UPPSC LT Grade Assistant Teacher (Mathematics) Official Paper (Held On: 29 Jul, 2018) (Previous Year Paper)

The correct answer is 1921.

Key Points

• The year 1921 is referred to as the 'Year of Great Divide' in India due to a significant demographic shift.
• It marked the first recorded decline in India's population growth rate, primarily due to high mortality caused by pandemics like the Spanish Flu and famines.
• Before 1921, India's population growth was largely stagnant or fluctuating due to lack of medical advancements and high death rates.
• Post-1921, with gradual improvements in healthcare, sanitation, and living conditions, India's population began to steadily grow.
• The 'Great Divide' signifies the transition from an era of fluctuating population growth to one of consistent and rapid growth.

Additional Information

• Spanish Flu Pandemic (1918-1920):
• It was one of the deadliest pandemics in history, claiming millions of lives globally, including a significant impact on India.
• India was one of the worst-hit countries, with an estimated death toll of over 12-18 million.
• Pre-1921 Mortality Factors:
• Lack of medical facilities, frequent famines, and high infant mortality rates contributed to stagnant population growth.
• Diseases such as plague, cholera, and malaria were widespread and uncontrolled.
• Post-1921 Demographic Trends:
• Gradual improvements in healthcare, vaccination programs, and sanitation led to a decline in death rates.
• India's population began to grow exponentially due to a combination of high birth rates and reduced mortality.
• Census in India:
• The first Census of India was conducted in 1872 under British rule, but it was incomplete.
• Starting from 1881, censuses were conducted every 10 years systematically.

The correct answer is paddy.

Key Points

• The System of Rice Intensification (SRI) is a method of cultivating paddy (rice) aimed at increasing yield and resource efficiency.
• SRI involves planting younger seedlings (8-12 days old) with wider spacing to allow better root and canopy growth.
• This method emphasizes reduced water usage by keeping the soil moist but not continuously flooded, unlike traditional paddy cultivation.
• SRI promotes the use of organic fertilizers and discourages excessive chemical inputs, making it eco-friendly and sustainable.
• Farmers using SRI have reported significant increases in yield with reduced costs, especially in water-scarce regions.

Additional Information

• Key Principles of SRI:
• Planting single seedlings instead of clumps to reduce competition for nutrients.
• Wide spacing of plants in a grid pattern enhances sunlight exposure and airflow.
• Intermittent irrigation (wetting and drying) rather than continuous flooding reduces water wastage.
• Benefits of SRI:
• Increased yields (up to 30-50% in some cases) due to better root and plant health.
• Reduced water usage (by 25-50%) compared to traditional methods.
• Minimized input costs by encouraging organic inputs and reducing dependency on chemical fertilizers.
• Environmental Impact: SRI helps in reducing greenhouse gas emissions by limiting the decomposition of organic matter in flooded conditions, which releases methane.
• Global Adoption: SRI has been successfully adopted in many countries, including India, Indonesia, Vietnam, and Madagascar, benefiting smallholder farmers.
• Challenges: Adoption of SRI requires skill development and labor-intensive practices, which can be barriers for large-scale farming or regions with limited labor availability.

The correct answer is Groundnut : Pod borer.

Key Points

• The Pod borer (Helicoverpa armigera) is primarily associated with crops like pulses and cotton, not groundnut.
• Groundnut crops are more commonly affected by pests like red hairy caterpillar (Amsacta albistriga) and aphids, rather than pod borers.
• In contrast, pod borers are known pests of gram (chickpea) and other leguminous crops.
• The other pairs listed in the options, such as gram with pod borer, paddy with banka, and maize with stem borer, are correctly matched based on pest-crop associations.
• Thus, the pair Groundnut : Pod borer is incorrectly matched in the given options.

Additional Information

• Pod borer (Helicoverpa armigera):
• A major pest affecting leguminous crops like chickpea, pigeon pea, and cotton.
• It damages the pods and seeds, leading to significant yield loss.
• Stem borer:
• A common pest for cereals like maize, rice, and sorghum.
• It bores into the stem, affecting nutrient and water transport in the plant.
• Banka (Rice gall midge):
• A pest associated with paddy (rice), causing the formation of silver shoots or gall-like structures.
• It stunts plant growth and reduces grain yield.
• Groundnut pests:
• Common pests include aphids, red hairy caterpillars, and thrips.
• Effective pest management includes cultural practices, crop rotation, and biological control methods.
• Integrated Pest Management (IPM):
• An approach combining biological, cultural, mechanical, and chemical methods to manage pests sustainably.
• IPM minimizes environmental impact and promotes long-term pest control.

The correct answer is 300%.

Key Points

• Crop rotation intensity refers to the number of times crops are planted and harvested on the same piece of land within a year.
• This specific rotation provides an effective balance of nutrients, improves soil health, and minimizes pest and disease buildup.
• The Maize-Potato-Mung bean rotation is commonly practiced in regions where conditions support all three crops in a single agricultural calendar year.
• Cropping intensity is calculated as the ratio of the total cropped area to the total cultivable area, expressed as a percentage. It indicates how intensively the land is being used for crop production over a year.
• The cropping intensity can be calculated by using the following formula:

$\text{Cropping Intensity} = \left( \frac{\text{Total Cropped Area}}{\text{Total Cultivable Area}} \right) \times 100$

In the case of Maize-Potato-Mung bean:

• Maize, Potato, and Mung bean are grown sequentially within a single year on the same field.
• This means the field is being used three times in a year.

So, for a given area of land, the total cropped area over the year would be:

$\text{Total Cropped Area} = (1 \text{ crop of Maize} + 1 \text{ crop of Potato} + 1 \text{ crop of Mung bean}) \times \text{Total Cultivable Area}$

$\text{Total Cropped Area} = 3 \times \text{Total Cultivable Area}$

Therefore, the cropping intensity would be

$\text{Cropping Intensity} = \left( \frac{3 \times \text{Total Cultivable Area}}{\text{Total Cultivable Area}} \right) \times 100$

Cropping Intensity = 300%

Additional Information

• Crop Rotation:
• It is the practice of growing different types of crops sequentially on the same field to improve soil fertility and health.
• It helps in breaking pest and disease cycles and optimizing the use of soil nutrients.
• Maize:
• A cereal crop known for its high carbohydrate content and suitability for various soil types.
• Usually sown in the summer (Kharif) season in India.
• Potato:
• A tuber crop rich in carbohydrates, typically grown in the winter (Rabi) season in India.
• It requires well-drained, sandy-loam soil with moderate temperatures.
• Mung Bean:
• A legume crop rich in protein, commonly grown in the summer or post-monsoon season.
• It improves soil fertility by fixing atmospheric nitrogen into the soil.
• Benefits of Crop Rotation:
• Enhances soil structure and organic matter content.
• Reduces reliance on chemical fertilizers and pesticides.
• Improves overall farm productivity and sustainability.

The correct answer is Groundnut : Kaushal.

