• The name "Dudhsagar" translates to "Sea of Milk" in English, owing to the waterfall's appearance of milky white water cascading down the rocks.
• It is one of the tallest waterfalls in India, with a height of approximately 310 meters (1017 feet).
• The waterfall is located in the Bhagwan Mahaveer Sanctuary and Molem National Park, which are rich in biodiversity.
• The site is famous as a tourist attraction and is particularly popular among trekkers and nature enthusiasts.
• The waterfall can be accessed via a railway line and is visible from the Chennai-Vasco railway route.
• The Mandovi River is often referred to as the "Lifeline of Goa" because it serves as a crucial water source for the region.

Additional Information

• Bhadra River
• The Bhadra River originates in the Western Ghats in Karnataka.
• It is a tributary of the Tungabhadra River, which eventually merges into the Krishna River.
• The river is famous for the Bhadra Wildlife Sanctuary, which is home to a variety of flora and fauna.
• Sharavati River
• The Sharavati River is located in Karnataka and is best known for forming the Jog Falls, one of India's highest waterfalls.
• This river is utilized for hydroelectric projects, contributing to the state's power supply.
• Ulhas River
• The Ulhas River is located in the state of Maharashtra.
• It flows westward and empties into the Arabian Sea.
• The river basin supports several towns and cities, including Kalyan, Thane, and Ulhasnagar.

The Correct answer is Sorghum.

Key Points

• Sorghum is widely known as the "Camel of Crops" due to its ability to survive in arid and semi-arid regions where other crops may fail.
• It is a cereal crop that is highly adaptable to drought conditions, extreme heat, and poor-quality soil, making it a vital food and fodder crop in many parts of the world.
• Sorghum is a staple food crop in many developing countries, especially in Africa and South Asia.
• The crop is rich in nutritional content and provides essential nutrients such as protein, dietary fiber, and minerals like iron and phosphorus.
• It is used for multiple purposes, including food (in the form of porridge, bread, and beverages), animal feed, and as a raw material for producing ethanol.
• Sorghum’s ability to grow with minimal water requirements and its resilience to harsh climates make it a critical crop for ensuring food security in water-scarce regions.
• Different varieties of sorghum are cultivated for grain production, syrup extraction, and biofuel production.

Additional Information

• Bajra
• Bajra, also known as Pearl Millet, is another drought-resistant crop commonly grown in arid and semi-arid regions.
• It is rich in fiber, protein, and essential minerals like magnesium and calcium, making it a healthy food choice.
• Bajra is primarily grown in India and Africa and is used to prepare flatbreads, porridge, and other traditional dishes.
• Ragi
• Ragi, also known as Finger Millet, is a staple food in many parts of India and Africa.
• It is a nutrient-rich crop, particularly high in calcium, making it beneficial for bone health.
• Ragi is widely used to make porridge, flatbreads, and malted beverages.
• Kodo
• Kodo Millet is a highly nutritious and drought-resistant millet grown in parts of India.
• It is rich in dietary fiber, protein, and antioxidants, making it a healthy choice for those with diabetes or gluten intolerance.
• Kodo is commonly used to prepare porridge, upma, and dosa.

The Correct answer is Vegetation.

Key Points

• Biodiversity hotspots are areas that are exceptionally rich in species diversity, particularly plant species, and are under threat of destruction.
• The concept of biodiversity hotspots was introduced by Norman Myers in 1988.
• One of the primary criteria for defining a biodiversity hotspot is the presence of at least 1,500 species of vascular plants as endemics, which means they are not found anywhere else in the world.
• Another important criterion is that the region must have lost at least 70% of its original vegetation, making the conservation of remaining vegetation crucial.
• Vegetation acts as the primary producer in ecosystems, supporting the survival of other organisms like animals and microbes, which is why it is the basis for defining biodiversity hotspots.
• Examples of biodiversity hotspots include the Himalayas, the Western Ghats, the Sundaland (including the Nicobar Islands), and the Indo-Burma region in India.
• Conservation efforts in biodiversity hotspots focus on protecting their vegetation as it directly impacts the survival of other species and the overall ecosystem health.

Additional Information

• All organisms
• Although biodiversity includes all organisms, the concept of biodiversity hotspots specifically focuses on plant species and their endemism, as they form the base for all ecosystems.
• Crop diversity
• Crop diversity refers to the variety of crops and their genetic traits, which is essential for agriculture but is not a criterion for defining biodiversity hotspots.
• Animals
• While animal species are an integral part of biodiversity, biodiversity hotspots are primarily defined based on vegetation and plant endemism.
• Animals depend on vegetation for food and habitat, making vegetation the most critical factor for biodiversity hotspots.

The correct answer is Option 1.

Key Points

• MAA (Mothers’ Absolute Affection): The primary objective of this scheme is the promotion of breastfeeding. It aims to create awareness among mothers about the benefits of breastfeeding and to provide support to ensure proper breastfeeding practices.
• LaQshya: This scheme focuses on the improvement in labour rooms and maternity operation theatres. It aims to enhance the quality of care during delivery and immediate postpartum care.
• SUMAN (Surakshit Matritva Aashwasan): This initiative provides free healthcare to pregnant women and newborns. It ensures respectful and quality maternity care with zero expense to the beneficiary.
• MCP Card (Mother and Child Protection Card): The MCP card is used for tracking the benefits under the Pradhan Mantri Matru Vandana Yojana (PMMVY). It records essential details of antenatal, postnatal care, and immunization for the mother and the child.
• The correct mapping of the schemes and their objectives is as follows:
• A - ii (MAA: Promotion of breastfeeding).
• B - iii (LaQshya: Improvement in labour room and maternity OT).
• C - iv (SUMAN: Free healthcare to pregnant women and newborn child).
• D - i (MCP Card: Benefits under PMMVY).

Additional Information

• MAA (Mothers’ Absolute Affection):
• This scheme is a part of the National Health Mission (NHM) and was launched by the Ministry of Health and Family Welfare.
• It emphasizes the importance of exclusive breastfeeding for the first six months of life and continued breastfeeding for up to two years or more.
• It includes awareness campaigns, counseling sessions, and training programs for healthcare providers.
• LaQshya:
• Launched under the National Health Mission (NHM), it is aimed at reducing maternal and neonatal mortality.
• This initiative ensures adherence to quality standards in labour rooms and maternity OTs.
• It focuses on improving infrastructure, infection control, and patient satisfaction.
• SUMAN (Surakshit Matritva Aashwasan):
• Launched in 2019 by the Government of India, this scheme provides assured, dignified, and respectful maternity care at no cost.
• It includes free tests, medicines, transport, and delivery services.
• The initiative also aims to reduce preventable maternal and newborn deaths.
• MCP Card (Mother and Child Protection Card):
• The card is a joint initiative of the Ministry of Women and Child Development and the Ministry of Health and Family Welfare.
• It serves as a tool for monitoring and ensuring the delivery of health and nutrition services to pregnant women and children.
• It is also used for availing benefits under schemes like PMMVY, Janani Suraksha Yojana (JSY), etc.

The Correct answer is 2001.

