Delhi Police Head Constable (Ministerial), 2025 Official Paper (Held On: 08 Jan, 2026 Shift 3) (Previous Year Paper)

The correct answer is A Sikh festival showcasing martial valor.

Key Points

• Hola Mohalla is a significant three-day Sikh festival tradition established by the tenth Sikh Guru, Guru Gobind Singh, in March 1701 at Anandpur Sahib.
• The term Hola is the masculine form of the feminine word Holi, while Mohalla refers to an organized procession or a military-style column.
• The primary objective of the festival was to strengthen the Sikh community by creating an occasion for military exercises and mock battles to prepare them against the Mughal Empire.
• It is celebrated annually in the lunar month of Chet, usually falling a day after the Hindu festival of Holi, shifting the focus from playing with colors to demonstrating physical strength and martial skills.
• The festivities culminate in a grand procession led by the Panj Pyaras, starting from Takht Sri Keshgarh Sahib, one of the five temporal seats (Takhts) of Sikhism.

 Additional Information

• Martial Arts and Nihangs:
• The Nihang Sikhs, also known as the Akalis, are the main attraction of the festival, dressed in traditional electric blue robes and large turbans.
• They perform Gatka (a traditional martial art), tent pegging, bareback horse riding, and other feats of bravery and skill.
• Religious Observances:
• Besides martial displays, the festival includes Kirtan (spiritual music), Path (scriptural reading), and Langar (community kitchen), where people of all castes and creeds eat together.
• The festival reinforces the core Sikh values of equality, community service, and readiness to protect the oppressed.

The correct answer is A is incorrect, but R is correct.

Key Points

• Assertion (A) claims that Terah Taali is performed by the Kalbelia tribe using swords and fire. In reality, this dance is the traditional folk art of the Kamada tribe of Rajasthan, and it is primarily performed by women rather than men. Hence, statement A is incorrect.
• Reason (R) explains that the dance involves sitting on the floor with multiple manjeeras (brass cymbals) tied to the body. The name Terah Taali literally translates to Thirteen Beats, referring to the thirteen cymbals used: nine are tied to the right leg, seven above the ankle and two at the knee, one on each elbow, one on each wrist, and two held in the hands. Hence, statement R is correct.

 Additional Information

• Kamada Tribe:
• A traditional nomadic community in Rajasthan that performs this dance as a ritualistic tribute to the folk deity Baba Ramdev.
• The performance is often accompanied by male singers playing the Tanpura, Chautara, and Dholak.
• Performance Elements:
• While the core involves manjeeras, performers may occasionally incorporate balancing a pot on their head or holding a sword in their teeth to demonstrate skill.
• The rhythm is complex, requiring the dancer to strike the body-tied cymbals with those held in their hands in a perfectly timed sequence.
• Kalbelia Dance:
• Distinct from Terah Taali, this is the Sapera (snake charmer) dance, characterized by swirling movements and UNESCO recognition as an intangible heritage.

The correct answer is Jiya Rai - para swimmer and endurance record-holder.

Key Points

• Jiya Rai, a 16-year-old Indian para-swimmer, created history in July 2024 by becoming the youngest and fastest female para-swimmer in the world to swim solo across the English Channel.
• The feat was achieved on July 28-29, 2024, where she covered a grueling distance of approximately 34 kilometers in a record time of 17 hours and 25 minutes.
• She commenced her swim from Abbott's Cliff in the United Kingdom and successfully reached Pointe de la Courte Dune in France, battling extreme cold and strong tidal currents.
• Jiya Rai is a person with Autism Spectrum Disorder (ASD), and her achievement is a testament to her extraordinary mental and physical resilience, supported by her father, Madan Rai, who serves in the Indian Navy.
• Her journey was facilitated by the Integrated Headquarters of Ministry of Defence (Navy) and the Swimming Federation of India, highlighting her role as an ambassador for para-sports in India.

 Additional Information

• Avani Lekhara:
• She is a world-renowned Para-shooter who won gold medals at both the Tokyo 2020 and Paris 2024 Paralympic Games.
• She specializes in 10m Air Rifle standing SH1 and is the first Indian woman to win two Paralympic gold medals.
• English Channel:
• Known as the Everest of Swimming, it is a part of the Atlantic Ocean that separates Southern England from Northern France.
• The Channel Swimming Association (CSA) is the official body that authenticates and records solo and relay swims across the channel.
• Anchal Gangwal:
• She is an officer in the Indian Air Force (IAF) who made headlines for becoming a fighter pilot after being the daughter of a tea seller in Madhya Pradesh.
• Her achievements are associated with military aviation and women's empowerment in the armed forces rather than endurance swimming.
• Jiya Rai's Previous Records:
• In March 2022, she became the youngest and fastest female para-swimmer to swim across the Palk Strait from Sri Lanka to India in 13 hours and 10 minutes.
• She has been honored with the Pradhan Mantri Rashtriya Bal Puraskar, the highest civilian honor for citizens under 18 in India.

The correct answer is Buddha.

Key Points

• The Jataka Tales are a voluminous body of ancient literature originating in India, specifically detailing the previous births (reincarnations) of Gautama Buddha before he achieved Mahaparinirvana.
• In these narratives, the Buddha appears in various forms, known as the Bodhisattva, which include both human incarnations (such as kings, priests, or outcasts) and animal incarnations (such as a monkey, elephant, or deer).
• There are traditionally 547 Jataka stories preserved in the Pali language, which are part of the Sutta Pitaka, specifically categorized under the Khuddaka Nikaya of the Tripitaka.
• Each story is intended to impart a moral lesson or illustrate the perfections (Paramitas) like generosity, patience, and wisdom that are necessary for attaining Enlightenment.
• Structurally, a Jataka usually contains a Paccuppannavatthu (story of the present), an Atitavatthu (story of the past/the Jataka proper), Gathas (verses), and a Samodhana (identification of the characters).

 Additional Information

• Mahavira:
• He was the 24^th Tirthankara of Jainism; his life and teachings are primarily recorded in the Jain Agamas and the Kalpa Sutra rather than the Jatakas.
• Rama and Krishna:
• These figures are central Avatars of Lord Vishnu in Hinduism, whose legends are found in the Ramayana and Mahabharata (including the Bhagavad Gita) respectively.
• Artistic Representations:
• The Jataka stories have been extensively depicted in ancient Indian art, including the stone carvings of Sanchi Stupa, Bharhut, and the intricate frescoes of the Ajanta Caves.
• Literary Influence:
• Many Jataka fables are precursors to stories found in the Panchatantra and have influenced folklore across Asia and the West, including potential links to Aesop's Fables.

The correct answer is National Action for Mechanised Sanitation Ecosystem.

Key Points

• The NAMASTE scheme is a Central Sector Scheme launched as a joint initiative by the Ministry of Social Justice and Empowerment (MoSJE) and the Ministry of Housing and Urban Affairs (MoHUA).
• It aims to achieve zero fatalities in sanitation work and ensure that all sanitation work is performed by skilled workers using mechanized equipment.
• The scheme focuses on the identification of Sewer and Septic Tank Workers (SSWs) across all Urban Local Bodies (ULBs) in India, providing them with a dignified livelihood and safety net.
• It replaces the earlier Self-Employment Scheme for Rehabilitation of Manual Scavengers (SRMS) to provide a more comprehensive approach toward mechanized sanitation.
• The NAMASTE portal was established to facilitate the profiling and registration of workers to enable targeted delivery of benefits.

 Additional Information

• Core Components of NAMASTE:
• Health Insurance: Workers and their families are provided with health cover under the Ayushman Bharat-Pradhan Mantri Jan Arogya Yojana (AB-PMJAY).
• Livelihood Assistance: Provision of capital subsidies and interest subvention for the procurement of sanitation machinery and vehicles for mechanized cleaning.
• Skill Development: Specialized training programs are conducted to equip workers with the technical skills required for safe mechanized operations.
• Personal Protective Equipment (PPE): Distribution of safety kits and standardized equipment to mitigate health risks associated with septic tank cleaning.
• Implementing Agency: The National Safai Karamcharis Finance and Development Corporation (NSKFDC) is the nodal agency responsible for the execution of the scheme.

The correct answer is 1-c, 2-a, 3-d, 4-b.

Key Points

• Puga Geothermal Field is located in the Puga Valley of the Ladakh Union Territory, known for its high-temperature springs and potential for large-scale geothermal power generation. Hence, Puga matches with Ladakh.
• Cambay province refers to the Cambay Graben in Gujarat, a region characterized by high heat flow and geothermal gradients linked to the Deccan Traps volcanism. Hence, Cambay matches with Gujarat.
• SONATA stands for the Son-Narmada-Tapti lineament, a major geological structure spanning central India; in the context of geothermal provinces, it is prominently associated with Madhya Pradesh. Hence, SONATA matches with Madhya Pradesh.
• Tattapani is a significant geothermal area located in the Balrampur district of Chhattisgarh, featuring several hot springs that discharge water at temperatures near the boiling point. Hence, Tattapani matches with Chhattisgarh.

 Additional Information

• Geothermal Energy in India:
• The Geological Survey of India (GSI) has identified approximately 340 hot spring localities across the country.
• India is estimated to have a geothermal potential of about 10,000 MW spread across seven major provinces.
• Key provinces include the Himalayan province, Sohana, Cambay, SONATA, West Coast, Godavari, and Mahanadi basins.
• The Ministry of New and Renewable Energy (MNRE) facilitates the development of these resources to achieve energy security and reduce carbon emissions.
• Geothermal energy is considered a renewable and base-load power source, meaning it provides a continuous supply of electricity regardless of weather conditions.