Key Points

• Kaushal is not a recognized or widely known variety of groundnut. It is incorrectly matched in the given list.
• Recognized varieties of groundnut include TMV-2, JL-24, K-6, and TAG-24, which are commonly cultivated in India.
• The other crop-variety pairs mentioned, such as Mustard : Vardan, Linseed : Chamatkar, and Gram : Udai, are correctly matched and are widely cultivated varieties.
• Crop-variety matching is crucial for agricultural development as it ensures the selection of the right seeds for specific soil and climatic conditions.
• Misidentification of crop varieties can lead to incorrect agricultural practices and lower productivity.

Additional Information

• Groundnut:
• Groundnut, also known as peanut, is a major oilseed crop grown in tropical and subtropical regions.
• It is rich in protein and oil, making it an important part of the agricultural economy.
• India is one of the largest producers of groundnut globally, with major cultivation in Gujarat, Andhra Pradesh, and Tamil Nadu.
• Crop Varieties:
• Crop varieties are developed through selective breeding and genetic improvement to enhance yield, disease resistance, and adaptability.
• Popular mustard varieties include Pusa Bold, Vardan, and Rohini, which are known for their high oil content and resistance to pests.
• Linseed varieties like Chamatkar are valued for their high oil and fiber content.
• Importance of Correctly Matched Crop Varieties:
• Ensures optimal yield and quality by matching crops with suitable soil types and climatic conditions.
• Reduces risk of crop failure by using varieties resistant to local pests and diseases.
• Promotes sustainable agricultural practices and contributes to food security.
• Agricultural Research in India:
• Organizations like the Indian Council of Agricultural Research (ICAR) play a key role in developing and disseminating improved crop varieties.
• State Agricultural Universities and research institutes collaborate to ensure farmers have access to high-quality seeds.

The correct answer is Measles.

Key Points

• Measles is a viral disease caused by the Measles morbillivirus, which cannot be treated with antibiotics as they are effective only against bacterial infections.
• Antibiotics target bacteria by disrupting their cell walls, protein synthesis, or DNA replication, but viruses like measles lack these structures, rendering antibiotics ineffective.
• The best preventive measure against measles is vaccination, particularly the Measles-Mumps-Rubella (MMR) vaccine, which provides long-term immunity.
• Management of measles focuses on supportive care, including hydration, antipyretics, and vitamin A supplementation to prevent complications like blindness and pneumonia.

Additional Information

• Antibiotics vs. Viruses: Antibiotics are ineffective against viruses because viruses replicate inside host cells and lack the cellular machinery targeted by antibiotics.
• Vaccination: Vaccines are the most effective way to prevent viral diseases. The MMR vaccine is highly effective against measles, with two doses providing around 97% immunity.
• Complications of Measles: Severe complications include pneumonia, encephalitis, and subacute sclerosing panencephalitis (SSPE), a rare but fatal brain disorder.
• Global Impact: Despite being preventable, measles remains a significant global health issue, causing over 140,000 deaths annually, primarily in unvaccinated children (WHO, 2022).
• Other Diseases in the List: Tuberculosis, tetanus, and cholera are bacterial infections and can be treated with appropriate antibiotics, unlike measles.

The correct answer is Radio : Karl Benz.

Key Points

• Charles Babbage is regarded as the "Father of the Computer" for conceptualizing the first mechanical computer known as the Analytical Engine.
• Radio was invented by Guglielmo Marconi, who successfully demonstrated wireless telegraphy and is often referred to as the "Father of Radio."
• E. Torricelli invented the barometer in 1643, making significant contributions to the field of atmospheric pressure measurement.
• Michael Faraday invented the dynamo, a device for converting mechanical energy into electrical energy, laying the foundation for modern electromagnetic technology.

Additional Information

• Charles Babbage: His Analytical Engine was the precursor to modern computers, integrating concepts such as an arithmetic logic unit and control flow.
• Guglielmo Marconi: He was awarded the Nobel Prize in Physics in 1909 for his pioneering work in wireless communication.
• Evangelista Torricelli: An Italian physicist and mathematician, Torricelli's invention of the barometer was a breakthrough in studying atmospheric pressure.
• Michael Faraday: Known for his discovery of electromagnetic induction, Faraday's work is fundamental to electrical engineering and physics.
• Karl Benz: He was the inventor of the first automobile powered by an internal combustion engine, but he is not associated with the invention of the radio.

The correct answer is geostationary.

Key Points

• Communication satellites are placed in a geostationary orbit, which means they remain fixed relative to a specific location on Earth's surface.
• A geostationary orbit is located at an altitude of approximately 35,786 kilometers (22,236 miles) above the equator.
• These satellites revolve around the Earth at the same rotational speed as the Earth’s rotation, achieving a stationary position relative to the ground.
• Geostationary satellites are essential for uninterrupted services such as telecommunications, television broadcasting, and weather forecasting.
• Examples of communication satellites in geostationary orbit include the INSAT series (India) and Intelsat series (global).

Additional Information

• Geostationary Orbit:
• It is a type of geosynchronous orbit directly above the Earth’s equator.
• Satellites in this orbit have an orbital period equal to the Earth’s rotational period (24 hours).
• Advantages of Geostationary Satellites:
• Continuous coverage of a specific area on Earth.
• Ideal for real-time communication services, such as satellite TV and internet.
• Other Satellite Orbits:
• Low Earth Orbit (LEO): Used for Earth observation and remote sensing (e.g., the International Space Station).
• Medium Earth Orbit (MEO): Primarily used for navigation systems like GPS.
• Geostationary Satellite Launch:
• These satellites are launched using powerful rockets to reach the required altitude.
• Once in orbit, their position is maintained using onboard thrusters.

The correct answer is Germanium.

Key Points

• Germanium is a semiconductor material, and its electrical conductivity increases with temperature due to the generation of more charge carriers (electrons and holes).
• In semiconductors like Germanium, the energy gap between the valence and conduction bands allows more electrons to jump to the conduction band at higher temperatures, enhancing conductivity.
• Unlike metals, where conductivity decreases with temperature, semiconductors exhibit an inverse relationship because of their unique band structure.
• Germanium is widely used in electronic devices such as transistors and diodes due to its temperature-dependent conductivity.
• This property makes Germanium useful in temperature-sensitive applications, including infrared optics and thermistors.

Additional Information

• Semiconductors:
• These are materials with a conductivity level between that of conductors and insulators, such as Germanium and Silicon.
• Their conductivity is highly dependent on temperature and doping (addition of impurities).
• Temperature Effect on Metals vs Semiconductors:
• In metals, conductivity decreases with temperature due to increased lattice vibrations causing electron scattering.
• In semiconductors, conductivity increases with temperature as more electrons gain energy to cross the energy gap into the conduction band.
• Energy Bands:
• The valence band contains electrons involved in bonding, while the conduction band contains free electrons that contribute to conductivity.
• The energy gap between these bands determines the material's electrical properties.
• Applications of Germanium:
• Germanium is used in electronics, infrared devices, fiber optics, and detectors due to its excellent semiconductor properties.
• Its temperature-dependent conductivity also makes it suitable for thermistors and temperature sensors.
• Other Semiconductors:
• Silicon is another widely used semiconductor, but it has a higher bandgap compared to Germanium, making it more suitable for high-temperature applications.