Key Points

• Sarva Shiksha Abhiyan (SSA) was launched in the year 2001 by the Government of India.
• It is a flagship program aimed at achieving the goal of universalization of elementary education in a time-bound manner.
• The program focuses on providing free and compulsory education to children in the age group of 6-14 years as per the provisions of Article 21A of the Indian Constitution.
• The SSA initiative is built to bridge the gaps in education by targeting disadvantaged groups, including girls, scheduled castes, scheduled tribes, and children with disabilities.
• The program is financed by both the Central and State governments under a shared funding pattern.
• SSA also emphasizes improving the quality of education, teacher training, and infrastructure development in schools.
• Later, SSA was integrated into the Samagra Shiksha Abhiyan in 2018 to consolidate various education-related programs.
• The initiative complements the objectives of the Right to Education (RTE) Act, 2009, ensuring every child receives access to quality education.

Additional Information

• 2007
• This option does not align with the launch of Sarva Shiksha Abhiyan. However, it is worth noting that during 2007, there was an emphasis on enhancing the SSA framework to improve access to education for marginalized communities.
• 2014
• In this year, the Government of India launched several new initiatives related to education, but it is not the year of the SSA launch. For instance, the Beti Bachao Beti Padhao scheme was introduced to promote girl child education.
• 2020
• In 2020, the National Education Policy (NEP 2020) was introduced, which brought transformative changes to the education sector. However, it is unrelated to the launch of SSA.

The Correct answer is Habeas Corpus.

Key Points

• The writ of Habeas Corpus is issued to secure the liberty of an individual.
• It is a Latin term that means "you may have the body."
• This writ is used to release a person who has been unlawfully detained or imprisoned.
• Under this writ, the court directs the person or authority who has detained another person to bring the detainee before the court.
• The purpose of this writ is to ensure that no individual is deprived of their liberty without legal justification.
• The Article 32 and Article 226 of the Indian Constitution empower the Supreme Court and High Courts, respectively, to issue this writ.
• It acts as a safeguard against arbitrary arrests and ensures protection of fundamental rights provided under the Constitution of India.

Additional Information

• Quo warranto
• The writ of Quo Warranto is issued to prevent a person from holding a public office without legal authority.
• It ensures that only a legally qualified person occupies a public office.
• The court can question the eligibility of the officeholder under this writ.
• Mandamus
• The writ of Mandamus is issued by a court to direct a public official or governmental body to perform a duty that they are obligated to perform.
• It cannot be issued against private individuals or entities.
• This writ ensures the proper functioning of public authorities.
• Prohibition
• The writ of Prohibition is issued by a higher court to a lower court or tribunal to prevent it from exceeding its jurisdiction or acting contrary to the law.
• It is issued when a lower court proceeds with a case that it does not have the authority to hear.
• This writ is preventive in nature.

The Correct answer is Wheat.

Key Points

• Wheat is not considered a coarse cereal crop (Millet).
• It is a staple food grain and is classified as a cereal crop, but not as a millet.
• Wheat is primarily grown in the temperate regions, requiring a cool climate during the growing season and bright sunshine at the time of maturity.
• It is rich in carbohydrates and a good source of dietary fiber and proteins, making it a significant food crop globally.
• Wheat is cultivated in several countries, with major producers being India, China, the USA, and Russia.
• Unlike millets, wheat is not drought-resistant and requires more irrigation and fertile soil for optimal growth.

Additional Information

• Maize
• Maize is a cereal crop often categorized as a coarse cereal.
• It is used both for human consumption and as livestock feed.
• Maize is a versatile crop that is grown in tropical, subtropical, and temperate regions.
• It is a major source of starch and is processed into various products like corn oil and corn syrup.
• Jowar
• Jowar, also called sorghum, is a major millet grown in India.
• It is highly drought-resistant and thrives in semi-arid regions.
• Jowar is a rich source of protein, fiber, and essential minerals.
• It is widely consumed in India and Africa as part of daily diets.
• Bajra
• Bajra, also known as pearl millet, is another common coarse cereal crop.
• It is well-adapted to arid and semi-arid regions due to its high resistance to drought.
• Bajra is rich in energy, protein, fiber, and essential nutrients like iron and magnesium.
• It is a staple food crop in Rajasthan, Gujarat, and Maharashtra.

The correct answer is Vietnam.

Key Points

• Vietnam abolished its Two-Child Policy in June 2025, marking a significant change in its population control measures.
• The Two-Child Policy was initially implemented to address concerns related to overpopulation and its impact on economic resources.
• In recent years, Vietnam faced challenges due to declining birth rates, leading to a shrinking workforce and an aging population.
• The abolition of the Two-Child Policy is aimed at encouraging citizens to have more children and thereby balancing the population structure.
• Vietnam's government has also been working on policies to address issues related to childcare, education, healthcare, and financial support for families.
• This decision aligns with global trends where countries like China have relaxed similar policies to combat demographic challenges.
• Vietnam has been implementing progressive measures to improve its socio-economic stability while adapting to changing demographic needs.

Additional Information

• Japan
• Japan has one of the world’s lowest birth rates and has implemented policies to encourage families to have more children, but it does not have a strict Two-Child Policy.
• The country faces challenges related to its aging population and workforce shortages.
• Japan is known for its advanced healthcare system and efforts to improve family welfare.
• India
• India does not have a nationwide Two-Child Policy, though some states promote policies to control population growth.
• The government focuses on measures like family planning, education, and awareness campaigns to manage population issues.
• India has a high population density and faces challenges related to resource allocation and infrastructure development.
• Australia
• Australia does not have a Two-Child Policy and encourages population growth through immigration and support for families.
• The country has a low population density and focuses on maintaining a balanced demographic structure.
• Australia is known for its progressive welfare system and support for families.

The Correct answer is A-ii, B-iv, C-iii, D-i.

Key Points

• Article 51A: This article in the Indian Constitution enumerates the Fundamental Duties of citizens. It highlights responsibilities such as protecting sovereignty, promoting harmony, safeguarding public property, and striving for excellence.
• Article 14: It guarantees equality before law and equal protection of the laws to all persons within the territory of India. This is a cornerstone of the principle of non-discrimination.
• Article 17: This article focuses on the Abolition of Untouchability. It prohibits the practice of untouchability in any form and declares such acts as offenses punishable by law.
• Article 18: It deals with the Abolition of Titles, aiming to prevent the creation of titles that promote inequalities, except for military and academic distinctions.
• The match between the articles and their subjects ensures a clear understanding of their respective significance in the Indian Constitution.

Additional Information

• Article 51A
• Introduced by the 42nd Amendment Act of 1976.
• It emphasizes duties such as respecting national symbols, protecting the environment, and upholding the spirit of brotherhood.
• There are 11 Fundamental Duties listed under this article.
• Article 14
• Forms the basis of the Right to Equality.
• It applies to both citizens and non-citizens within Indian territory.
• It ensures that no person is above the law and mandates fairness in legal proceedings.
• Article 17
• Prohibits the practice of untouchability in any form.
• Ensures equality and dignity to individuals irrespective of their caste.
• Legal provisions to enforce this article are implemented through the Protection of Civil Rights Act, 1955.
• Article 18
• Abolishes titles like Raja, Maharaja, Sir, etc., except for military and academic distinctions.
• Aims to promote equality and discourage hereditary privileges.
• Ensures that titles do not interfere with democratic ideals.