The correct answer is It is based on the imagined life of the Tamil Bhakti poet Andal..

Key Points

• The Queen of Jasmine Country is a historical fiction novel written by Sharanya Manivannan, published in 2018, which provides a lyrical reimagining of the life of the 9^th-century poet-saint Andal. Hence, statement 1 is correct.
• The novel explores the transition of a young girl named Kodhai into the mystic saint Andal, focusing on her refusal to marry a mortal man and her spiritual dedication to Lord Vishnu. Hence, statement 1 is correct.
• The book is an original work written in English by a Chennai-based author; it is not a translation of Malayalam fiction nor has it been cited as a winner of the Sahitya Akademi Award for that specific category. Hence, statement 2 is incorrect.
• The narrative is not a translation of a Kannada novel by Tejaswini Niranjana; it is a contemporary Indian English literary work. Hence, statement 3 is incorrect.
• The setting of the book is Srivilliputhur in Tamil Nadu during the medieval period, specifically during the peak of the Bhakti Movement, and is not a political biography set in British Raj Bengal. Hence, statement 4 is incorrect.

 Additional Information

• Andal (Kodhai):
• She is the only female among the twelve Alwars, the medieval Vaishnavite saints of South India who popularized Bhakti through Tamil hymns.
• She is credited with composing two seminal works of Tamil literature: the Thiruppavai (a collection of 30 stanzas) and the Nachiar Tirumozhi (143 verses).
• Her poetry is renowned for Madhura Bhakti (erotic mysticism), where she expresses her intense desire for a divine union with Lord Ranganatha.
• Sharanya Manivannan:
• She is a multifaceted writer whose works often deal with themes of femininity, mythology, and spirituality.
• In The Queen of Jasmine Country, she utilizes a first-person narrative to humanize the saint, focusing on the sensory experiences of 9^th-century South India.
• The Bhakti Movement:
• This was a significant religious revolution in Medieval India that emphasized individual devotion to a personal god, bypassing rigid caste hierarchies and rituals.
• The Alwars (Vishnu devotees) and Nayanars (Shiva devotees) in the Tamil region were the early pioneers of this movement.

The correct answer is A is not guilty, as the act was done in good faith for the child's benefit.

Key Points

• Under Section 27 of the Bharatiya Nyaya Sanhita, 2023 (which corresponds to Section 89 of the Indian Penal Code), nothing which is done in good faith for the benefit of a person under eighteen years of age, or of unsound mind, by or by consent of the guardian or other person having lawful charge of that person, is an offence by reason of any harm which it may cause.
• The provision explicitly protects individuals who authorize risky medical procedures if the intent is to prevent death or cure a serious disease or infirmity. In this case, 'A' is acting to cure the child of a condition that requires a risky operation.
• The law recognizes that a child (minor) is legally incapable of giving valid consent for complex medical decisions; therefore, the guardian is empowered to provide consent on their behalf to ensure the child's welfare.
• The proviso to this section clarifies that the exception does not extend to the intentional causing of death, but it does cover acts done for the purpose of preventing death or grievous hurt, even if the person knows the act is likely to cause death.

 Additional Information

• Good Faith:
• Defined in Section 2(10) of the BNS 2023, it implies that the act was done with due care and attention.
• The absence of malice and the presence of a benevolent motive are central to establishing this defense.
• General Exceptions:
• These are found in Chapter III of the Bharatiya Nyaya Sanhita, 2023.
• Section 26 (BNS) covers acts done by consent in good faith for a person's benefit (formerly Section 88 IPC).
• Section 30 (BNS) addresses acts done in good faith for the benefit of a person without consent (formerly Section 92 IPC) in emergency situations where obtaining consent is impossible.
• Legal Liability:
• Options involving culpable homicide or grievous hurt are incorrect because the mens rea (guilty mind) is absent; the intention was to save the life rather than cause harm.
• The Bharatiya Nyaya Sanhita, 2023 preserves these protections to ensure that medical professionals and guardians can take necessary life-saving risks without the fear of criminal prosecution.

The correct answer is Lachit Barphukan.

Key Points

• Lachit Borphukan was a distinguished commander and Borphukan (General) of the Ahom Kingdom in present-day Assam, who famously thwarted the Mughal Empire’s expansion under Aurangzeb.
• He is best known for his leadership in the Battle of Saraighat in 1671, which is regarded as one of the most significant naval battles ever fought on a river in India.
• The battle took place on the Brahmaputra River at Guwahati, where the Ahom army used superior knowledge of the terrain and guerrilla warfare tactics to overcome the larger Mughal fleet.
• Despite being severely ill, Lachit Barphukan inspired his soldiers by boarding a boat and leading the charge against the forces of Raja Ram Singh I, who was leading the Mughal army on behalf of the Kachwaha Dynasty of Amber.
• His strategic brilliance in using the narrow river stretch at Saraighat neutralized the Mughal advantage in cavalry and heavy artillery, resulting in a decisive victory that preserved the independence of the Ahom Kingdom.
• To honor his legacy, Lachit Divas is celebrated annually on November 24^th in Assam, and the National Defence Academy (NDA) awards the Lachit Borphukan Gold Medal to its best cadet.

 Additional Information

• Srimanta Sankardeva:
• A 15^th-16^th century polymath, saint, and social-religious reformer who founded the Neo-Vaishnavite movement in Assam.
• He introduced the Sattras (monasteries) and developed the Sattriya dance, which is now one of the eight classical dances of India.
• Rani Durgavati:
• The ruling Queen of Gondwana from 1550 to 1564, she is remembered for her heroic defense against the Mughal forces of Akbar led by Asaf Khan.
• She belonged to the Chandel Dynasty and died in battle near Jabalpur.
• Sant Tukaram:
• A prominent 17^th-century Hindu saint and poet of the Bhakti Movement in Maharashtra.
• He was part of the Varkari tradition and is celebrated for his devotional poetry known as Abhangas.

The correct answer is I, II and III all are correct.

Key Points

• Electric generators are devices that convert mechanical energy into electrical energy. They operate on the principle of electromagnetic induction, where a change in magnetic flux linked with a coil induces an electromotive force (EMF). Hence, statement I is correct.
• An AC generator (alternator) uses two full, continuous slip rings connected to the ends of the armature coil. These rings rotate with the coil and maintain constant contact with carbon brushes, allowing the induced current to change its direction periodically, producing Alternating Current. Hence, statement II is correct.
• A DC generator utilizes a split-ring commutator to ensure the output current flows in only one direction. The commutator acts as a mechanical rectifier, reversing the connections of the armature coil to the external circuit at the exact moment the current in the coil reverses. Hence, statement III is correct.

 Additional Information

• Electromagnetic Induction:
• Discovered by Michael Faraday in 1831, it forms the fundamental basis for most electrical power generation.
• The magnitude of the induced EMF is governed by Faraday's Law, which states it is directly proportional to the rate of change of magnetic flux.
• The direction of the induced current in the generator's coil is determined using Fleming's Right-Hand Rule.
• Fleming's Right-Hand Rule:
• Thumb: Indicates the direction of motion of the conductor relative to the magnetic field.
• Forefinger: Points in the direction of the magnetic field (North to South).
• Middle Finger: Shows the direction of the induced current.
• Commutators vs. Slip Rings:
• Slip rings are used when a smooth, sinusoidal AC output is required, commonly found in hydroelectric and thermal power plants.
• Split rings are essential for DC motors and DC generators to maintain consistent torque or unidirectional current respectively.

The correct answer is 130.

Key Points

• The Human Development Index (HDI) is a composite statistical tool used to measure a country's overall achievement in its social and economic dimensions.
• According to the data for 2025, India secured the 130^th position, reflecting its global standing in terms of human welfare and development.
• The ranking is released by the United Nations Development Programme (UNDP) as part of its annual Human Development Report (HDR).
• The index provides a more comprehensive view of a nation's status than GDP alone, as it integrates health and education metrics alongside economic output.
• India's rank of 130 is derived from a calculated score that considers the country's progress in expanding functional capabilities and improving the quality of life for its citizens.

 Additional Information

• Core Dimensions of HDI:
• A Long and Healthy Life: This is assessed through Life Expectancy at Birth, which indicates the average number of years a newborn is expected to live.
• Knowledge and Education: Measured by two indicators: Mean Years of Schooling for adults aged 25 and older, and Expected Years of Schooling for children entering the education system.
• A Decent Standard of Living: This is measured using Gross National Income (GNI) per capita adjusted for Purchasing Power Parity (PPP) to account for cost of living differences.
• Ranking Tiers:
• Countries are categorized into four groups based on their HDI scores: Very High, High, Medium, and Low human development.
• India typically falls within the Medium Human Development category, alongside several other emerging economies.
• The HDI value ranges from 0 to 1; a value closer to 1 signifies a higher level of human development.

The correct answer is Only 1 and 3.