Given:

Side of hexagon (a) = 2√3 cm

Formula used:

Area of hexagon = (3√3/2) × a²

Calculation:

Area = (3√3/2) × (2√3)²

⇒ Area = (3√3/2) × (4 × 3)

⇒ Area = (3√3/2) × 12

⇒ Area = 18√3 cm²

∴ The correct answer is option (4).

Given:

2x + (2/x) = 3

Formula Used:

To find the value of x³ + (1/x³) + 2, we use the following identity:

[x + (1/x)]³ = x³ + (1/x³) + 3[x + (1/x)]

Calculations:

2x + (2/x) = 3

x + (1/x) = 3/2

⇒ Use the identity:

[x + (1/x)]3 = x3 + (1/x3) + 3[x + (1/x)]

⇒ (3/2)3 = x3 + (1/x3) + 3 × (3/2)

⇒ 27/8 = x3 + (1/x3) + 9/2

⇒ x3 + (1/x3) = (27/8) - (9/2)

⇒ x3 + (1/x3) = -9/8

⇒ Add 2 to the result:

x3 + (1/x3) + 2 = (-9/8) + 2

⇒ x3 + (1/x3) + 2 = 7/8

∴ The correct answer is option (4).

Given:

Quadratic equation: 2x² + px + 4 = 0

One root (x₁) = 2

Formula used:

If the roots of a quadratic equation are x₁ and x₂, then:

Sum of roots = -b/a

Product of roots = c/a

Calculations:

Given that one root is x₁ = 2, let the other root be x₂.

Using the formula for product of roots:

⇒ x₁ × x₂ = c/a

⇒ 2 × x₂ = 4/2

⇒ x₂ = 2 / 2

⇒ x₂ = 1

∴ The other root is 1, and the correct answer is option (3).

The correct answer is Rajasthan.

Key Points

• The military exercise 'Vijay Prahar' was conducted by the Indian Army in May 2018 in Rajasthan, specifically in the Mahajan Field Firing Range.
• This exercise involved over 25,000 troops of the South Western Command, showcasing advanced combat strategies and readiness.
• 'Vijay Prahar' focused on testing integrated firepower and maneuverability in a joint operational scenario involving infantry, artillery, armored units, and air support.
• The exercise aimed to validate offensive strategies to ensure preparedness for future conflicts.
• It also emphasized the use of modern warfare techniques and technologies such as precision targeting and network-centric operations.

Additional Information

• South Western Command:
• One of the seven operational commands of the Indian Army, headquartered in Jaipur, Rajasthan.
• Responsible for securing India's western borders and conducting strategic military exercises.
• Military Exercises:
• India conducts various military exercises, both domestic and international, to enhance combat readiness and foster cooperation with other nations.
• Examples include 'Operation Gagan Strike,' 'Yudh Abhyas,' and 'Indra' among others.
• Precision Targeting:
• A modern warfare technique that uses advanced technologies like drones, satellite imagery, and precision-guided munitions.
• Helps in minimizing collateral damage and increasing operational efficiency.
• Network-Centric Operations:
• A military strategy that leverages information technology to link forces, enabling real-time data sharing and enhanced decision-making.
• It plays a crucial role in modern warfare for achieving faster and more coordinated responses.

The correct answer is Saina Nehwal.

Key Points

• Saina Nehwal won the Women Singles Title in Badminton at the Commonwealth Games 2018 held in Gold Coast, Australia.
• She defeated fellow Indian shuttler P. V. Sindhu in the finals with a score of 21-18, 23-21.
• It was a historic moment as it marked an all-Indian final in the Women’s Singles event of Badminton at the Commonwealth Games.
• This victory added to Saina Nehwal's illustrious career, making her one of the most celebrated badminton players in India.
• Saina had previously won the same title at the 2010 Commonwealth Games in New Delhi, making her a two-time Commonwealth Games gold medalist in this category.

Additional Information

• Commonwealth Games 2018:
• The 2018 Commonwealth Games were held in Gold Coast, Australia, from April 4 to April 15, 2018.
• India secured an impressive tally of 66 medals, including 26 gold, 20 silver, and 20 bronze, finishing 3rd in the overall medal rankings.
• Badminton contributed significantly to India’s medal count, with players like Saina Nehwal, P. V. Sindhu, and Kidambi Srikanth excelling.
• Badminton in Commonwealth Games:
• Badminton has been a part of the Commonwealth Games since 1966.
• India has consistently been one of the top-performing nations in badminton at the Commonwealth Games.
• Saina Nehwal's Career Highlights:
• She is the first Indian shuttler to win an Olympic medal (bronze at the London 2012 Olympics).
• She became the first Indian woman to achieve the World No. 1 ranking in badminton in 2015.
• Saina has won numerous BWF titles, including the prestigious All England Open runner-up position in 2015.
• Women's Singles Final (2018 CWG):
• The final match between Saina Nehwal and P. V. Sindhu was closely contested, showcasing the high level of talent in Indian badminton.
• Saina's experience and aggressive gameplay were key factors in her victory over Sindhu.

The correct answer is 138th

• India’s rank in the 2018 World Press Freedom Index 138th.

Key Points

• The Index is compiled and published by Reporters without borders.
• Reporters without borders is a Paris based non-profit organisation.
• India ranked 142nd in the 2020 World Press Freedom Index
• Around 180 countries were ranked in the index.
• The World Press Freedom Index 2018 was topped by Norway followed by Finland, Sweden, Netherlands and Denmark.

Additional Information

• According to the 2020 World Press Freedom Index, Pakistan ranked 145 dropping three places, Bangladesh ranked 151, North Korea was at 180th

The correct answer is Shatapatha Brahmana.

Key Points

• The term 'Mleccha' was used in ancient Indian texts to describe people who could not speak Sanskrit properly or adhered to non-Vedic practices.
• The Shatapatha Brahmana, a prose text associated with the Shukla Yajurveda, provides references to the term 'Mleccha' in the context of linguistic and cultural distinctions.
• The Shatapatha Brahmana categorizes those outside the Vedic cultural framework, including those with different linguistic practices, as 'Mlecchas.'
• This text highlights the importance of Sanskrit as the language of Vedic rituals and knowledge, often contrasting it with non-Sanskrit speakers.
• The term 'Mleccha' was not necessarily derogatory but was used to mark linguistic and cultural differences.