The Correct answer is Football.

Key Points

• The Durand Cup is one of the oldest football tournaments in the world and the oldest in India.
• It was first held in the year 1888 and is named after Mortimer Durand, who was then the Foreign Secretary in British India.
• This tournament is a symbol of India’s football heritage and has been a platform for showcasing football talent.
• The Indian Army has been instrumental in organizing the Durand Cup for many years.
• It features participation from some of India’s top football teams, including those from the Indian Super League (ISL), I-League, and other domestic leagues.
• The tournament usually takes place annually and includes teams from both professional and amateur circuits.
• The Durand Cup is often seen as a testing ground for young and emerging football talent in India.
• The winner of the Durand Cup is awarded a set of three trophies: the President’s Cup, the Durand Cup, and the Shimla Trophy.

Additional Information

• Cricket
• Cricket is the most popular sport in India and is governed by the Board of Control for Cricket in India (BCCI).
• Famous tournaments associated with cricket in India include the Indian Premier League (IPL), Ranji Trophy, and Vijay Hazare Trophy.
• Cricket is played in different formats such as Test Matches, One-Day Internationals (ODIs), and T20s.
• Basketball
• Basketball is governed by the Basketball Federation of India (BFI).
• The sport is gaining popularity in India, especially in schools and colleges.
• India participates in international basketball tournaments, including the FIBA Asia Cup.
• Hockey
• Hockey is India’s national sport and has a rich history, with the Indian men’s hockey team winning multiple Olympic gold medals.
• Major tournaments include the Hockey India League (HIL) and the Sultan Azlan Shah Cup.
• The Indian women’s hockey team has also gained prominence in recent years, performing well in international events like the Olympics and World Cup.

The Correct answer is Panch Mahal.

Key Points

• Panch Mahal is a five-storied architectural marvel built by Emperor Akbar in Fatehpur Sikri.
• It was constructed in the 16th century and served as a place for the emperor to relax and enjoy the cool breeze.
• The structure is also referred to as the "Badgir" or "wind catcher" tower due to its design that allows natural ventilation.
• The building is built using red sandstone, a signature material in Mughal architecture.
• The five stories of the Panch Mahal are in diminishing size as they ascend, giving it a pyramidal structure.
• Each story is supported by pillars, with the ground floor having the maximum number of pillars (84).
• The structure was primarily used as a pleasure palace and as a place for informal gatherings.
• It is a reflection of the blend of Persian, Islamic, and Indian architectural styles that flourished during Akbar's reign.

Additional Information

• Moti Mahal
• Moti Mahal translates to the "Pearl Palace" and is a historical structure found in various locations in India, such as Rajasthan and Madhya Pradesh.
• It is not located in Fatehpur Sikri nor was it built by Emperor Akbar.
• Heera Mahal
• The term "Heera Mahal" means "Diamond Palace," but it is not associated with Akbar's architectural works or Fatehpur Sikri.
• Rang Mahal
• Rang Mahal, meaning "Painted Palace," is a common name for palaces found in several parts of India, especially in Rajasthan.
• It was not built by Akbar nor is it located in Fatehpur Sikri.

The correct answer is August 20, 2025.

Key Points

• The bill for the promotion and regulation of online gaming was introduced in the Lok Sabha on August 20, 2025.
• The Government disclosed that people lose approximately ₹20,000 crore every year due to online gaming activities, which highlights the significance of regulating this sector.
• The bill aims to establish specific rules and regulations to ensure fair practices in online gaming and prevent financial losses or exploitation.
• The introduction of this bill is a part of the Government's efforts to address the economic and social consequences of unregulated online gaming activities in India.
• This initiative aligns with the broader goal of promoting safe and responsible gaming practices while preventing illegal activities such as fraud and gambling addiction.

Additional Information

• August 30, 2025
• This date does not align with the timeline of the bill's introduction in the Lok Sabha.
• Thu Jul 17 2025 00:00:00 GMT+0530 (India Standard Time)
• This date refers to a day in July 2025, which does not correspond to the actual date of the bill's introduction.
• Tue Sep 02 2025 00:00:00 GMT+0530 (India Standard Time)
• This date is in September 2025, whereas the correct date of introduction was in August 2025.

The Correct answer is Option 4.

Key Points

• White Revolution is associated with Milk production. This revolution was spearheaded by the visionary leader Dr. Verghese Kurien, also known as the "Father of the White Revolution" in India.
• Blue Revolution refers to the growth and development of Fish production. It aimed to increase the production of aquatic organisms and ensure sustainable fisheries.
• Yellow Revolution is linked to the production of Oil seeds. This revolution aimed to make India self-reliant in edible oils production.
• Green Revolution is connected to the increased production of Food grains, particularly wheat and rice, through the use of high-yield variety seeds, irrigation, fertilizers, and pesticides. It was initiated in the 1960s under the leadership of M.S. Swaminathan.
• The correct matching is:
• A - iii (White - Milk)
• B - iv (Blue - Fish)
• C - i (Yellow - Oil seeds)
• D - ii (Green - Food grains)

Additional Information

• White Revolution
• The White Revolution in India was launched in 1970 by the National Dairy Development Board (NDDB).
• It significantly increased milk production, making India the largest producer of milk and dairy products globally.
• The program associated with this revolution was called Operation Flood.
• Blue Revolution
• The Blue Revolution was aimed at increasing aquaculture and fish production.
• It includes both marine and inland fisheries.
• It promotes new technologies, sustainable fishing practices, and infrastructure development.
• Yellow Revolution
• The Yellow Revolution primarily focused on increasing the production of edible oils like mustard, sunflower, and groundnut oil.
• The revolution gained momentum in the 1980s.
• Green Revolution
• The Green Revolution was initiated in the 1960s to achieve self-sufficiency in food grain production.
• It introduced high-yield variety (HYV) seeds, improved irrigation facilities, and modern agricultural techniques.
• The revolution initially focused on the states of Punjab, Haryana, and Uttar Pradesh.

The Correct answer is Parthenium.

Key Points

• Parthenium is commonly known as "Gajar Ghas" or "Congress Grass" in India.
• It is an invasive weed species introduced in India accidentally in the 1950s through imported wheat from the United States.
• This weed is known for causing severe health issues such as skin allergies, respiratory problems, and asthma in humans.
• It also has a detrimental impact on agriculture as it reduces soil fertility, affects crop yields, and competes with native vegetation.
• Parthenium is classified as a noxious weed due to its rapid spread and adverse effects on ecosystems.
• It is primarily found in open lands, roadsides, and agricultural fields in India.
• The weed produces chemicals such as parthenin, which suppresses the growth of other plants nearby—a phenomenon known as allelopathy.
• Efforts to control Parthenium include mechanical removal, biological control using insects like the Mexican beetle (Zygogramma bicolorata), and chemical herbicides.