Key Points

• India is one of the largest producers of cotton in the world. Statistically, India consistently ranks as the second-largest producer globally, closely following China, and it possesses the largest area under cotton cultivation in the world, accounting for approximately 25% of the total global cotton area. Hence, statement 1 is correct.
• Cotton is a Kharif crop that requires specific soil and moisture conditions. It thrives best in the Black Cotton Soil (also known as Regur Soil) found predominantly in the Deccan Plateau region. While it can be grown in alluvial soils, heavy rainfall is detrimental to the crop as it causes waterlogging and damages the lint; cotton needs moderate, well-distributed rainfall. Hence, statement 2 is incorrect.
• The physiological development of the cotton plant is highly sensitive to climate. It requires a warm climate with temperatures ranging between 21 and 30 degrees Celsius. Crucially, for the fiber to mature and the bolls to burst properly, the plant requires a minimum of 210 frost-free days and clear, bright sunshine during the ripening period. Hence, statement 3 is correct.

 Additional Information

• Major Cotton Producing States:
• Gujarat: Often the leading producer in India due to favorable soil and irrigation.
• Maharashtra: Has the largest area under cultivation, primarily in the Vidarbha and Marathwada regions.
• Telangana and Rajasthan: Significant contributors to the national output through both rain-fed and irrigated systems.
• Technical Aspects:
• BT Cotton: India has seen a massive adoption of genetically modified Bacillus thuringiensis (BT) cotton to resist pests like the Pink Bollworm.
• Species: India is unique for commercially cultivating all four species of cotton: Gossypium arboreum, herbaceum, hirsutum, and barbadense.
• Economic Role: The cotton sector supports the Textile Industry, which is one of the largest employers and exporters in the Indian economy.

The correct answer is Maldives.

Key Points

• The Bay of Bengal Initiative for Multi-Sectoral Technical and Economic Cooperation (BIMSTEC) is a unique regional group that bridges South Asia and Southeast Asia, focusing on countries dependent on the Bay of Bengal.
• The organization was established on 6 June 1997 following the Bangkok Declaration, originally involving four member states under the acronym BIST-EC.
• Currently, the grouping consists of seven members: five from South Asia (Bangladesh, Bhutan, India, Nepal, and Sri Lanka) and two from Southeast Asia (Myanmar and Thailand).
• Maldives is not a part of BIMSTEC; it is a member of the South Asian Association for Regional Cooperation (SAARC) and the Indian Ocean Rim Association (IORA).
• The BIMSTEC headquarters or Secretariat is located in Dhaka, Bangladesh.

 Additional Information

• Bangladesh:
• As a founding member, it plays a central role by hosting the permanent Secretariat and leading the Trade, Investment and Development sector within the organization.
• Myanmar:
• It officially joined the group on 22 December 1997, adding a critical geographical link to the ASEAN region and contributing to energy and agriculture cooperation.
• Bhutan:
• It became a member in 2004, focusing its efforts on environment and climate change initiatives to leverage collective regional support for its landlocked economy.

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

Key Points

• The GDP Deflator is a comprehensive measure of inflation that tracks the prices of all newly produced, domestic, final goods and services in an economy, serving as a more comprehensive indicator than consumer-specific indices. Hence, A matches with 2.
• Inflation is defined as a sustained and persistent increase in the general price levels of commodities over a specific period, which results in the erosion of the purchasing power of a currency. Hence, B matches with 3.
• Demand Forecasting is a strategic management process that uses historical data and analytical models to predict future customer requirements, acting as an anticipatory tool for effective resource and production planning. Hence, C matches with 1.

 Additional Information

• Economic Indicators and Tools:
• The GDP Deflator is calculated using the formula: (Nominal GDP / Real GDP) × 100. Unlike the Consumer Price Index (CPI), it is not restricted to a fixed basket of goods and reflects current consumption patterns.
• Inflation can be categorized into Demand-Pull Inflation (rising demand) and Cost-Push Inflation (rising input costs). In India, it is primarily monitored through the CPI and Wholesale Price Index (WPI).
• Demand Forecasting techniques include Qualitative Methods (like the Delphi Method and Market Surveys) and Quantitative Methods (like Time Series Analysis and Regression models). It is essential for minimizing inventory costs and optimizing the supply chain.
• The Ministry of Statistics and Programme Implementation (MoSPI) in India is responsible for releasing data related to GDP and national accounts.

The correct answer is Integrating with world economy.

Key Points

• Globalisation in the Indian context refers to the process of opening up the domestic economy to the rest of the world, fostering economic integration and interdependence.
• It was a vital pillar of the 1991 New Economic Policy (NEP), launched by the government of P.V. Narasimha Rao and then Finance Minister Dr. Manmohan Singh.
• The primary objective was to overcome the Balance of Payments (BoP) crisis by encouraging the free flow of goods, services, capital, and technology across international borders.
• Key measures included the significant reduction of customs duties and the removal of quantitative restrictions (quotas) on imports to facilitate international trade.
• It aimed to make Indian industries globally competitive by allowing them to access foreign raw materials, capital goods, and advanced technical expertise.
• Globalisation also involved making the Indian Rupee partially convertible on the current account to streamline international transactions.

 Additional Information

• LPG Reforms:
• Liberalisation: Replaced the restrictive License Raj with a deregulated environment, allowing private businesses to expand without excessive government interference.
• Privatisation: Involved disinvestment in Public Sector Undertakings (PSUs), transferring management and ownership to the private sector to improve efficiency.
• Incorrect Concepts:
• Import Duties: Rather than increasing them, the reforms focused on lowering tariffs to align Indian prices with global market rates.
• Foreign Direct Investment (FDI): The policy actively sought to increase FDI and Foreign Institutional Investment (FII) inflows by relaxing equity caps in sectors like telecommunications and insurance.
• Foreign Banks: Globalisation encouraged the entry of foreign banks and the expansion of private sector banking to modernize the financial system.
• Legislative Changes: The government replaced the rigid Foreign Exchange Regulation Act (FERA) with the more flexible Foreign Exchange Management Act (FEMA) in 1999 to support the globalised economic framework.

The correct answer is Wheat and rice, through High Yielding Variety (HYV) seeds.

Key Points

• The Green Revolution in India was launched in the mid-1960s, specifically starting around 1965-66, to address chronic food shortages and achieve national food security.
• The movement was spearheaded by M.S. Swaminathan, known as the Father of Green Revolution in India, who adapted the scientific breakthroughs of Nobel Laureate Norman Borlaug to Indian conditions.
• The primary focus was the introduction of High Yielding Variety (HYV) seeds, which were genetically modified to be semi-dwarf and highly responsive to chemical fertilizers.
• The revolution initially targeted Wheat through varieties like Lerma Rojo and Sonora 64, later expanding significantly into Rice production with the IR8 variety.
• Key technological components included the expansion of irrigation infrastructure, the use of synthetic fertilizers, pesticides, and the mechanization of farm labor through tractors and tillers.
• Geographically, the first phase was concentrated in the states of Punjab, Haryana, and Western Uttar Pradesh due to their existing irrigation facilities.

 Additional Information

• High Yielding Variety Program (HYVP):
• This policy was implemented during the Fourth Five-Year Plan to provide farmers with subsidised inputs and technical assistance.
• It fundamentally shifted Indian agriculture from subsistence farming to commercial farming.
• Impact on Food Grains:
• India transitioned from a food-deficit nation dependent on imports (like the PL-480 program from the USA) to a food-surplus nation.
• The Food Corporation of India (FCI) was strengthened during this era to maintain buffer stocks and ensure Price Support Operations.
• Limitations and Subsequent Phases:
• The initial Green Revolution largely bypassed Pulses and Oilseeds, which led to the later launch of the Yellow Revolution (oilseeds) and Blue Revolution (fisheries).
• Contemporary efforts focus on an Evergreen Revolution, aiming for ecological sustainability alongside high productivity.

The correct answer is Major Dhyan Chand.

Key Points

• National Sports Day, also known as Rashtriya Khel Divas, is celebrated across India annually on August 29.
• The day commemorates the birth anniversary of Major Dhyan Chand, a legendary field hockey player born in 1905 in Allahabad (now Prayagraj).
• Dhyan Chand is widely regarded as the greatest hockey player of all time, earning nicknames like 'The Wizard' and 'The Magician' due to his superlative ball control and goal-scoring prowess.
• He played a central role in India winning three consecutive Olympic Gold Medals in field hockey during the years 1928 (Amsterdam), 1932 (Los Angeles), and 1936 (Berlin).
• The Government of India officially designated this day as National Sports Day in 2012 to inspire the youth and promote a culture of sports and physical fitness.
• On this occasion, the President of India confers prestigious National Sports Awards, including the Major Dhyan Chand Khel Ratna Award, Arjuna Award, and Dronacharya Award, at Rashtrapati Bhavan.

 Additional Information

• Milkha Singh:
• Commonly known as 'The Flying Sikh', he was a world-class sprinter who won gold at the 1958 Commonwealth Games and finished fourth in the 400m final at the 1960 Rome Olympics.
• Sachin Tendulkar:
• Known as the 'Master Blaster', he is the highest run-scorer in international cricket and the first sportsperson to be honored with the Bharat Ratna, India's highest civilian award.
• Kapil Dev:
• A legendary all-rounder and former captain who led the Indian Cricket Team to its historic first World Cup title in 1983 by defeating the West Indies.
• Major Dhyan Chand Khel Ratna:
• In August 2021, the Rajiv Gandhi Khel Ratna Award, India's highest sporting honor, was renamed the Major Dhyan Chand Khel Ratna Award to honor the hockey icon's legacy.