Additional Information

• Shatapatha Brahmana:
• One of the most significant Brahmana texts associated with the Shukla Yajurveda.
• It provides detailed explanations of Vedic rituals, ceremonies, and their symbolic meanings.
• The text is divided into several chapters (Adhyayas) and sections (Kandas).
• Mleccha:
• A term used in ancient India to denote people outside the Vedic cultural sphere.
• It was often associated with non-Sanskrit speakers or those who did not follow Vedic practices.
• Sanskrit Language:
• Considered the sacred language of the Vedas and Vedic rituals.
• Proficiency in Sanskrit was seen as a marker of adherence to Vedic traditions.
• Vedic Literature:
• Comprises four Vedas (Rigveda, Yajurveda, Samaveda, Atharvaveda), Brahmanas, Aranyakas, and Upanishads.
• Brahmanas like the Shatapatha Brahmana provide explanations and instructions for Vedic rituals.
• Cultural Significance:
• The distinction between 'Arya' (followers of Vedic culture) and 'Mleccha' reflects the cultural and linguistic diversity of ancient India.
• These categorizations were used to understand and organize the complex social fabric of the time.

The correct answer is A-3, B-1, C-2, D-4.

Key Points

• Chandragupta I was married to Kumara Devi, a princess of the Lichchhavi dynasty, which helped establish the Gupta Empire.
• Samudragupta was married to Dutta Devi, who is mentioned in inscriptions as his queen.
• Chandragupta II was married to Kuberanaga, a Naga princess, which strengthened alliances and expanded the empire.
• Kumaragupta I was married to Ananta Devi, as per historical records and inscriptions from the Gupta period.

Additional Information

• Gupta Empire: The Gupta dynasty is considered one of the golden ages of Indian history, marked by advancements in art, science, and culture.
• Samudragupta: Known as the "Napoleon of India," Samudragupta expanded the empire significantly and was an accomplished poet and musician.
• Chandragupta I's marriage: His marriage to Kumara Devi of the Lichchhavi dynasty played a crucial role in the Gupta Empire's rise to power.
• Gupta inscriptions: Many details about Gupta rulers and their spouses are derived from inscriptions, such as the Allahabad Pillar Inscription.
• Political alliances: Marriages in the Gupta era were often used to strengthen political alliances and consolidate power within and outside the empire.

The correct answer is 1 only.

Key Points

• The Arthashastra, authored by Kautilya (Chanakya), is considered the oldest known treatise on Indian State Policy, providing insights into governance, administration, and economics.
• It was composed during the Mauryan period, around the 3rd century BCE, and is regarded as a masterpiece in ancient political and economic thought.
• The book extensively describes the Mauryan empire's administrative structure, including the duties of the king, ministers, and other officials, as well as strategies for internal and external security.
• Statement 2 is incorrect as the Arthashastra does provide a detailed description of the Mauryan empire and its administration, which aligns closely with historical accounts of the period.
• Thus, only statement 1 is correct, making the correct answer 1 only.

Additional Information

• Kautilya (Chanakya): A key figure in Indian history who served as an advisor and prime minister to Chandragupta Maurya. His ideas laid the foundation for the Mauryan Empire's administration.
• Structure of Arthashastra: The book is divided into 15 sections (adhikaranas) and covers topics such as statecraft, military strategy, economic policy, and social welfare.
• Core Concepts: The Arthashastra emphasizes concepts like dharma (moral duty), danda (punishment or enforcement), and niti (policy or strategy) as the pillars of governance.
• Comparison with Other Texts: Unlike religious texts like the Vedas or epics like the Mahabharata, the Arthashastra is a secular and pragmatic guide to governance and administration.
• Rediscovery: The text was lost for centuries and was rediscovered in 1905 by R. Shamasastry, who translated and published it, bringing it back to prominence in modern times.

The correct answer is Ibn Batuta.

Key Points

• Ibn Batuta, a Moroccan traveler and scholar, visited India during the reign of Muhammad bin Tughlaq (14th century).
• He served as the Qadi (judge) in the court of Muhammad bin Tughlaq and documented his travels in his work, "Rihla" (The Journey).
• In his writings, Ibn Batuta described Delhi as one of the greatest and most magnificent cities of the world during that period.
• He marveled at the city’s architectural brilliance, bustling markets, and the grandeur of the Sultan’s court.
• Ibn Batuta's account remains a valuable source of information about the socio-economic and cultural life of medieval India.

Additional Information

• Rihla (The Journey): This is the travelogue of Ibn Batuta, documenting his travels across the Islamic world, including North Africa, the Middle East, India, and China.
• Delhi under Muhammad bin Tughlaq: The city was an important cultural and political hub during the Tughlaq Dynasty, known for its ambitious projects and administrative reforms.
• Medieval Travelers: Along with Ibn Batuta, other notable travelers to India include Marco Polo, Al-Biruni, and Niccolò de' Conti, who documented various aspects of Indian life.
• Qadi Role: A Qadi is a judge ruling in accordance with Islamic law (Sharia). Ibn Batuta’s appointment as Qadi signifies his deep understanding of Islamic jurisprudence.
• Architectural Brilliance of Delhi: During the medieval period, Delhi was adorned with iconic structures like the Qutub Minar, Alai Darwaza, and tombs, showcasing Indo-Islamic architecture.

The correct answer is Lord Ripon.

Key Points

• Lord Ripon is known as the Father of Local Self-Government in India due to his significant reforms in the local governance system.
• In 1882, Lord Ripon introduced the Local Self-Government Resolution, which laid the foundation for democratic decentralization in India.
• His reforms aimed at empowering local bodies like municipalities and district boards, making them responsible for public services such as education, health, and sanitation.
• He emphasized the importance of local participation in governance and entrusted local bodies with financial and administrative autonomy.
• The resolution is considered a landmark in Indian administrative history, as it promoted the idea of grassroots-level governance.

Additional Information

• Lord Ripon's Governance:
• He served as the Viceroy of India from 1880 to 1884 under Queen Victoria's reign.
• He is also known for repealing the Vernacular Press Act, which was seen as oppressive by Indians.
• Local Self-Government:
• It refers to the administration of local areas by locally elected representatives, ensuring decentralization of power.
• This concept is crucial for grassroots democracy and promoting citizen participation in governance.
• Historical Context:
• Before Lord Ripon, local governance was largely controlled by colonial officials, with minimal participation from Indians.
• Ripon's reforms were a step towards self-rule and inspired future movements for independence.
• Impact of Ripon's Reforms:
• The resolution laid the foundation for the current Panchayati Raj system in rural areas and municipal governance in urban areas.
• It also emphasized the need for financial independence of local bodies, which remains a key aspect of local governance today.

The correct answer is Option 2 (14 days).

Key Points

• The impeachment process of the President of India requires a prior notice of at least 14 days.
• The notice must be signed by one-fourth of the total members of the house initiating the motion.
• The impeachment motion is based on the grounds of violation of the Constitution.
• After the notice is given, the motion must be supported by a two-thirds majority of the total membership of both houses of Parliament.
• The process for impeachment is detailed under Article 61 of the Indian Constitution.

Additional Information

• Impeachment: The President of India can be removed from office through impeachment for the violation of the Constitution.
• Article 61: This article outlines the procedure for impeachment, which includes investigation and voting by Parliament.
• Grounds for Impeachment: Violation of the Constitution is the only explicit ground for impeachment mentioned in the Indian Constitution.
• Two-thirds Majority: The motion for impeachment must be passed by a two-thirds majority of the total membership of each house of Parliament.
• Role of Judiciary: The charges against the President are investigated by a committee formed for this purpose before the voting process in Parliament takes place.