Additional Information

• Bathua
• Bathua is an edible leafy vegetable commonly consumed in India.
• It is rich in vitamins, minerals, and antioxidants, making it highly nutritious.
• Bathua is often used in preparing dishes like parathas, soups, and curries.
• Unlike Parthenium, Bathua is not harmful but rather beneficial for human consumption.
• Gokharoo (Xanthium)
• Gokharoo, also known as Xanthium, is a medicinal plant used in Ayurveda.
• It is often used for treating urinary tract infections, kidney stones, and skin diseases.
• Gokharoo produces spiny fruits that can stick to animal fur or clothing for seed dispersal.
• Junglee Jowar
• Junglee Jowar refers to wild sorghum, a type of grass.
• It grows naturally in grasslands and agricultural areas and is known for its drought tolerance.
• Unlike Parthenium, Junglee Jowar is not invasive or harmful; it has limited agricultural significance.

The Correct answer is Karpoori Thakur.

Key Points

• Karpoori Thakur was a prominent Indian politician and a social reformer from Bihar who forcefully raised the issue of reservation for Backward Castes in the state for the first time.
• He was the Chief Minister of Bihar during two different terms: 1970-71 and 1977-79.
• Thakur is known for implementing the reservation policy for the Backward Classes in Bihar, which was a landmark decision in social justice.
• He introduced a quota system for the Backward Classes in government jobs and educational institutions, ensuring their representation and opportunities.
• His efforts were aimed at addressing the socio-economic inequalities faced by marginalized communities.
• Karpoori Thakur was also a staunch supporter of socialism and worked extensively for the upliftment of the underprivileged sections of society.
• He is remembered as a leader who advocated for the rights of the oppressed and worked tirelessly to bring about social and political reforms.

Additional Information

• Sharad Yadav
• Sharad Yadav was a senior Indian politician and member of the Lok Sabha.
• He was associated with issues concerning social justice, but he is not credited with raising the reservation issue in Bihar for the first time.
• Lalu Prasad Yadav
• Lalu Prasad Yadav, a former Chief Minister of Bihar, is well-known for his support of social justice and empowerment of Backward Castes.
• However, he addressed the reservation issue after Karpoori Thakur's initial efforts.
• Mulayam Singh Yadav
• Mulayam Singh Yadav was a prominent leader from Uttar Pradesh and former Chief Minister of the state.
• His work largely focused on social justice for Backward Castes in Uttar Pradesh rather than Bihar.

The correct answer is B.R. Ambedkar.

Key Points

• B.R. Ambedkar, the principal architect of the Indian Constitution, was skeptical of the idea of strengthening local government bodies, particularly during the formative years of India's independence.
• He believed that local bodies lacked the capacity, organization, and infrastructure to handle governance effectively.
• Ambedkar emphasized the need for a strong central government to ensure uniformity and equity in governance across the country.
• He was particularly concerned about the caste-based inequalities prevalent in rural areas, which could perpetuate discrimination if too much power was given to local institutions.
• Ambedkar advocated for the protection of marginalized groups, such as Scheduled Castes and Scheduled Tribes, which he believed could be best ensured through a centralized framework.
• He also opined that strengthening local governance prematurely could lead to poor implementation of policies and hinder socio-economic reforms.

Additional Information

• Vinoba Bhave
• Vinoba Bhave was a disciple of Mahatma Gandhi and is known for his role in the Bhoodan Movement, aimed at redistributing land to the landless.
• He supported decentralization and believed in empowering local communities through self-governance.
• Mahatma Gandhi
• Mahatma Gandhi was a staunch advocate of Panchayati Raj and believed in the importance of strengthening local self-government.
• He envisioned a decentralized political structure where villages would function as self-reliant units.
• M.A. Ayyangar
• M.A. Ayyangar was the second Speaker of the Lok Sabha in India and played an active role in parliamentary procedures.
• He was not directly associated with the debate on local governance but focused more on legislative and parliamentary governance.

The Correct answer is A-iii, B-ii, C-i, D-iv.

Key Points

• The First Buddhist Council was held at Rajgrih, shortly after the death of Gautama Buddha, around 483 BCE. It was presided over by Mahakassapa and aimed at compiling Buddha's teachings.
• The Second Buddhist Council was convened at Vaishali, approximately 100 years after Buddha's death. It focused on resolving disputes regarding monastic practices and the interpretation of the Vinaya.
• The Third Buddhist Council took place at Pataliputra under the patronage of Emperor Ashoka around 250 BCE. It aimed at purifying the Buddhist teachings and removing heretical views.
• The Fourth Buddhist Council was held in Kashmir during the reign of King Kanishka, a Kushan ruler, around the 1st century CE. This council focused on the division of Buddhism into two sects: Mahayana and Theravada.
• These councils were crucial in shaping the development of Buddhism and its spread across the Indian subcontinent and beyond.

Additional Information

• Rajgrih
• Rajgrih was the ancient capital of the Magadha kingdom and is located in present-day Bihar, India.
• It is known for its association with both Buddhism and Jainism.
• Rajgrih is also home to various ancient Buddhist sites such as the Griddhakuta Hill, where Buddha delivered many sermons.
• Vaishali
• Vaishali was an ancient city located in modern-day Bihar, India.
• It holds significance in both Buddhist and Jain traditions.
• Vaishali is considered the birthplace of Lord Mahavira, the 24th Tirthankara of Jainism.
• Pataliputra
• Pataliputra, present-day Patna, was a major city in ancient India and the capital of the Mauryan Empire under Emperor Ashoka.
• This city was a hub of administrative, cultural, and religious activities during ancient times.
• It played a pivotal role in the spread of Buddhism across India and beyond.
• Kashmir
• Kashmir served as a significant center for Buddhism during the reign of King Kanishka.
• It is noted for its role in the establishment and growth of the Mahayana sect of Buddhism.
• The Fourth Buddhist Council held here led to significant developments in Buddhist philosophy and literature.

The correct answer is Fat tissue.

Key Points

• The hump of a camel is primarily made up of fat tissue, which serves as an energy reserve.
• It helps camels survive in desert climates where food and water are scarce.
• When food is not available, the fat in the hump is metabolized for energy and water.
• The hump may shrink and droop if the camel uses up the stored fat during extended periods of food scarcity.
• Contrary to popular belief, the hump does not store water but contributes indirectly to a camel's survival in arid conditions.
• The ability of camels to metabolize fat efficiently from their hump is an adaptation to their desert environment.
• The hump also helps in thermoregulation by reducing heat exposure to the rest of the body.

Additional Information

• Cartilage
• Cartilage is a flexible connective tissue found in areas such as joints, the ear, and the nose.
• It provides structural support and reduces friction in joints but is not present in the camel's hump.
• Lymph
• Lymph is a colorless fluid that circulates in the lymphatic system, playing a role in immune responses.
• It is not related to the structure of a camel’s hump.
• Collagen fibers
• Collagen fibers are a major component of connective tissues and are responsible for providing strength and structure.
• They are found in skin, tendons, and ligaments but not in the camel’s hump.

The Correct answer is Walter G. Rosen.