The correct answer is Guerilla warfare tactics and a deep understanding of Deccan topography..

Key Points

• Chhatrapati Shivaji Maharaj pioneered a strategic military doctrine known as Ganimi Kava, which translates to guerilla warfare, focusing on surprise attacks and high mobility.
• The Deccan topography, defined by the rugged Sahyadri mountain ranges, dense forests, and narrow valleys, served as a natural fortress for the Maratha army against the numerically superior Mughals.
• His forces utilized hit-and-run tactics, which involved attacking Mughal supply lines and camps at night or in difficult terrain and retreating to the safety of hill forts before the enemy could retaliate.
• By avoiding pitched battles on open plains, Shivaji neutralized the Mughal heavy cavalry and artillery, forcing them into narrow mountain passes where their sheer numbers became a disadvantage.
• The Mawal infantry, recruited from local hill tribes, possessed an intimate knowledge of the Western Ghats, allowing them to navigate treacherous paths that were inaccessible to the Mughal imperial forces.

 Additional Information

• Hill Forts:
• Shivaji controlled over 300 forts at the time of his death, including Raigad, Pratapgad, and Sinhagad, which acted as administrative and military hubs.
• These forts were strategically located to monitor trade routes and provide refuge during Mughal invasions.
• Naval Strategy:
• Known as the Father of the Indian Navy, he established a coastal fleet to counter the Siddis and European powers, but this was secondary to his land-based guerilla tactics against Aurangzeb.
• Direct Confrontation:
• Unlike the Rajputs, who often engaged in direct frontal assaults, Shivaji strategically avoided open-field combat where the Mughal cavalry was most effective.
• Siege Warfare:
• While European siege warfare relied on heavy bombardment, Shivaji focused on swift escalation and internal intelligence to capture forts, often using Ghorpads (monitor lizards) or expert climbers for stealth entry.

The correct answer is Swimming Federation of India.

Key Points

• The 77^th Senior National Aquatic Championships 2024 was successfully organized by the Swimming Federation of India (SFI), the national governing body for aquatic sports in the country.
• The prestigious event was held from September 10 to September 13, 2024, at the Sandeep Sejwal City Aquatics Centre located in Yelahanka, Bengaluru.
• The championship featured top-tier competitions across three major disciplines: Swimming, Diving, and Water Polo, serving as a platform for athletes to qualify for international meets.
• The Swimming Federation of India is a member of World Aquatics (formerly known as FINA) and the Asia Aquatics, ensuring that the competition adheres to international technical standards.
• Prominent teams such as the Services Sports Control Board (SSCB), Railway Sports Promotion Board (RSPB), and various state units participated, with Karnataka often being a dominant force in the medal tally.

 Additional Information

• Athletics Federation of India (AFI):
• Founded in 1946, it is the apex body for athletics in India and is responsible for managing the National Inter-State Senior Athletics Championships.
• It is affiliated with World Athletics and the Asian Athletics Association.
• All India Football Federation (AIFF):
• Established in 1937 and headquartered in New Delhi, it governs the sport of football, including the Santosh Trophy and the Indian National Team.
• It received FIFA affiliation in 1948, shortly after India's independence.
• Wrestling Federation of India (WFI):
• The governing body for wrestling in India, it promotes both Freestyle and Greco-Roman wrestling styles.
• It is responsible for selecting athletes for the Olympic Games and World Championships in coordination with the United World Wrestling (UWW).

The correct answer is Handles anti-doping enforcement in India.

Key Points

• The National Anti-Doping Agency (NADA) is the primary organization in India responsible for promoting, coordinating, and monitoring the anti-doping program in all forms of sports.
• It was established as a registered society on November 24, 2005, under the Societies Registration Act of 1860 to ensure a drug-free environment in Indian sports.
• NADA operates as an autonomous body under the Ministry of Youth Affairs and Sports, Government of India.
• The primary mandate of the agency is the implementation of anti-doping rules that are consistent with the World Anti-Doping Agency (WADA) code.
• Its core functions include sample collection (both urine and blood), conducting testing, monitoring athletes through Whereabouts information, and investigating anti-doping rule violations (ADRVs).
• NADA is also responsible for the Therapeutic Use Exemption (TUE) process, allowing athletes to use necessary medications that might otherwise be on the prohibited list.

 Additional Information

• World Anti-Doping Agency (WADA):
• Established in 1999 and headquartered in Montreal, Canada, WADA is the international independent agency that sets the global standards for anti-doping.
• It maintains the Prohibited List, which is updated annually to include substances and methods banned in sports.
• The National Anti-Doping Act, 2022:
• This landmark legislation provides statutory authority to NADA and the National Dope Testing Laboratory (NDTL).
• It empowers NADA to conduct searches and seizures and establishes a legal framework for consequences for athletes who violate doping norms.
• National Dope Testing Laboratory (NDTL):
• Located in New Delhi, it is the premier analytical testing laboratory in India, accredited by WADA for testing sport samples.
• Anti-Doping Administration and Management System (ADAMS):
• NADA utilizes this web-based database to coordinate anti-doping activities and track athlete data globally.

Shortcut Trick

The unequal side (base) b = 36 − (2 × 15) = 6 cm.

Divide the triangle into two right-angled triangles; the semi-base is 3 cm.

Height (h) = √(15² − 3²) = √(225 − 9) = √216 = 6√6 cm.

Area = (1/2) × base × height = (1/2) × 6 × 6√6 = 18√6 cm².

∴ The correct answer is 18√6 cm².

Alternate Method

Given:

Perimeter of the isosceles triangle = 36 cm

Length of equal sides (a) = 15 cm

Formula Used:

Heron's Formula: Area = √(s × (s − a) × (s − b) × (s − c))

Where s is the semi-perimeter = (a + b + c) ÷ 2

Calculations:

⇒ Let the sides be a = 15 cm, b = 15 cm, and c (unequal side).

⇒ c = 36 − (15 + 15) = 6 cm

⇒ Semi-perimeter (s) = 36 ÷ 2 = 18 cm

⇒ Area = √(18 × (18 − 15) × (18 − 15) × (18 − 6))

⇒ Area = √(18 × 3 × 3 × 12)

⇒ Area = √(162 × 12) = √1944

⇒ Area = √(324 × 6) = 18√6 cm²

∴ The correct answer is 18√6 cm².

Additional Information

Area of Isosceles Triangle

If equal sides are 'a' and base is 'b', Area = (b/4) × √(4a² − b²).

Altitude Property

In an isosceles triangle, the altitude drawn from the vertex to the unequal base bisects the base at 90°.

Heron's Formula

Used for any triangle with sides a, b, c: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.

Shortcut Trick

The circumference (C) of a circle is directly proportional to its radius (r) because C = 2πr.

In any linear relationship (where power of variable is 1), the percentage change in the result is exactly the same as the percentage change in the variable.

Since the radius increases by 20%, the circumference must also increase by 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Percentage increase in radius = 20%

Formula Used:

Circumference of a circle (C) = 2 × π × r

Calculations:

⇒ Let the initial radius of the circle be r.

⇒ Initial circumference (C₁) = 2 × π × r

⇒ New radius (r_new) = r + 20% of r = r + 0.2r = 1.2r

⇒ New circumference (C₂) = 2 × π × (1.2r) = 2.4πr

⇒ Increase in circumference = C₂ − C₁ = 2.4πr − 2πr = 0.4πr

⇒ Percentage Change = (Increase / Original Circumference) × 100

⇒ Percentage Change = (0.4πr / 2πr) × 100 = 0.2 × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Change in Area of Circle

Area (A) = πr². If the radius increases by x%, the Area increases by (2x + x²/100)%. For 20%, Area increase = 44%.

Linear Dimension Property

In any 2D figure, if all linear dimensions (radius, side, diagonal, perimeter) are scaled by k%, the perimeter or circumference also changes by k%.

Ratio Method

If radius ratio is r₁:r₂, then circumference ratio is also r₁:r₂, but area ratio is r₁²:r₂².

Shortcut Trick

Area of a square = (side)² = 28 × 28 = 784 m².

Area of the inscribed circle = (π/4) × Area of Square.

Remaining Area = (1 − π/4) × Area of Square.

Remaining Area = (1 − 3.14/4) × 784 = (1 − 0.785) × 784 = 0.215 × 784 = 168.56 m².

∴ The correct answer is 168.56 m².

Alternate Method

Given:

Side of square courtyard (s) = 28 m

Value of π = 3.14

Formula Used:

Area of Square = s²

Area of Circle = πr²

Where r is the radius of the circle.

Calculations:

⇒ Area of the square courtyard = 28 × 28 = 784 m²

⇒ Since the circle touches all four sides, its diameter = side of the square = 28 m

⇒ Radius (r) of the circular fountain = 28 ÷ 2 = 14 m

⇒ Area of the circular fountain = 3.14 × 14 × 14

⇒ Area of the circular fountain = 3.14 × 196 = 615.44 m²

⇒ Remaining area = Area of square − Area of circle

⇒ Remaining area = 784 − 615.44 = 168.56 m²

∴ The correct answer is 168.56 m².

Additional Information

Inscribed Circle (Incircle) of a Square

When a circle is inscribed in a square of side 'a', its radius r = a/2 and its area is πa²/4.

Circumscribed Circle of a Square

When a circle passes through all vertices of a square of side 'a', its radius R = a/√2 and its area is πa²/2.