The correct answer is The Chief Justice of the High Court of the State.

Key Points

• The Governor of a State in India takes the oath of office and secrecy administered by the Chief Justice of the High Court of the State.
• The oath is in accordance with the provisions of Article 159 of the Indian Constitution.
• The oath includes allegiance to the Constitution of India, upholding sovereignty and integrity, and performing duties faithfully.
• If the Chief Justice of the High Court is unavailable, the senior-most judge of the High Court administers the oath.
• The process ensures the Governor's constitutional role and accountability as the nominal head of the state.

Additional Information

• Article 159 of the Indian Constitution: It specifies the oath or affirmation taken by the Governor while entering office.
• Governor's Role: The Governor serves as the constitutional head of a state and exercises powers as per the Constitution.
• Appointment of Governor: The Governor is appointed by the President of India under Article 155.
• Tenure: The Governor holds office for a term of five years but can be removed earlier by the President or resign voluntarily.
• High Court Structure: Each state has a High Court, and its Chief Justice is the senior-most judicial authority in the state.

The correct answer is Part IX.

Key Points

• Part IX of the Indian Constitution is titled "The Panchayats" and provides for a three-tier system of Panchayati Raj for rural self-governance.
• It was added to the Constitution through the 73rd Constitutional Amendment Act, 1992, which came into effect on April 24, 1993.
• The three tiers include Gram Panchayat (village level), Panchayat Samiti (block level), and Zila Parishad (district level).
• The Panchayati Raj system is aimed at decentralization of power to ensure greater participation of people in governance at the grassroots level.
• Articles 243 to 243-O under Part IX deal with the structure, powers, and functioning of Panchayati Raj institutions (PRIs).

Additional Information

• 73rd Constitutional Amendment Act, 1992: This amendment provided a constitutional status to Panchayati Raj institutions and aimed at strengthening local self-governance in rural areas.
• Eleventh Schedule: This schedule of the Constitution contains 29 subjects that are to be devolved to Panchayati Raj institutions, such as agriculture, rural housing, health, and education.
• Gram Sabha: It is a body consisting of persons registered in the electoral rolls of a village within the area of a Gram Panchayat. It acts as the foundation of the Panchayati Raj system.
• State Election Commission: The 73rd Amendment mandates the establishment of a State Election Commission to conduct free and fair elections to Panchayati Raj institutions.
• Reservation of Seats: The Act provides for the reservation of seats for Scheduled Castes (SCs), Scheduled Tribes (STs), and women (not less than one-third of the total seats) in every Panchayat.

The correct answer is Chhattisgarh.

Key Points

• Chhattisgarh does not have any operational oil refinery as of now, while states like Gujarat, Kerala, and West Bengal have established refineries.
• Gujarat hosts major refineries like the Reliance Jamnagar Refinery, which is the largest refinery in the world.
• Kerala has the Kochi Refinery, a major petroleum refinery in southern India.
• West Bengal has the Haldia Refinery, which is a key facility under the Indian Oil Corporation.

Additional Information

• Oil Refinery: An industrial facility that processes crude oil into usable petroleum products like gasoline, diesel, jet fuel, and other petrochemicals.
• Major Oil Refining States in India: States like Gujarat, Maharashtra, Tamil Nadu, and Assam are home to significant refinery operations.
• Jamnagar Refinery: Located in Gujarat, it is the largest oil refinery complex in the world, operated by Reliance Industries.
• Strategic Importance: Oil refineries are critical for energy security and economic growth as they cater to domestic and industrial fuel needs.
• Indian Oil Corporation (IOC): A leading public sector company in India that operates many refineries across the country, including the Haldia Refinery in West Bengal.

The correct answer is Orange River.

Key Points

• The Orange River is located in southern Africa and does not flow in Australia.
• It is one of the longest rivers in Africa, originating in the Drakensberg Mountains of Lesotho and flowing westward into the Atlantic Ocean.
• Rivers such as the Hunter River, Flinders River, and Gilbert River are located in Australia, making them incorrect options.
• The Orange River is vital for water supply and irrigation in the region but is geographically limited to the African continent.
• This exclusion of the Orange River highlights its distinct location compared to the rivers mentioned in the question.

Additional Information

• Hunter River: Located in New South Wales, Australia, it is known for its significance in agriculture and coal mining.
• Flinders River: The longest river in Queensland, Australia, it flows into the Gulf of Carpentaria and is critical for regional ecosystems.
• Gilbert River: Also located in Queensland, Australia, it contributes to the Gulf Country’s water systems and supports cattle grazing industries.
• Orange River Basin: Covers regions in South Africa, Namibia, and Botswana, providing water resources for mining, agriculture, and energy production.
• River Classification: Rivers are classified based on geography, ecosystem roles, and their contribution to human activities such as irrigation, transportation, and energy generation.

The correct answer is Nagaland.

Key Points

• Nagaland was the only Indian state to record a decrease in its population according to the 2011 Census data.
• The population of Nagaland declined from 1,988,636 in 2001 to 1,978,502 in 2011, marking a negative growth rate of -0.58%.
• The decline was attributed to migration, discrepancies in earlier census data, and other socio-economic factors.
• Population changes in Nagaland raised concerns about the accuracy of enumeration processes in the region.
• This negative population growth is a unique case among Indian states during the 2011 Census.

Additional Information

• Census of India:
• The Census of India is conducted every 10 years and is the most comprehensive source of demographic data in the country.
• The 2011 Census was the 15th National Census since 1872 and covered all states and union territories.
• It provides data on population size, density, literacy rate, sex ratio, and other socio-economic indicators.
• Population Growth Rate:
• Population growth rate refers to the percentage change in the population over a specific time period.
• A negative growth rate indicates a decline in the population size during the period.
• Factors Affecting Population Decline:
• Migration: Movement of people to other regions for better opportunities can lead to population decline.
• Errors in census enumeration can also affect population data accuracy.
• Socio-political factors, such as conflicts, may influence demographic trends.
• Significance of Census Data:
• Census data is crucial for policy-making, development planning, and resource allocation.
• It helps identify trends in population dynamics and socio-economic changes over time.

The correct answer is War.

Key Points

• According to Thomas Malthus, war is one of the key "positive checks" that limits population growth by reducing the population through mortality.
• Malthus categorized population control measures into two types: preventive checks (e.g., moral restraint, delayed marriage) and positive checks (e.g., famine, disease, war).
• War was emphasized as a natural and inevitable mechanism to bring population levels back in alignment with the available resources.
• Malthus argued that without such "positive checks," human population growth would outpace food production, leading to widespread scarcity and suffering.
• His theory is based on the principle that population grows geometrically while food supply grows arithmetically, creating a gap that needs to be corrected by such checks.