Key Points

• The term "Biodiversity" was first introduced by Walter G. Rosen in 1985.
• He used this term during the planning for the "National Forum on BioDiversity" held in Washington D.C., USA.
• The term "Biodiversity" is a short form of Biological Diversity, which refers to the variety of life forms on Earth, including plants, animals, microorganisms, and their ecosystems.
• Rosen's contribution helped in emphasizing the importance of conserving the Earth’s vast variety of life forms and their interactions.
• Later, the term gained global recognition and became widely used in the fields of ecology, environmental science, and conservation biology.
• Today, biodiversity is a critical concept in tackling issues like climate change, habitat destruction, species extinction, and ecosystem degradation.

Additional Information

• R.H. Whittaker
• R.H. Whittaker was a prominent ecologist who proposed the Five Kingdom Classification in 1969.
• The classification includes the kingdoms Monera, Protista, Fungi, Plantae, and Animalia.
• He is known for his contributions to the study of plant ecology and the concept of biomes.
• R.D. Barnes
• R.D. Barnes was a zoologist and is widely recognized for his work on invertebrate zoology.
• He authored the book "Invertebrate Zoology", which is a comprehensive study of invertebrates.
• N. Myers
• N. Myers is an environmentalist known for introducing the concept of "biodiversity hotspots" in 1988.
• Biodiversity hotspots are regions with significant levels of species richness and are under threat due to human activities.
• His work has been instrumental in focusing conservation efforts on ecologically critical areas.

Given:

Number of sides of hexagon (n) = 6

Formula used:

Number of diagonals = n(n − 3) ÷ 2

Calculations:

⇒ Number of diagonals = 6(6 − 3) ÷ 2

⇒ = 6 × 3 ÷ 2

⇒ = 18 ÷ 2

⇒ = 9

∴ The number of diagonals in a hexagon is 9.

Given:

20% of a number = 50

Formula used:

Percentage = (Part ÷ Whole) × 100

Calculations:

⇒ 20% = 50

⇒ 20 ÷ 100 × Number = 50

⇒ Number = 50 × 100 ÷ 20

⇒ Number = 250

∴ The required number is 250.

The Correct answer is 8th August, 1942.

Key Points

• The famous Quit India resolution was passed on 8th August, 1942 during a meeting of the All India Congress Committee (AICC) held in Bombay.
• This resolution marked the beginning of the Quit India Movement, a major campaign led by the Indian National Congress against British rule in India.
• The movement was initiated during World War II, when Indians demanded an end to British colonial rule.
• The resolution was introduced by Mahatma Gandhi, who called for a mass civil disobedience campaign that included non-cooperation and peaceful protests.
• The primary slogan of the movement was “Do or Die”, urging Indians to fight for independence with determination.
• Key leaders of the Indian National Congress, including Jawaharlal Nehru, Sardar Patel, and Maulana Azad, supported the resolution.
• As a result of this movement, the British government arrested most of the Congress leaders, including Mahatma Gandhi, leading to widespread protests across India.
• The Quit India Movement is considered one of the final steps toward India's independence, which was achieved on 15th August, 1947.

Additional Information

• Key Characteristics of the Quit India Movement
• The movement involved participation from various sections of Indian society, including farmers, workers, students, and women.
• The British government responded with severe repression, including arrests, violence, and censorship.
• Despite the challenges, the movement united Indians in their demand for independence and showcased the strength of nonviolent resistance.
• It inspired similar movements in other colonies seeking independence from imperial rule.
• Historical Context of 1942
• The Quit India Movement was launched during World War II, when the British were facing challenges both internationally and domestically.
• The movement highlighted the growing impatience among Indians for independence.
• It also exposed the weakening grip of the British Empire on its colonies.

Given:

LCM of two numbers = 2079

HCF of two numbers = 27

One number = 189

Formula used:

Product of two numbers = LCM × HCF

Calculations:

⇒ Product of two numbers = 2079 × 27

⇒ Product = 56133

⇒ Other number = Product ÷ Given number

⇒ Other number = 56133 ÷ 189

⇒ Other number = 297

∴ The other number is 297.

The Correct answer is Sodium.

Key Points

• Sodium is a highly reactive metal that reacts vigorously with water at normal temperature.
• The reaction of sodium with water produces sodium hydroxide (NaOH) and releases hydrogen gas, along with a significant amount of heat.
• Sodium reacts so intensely that it can cause fire or explosions when exposed to water.
• To prevent accidental reactions, sodium is usually stored in substances like kerosene oil or mineral oil, which keep it away from moisture.
• This reaction is an example of the property of alkali metals, which are known for their high reactivity with water.
• The vigorous reaction of sodium with water is often utilized in educational demonstrations to explain chemical reactivity.

Additional Information

• Mercury
• Mercury is a liquid metal at room temperature and does not react with water under normal conditions.
• It has a wide range of industrial applications, including in thermometers, barometers, and electrical switches.
• Mercury is toxic and requires careful handling due to its poisonous effects on humans and the environment.
• Aluminium
• Aluminium is a metal that does not react with water at room temperature due to the presence of a protective oxide layer on its surface.
• It is widely used in applications like construction, aerospace, packaging, and electrical components.
• Magnesium
• Magnesium reacts with water, but this reaction occurs slowly at room temperature and is much faster in hot water.
• The reaction produces magnesium hydroxide and releases hydrogen gas.
• Magnesium is commonly used in alloys, fireworks, and lightweight materials.

The Correct answer is 9 phases.

Key Points

• The 2024 Lok Sabha elections in India were conducted in 9 phases, as per the Election Commission of India.
• Conducting the elections in multiple phases ensures a smooth and fair election process in the world's largest democracy.
• Each phase covers different constituencies, allowing the Election Commission to efficiently manage resources like security forces and polling personnel.
• Multi-phase elections are crucial in India due to its vast geographical diversity and population size.
• The 9-phase election process allows for better monitoring, reducing the chances of malpractices and ensuring transparency.
• In previous elections, such as the 2019 Lok Sabha elections, a similar multi-phase strategy was adopted.
• Important factors like law and order, logistics, and political climate are considered while deciding the number of phases.
• The phased approach also ensures that voters in each region have adequate time to prepare and participate in the electoral process.

Additional Information

• Importance of Multi-Phase Elections
• Multi-phase elections help maintain law and order in sensitive regions by deploying security forces effectively.
• It provides enough time for electoral officials to prepare and ensures a more organized voting process.
• This approach also allows for better voter turnout as people have sufficient time to plan their participation.
• Single-Phase vs Multi-Phase Elections
• Single-phase elections are quicker but are usually preferred in smaller countries or regions with fewer logistical challenges.
• Multi-phase elections are ideal for large democracies like India, where organizing elections in a single day is not feasible.

The Correct answer is IV, II, I, III.

Key Points

• The chronological order of the Acts is as follows:
• Regulating Act (1773): This was the first attempt by the British Parliament to regulate the affairs of the East India Company in India. It introduced the post of the Governor-General of Bengal and established a Supreme Court at Calcutta.
• Pitt's India Act (1784): Named after William Pitt the Younger, this act distinguished between the commercial and political functions of the East India Company. It established a Board of Control to supervise civil, military, and revenue affairs.
• Charter Act (1813): This act marked the beginning of the era of British government control over the East India Company. It ended the monopoly of the company over trade, allowing other British merchants to trade with India, except for tea and trade with China.
• The Indian Councils Act (1861): This act introduced non-official Indian members in the legislative councils and laid the foundation for the parliamentary system in India.
• The correct chronological order is: Regulating Act (1773), Pitt's India Act (1784), Charter Act (1813), The Indian Councils Act (1861).