Area of Remaining Corners

The area of the four corners outside the incircle of a square is approximately 21.5% of the total square area (when π ≈ 3.14).

Shortcut Trick

Angle in radians is directly given by the ratio of Arc length to Radius.

θ = Arc / Radius

θ = 10 / 75 = 2/15 rad

∴ The correct answer is 2/15 rad.

Alternate Method

Given:

Length of the pendulum (Radius, r) = 75 cm

Length of the arc (s) = 10 cm

Formula Used:

Arc length (s) = Radius (r) × Angle (θ) [where θ is in radians]

Calculations:

⇒ s = r × θ

⇒ 10 = 75 × θ

⇒ θ = 10 / 75

⇒ θ = 2 / 15 rad

∴ The correct answer is 2/15 rad.

Additional Information

Radian to Degree Conversion

To convert radians to degrees, multiply the value by 180/π. Conversely, to convert degrees to radians, multiply by π/180.

Area of a Sector

The area of the sector formed by the swing is given by the formula A = 1/2 × r² × θ, where θ is in radians.

Pendulum Motion

For small angles, the motion of a simple pendulum is approximately Simple Harmonic Motion (SHM), and the length of the string acts as the radius of the circular path.

Shortcut Trick

When a solid is melted into n equal smaller solids, the relation between the radii is: R = r × n^1/3.

Here, R = 8 cm and n = 8.

8 = r × 8^1/3

8 = r × 2

r = 8 ÷ 2 = 4 cm.

∴ The correct answer is 4 cm.

Alternate Method

Given:

Radius of the original large hemisphere (R) = 8 cm

Number of smaller hemispheres (n) = 8

Formula Used:

Volume of a hemisphere = (2/3) × π × radius³

Calculations:

The total volume remains constant during recasting.

⇒ Volume of Original Hemisphere = 8 × Volume of each Smaller Hemisphere

⇒ (2/3) × π × (8)³ = 8 × (2/3) × π × r³

⇒ 512 = 8 × r³

⇒ r³ = 512 ÷ 8

⇒ r³ = 64

⇒ r = ³√64

⇒ r = 4 cm

∴ The correct answer is 4 cm.

Additional Information

Total Surface Area of Hemisphere

The total surface area (TSA) includes the curved base and the flat circular top: TSA = 3 × π × r².

Curved Surface Area of Hemisphere

The area of the curved part only is exactly half of a full sphere: CSA = 2 × π × r².

Conservation of Volume

When any 3D shape is melted and recast into another shape, the total volume remains unchanged provided there is no wastage.

Volume of a Sphere

A full sphere has exactly double the volume of a hemisphere of the same radius: V = (4/3) × π × r³.

Shortcut Trick

For a hollow hemisphere with outer radius R and inner radius r:

Total Surface Area (TSA) = π(3R² + r²)

Given: R = 10 cm, r = 6 cm, π = 22/7

TSA = (22/7) × (3 × 10² + 6²) = (22/7) × (300 + 36) = (22/7) × 336

TSA = 22 × 48 = 1056 cm²

∴ The correct answer is 1056 cm².

Alternate Method

Given:

Outer radius (R) = 10 cm

Inner radius (r) = 6 cm

Value of π = 22/7

Formula Used:

1. Outer Curved Surface Area = 2πR²
2. Inner Curved Surface Area = 2πr²
3. Area of ring-shaped base = π(R² − r²)

Calculations:

⇒ Total Surface Area = Outer CSA + Inner CSA + Base Area

⇒ TSA = 2πR² + 2πr² + π(R² − r²)

⇒ TSA = π [2R² + 2r² + R² − r²]

⇒ TSA = π [3R² + r²]

⇒ TSA = (22/7) × [3(10)² + (6)²]

⇒ TSA = (22/7) × [300 + 36]

⇒ TSA = (22/7) × 336

⇒ TSA = 22 × 48

⇒ TSA = 1056 cm²

∴ The correct answer is 1056 cm².

Additional Information

Volume of Hollow Hemisphere

The volume of the material in a hollow hemisphere is (2/3)π(R³ − r³), where R is the outer radius and r is the inner radius.

Surface Area of Solid Hemisphere

For a solid hemisphere of radius r, the Curved Surface Area is 2πr² and the Total Surface Area (including flat base) is 3πr².

Area of a Ring (Annulus)

The area between two concentric circles is π(R² − r²). In a hollow hemisphere, this forms the area of the base ring.

Shortcut Trick

The Total Surface Area (TSA) of a pyramid is simply the sum of its Lateral Surface Area (LSA) and the Base Area.

LSA = TSA − Base Area

LSA = 784 − 400 = 384 cm²

∴ The correct answer is 384 cm².

Alternate Method

Given:

Total Surface Area (TSA) of square pyramid = 784 cm²

Area of the square base = 400 cm²

Formula Used:

Total Surface Area (TSA) = Lateral Surface Area (LSA) + Area of Base

Calculations:

⇒ TSA = LSA + Base Area

⇒ 784 = LSA + 400

⇒ LSA = 784 − 400

⇒ LSA = 384 cm²

∴ The correct answer is 384 cm².

Additional Information

Volume of a Square Pyramid

The volume is given by (1/3) × Base Area × Vertical Height (h). V = (1/3) × a² × h.

Lateral Surface Area (LSA) Formula

LSA = 2 × side (a) × slant height (l), or (1/2) × Perimeter of base × Slant height.

Slant Height Relation

For a square pyramid, the relationship is l² = h² + (a/2)², where l is slant height and h is vertical height.

Shortcut Trick

The ratio of Height to Shadow remains constant for objects at the same time.

Ratio = 6 / 4 = 1.5

Tower Height = 1.5 × Tower Shadow

Tower Height = 1.5 × 28 = 42 m

∴ The correct answer is 42 m.

Alternate Method

Given:

Stick Height (h₁) = 6 m

Stick Shadow (s₁) = 4 m

Tower Shadow (s₂) = 28 m

Formula Used:

Based on the similarity of triangles (AAA criteria), the ratio of corresponding sides is equal:

Height₁ / Shadow₁ = Height₂ / Shadow₂

Calculations:

⇒ 6 / 4 = H / 28

⇒ H = (6 ÷ 4) × 28

⇒ H = 1.5 × 28

⇒ H = 42 m

∴ The correct answer is 42 m.

Additional Information

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.

Sun Altitude

At any given time, the Sun's rays strike the ground at the same angle for all objects in the same location, creating similar right-angled triangles.

Properties of Ratios

If a : b = c : d, then the product of extremes equals the product of means (a × d = b × c).

Shortcut Trick

Assertion (A) represents a fundamental Circle Theorem: A line from the center to the midpoint of a chord is always perpendicular (90°) to it.

Reason (R) claims every intersection between a radius and a chord is 90°, which is False because this only happens at the midpoint.

Since the theorem correctly validates A and the generalization in R is incorrect, A is true and R is false.

∴ The correct answer is A is true, but R is false.

Alternate Method

Given: A circle with center O and a chord AB. Let M be the midpoint of AB.

Property Used: SSS (Side-Side-Side) Congruence Criterion.

Step-by-step validation of Assertion (A):

⇒ Consider ΔOAM and ΔOBM.

⇒ OA = OB (Radii of the same circle).

⇒ AM = MB (Given M is the midpoint of chord AB).

⇒ OM = OM (Common side).

⇒ By SSS congruence, ΔOAM ≅ ΔOBM.

⇒ ∴ ∠OMA = ∠OMB (By CPCT).

⇒ Since ∠OMA + ∠OMB = 180° (Linear pair), each angle must be 90°.

⇒ This proves Assertion (A) is true.

Validation of Reason (R):

⇒ A radius can intersect a chord at any point (not just the midpoint).

⇒ If the intersection is not at the midpoint, the angle is not 90°.

⇒ ∴ Reason (R) is false.

∴ The correct answer is A is true, but R is false.

Additional Information

Perpendicular Bisector Property

The perpendicular bisector of any chord of a circle always passes through the center of the circle.

Equal Chords and Distance

Chords of a circle which are equidistant from the center are equal in length.

Angle in a Semicircle

A chord that is also a diameter subtends a right angle (90°) at any point on the circumference.

Radius-Tangent Relationship

A radius is always perpendicular to the tangent at the point of contact, which is different from a chord intersection.

Shortcut Trick

The radius, tangent, and distance from center form a right-angled triangle.

Identify the Pythagorean Triplet: (7, 24, 25).

Given: Radius = 7 cm and Hypotenuse (distance) = 25 cm.

The missing side (tangent length) is 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given:

Radius of the circle (OA) = 7 cm

Distance of point P from center (OP) = 25 cm

Formula Used:

In a right-angled triangle ΔOAP, by Pythagoras Theorem:

OP² = OA² + AP²

Calculations:

⇒ AP² = OP² − OA²

⇒ AP² = 25² − 7²

⇒ AP² = 625 − 49

⇒ AP² = 576

⇒ AP = √576

⇒ AP = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Tangent-Radius Property

A tangent at any point of a circle is perpendicular to the radius through the point of contact (Angle OAP = 90°).

Length of Tangents

The lengths of tangents drawn from an external point to a circle are always equal.

Common Pythagorean Triplets

Frequently used triplets in exams include (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (9, 40, 41).