Additional Information

• Thomas Malthus and his Theory:
• Thomas Robert Malthus was an 18th-century British economist and demographer.
• In his famous work, An Essay on the Principle of Population (1798), he proposed that population growth, if unchecked, would lead to resource depletion.
• He introduced the concepts of "preventive checks" and "positive checks" to manage population growth.
• Preventive Checks:
• These are measures to reduce birth rates, such as moral restraint, delayed marriage, and birth control.
• Malthus supported moral restraint as the most ethical preventive measure.
• Positive Checks:
• These are natural and man-made factors that increase death rates, including famine, disease, and war.
• Malthus believed these checks were necessary to prevent overpopulation and maintain the balance between population and resources.
• Criticism of Malthusian Theory:
• Modern economists argue that technological advancements in agriculture and resource management have disproved Malthus's dire predictions of widespread famine and resource scarcity.
• Critics also highlight that social, economic, and political factors greatly influence population dynamics beyond Malthus's simplistic model.

The correct answer is Ecosystem.

Key Points

• A biome is a large ecological area on the Earth's surface, classified based on its climate, flora, and fauna. Examples include desert, tundra, and grassland.
• An ecosystem, on the other hand, refers to a community of living organisms interacting with their physical environment, such as a pond or forest.
• Ecosystem is not a biome; it is a smaller functional unit within a biome where biotic (living) and abiotic (non-living) components interact.
• Biomes are broader classifications, encompassing multiple ecosystems. For example, the desert biome includes various ecosystems such as sand dunes, oases, and rocky deserts.

Additional Information

• Definition of Biome: A biome is characterized by specific climate conditions and distinct types of plants and animals adapted to it. Common examples include forests, grasslands, deserts, and tundras.
• Definition of Ecosystem: An ecosystem includes all living organisms and their interactions with the non-living environment within a specific location. It can vary in size, from a small pond to a large forest.
• Difference Between Biome and Ecosystem: Biomes are larger geographical areas with similar climatic conditions, while ecosystems are smaller units within biomes where organisms interact with their environment.
• Major Biomes: Earth’s major biomes include tropical rainforests, savannas, deserts, grasslands, tundras, and aquatic biomes like freshwater and marine ecosystems.
• Examples of Ecosystems: Examples include coral reefs, mangroves, wetlands, and urban ecosystems, each functioning uniquely within a biome.

The correct answer is Uttar Pradesh.

Key Points

• Dudhwa National Park is located in the Lakhimpur Kheri district of Uttar Pradesh, India.
• It was established in 1977 and later designated as a tiger reserve in 1987 under Project Tiger.
• The park spans an area of approximately 490 square kilometers and is part of the Terai ecosystem.
• Dudhwa is renowned for its population of Bengal tigers, swamp deer (Barasingha), and Indian rhinoceros.
• The park is also a haven for birdwatchers, with over 400 species of resident and migratory birds recorded.

Additional Information

• Terai Ecosystem: The park is part of the Terai belt, a marshy grassland ecosystem found in the foothills of the Himalayas, known for its rich biodiversity.
• Fauna: Key species include the Bengal tiger, Indian rhinoceros, Asian elephant, Barasingha, and Gangetic dolphin (in nearby rivers).
• Flora: The park consists of dense sal forests, grasslands, and wetlands, providing a diverse habitat for wildlife.
• Conservation Challenges: The park faces threats such as habitat destruction, human-wildlife conflict, and poaching, prompting ongoing conservation efforts.
• Tourism: Dudhwa National Park attracts eco-tourists and wildlife enthusiasts, with safaris and guided tours offered to explore its unique biodiversity.

The correct answer is 2.2.

Key Points

• The Total Fertility Rate (TFR) in India according to the Fourth Round of the National Family Health Survey (NFHS-4) was reported to be 2.2 children per woman.
• A TFR of 2.2 indicates that India is moving towards achieving replacement-level fertility (2.1 children per woman).
• The decline in TFR is attributed to increased awareness of family planning, better access to contraception, and socio-economic improvements.
• The TFR varies significantly across states, with some states like Kerala and Tamil Nadu having a TFR below 2, while others like Bihar and Uttar Pradesh report higher rates.
• The reduction in TFR is a critical factor in stabilizing population growth and achieving sustainable development goals (SDGs).

Additional Information

• Total Fertility Rate (TFR):
• TFR is the average number of children that would be born to a woman over her lifetime, based on current birth rates.
• A TFR of 2.1 is considered the replacement level, where the population remains stable without migration.
• National Family Health Survey (NFHS):
• NFHS is a large-scale, multi-round survey conducted in a representative sample of households across India.
• It provides essential data on health, nutrition, fertility, family planning, and mortality.
• Factors Influencing TFR Decline:
• Improved access to education, particularly for women.
• Increased availability and use of contraceptives.
• Rising urbanization and economic development.
• Government initiatives promoting family planning and awareness campaigns.
• Implications of Low TFR:
• Lower TFR leads to slower population growth, reducing the strain on resources and public services.
• However, prolonged low TFR can result in an aging population, leading to a higher dependency ratio in the future.

Calculation:

If $\theta \in \mathbb{R}$ then trigonometric functions with imaginary arguments relate to hyperbolic functions using the identities:

$\cos(i\theta) = \cosh(\theta)$

$\sin(i\theta) = i\sinh(\theta)$

$\tan(i\theta) = i\tanh(\theta)$

$\cot(i\theta) = -i\coth(\theta)$

Now, analyzing the options given:

1. $\cos(i\theta) = i\cosh(\theta)$ → Incorrect, correct form is $\cos(i\theta) = \cosh(\theta)$

2. $\sin(i\theta) = i\sinh(\theta)$ → Correct

3. $\tan(i\theta) = \tanh(\theta)$ → Incorrect, correct form is $\tan(i\theta) = i\tanh(\theta)$

4. $\cot(i\theta) = i\coth(\theta)$ → Incorrect, correct form is $\cot(i\theta) = -i\coth(\theta)$

Hence, the correct answer is Option 2.

Additional Information

Important Hyperbolic Identities:

$\cosh^2 x - \sinh^2 x = 1$

Even/Odd Properties of Hyperbolic Functions:

$\sinh(-x) = -\sinh(x) \quad \text{(odd function)}$

$\cosh(-x) = \cosh(x) \quad \text{(even function)}$

$\tanh(-x) = -\tanh(x) \quad \text{(odd function)}$

$\coth(-x) = -\coth(x) \quad \text{(odd function)}$

Inverse Hyperbolic Functions:

$\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})$

$\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1})$

$\tanh^{-1}x = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right)$

Calculation:

$\left| \frac{z - 3}{z + 3} \right| = 2$, where z = x + iy 

$\Rightarrow \frac{|z - 3|}{|z + 3|} = 2 \Rightarrow \frac{√{(x - 3)^2 + y^2}}{√{(x + 3)^2 + y^2}} = 2$

$\Rightarrow \frac{(x - 3)^2 + y^2}{(x + 3)^2 + y^2} = 4$

$\Rightarrow x^2 - 6x + 9 + y^2 = 4(x^2 + 6x + 9 + y^2)$

$\Rightarrow x^2 - 6x + 9 + y^2 = 4x^2 + 24x + 36 + 4y^2$

$\Rightarrow -3x^2 - 30x - 27 - 3y^2 = 0$

$\Rightarrow x^2 + 10x + y^2 + 9 = 0$

$\Rightarrow x^2 + 10x + 25 + y^2 = 16$ (completing the square)

$\Rightarrow (x + 5)^2 + y^2 = 16$

Comparing $(x + 5)^2 + y^2 = 16$ with the standard form, we have:

Centre = ( -5, 0)

Radius = √16 = 4

Hence, the correct answer is Option 4.