Additional Information

• Regulating Act (1773):
• The act established the Office of Governor-General of Bengal, with Warren Hastings being the first to hold the position.
• The Supreme Court of Judicature was set up in Calcutta, consisting of one chief justice and three judges.
• It was the first act to assert the right of the British Parliament to regulate the East India Company.
• Pitt's India Act (1784):
• This act created the Board of Control to oversee the political affairs of the East India Company.
• It allowed the British government to have direct control over Indian administration while leaving the company with administrative and commercial functions.
• Charter Act (1813):
• The act ended the trade monopoly of the East India Company in India, except for trade in tea and trade with China.
• It allowed Christian missionaries to enter India for the purpose of religious and social reforms.
• The Indian Councils Act (1861):
• This act marked the beginning of the legislative councils in India and introduced the concept of representative institutions.
• It provided for the inclusion of Indians in the legislative process, laying the foundation for later reforms.

The correct answer is Tamil Nadu.

Key Points

• Tamil Nadu receives floods during the winter months, primarily due to the Northeast Monsoon, which occurs between the months of October and December.
• The state is geographically located in such a way that it receives most of its rainfall during the retreating monsoon season, unlike other parts of India that primarily depend on the Southwest Monsoon.
• Major rivers such as the Cauvery, Vaigai, and Thamirabarani often overflow during heavy rains, contributing to flooding in several parts of the state.
• Several districts, including Chennai, Cuddalore, and Nagapattinam, are highly vulnerable to floods during this period due to their low-lying nature and proximity to the coast.
• Urban areas like Chennai often face severe waterlogging because of inadequate drainage infrastructure and high-intensity rainfall.
• Floods during this period significantly affect agriculture, infrastructure, and livelihoods in Tamil Nadu, making disaster preparedness crucial.
• The Tamil Nadu government implements strategies such as early warning systems and flood mitigation projects to minimize the impact of floods during winter.

Additional Information

• Gujarat
• Gujarat primarily receives rainfall during the Southwest Monsoon from June to September. It is not prone to winter floods.
• Occasionally, Gujarat experiences cyclonic storms from the Arabian Sea, which may lead to flooding, but this generally occurs outside the winter season.
• Madhya Pradesh
• Madhya Pradesh is a landlocked state and relies predominantly on the Southwest Monsoon for its rainfall.
• Winter floods are uncommon in Madhya Pradesh as the state does not experience significant rainfall during this period.
• Maharashtra
• Maharashtra also depends on the Southwest Monsoon for its primary rainfall.
• Floods in Maharashtra usually occur during the monsoon season, particularly in regions like Konkan, which includes Mumbai, but winter floods are not prevalent.

Calculation:

The sum of the coefficients of a polynomial P(x) is found by setting the variable

x= 1

Let $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

Then the sum of coefficients is $P(1) = a_n + a_{n-1} + \dots + a_1 + a_0$

Given the polynomial:

$P(x) = (1 + x - 3x^2)^{15}$

Substitute $x = 1$ into the expression:

$P(1) = (1 + 1 - 3(1)^2)^{15}$

$P(1) = (1 + 1 - 3)^{15}$

$P(1) = (2 - 3)^{15}$

$P(1) = (-1)^{15}$

Since the power is odd:

$P(1) = -1$

The sum of coefficients is -1

Calculation:

Given complex numbers z₁ and z₂ such that $|z_1| < 1$ and $|z_2| < 1$

We consider the expression $\left|\frac{z_1 - z_2}{1 - \overline{z_1}z_2}\right|$

For any two points inside the unit disk, the following inequality holds:

$\left|\frac{z_1 - z_2}{1 - \overline{z_1}z_2}\right| < 1$

This is a standard result from complex analysis (Schwarz–Pick inequality)

∴  The value of $\left|\frac{z_1 - z_2}{1 - \overline{z_1}z_2}\right|$ is strictly less than 1

Calculation:

We study the convergence of the following two series:

I. $\sum_{n=1}^{\infty} \cos\left(\frac{1}{n}\right)$

II. $\sum_{n=1}^{\infty} \frac{1}{n}\sin\left(\frac{1}{n}\right)$

Series I:

As $n \to \infty$, we have $\frac{1}{n} \to 0$ and hence $\cos\left(\frac{1}{n}\right) \to \cos 0 = 1$

Since the general term does not tend to zero, the series $\sum \cos\left(\frac{1}{n}\right)$ is divergent

Series II:

For small values of x, we know that $\sin x \approx x$

So for large n, $\sin\left(\frac{1}{n}\right) \approx \frac{1}{n}$

Hence, $\frac{1}{n}\sin\left(\frac{1}{n}\right) \approx \frac{1}{n^2}$

Since $\sum \frac{1}{n^2}$ is a convergent series, $\sum \frac{1}{n}\sin\left(\frac{1}{n}\right)$ also converges by Limit Comparison Test.

Series I diverges and Series II converges.

∴ The correct answer is II converges but I does not

Calculation:

Give equation of parabola  y2 = 4ax

⇒focus (a, 0)

So, the equation of the latus rectum is x = a

Since, the parabola is symmetric about the x-axis, the required area is twice the area bounded by the curve in the first quadrant from x = 0 to x = a

From the equation $y^2 = 4ax$, we get $y = 2\sqrt{a}\sqrt{x}$ for the upper half:

Required Area A = $2 \int_{0}^{a} y , dx$

$A = 2 \int_{0}^{a} 2\sqrt{a} x^{1/2} , dx$

$A = 4\sqrt{a} \int_{0}^{a} x^{1/2} , dx$

Using the power rule for integration:

$A = 4\sqrt{a} \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{a}$

$A = 4\sqrt{a} \times \frac{2}{3} \left[ a^{3/2} - 0 \right]$

$A = \frac{8}{3}\sqrt{a} \cdot a^{3/2}$

Since $\sqrt{a} = a^{1/2}$:

$A = \frac{8}{3} a^{(1/2 + 3/2)}$

$A = \frac{8}{3} a^2$

∴ The Correct Option is  $\frac{8}{3} a^2$

Concept:

There are two types of convergence of a series:

1. Absolute convergence: $\sum |u_n|$ is convergent.
2. Conditional convergence: $\sum u_n$ is convergent but $\sum |u_n|$ is divergent.

Calculation:

Statement I:

$\sum u_n$ is convergent $\Rightarrow \sum |u_n|$ is convergent.

This statement is false

Counter Example: $\sum \frac{(-1)^{n+1}}{n }$ is convergent, but $\sum |\frac{(-1)^{n+1}}{n} | = \sum \frac{1}{n}$ is divergent

So, convergence of $\sum u_n$ does not guarantee convergence of $\sum |u_n|$

Statement II:

$\sum |u_n|$ is convergent $\Rightarrow \sum u_n$ is convergent.