Shortcut Trick

Identify Pythagorean Triplets and Quadrants:

For θ (Q1): sin θ = 3/5 → (3, 4, 5) triplet ⇒ cos θ = 4/5 (positive in Q1).

For φ (Q3): cos φ = −5/13 → (5, 12, 13) triplet ⇒ sin φ = −12/13 (negative in Q3).

Apply expansion: sin(θ + φ) = sin θ cos φ + cos θ sin φ = (3/5 × −5/13) + (4/5 × −12/13) = −15/65 − 48/65 = −63/65.

∴ The correct answer is −63/65.

Alternate Method

Given: sin θ = 3/5 (1^st Quadrant), cos φ = −5/13 (3^rd Quadrant).

Formula: sin(A + B) = sin A cos B + cos A sin B

Step 1: Calculate cos θ and sin φ

⇒ In 1^st Quadrant (Aircraft A): All ratios are positive.

⇒ cos θ = √(1 − sin²θ) = √(1 − (3/5)²) = √(1 − 9/25) = √(16/25) = 4/5.

⇒ In 3^rd Quadrant (Aircraft B): sin and cos are negative.

⇒ sin φ = −√(1 − cos²φ) = −√(1 − (−5/13)²) = −√(1 − 25/169) = −√(144/169) = −12/13.

Step 2: Substitute in Addition Formula

⇒ sin(θ + φ) = (sin θ × cos φ) + (cos θ × sin φ)

⇒ sin(θ + φ) = (3/5 × −5/13) + (4/5 × −12/13)

⇒ sin(θ + φ) = (−15/65) + (−48/65)

⇒ sin(θ + φ) = −63/65.

∴ The correct answer is −63/65.

Additional Information

Trigonometric Signs (ASTC Rule)

Quadrant 1: All positive | Quadrant 2: sin positive | Quadrant 3: tan positive | Quadrant 4: cos positive.

Compound Angle Formulas

sin(A ± B) = sin A cos B ± cos A sin B | cos(A ± B) = cos A cos B ∓ sin A sin B.

Standard Pythagorean Triplets

Common integer sides for right triangles: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), and (9, 40, 41).

Shortcut Trick

When two equal distances are covered at different speeds v₁ and v₂, the Average Speed is the harmonic mean of the speeds.

Average Speed = (2 × v₁ × v₂) ÷ (v₁ + v₂)

⇒ (2 × 5 × 6) ÷ (5 + 6) = 60 ÷ 11

⇒ 5.4545... km/h

∴ The correct answer is 5.45 km/h.

Alternate Method

Given:

Distance going = 30 km, Speed going (v₁) = 5 km/h

Distance returning = 30 km, Speed returning (v₂) = 6 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Time taken going (t₁) = Distance ÷ Speed = 30 ÷ 5 = 6 hours

⇒ Time taken returning (t₂) = Distance ÷ Speed = 30 ÷ 6 = 5 hours

⇒ Total Distance = 30 + 30 = 60 km

⇒ Total Time = 6 + 5 = 11 hours

⇒ Average Speed = 60 ÷ 11

⇒ Average Speed ≈ 5.4545 km/h

∴ The correct answer is 5.45 km/h.

Additional Information

Average Speed (Different Distances)

When distances are unequal, always use the fundamental formula: Average Speed = (d₁ + d₂ + ...) ÷ (t₁ + t₂ + ...).

Relative Speed

If two objects move toward each other, Relative Speed = v₁ + v₂. If they move in the same direction, Relative Speed = |v₁ − v₂|.

Conversion of Units

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Shortcut Trick

Total Price = 2 × 400 = ₹800

Net Payable = 95% of Total Price (since discount is 5%)

Payable Amount = 0.95 × 800 = ₹760

∴ The correct answer is ₹760.

Alternate Method

Given:

Price of one screen guard = ₹400

Number of items bought = 2

Discount rate = 5% on the total bill

Formula Used:

Total Bill = Price × Quantity

Final Amount = Total Bill − (Discount % × Total Bill)

Calculations:

⇒ Total Bill of two screen guards = 2 × 400 = ₹800

⇒ Discount Amount = 5% of 800

⇒ Discount Amount = (5 ÷ 100) × 800 = ₹40

⇒ Amount to pay = 800 − 40 = ₹760

∴ The correct answer is ₹760.

Additional Information

Selling Price and Discount

The price after a discount is the Selling Price (SP). It is calculated as SP = Marked Price (MP) − Discount.

Discount Percentage Formula

To find the Discount %, use the formula: Discount % = (Discount Amount ÷ Marked Price) × 100.

Successive Discounts

If two discounts d₁% and d₂% are applied one after another, the equivalent single discount is (d₁ + d₂ − (d₁ × d₂) ÷ 100)%.

Shortcut Trick

Directly divide the corresponding numbers: 9.5 ÷ 0.19 = 50 and 0.085 ÷ 0.0017 = 50.

The expression simplifies to √(50 × 50).

The square root of (50²) is 50.

∴ The correct answer is 50.

Alternate Method

Given: A pharmaceutical quality analyst is verifying a concentration index using the expression: √((9.5 × 0.085) / (0.0017 × 0.19))

Formula Used: Decimal removal and √(a × b) properties.

⇒ √((9.5 × 0.085) / (0.19 × 0.0017))

⇒ √((9.5 / 0.19) × (0.085 / 0.0017))

⇒ To remove decimals, shift the decimal points: (950 / 19) × (850 / 17)

⇒ √(50 × 50)

⇒ √2500

⇒ 50

∴ The correct answer is 50.

Additional Information

Decimal Adjustment Rule

When dividing decimals, multiply both the numerator and denominator by 10ⁿ, where n is the number of decimal places, to convert them into whole numbers.

Square Root of Products

The property √(a × b) = √a × √b allows for simplifying complex expressions by calculating roots of individual factors.

Uniformity in Dosage

In pharmaceutical production, maintaining a consistent concentration index is vital for ensuring that every tablet in a batch meets safety and efficacy standards.

Shortcut Trick

Use the direct formula for two people working together: (A × B) ÷ (A + B).

Here, Amit (A) = 30 and Babli (B) = 45.

Required time = (30 × 45) ÷ (30 + 45) = 1350 ÷ 75 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given:

Time taken by Amit = 30 days

Time taken by Babli = 45 days

Formula Used:

Efficiency = Total Work ÷ Time

Total Work = LCM of individual days

Calculations:

⇒ Total Efficiency of Amit and Babli = 3 + 2 = 5 units/day

⇒ Time taken to complete the work together = Total Work ÷ Total Efficiency

⇒ Time = 90 ÷ 5 = 18 days

∴ The correct answer is 18 days.

Additional Information

Pipes and Cisterns

The same LCM logic is applied here. If an outlet pipe (leak) is present, its efficiency is subtracted from the total.

Work and Wages

Total wages are always distributed in the ratio of the efficiency (or actual work done) of the individuals.

Efficiency and Time Relation

Efficiency is inversely proportional to the Time taken. If the ratio of time taken by two people is x : y, then their efficiency ratio is y : x.

Shortcut Trick

Change in average weight = 65 − 64 = 1 kg (Increase).

Total students after new student joins = 6 + 1 = 7.

Total weight increase across all students = 7 × 1 = 7 kg.

Weight of new student = Old Average + Total Increase = 64 + 7 = 71 kg.

∴ The correct answer is 71 kg.

Alternate Method

Given:

Initial number of students = 6

Initial average weight = 64 kg

New average weight = 65 kg

Formula:

Total Sum = Average × Number of observations

New Observation = (New Average × New Count) − (Old Average × Old Count)

Calculations:

⇒ Total weight of 6 students = 6 × 64 = 384 kg

⇒ Total weight of 7 students (including new student) = 7 × 65 = 455 kg

⇒ Weight of new student = Total weight (7 students) − Total weight (6 students)

⇒ Weight of new student = 455 − 384

⇒ Weight of new student = 71 kg

∴ The correct answer is 71 kg.

Additional Information

New Member Formula

Value of new member = New Average + [Number of old members × Change in Average].

Member Leaving Formula

Value of member who left = Old Average − [Number of new members × Change in Average].

Weighted Average

Weighted Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), used when two groups with different averages are combined.

Shortcut Trick

Total Failed Percentage = (% Failed in Physics + % Failed in Chemistry) − % Failed in Both

⇒ Total Failed = (50% + 40%) − 10% = 80%

⇒ Percentage of candidates who Passed in Both = 100% − 80% = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

Failed in Physics, n(P) = 50%

Failed in Chemistry, n(C) = 40%

Failed in Both, n(P ∩ C) = 10%

Formula Used:

Total Failed Percentage, n(P ∪ C) = n(P) + n(C) − n(P ∩ C)

Calculations:

⇒ Candidates failed in Physics only = 50% − 10% = 40%

⇒ Candidates failed in Chemistry only = 40% − 10% = 30%

⇒ Total percentage of candidates who failed in at least one subject = 40% (Physics only) + 30% (Chemistry only) + 10% (Both) = 80%

⇒ Percentage of candidates who passed in both sections = 100% − 80% = 20%

∴ The correct answer is 20%.

Additional Information

Inclusion-Exclusion Principle

For any two finite sets A and B, the cardinality of the union is given by n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

De Morgan's Laws

The percentage of those who passed in both subjects is the complement of the union of those who failed in either subject: n(Pass Both) = 100 − n(Fail Physics ∪ Fail Chemistry).