Calculation:

Given expression:

$\left(1 - \omega + \omega^2\right)^5 + \left(1 + \omega - \omega^2\right)^5 - 32$

Using the identity for cube roots of unity:

⇒ $1 + \omega + \omega^2 = 0 \Rightarrow \omega^2 + \omega = -1$

Simplify each term:

⇒ $1 - \omega + \omega^2 = -2\omega$

⇒ $1 + \omega - \omega^2 = -2\omega^2$

Substitute into the original expression:

⇒ $(-2\omega)^5 + (-2\omega^2)^5 - 32$

Calculate powers:

$= (-2)^5 \omega^5 + (-2)^5 (\omega^2)^5 - 32 = -32 \omega^5 - 32 \omega^{10} - 32$

Use $\omega^3 = 1$to reduce powers:

⇒ $\omega^5 = \omega^2, \quad \omega^{10} = \omega$

So expression becomes:

⇒ $-32 \omega^2 - 32 \omega - 32 = -32 (\omega^2 + \omega + 1)$

Since $\omega^2 + \omega + 1 = 0$ we have:

⇒ $$-32 \times 0 = 0$$

Hence, the correct answer is Option 1.

Calculation:

$z = 3 - 4i \Rightarrow \text{Re}(z) = 3, \text{Im}(z) = -4$

$|z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

$\sqrt{z} = \pm \left( \sqrt{ \frac{|z| + \text{Re}(z)}{2} } + i \cdot \operatorname{sgn}(\text{Im}(z)) \cdot \sqrt{ \frac{|z| - \text{Re}(z)}{2} } \right)$

$\Rightarrow \sqrt{3 - 4i} = \pm \left( \sqrt{ \frac{5 + 3}{2} } - i \cdot \sqrt{ \frac{5 - 3}{2} } \right)$

$= \pm \left( \sqrt{4} - i \cdot \sqrt{1} \right) = \pm (2 - i)$

Hence, the correct answer is Option 4.

Calculation:

$\cos(x + iy) = \cos \alpha + i \sin \alpha$

$\Rightarrow \cos(x + iy) = \cos x \cosh y - i \sin x \sinh y$

$\Rightarrow \cos x \cosh y = \cos \alpha, \quad -\sin x \sinh y = \sin \alpha$

$\Rightarrow (\cos x \cosh y)^2 + (\sin x \sinh y)^2 = \cos^2 \alpha + \sin^2 \alpha = 1$

$\Rightarrow \cos^2 x \cosh^2 y + \sin^2 x \sinh^2 y = 1$

$\Rightarrow \cos^2 x (1 + \sinh^2 y) + \sin^2 x \sinh^2 y = 1$

$\Rightarrow \cos^2 x + \sinh^2 y (\cos^2 x + \sin^2 x) = 1$

$\Rightarrow \cos^2 x + \sinh^2 y = 1 \Rightarrow \sinh^2 y = 1 - \cos^2 x$

$\Rightarrow \cosh 2y = 2 \sinh^2 y + 1, \quad \cos 2x = 2 \cos^2 x - 1$

$\Rightarrow \cosh 2y + \cos 2x = 2(1 - \cos^2 x) + 1 + 2 \cos^2 x - 1 = 2$

Hence, the correct answer is Option 2.

Calculation:

$z = -8i = 8e^{i\frac{3\pi}{2}}$

$\text{Let } z = re^{i\theta}, \text{ then the cube roots are given by } \sqrt[3]{z} = \sqrt[3]{r} \cdot e^{i\left(\frac{\theta + 2\pi k}{3}\right)} \text{ for } k = 0, 1, 2$

$\Rightarrow \sqrt[3]{8e^{i\frac{3\pi}{2}}} = 2e^{i\left(\frac{3\pi/2 + 2\pi k}{3}\right)}$

For k = 0:

$z_0 = 2e^{i\frac{\pi}{2}} = 2i$

For k = 1:

$z_1 = 2e^{i\frac{7\pi}{6}} = 2\left(\cos\frac{7\pi}{6} + i\sin\frac{7\pi}{6}\right) = 2\left(-\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) = -\sqrt{3} - i$

For k = 2:

$z_2 = 2e^{i\frac{11\pi}{6}} = 2\left(\cos\frac{11\pi}{6} + i\sin\frac{11\pi}{6}\right) = 2\left(\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) = \sqrt{3} - i$

∴ the cube roots of $z = -8i$ are $2i,\ -\sqrt{3} - i,\ \sqrt{3} - i$.

Hence, the correct answer is Option 1.

Calculation:

$z = x + iy \Rightarrow z - 1 = x - 1 + iy,\quad 2z + 1 = 2x + 1 + 2iy$

⇒ $\frac{z - 1}{2z + 1} = \frac{(x - 1) + iy}{(2x + 1) + 2iy}$

⇒ $\text{Im} \left( \frac{a + ib}{c + id} \right) = \frac{bc - ad}{c^2 + d^2}$

$\text{Let } a = x - 1, \quad b = y, \quad c = 2x + 1, \quad d = 2y$

$\Rightarrow \text{Im} = \frac{y(2x + 1) - (x - 1)(2y)}{(2x + 1)^2 + 4y^2} = \frac{2xy + y - 2xy + 2y}{(2x + 1)^2 + 4y^2} = \frac{3y}{(2x + 1)^2 + 4y^2}$

$\text{Given: } \frac{3y}{(2x + 1)^2 + 4y^2} = -4$

$\Rightarrow 3y = -4\left((2x + 1)^2 + 4y^2\right)$

$\Rightarrow 3y = -4(4x^2 + 4x + 1 + 4y^2)$

$\Rightarrow 3y = -16x^2 - 16x - 4 - 16y^2$

$\Rightarrow 16x^2 + 16x + 16y^2 + 3y + 4 = 0$

This is a general equation of a circle since  x² and y² have equal coefficients and the same signs

Hence, the correct answer is Option 4.

Calculation:

$f(z) = (x^2 + a y^2) + i b x y$ is analytic where $z = x + i y$.

Let $u = x^2 + a y^2$ and $v = b x y$ be the real and imaginary parts of $f(z)$.