This statement is true

If a series converges absolutely, then it always converges

∴ The correct option is Only II

Concept:

For the sphere $x^2 + y^2 + z^2 = a^2$, the polar plane of a point $(x_1, y_1, z_1)$ is

$xx_1 + yy_1 + zz_1 = a^2$

If this polar plane coincides with the given plane

lx + my + nz = p

then comparing coefficients, we get

$x_1 = \frac{a^2 l}{p}, \quad y_1 = \frac{a^2 m}{p}, \quad z_1 = \frac{a^2 n}{p}$

Hence**,** the pole of the plane  lx +my + nz = p with respect to the sphere $x^2 + y^2 + z^2 = a^2$ is

$\rm \left(\frac{l}{p} a^{2}, \frac{m}{p} a^{2}, \frac{n}{p} a^{2}\right)$

∴ The correct option is $\rm \left(\frac{l}{p} a^{2}, \frac{m}{p} a^{2}, \frac{n}{p} a^{2}\right)$

Calculation:

Consider the sequence $S_n = \left(1+\frac{1}{n}\right)^n$

It is a well-known standard result that

$\displaystyle \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n = e$

The number e is known as Euler’s number

It is an established fact that e is an irrational number

Hence, the given sequence converges and its limit is an irrational number

∴ The Correct answer is  Converges to an irrational number

Calculation:

The equation of the family of parabolas having vertices at the origin and axes along the positive direction of the x-axis is given by:

$y^2 = 4ax$ ... (1)

where a is an arbitrary parameter.

To form the differential equation, differentiate (1)

$2y \frac{dy}{dx} = 4a$ ... (2)

Now, we substitute the value of 4a from equation (2) back into equation (1) to eliminate the arbitrary constant:

$y^2 = \left( 2y \frac{dy}{dx} \right) x$

$\Rightarrow y^2 = 2xy \frac{dy}{dx}$

Rearranging the terms, we get:

$y^2 - 2xy \frac{dy}{dx} = 0$

∴   The Correct Answer is $\rm y^{2}-2 x y \frac{d y}{d x}=0$

Calculation:

A first order linear differential equation in x(y) of the form $\dfrac{dx}{dy} + P(y)x = Q(y)$ has integrating factor $\mu(y) = e^{\int P(y),dy}$

We have  $(1-y^2)\dfrac{dx}{dy} + yx = ay$, with $|y|<1$

Divide throughout by 1-y² we get:

$\dfrac{dx}{dy} + \dfrac{y}{1-y^2}x = \dfrac{ay}{1-y^2}$

So $P(y) = \dfrac{y}{1-y^2}$

⇒ $\mu(y) = e^{\int \frac{y}{1-y^2},dy}$

Evaluate the integral using $u = 1-y^2 \Rightarrow du = -2y,dy$:

$\displaystyle \int \frac{y}{1-y^2},dy = -\frac12 \int \frac{du}{u} = -\frac12 \ln|u| = -\frac12 \ln(1-y^2)$

(since $|y|<1 \Rightarrow 1-y^2>0$)

∴  $\mu(y) = e^{-\frac12 \ln(1-y^2)} = (1-y^2)^{-1/2} = \dfrac{1}{\sqrt{1-y^2}}$

∴ The correct option is $\rm \frac{1}{\sqrt{1-y^{2}}}$

Calculation:

We are given the differential equation

(1 + x) y dx + (1 − y) x dy = 0

Divide by x y:

((1 + x)/x) dx + ((1 − y)/y) dy = 0

⇒ (1/x + 1) dx + (1/y − 1) dy = 0

Integrate both terms:

∫(1/x + 1) dx + ∫(1/y − 1) dy = 0

⇒ ln|x| + x + ln|y| − y = C

⇒ ln|x| + ln|y| + x − y = C

⇒ ln|xy| + x − y = C

⇒ ln|xy| = C − x + y

Taking exponential on both sides:

xy = C e^{y − x}

∴ The Correct Answer is : xy = Ce− x+y

Calculation:

An even function satisfies $f(x) = f(-x)$ for all x

Differentiate both sides with respect to x :

$f'(x) = -f'(-x)$

Now put $x = 0$:

$f'(0) = -f'(0)$

⇒ $2f'(0) = 0$

⇒ $f'(0) = 0$

∴  If $f(x)$ is even and $f'(0)$ exists, then $f'(0) = 0$

∴ The correct answer is 0

Calculation:

Statement I: General value of log √i

First write i in exponential form:

i = e^{ i(π/2 + 2πk) }

Taking square root:

√i = e^{ i(π/4 + πk) }

Now take logarithm:

log √i = ln|√i| + i(π/4 + πk + 2πn)

Since |√i| = 1, ln|√i| = 0

So, log √i = i(π/4 + πm) = πi(4m + 1)/4

This does not match the given form (1/2)(8n + 1)πi.

Hence, Statement I is incorrect.

Statement II: Value of cos(log i^-i)

Using principal values, i = e^iπ/2

i^-i = (e^iπ/2)^-i = e^-i2π/2 = e^π/2

Now, log i^-i = log(e^π/2) = π/2

cos(log i^-i) = cos(π/2) = 0

Hence, Statement II is correct.

So, Only Statement II is correct.

∴ The correct answer is: Only II

Concept Used:

In a group, n⁻¹ satisfies  n ☆ n⁻¹ = e

where e is identity element

Calculation:

Step 1: Identity element

Let identity = e

⇒ n ☆ e = n

⇒ n + e + ne = n

⇒ e(1 + n) = 0

⇒ e = 0

∴ Identity element = 0

Step 2: Inverse of n

Let inverse of n = n⁻¹

⇒ n ☆ n⁻¹ = 0

⇒ n + n⁻¹ + nn⁻¹ = 0

⇒ n⁻¹(1 + n) = −n

⇒ n⁻¹ = $\rm -\frac{n}{n+1}$

∴ The correct answer is $\rm -\frac{n}{n+1}$

Calculation:

|a| = |b| = |c| = 1 , |a + b|2 = 4 ,|b + c|2 = 4 ,|c + a|2 = 4

|a + b|² = 1 + 1 + 2a·b

⇒ 4 = 2 + 2a·b ⇒ a·b = 1 …… (1)

|b + c|² = 1 + 1 + 2b·c

⇒ 4 = 2 + 2b·c ⇒ b·c = 1 …… (2)

|c + a|² = 1 + 1 + 2c·a

⇒ 4 = 2 + 2c·a ⇒ c·a = 1 …… (3)

|2a + 3b + 4c|² = 4|a|² + 9|b|² + 16|c|² +12a·b + 16a·c + 24b·c

= 4 + 9 + 16 + 12(1) + 16(1) + 24(1) = 29 + 52 = 81

⇒ |2a + 3b + 4c| = √81

∴ |2a** + 3b + 4c| = 9**

∴ The Correct Answer is 9

Given:

(1 + x²) dy/dx + 2x y = cos x

Formula used:

Linear differential equation:

dy/dx + P(x) y = Q(x)

Integrating factor = e^{∫P(x) dx}

Calculation:

⇒ Divide whole equation by (1 + x²)

⇒ dy/dx + (2x / (1 + x²)) y = cos x / (1 + x²)

⇒ P(x) = 2x / (1 + x²)

⇒ IF = e^{∫ 2x / (1 + x2) dx}

⇒ IF = e^{log(1 + x2)}

⇒ IF = 1 + x²

∴ The correct answer is 1 + x².