Venn Diagram Regions

In a two-circle Venn diagram, the region outside both circles represents elements that do not belong to either set (e.g., passing in both when circles represent failures).

Shortcut Trick

Surface Area = 4πr² = 616 ⇒ r² = 616 ÷ (4 × 22/7) = 49.

Radius (r) = 7 cm.

Volume = (4/3)πr³ = (4/3) × (22/7) × 7 × 7 × 7 = 4312/3 cm³.

∴ The correct answer is 4312/3 cm³.

Alternate Method

Given:

Surface Area of the Sphere = 616 cm²

Formula Used:

1. Surface Area of a Sphere = 4πr²
2. Volume of a Sphere = (4/3)πr³

Take π = 22/7

Calculations:

⇒ 4 × (22/7) × r² = 616

⇒ (88/7) × r² = 616

⇒ r² = (616 × 7) ÷ 88

⇒ r² = 7 × 7 = 49

⇒ r = √49 = 7 cm

Now, finding the Volume:

⇒ Volume = (4/3) × (22/7) × 7³

⇒ Volume = (4/3) × (22/7) × 7 × 7 × 7

⇒ Volume = (4/3) × 22 × 49

⇒ Volume = (4 × 1078) ÷ 3

⇒ Volume = 4312/3 cm³

∴ The correct answer is 4312/3 cm³.

Additional Information

Volume of Hemisphere

The volume of a hemisphere is exactly half that of a sphere: V = (2/3)πr³.

Total Surface Area of Hemisphere

It includes the curved surface and the flat circular base: TSA = 3πr².

Curved Surface Area of Hemisphere

The CSA of a hemisphere is half the surface area of a sphere: CSA = 2πr².

Sphere Diameter

The diameter is the longest distance through the center: d = 2r.

Shortcut Trick

Find Radius (r) using Base Area: r = √(616 × 7 ÷ 22) = √(28 × 7) = 14 cm.

Use Curved Surface Area (CSA) formula: h = CSA ÷ (2 × π × r).

h = 1320 ÷ (2 × 22/7 × 14) = 1320 ÷ 88 = 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

Given:

Base Area of Cylinder = 616 cm²

Curved Surface Area (CSA) = 1,320 cm²

π = 22/7

Calculations:

⇒ Base Area of Cylinder = πr²

⇒ 616 = (22/7) × r²

⇒ r² = (616 × 7) ÷ 22

⇒ r² = 28 × 7 = 196

⇒ Radius (r) = √196 = 14 cm

⇒ Curved Surface Area (CSA) = 2πrh

⇒ 1320 = 2 × (22/7) × 14 × h

⇒ 1320 = 2 × 22 × 2 × h

⇒ 1320 = 88 × h

⇒ Height (h) = 1320 ÷ 88

⇒ h = 15 cm

∴ The correct answer is 15 cm.

Additional Information

Total Surface Area (TSA) of a Cylinder

The Total Surface Area includes the curved surface and the areas of the two circular bases: TSA = 2πr(h + r).

Volume of a Cylinder

The Volume represents the capacity or space occupied by the cylinder: Volume = πr²h.

Base Circumference

The perimeter of the base of the cylinder is given by the formula for the Circumference of a circle: C = 2πr.

Shortcut Trick

Find the Radius squared (r²): r² = (3 × Volume) ÷ (π × h)

r² = (3 × 3300) ÷ (22/7 × 14) = 9900 ÷ 44 = 225

Calculate Slant height (l) using l = √(r² + h²)

l = √(225 + 14²) = √(225 + 196) = √421 cm

∴ The correct answer is √421 cm.

Alternate Method

Given:

Height (h) of the cone = 14 cm

Volume (V) of the cone = 3300 cm³

Formula Used:

Volume of a right circular cone = (1/3) × π × r² × h

Slant height (l) = √(r² + h²)

Calculations:

⇒ 3300 = (1/3) × (22/7) × r² × 14

⇒ 3300 = (22 × 2 × r²) ÷ 3

⇒ 3300 = (44 × r²) ÷ 3

⇒ r² = (3300 × 3) ÷ 44

⇒ r² = 75 × 3 = 225

⇒ Radius (r) = √225 = 15 cm

⇒ Slant height (l) = √(15² + 14²)

⇒ l = √(225 + 196)

⇒ l = √421 cm

∴ The correct answer is √421 cm.

Additional Information

Curved Surface Area (CSA)

The area of the curved surface of a cone is given by π × r × l.

Total Surface Area (TSA)

The total area including the base is π × r × (l + r).

Relation between Dimensions

In a right circular cone, the slant height, radius, and height form a right-angled triangle: l² = r² + h².

Logic followed here is,

3 + 2² = 3 + 4 = 7

7 + 3² = 7 + 9 = 16

16 + 4² = 16 + 16 = 32

32 + 5² = 32 + 25 = 57

Hence, the correct answer is “Option 2”.

Given diagram is 8 × 8 Grid.

• There are 9 vertical gird lines(Including the left and right borders)
• There are 9 horizontal gird lines(Including the top and bottom borders)

Rectangles are determined by choosing

• Any two of the vertical lines, and
• Any two of horizontal lines

So, the total number of rectangles  = ⁹C₂ × ⁹C₂ 

⇒ (9! \ 7! 2!) × (9! \ 7! 2!)

⇒ 36 × 36

⇒ 1296

​There are 1296 rectangles present in the given figure.

Hence, the correct answer is “Option 1”.

Here,​

Figures A, C, and D share the same property of being linear shapes. like, +,  -,  ×.

While,

Figure B  stands out as the only non-linear/curved shape.

Therefore, the figure that is different from the others is B.

Hence, the correct answer is “Option 2”.

The logic followed here is :

A "Knife" is used to "Cut".

Similarly,

A "Pen" is used to "Write".

Hence, the correct answer is "option (1)".

Additional Information 

Pattern followed here is,

Hence, the correct answer is “Option 2”.

Given:

The final result after the increase = 18

The value of increase = 5

Calculations:

Let the required number be x.

According to the algebraic condition:

⇒ x + 5 = 18

By transposing 5 to the right-hand side (RHS):

⇒ x = 18 − 5

⇒ x = 13

Hence, the correct answer is “Option 3”.

Mirror image of given figure,

Hence, the correct answer is “Option 1”.

Pattern followed here is,

All the elements rotates clockwise.

Hence, the correct answer is “Option 2”.

BODMAS Table: 

Given equation: 40 B 8 T 6 M 3 K 4 = ?

Now, replacing the symbols:

⇒ 40 ÷ 8 - 6 + 3 × 4 = ?

⇒ 40 ÷ 8 - 6 + 3 × 4 = ?

⇒ 5 - 6 + 3 × 4 = ?

⇒ 5 - 6 + 12 = ?

⇒ 5 - 6 + 12 = ?

⇒ -1 + 12 = ?

⇒ 11.

Hence, the correct answer is "Option 2".

Triangles in the given figure,

There are 44 triangles present n the given figure.

Hence, the correct answer is “Option 3”.

Family Chart:

Family tree,

That lady is Daughter of Ravi.

Hence, the correct answer is “Option 3”.

OCCASION is coded as RFFDWMSR,

Similarly,

MARRIAGE coded as,

Hence, the correct answer is “Option 2”.

Pattern: (First number)² ÷ 2 = Second number

For 16 : 128 →

(16)² ÷ 2 = 256 ÷ 2 = 128

For 18 : 162 →

(18)2 ÷ 2 = 324 ÷ 2 = 162

Similarly,

For 22 : ? →

(22)2 ÷ 2 = 484 ÷ 2 = 242

Therefore, the missing number in the sequence is 242.

Hence, the correct answer is “Option 2”.

Here,

Three of the words Inclination, Preference, and Liking are synonyms or belong to the same category of positive interest or predisposition.

While,

Apathy represents the absence of feeling or interest.

Therefore, the word that does not belong to the group is Apathy.

Hence, the correct answer is “Option 4”.

Additional Information

For 256: (16)² = 256

For 49: 7² = 49

For 64: 8² = 64

For 243: It is not a perfect square.

Therefore, 243 is the odd one out in the group.

Hence, the correct answer is “Option 2”.

Pattern followed here is,

Hence, the correct answer is “Option 1”.

Given equation:

6 - 20 ÷ 12 × 7 + 1 = 70

Let's check each option one by one:

Option 1) ÷ and +

After interchanging ÷ and +:

6 - 20 + 12 × 7 ÷ 1 = 70

= 6 - 20 + 84 ÷ 1

= 6 - 20 + 84

= 70

Since 70 = 70, this option is correct.

Option 2) × and -

After interchanging × and -:

6 × 20 ÷ 12 - 7 + 1 = 70

= 120 ÷ 12 - 7 + 1

= 10 - 7 + 1

= 4

Since 4 ≠ 70, this option is incorrect.

Option 3) × and +

After interchanging × and +:

6 - 20 ÷ 12 + 7 × 1 = 70

= 6 - 1.67 + 7

= 11.33 (approx)

Since 11.33 ≠ 70, this option is incorrect.

Option 4) ÷ and ×

After interchanging ÷ and ×:

6 - 20 × 12 ÷ 7 + 1 = 70

= 6 - 34.29 + 1

= -27.29 (approx)

Since -27.29 ≠ 70, this option is incorrect.

Hence, the correct answer is "Option 1".