By Cauchy-Riemann equations,

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Calculate partial derivatives:

$\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = 2 a y$

$\frac{\partial v}{\partial x} = b y, \quad \frac{\partial v}{\partial y} = b x$

From Cauchy-Riemann equations,

$2x = b x \implies b = 2$

$2 a y = - b y \implies 2 a = -b \implies 2 a = -2 \implies a = -1$

Therefore,

$a + b = -1 + 2 = 1$

Hence, the correct answer is Option 2.

Calculation:

$(a)\ f(z) = \bar{z}$

$\bar{z}$ is the complex conjugate of z . It does not satisfy the Cauchy-Riemann equations anywhere in $\mathbb{C} .$

Therefore, it is nowhere analytic.

$(b)\ f(z) = z^2$

This is a polynomial function. All polynomials in z are entire functions, i.e., analytic everywhere on $\mathbb{C}.$

Therefore, it is analytic everywhere.

$(c)\ f(z) = |z|^2 = z\bar{z}$

We analyze this at z = 0 . Let z = x + iy , then:

$f(z) = |z|^2 = x^2 + y^2$

This is a real-valued function and depends on both z and $\bar{z}$, so it is not holomorphic. Let's apply the Cauchy-Riemann equations:

Let $u(x, y) = x^2 + y^2$,$v(x, y) = 0$

$u_x = 2x,\quad u_y = 2y,\quad v_x = 0,\quad v_y = 0$

At (0,0): $u_x = 0$, u_y = 0 , so Cauchy-Riemann equations hold at the point, but the function is still not complex differentiable in any neighborhood of 0. Hence, it is not analytic at z = 0 

$(d)\ f(z) = e^z$

This is the complex exponential function, which is entire (analytic everywhere in $\mathbb{C}$

Therefore, it is analytic everywhere.

Hence, the correct answer is Option (c).

Calculation:

$z = x + iy$

⇒ $|z + i| = \sqrt{x^2 + (y+1)^2}$

⇒ $|z - i| = \sqrt{x^2 + (y-1)^2}$

⇒ $|z + i| > |z - i| \Rightarrow \sqrt{x^2 + (y+1)^2} > \sqrt{x^2 + (y-1)^2}$

$\Rightarrow x^2 + (y+1)^2 > x^2 + (y-1)^2$

$\Rightarrow (y+1)^2 > (y-1)^2$

$\Rightarrow y^2 + 2y + 1 > y^2 - 2y + 1$

$\Rightarrow 2y > -2y \Rightarrow 4y > 0 \Rightarrow y > 0$

∴  the inequality  |z + i| > |z - i|  holds true if and only if $\ \operatorname{Im} z > 0$

Hence, the correct answer is Option 4.

Calculation:

We are asked to evaluate the definite integral:

$\int_{-3}^{3} \frac{x^2}{1 + 3^x} , dx$

Let $f(x) = \frac{x^2}{1 + 3^x}$. We analyze the symmetry of the function:

$f(-x) = \frac{(-x)^2}{1 + 3^{-x}} = \frac{x^2}{1 + \frac{1}{3^x}} = \frac{x^2 \cdot 3^x}{3^x + 1}$

So,

$f(x) + f(-x) = \frac{x^2}{1 + 3^x} + \frac{x^2 \cdot 3^x}{1 + 3^x} = x^2$

Therefore,

$\int_{-3}^{3} f(x) , dx = \int_{-3}^{3} \frac{x^2}{1 + 3^x} , dx = \int_{-3}^{3} \frac{f(x) + f(-x)}{2} , dx = \int_{-3}^{3} \frac{x^2}{2} , dx$

Since $x^2$ is even, we simplify:

$\int_{-3}^{3} \frac{x^2}{2} , dx = 2 \cdot \int_{0}^{3} \frac{x^2}{2} , dx = \int_{0}^{3} x^2 , dx$

$= \left[ \frac{x^3}{3} \right]_0^3 = \frac{27}{3} = 9$

Hence, the correct answer is Option (d): 9.

Calculation:

We are given then curves  $y=sinx, y=cosx$ and  y-axis is

First, find the point of intersection $\sin x = \cos x \Rightarrow \tan x = 1 \Rightarrow x = \frac{\pi}{4}$

We need to evaluate the area between the curves from $x = 0 \text{ to } x = \frac{\pi}{4}.$

$A = \int_{0}^{\frac{\pi}{4}} (\cos x - \sin x) , dx$

$= \left[ \sin x + \cos x \right]_0^{\frac{\pi}{4}}$

$= \left( \sin \frac{\pi}{4} + \cos \frac{\pi}{4} \right) - \left( \sin 0 + \cos 0 \right)$

$= \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \right) - (0 + 1)$

$= \sqrt{2} - 1$

Hence, the correct answer is Option (b).

Calculation:

$\lim_{x \to 2} \frac{\sqrt{ax + b} - 3}{x - 2} = \frac{1}{2}$

To form an indeterminate $\frac{0}{0},\ \sqrt{2a + b} = 3 \Rightarrow 2a + b = 9 \quad \text{(1)}$

Multiply numerator and denominator by the conjugate $\frac{\sqrt{ax + b} - 3}{x - 2} \cdot \frac{\sqrt{ax + b} + 3}{\sqrt{ax + b} + 3}$

$= \frac{(ax + b) - 9}{(x - 2)(\sqrt{ax + b} + 3)}$

$\text{Using (1), we write: } ax + b - 9 = a(x - 2)$

$\Rightarrow \frac{a(x - 2)}{(x - 2)(\sqrt{ax + b} + 3)} = \frac{a}{\sqrt{ax + b} + 3}$

$\lim_{x \to 2} \frac{a}{\sqrt{ax + b} + 3} = \frac{a}{6} {=} \frac{1}{2}$

$\Rightarrow a = 3$

$\text{Substitute in (1): } 2(3) + b = 9 \Rightarrow b = 3$

Hence, the correct answer is Option 1:

Calculation:

Check differentiability of $y = |x|$ at $x=0$:

⇒ $y = \begin{cases} x, & x \geq 0 \ -x, & x < 0 \end{cases}$

Left-hand derivative at 0:

⇒ $\lim_{h \to 0^-} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h - 0}{h} = -1$

Right-hand derivative at 0:

⇒ $\lim_{h \to 0^+} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0^+} \frac{h - 0}{h} = 1$

Since $-1 \neq 1$, $y = |x|$ is not differentiable at $x=0$.

Now check differentiability of $y = x|x|$:

⇒ $y = \begin{cases} x^2, & x \geq 0 \ -x^2, & x < 0 \end{cases}$

Derivative for $x > 0$:

⇒ $y' = 2x$

Derivative for $x < 0$:

⇒ $y' = -2x$

Left-hand derivative at 0:

⇒ $\lim_{h \to 0^-} \frac{-h^2 - 0}{h} = \lim_{h \to 0^-} -h = 0$

Right-hand derivative at 0:

⇒ $\lim_{h \to 0^+} \frac{h^2 - 0}{h} = \lim_{h \to 0^+} h = 0$

Since left and right derivatives are equal and differentiable elsewhere, $y = x|x|$ is differentiable everywhere.

∴ only Statement II is true.

Hence, the correct answer is Option 2.