Formula Used:

Parametric point: (a sec θ, b tan θ)

Derivative slope: $\frac{dy}{dx}$

Normal slope = −1 / (slope of tangent)

Calculation:

Given Equation of  Hyperbola :$\rm \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}$ = 1

The Parametric coordinates: x = a sec θ,   y = b tan θ

Slope of tangent: $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$

 $\Rightarrow \frac{dy}{dx} = \frac{b\sec^2\theta}{a\sec\theta\tan\theta} = \frac{b\sec\theta}{a\tan\theta}$

Slope of normal = − reciprocal of tangent slope

⇒ $m_{normal} = -\frac{a\tan\theta}{b\sec\theta}$

Point on hyperbola = (a sec θ, b tan θ)

Therefore the equation of normal becomes 

$y - b\tan\theta = -\frac{a\tan\theta}{b\sec\theta} (x - a\sec\theta)$

Multiply both sides by b sec θ:

⇒ b sec θ y − b² sec θ tan θ = − a tan θ x + a² sec θ tan θ

⇒ b y(sec θ ) +  a x(tan θ) x =  a2 sec θ tan θ + b2 sec θ tan θ

Rearranging terms:$\rm \frac{a x}{\sec \theta}+\frac{\text { by }}{\tan \theta}=a^{2}+b^{2}$ 

∴ The correct answer is $\rm \frac{a x}{\sec \theta}+\frac{\text { by }}{\tan \theta}=a^{2}+b^{2}$

Concept Used:

Scalar triple product $\rm [\vec{A}, \vec{B}, \vec{C}]$= determinant of matrix 

Calculation:

The given vectors are  

$\vec{A}$ = î − k̂ = (1, 0, −1) 

$\vec{B}$ = xî + ĵ + (1 − x)k̂ = (x, 1, 1 − x)

$\vec{C}$ = yî + xĵ + (1 + x − y)k̂ = (y, x, 1 + x − y)

⇒$\rm [\vec{A}, \vec{B}, \vec{C}]$  =$\begin{vmatrix} 1 & 0 & -1 \ x & 1 & 1 - x \ y & x & 1 + x - y \end{vmatrix}$

 = (1 + x − y) − (x − x2) − (x2 − y)

 = 1 + x − y − x+ x2 − x2 + y = 1

No x, y present in final value.

∴ The Correct Answer isoption (1)Neither on x nor on y

Concept Used:

Shift of origin transformation

If new origin is (h, k), then:

x = X + h,   y = Y + k

Calculation:

Given Equation of line: $\rm \frac{x}{a}+\frac{y}{b}$ = 2

Here, h = a, k = b

⇒ x = X + a

⇒ y = Y + b

Substitute into given equation:

⇒ $\frac{(X + a)}{a }+ \frac{(Y + b)}{b }= 2$

⇒ $\frac{X }{a } +1+ \frac{Y }{b } +1= 2$

⇒$\frac{X }{a } + \frac{Y }{b } = 0$

Hence, Final Transformed Equation: $\frac{X }{a } + \frac{Y }{b } = 0$

∴ The Correct answer is $\rm \frac{x}{a}+\frac{y}{b} =0$

Concept Used:

Probability = (Favourable outcomes) / (Total outcomes)

Both balls same colour = both white OR both black

Formula Used:

nCr = n! / r!(n − r)!

Calculation:

Number of white balls = 4

Number of black balls = 3

Total balls = 7

Balls drawn = 2

⇒ Total ways to choose 2 balls = 7C2 = 21

⇒ Ways to choose 2 white balls = 4C2 = 6

⇒ Ways to choose 2 black balls = 3C2 = 3

⇒ Favourable outcomes = 6 + 3 = 9

⇒ Probability = 9 / 21

⇒ Probability = 3 / 7

Hence, The probability that both balls are of the same colour = 3/7

∴The Correct Answer is 3/7

Formula Used:

eccentricity e: = √(1 − b2/a2)

Ellipse standard form:

(x − h)²/a² + (y − k)²/b² = 1

Calculation:

Given Ellipse equation is 4x2 + y2 − 8x + 2y + 1 = 0

⇒ 4x² − 8x + y² + 2y + 1 = 0

⇒ 4(x² − 2x) + (y² + 2y) + 1 = 0

⇒ 4[(x − 1)² − 1] + [(y + 1)² − 1] + 1 = 0

⇒ 4(x − 1)² + (y + 1)² − 4 = 0

⇒ 4(x − 1)² + (y + 1)² = 4

⇒ (x − 1)²/1 + (y + 1)²/4 = 1

⇒ a² = 4,   b² = 1

Now, eccentricity $e = √(1 − \frac{b2}{a2}) = √(1 − \frac{1}{4}) = √\frac{3}{4} = √\frac{3 }{2}$

∴ The Correct Answer is $\frac{\sqrt{3}}{2}$

Formula Used :

For the differential Equation dy/dx + Py = Q

Integrating Factor (I.F.) = e∫P dx

Solution: y × I.F. = ∫(Q × I.F.) dx + C

Calculation:

Given equation is 

dy/dx + y = e^x

Here, P = 1 and Q = e^x.

The integrating factor, I(x):

I(x) = e^(∫1 dx) = e^x

 Multiply through by the integrating factor:

e^x * dy/dx + e^x * y = e^x * e^x

⇒(d/dx)(e^x * y) = e^2x

Integrate both sides:

⇒∫(d/dx)(e^x * y) dx = ∫e^2x dx

⇒e^x * y = ($\frac{1}{2}$)e^2x + C

⇒y = ($\frac{1}{2}$)e^x + Ce^-x

 Apply the initial condition y(0) = 1/2:

$\frac{1}{2}$ = ($\frac{1}{2}$)e⁰ + Ce⁰

$\frac{1}{2}$ = $\frac{1}{2}$ + C

C = 0

Thus, the solution is:y = $\frac{1}{2}$e^x

∴ The correct answer is y = $\frac{1}{2}$ex

Formula Used:

⇒ ∫ x^a dx = x^a+1 / (a + 1)

Calculation:

⇒ I =$\int^1_0 x(1 − x)n dx$

⇒ Substitute u = 1 − x

⇒ x = 1 − u, dx = −du

⇒ Limits: x = 0 → u = 1, x = 1 → u = 0

⇒ I =$\int^1_0$ (1 − u)uⁿ du

⇒ I =  $\int^1_0$  (uⁿ − uⁿ⁺¹) du

⇒$| \frac{u^{n+1}}{ n + 1} − \frac{u^{n+2} }{n + 2} \left.\right|_0^1$

⇒ I = $\frac{1}{n+1} - \frac{1}{n+2}$ 

⇒ I =$\frac{1}{ (n + 1)(n + 2)}$

∴ The Correct Answer is  $\frac{1}{ (n + 1)(n + 2)}$