Pattern followed here is,

• Triangles and Minus sign interchange their positions.
• Multiplication sign moved in place of circle.
• Circles moved in place of Plus sign.
• Plus sign moved in place of multiplication sign.

Similarly,

• Plus sign and circles interchange their positions.
• Triangles moved in place of Multiplication sign.
• Multiplication sign moved in place of minus sign.
• Minus sign moved in place of triangles.

Hence, the correct answer is “Option 4”.

Here,

Heart, Brain, and Lungs are internal organs.

While,

Finger is a external body parts of the human body.

Therefore, the odd one out is Finger.

Hence, the correct answer is “Option 3”.

BODMAS Table: 

Given expression: [15 + (3 - (2 * 4))] / 2

Now, replacing the symbols:

⇒ [15 - (3 * (2 / 4))] + 2

⇒ [15 - (3 * (2 / 4))] + 2

⇒ [15 - (3 * 0.5)] + 2

⇒ [15 - 1.5] + 2

⇒ 13.5 + 2

⇒ 15.5

Hence, the correct answer is "Option 1".

Family Chart:

Family tree,

Sita is Grandmother of Arjun.

Hence, the correct answer is “Option 2”.

BODMAS Table: 

Given expression: 40 ÷ 8 × 4 - 2

Now, replacing the symbols:

⇒ 40 + 8 ÷ 4 × 2

⇒ 40 + 8 ÷ 4 × 2

⇒ 40 + 2 × 2

⇒ 40 + 4

⇒ 40 + 4

⇒ 44

Hence, the correct answer is "Option 3".

Triangles in the given figure,

There are 11 triangle present in the given figure.

Hence, the correct answer is “Option 3”.

The logic followed here is:

For BDF to CEG

The common pattern is a constant letter shift of +1 for each letter.

Applying the same logic to MOQ

The resulting word is NPR, which corresponds to Option 4.

Hence, the correct answer is 'Option 4'.

Pattern followed here is,

Hence, the correct answer is “Option 1”.

The correct answer is: Dubiously.

Key Points

• The word "regrow" in the context of the sentence refers to the process of forests growing back after trees have been lost, implying restoration. (पुनः बढ़ना)
• Example: The vegetation began to regrow rapidly after the rainy season.
• The word "dubiously" means something that is doubtful, uncertain, or not to be trusted. (संदिग्ध रूप से)
• Example: He looked at the offer dubiously, unsure if it was genuine.
• In the context of the given sentence, "dubiously" does not align with the intended meaning of significant reduction, making it the correct answer.

Therefore, the correct answer is: Dubiously.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Notable (उल्लेखनीय): Something that is worthy of attention or remarkable.
• Example: Her notable achievements in the field of science earned her a prestigious award.
• Substantially (काफी हद तक): To a great or significant extent.
• Example: The company’s profits have substantially increased over the past year.
• Considerably (काफी): By a notably large amount or to a significant degree.
• Example: The weather has improved considerably since last week.

The correct answer is: Option 1 i.e. capitulate.

Key Points

• The word "concede" means to admit or acknowledge something, often reluctantly, or to surrender or yield in a dispute. (स्वीकार करना / हार मान लेना)
• Example: After a long argument, she finally conceded that she was wrong.
• "Capitulate" means to cease resisting an opponent or demand, and surrender. It is synonymous with "concede." (आत्मसमर्पण करना)
• Example: The enemy was forced to capitulate after weeks of intense battle.
• Thus, the synonym for 'concede' is 'capitulate'.

Therefore, the correct answer is: Option 1 i.e. capitulate.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Contradict (विरोध करना): To deny the truth of something, especially by asserting the opposite.
• Example: His actions contradict his earlier statements.
• Repudiate (अस्वीकार करना): To reject or refuse to accept something.
• Example: The politician repudiated claims of corruption against him.
• Counterfeit (नकली): Something made in imitation of something valuable with the intent to deceive.
• Example: The shopkeeper was caught selling counterfeit currency.

The correct answer is: 3) obscure.

Key Points

• The word "famous" means being widely known and recognized by many people for achievements, qualities, or characteristics. (प्रसिद्ध)
• Example: Sachin Tendulkar is a famous cricketer known for his exceptional batting skills.
• The word "obscure" means not well-known, unclear, or not easily understood. It is the opposite of "famous." (अप्रसिद्ध / अस्पष्ट)
• Example: The author wrote an obscure book that only a few people have read.
• Hence, the correct antonym of 'famous' is 'obscure.'

Therefore, the correct answer is: 3) obscure.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Boring (उबाऊ): Something that is not interesting or exciting.
• Example: The movie was so boring that I almost fell asleep in the theater.
• Eminent (प्रख्यात): Someone who is famous and respected in a particular field.
• Example: Dr. Abdul Kalam was an eminent scientist and a beloved President of India.
• Untalented (अप्रतिभाशाली): Lacking talent, skill, or natural ability.
• Example: The untalented singer struggled to impress the judges during the audition.

The correct answer is: Option 1, i.e., Praise.

Key Points

• The word "Stigmatize" means to label someone or something in a way that is disgraceful, disapproving, or shameful. (कलंकित करना)
• Example: Society often stigmatizes people who have been to prison, making it harder for them to reintegrate.
• The word "Praise" refers to expressing approval or admiration for someone or something. (प्रशंसा करना)
• Example: The teacher praised the students for their excellent performance in the competition.
• Hence, the opposite of "Stigmatize" is "Praise".

Therefore, the correct answer is: Option 1, i.e., Praise.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Protect (सुरक्षा करना): To keep someone or something safe from harm or injury.
• Example: Parents always do their best to protect their children from danger.
• Spoil (बिगाड़ना): To damage or ruin something; to harm the character of someone by being too lenient or indulgent.
• Example: Excessive pampering can spoil a child’s behavior.
• Complain (शिकायत करना): To express dissatisfaction or annoyance about something.
• Example: She complained about the noise coming from her neighbor’s house.

The correct answer is: Option 2 i.e. Tariff.

Key Points

• The word "Tariff" refers to a tax or duty imposed by a government on imports or exports. (शुल्क)
• Example: The government increased the tariff on imported goods to protect local industries.
• This term is commonly used in the context of international trade, where countries levy taxes on products crossing their borders.

Therefore, the correct answer is: Option 2 i.e. Tariff.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Customs (सीमा शुल्क): Refers to the duties levied on goods imported into or exported out of a country, and also encompasses the government agency responsible for collecting these duties.
• Example: The customs officer inspected the luggage for any prohibited items.
• Excise tax (उत्पाद कर): A tax levied on certain goods produced or sold within a country.
• Example: The government imposes an excise tax on tobacco products to discourage smoking.
• Penalty (दंड): A punishment, usually a fine or other sanction, imposed for breaking a law or rule.
• Example: The company had to pay a huge penalty for violating environmental regulations.

The correct answer is: Xenophobia.

Key Points

• The word "Xenophobia" means a fear or dislike of people from other countries or cultures. (विदेशियों या अन्य संस्कृतियों के प्रति भय या नापसंदगी)
• Example: His xenophobia prevented him from traveling to other countries or interacting with foreigners.
• It reflects a biased or prejudiced attitude towards those who are perceived as outsiders or strangers. (अन्य लोगों के प्रति पूर्वाग्रह)

Therefore, the correct answer is: Xenophobia.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Agoraphobia (एगोराफोबिया): Fear of open spaces or crowded places.
• Example: Due to her agoraphobia, she avoids going to crowded shopping malls.
• Nyctophobia (निक्तोफोबिया): Fear of the dark.
• Example: Children with nyctophobia often require a nightlight in their room to feel secure.
• Hemophobia (हेमोफोबिया): Fear of blood.
• Example: His hemophobia made it difficult for him to handle injuries or medical procedures.

The correct answer is '4' i.e. 'is, drank'.

 

Key Points

• The verb "is" is used to describe the state of the milk in the carton, which is singular and present tense.
• The verb "drank" is the correct past tense form of "drink" and is used to describe an action that occurred in the past ("last night").
• Options 1, 2, and 3 are incorrect because they either use incorrect verb forms or do not match the grammatical structure required for the sentence.

Therefore, the correct answer is Option 4.

 

Complete Sentence: "The milk in the carton is almost over. Someone drank it last night."

 

Additional Information

• Option 1: "has, drank" is incorrect because "has" is not suitable for describing the state of the milk, and "drank" does not pair with "has."
• Option 2: "have, have drank" is incorrect because "have drank" is grammatically incorrect; the correct form is "have drunk."
• Option 3: "are, has drunk" is incorrect because "are" is used for plural subjects and does not match with "milk," which is singular.

The correct answer is '2' i.e. break.

 

Key Points

• The word "break" is a noun that refers to a short rest or pause from work or activity, making it the correct choice in this context.
• The sentence implies that the students needed a brief respite after the long lecture, so the word "break" fits appropriately.
• Option 1: "brake" refers to a device used to stop a vehicle, which is unrelated to the context of the sentence.
• Options 3 and 4: "braek" and "braik" are misspelled and not valid words in English.

Therefore, the correct answer is Option 2.

 

Complete Sentence: "After the long lecture, the students needed a short break."

 

Additional Information

• Option 1: "brake" is used in the context of vehicles or machinery and is unrelated to the sentence meaning.
• Option 3: "braek" is a typographical error and does not exist in English.
• Option 4: "braik" is also a misspelling and not a recognized word in English.