Delhi Police Head Constable (Ministerial), 2025 Official Paper (Held On: 12 Jan, 2026 Shift 3) (Previous Year Paper)

The correct answer is Arundhati Roy - The God of Small Things.

Key Points

• Arundhati Roy became the first Indian woman and the first Indian citizen to win the prestigious Man Booker Prize for Fiction in 1997.
• Her debut novel, The God of Small Things, is a semi-autobiographical story set in Ayemenem, Kerala, which intricately weaves together personal stories with the socio-political context of India.
• The narrative focuses on the childhood experiences of fraternal twins, Estha and Rahel, exploring how small things affect people's lives and their fates, while challenging the 'Love Laws' of the traditional Indian caste system.
• The novel achieved significant commercial success, selling over six million copies worldwide and being translated into more than 40 languages, establishing Roy as a leading voice in global literature.
• Before Roy’s win, Indian-born authors like V.S. Naipaul (1971) and Salman Rushdie (1981) had won the prize, but they held British citizenship at the time of their awards.

 Additional Information

• Kiran Desai - The Inheritance of Loss:
• Kiran Desai, the daughter of Anita Desai, won the Man Booker Prize in 2006 for this novel.
• The story deals with the themes of post-colonialism, globalization, and identity, set in the Himalayas and New York City.
• Jhumpa Lahiri - Interpreter of Maladies:
• She is a famous Indian-American author who won the Pulitzer Prize for Fiction in 2000 for this collection of short stories.
• The book explores the lives of Indians and Indian Americans caught between their roots and the New World.
• Anita Desai - Clear Light of Day:
• Anita Desai is a veteran Indian novelist who has been shortlisted for the Booker Prize three times but has not won it.
• Clear Light of Day (1980) is one of her most acclaimed works, focusing on family dynamics in Old Delhi against the backdrop of the Partition.

The correct answer is The Constitution permits preventive detention under Articles 22(3) to (7), but requires that the detained person must be informed of grounds of detention and allowed to make a representation, and the Supreme Court has upheld these laws subject to procedural safeguards..

Key Points

• Article 22 of the Indian Constitution provides protection against arrest and detention. While Articles 22(1) and 22(2) deal with punitive detention, Articles 22(3) to 22(7) specifically govern preventive detention, which is detention without trial based on suspicion. Hence, statement 3 is correct.
• The Supreme Court in the landmark A.K. Gopalan v. State of Madras (1950) case upheld the validity of the Preventive Detention Act, stating it does not violate Article 21 as it follows a procedure established by law. Hence, statement 1 is incorrect.
• Under Article 22(3), the fundamental right to be produced before a Magistrate within 24 hours (not 48 hours) does not apply to a person detained under preventive detention laws. Hence, statement 2 is incorrect.
• There is no constitutional requirement for prior approval from the Supreme Court or any court before a detention order is passed; it is an executive decision usually reviewed by an Advisory Board after the detention. Hence, statement 4 is incorrect.
• Constitutional safeguards include the right of the detainee to be informed of the grounds of detention as soon as possible and the right to make the earliest possible representation against the detention order to the authorities. Hence, statement 3 is correct.

 Additional Information

• Preventive Detention Laws:
• Legislative Authority: Parliament has exclusive power for matters like defense and foreign affairs, while both Parliament and State Legislatures have concurrent power for matters like public order.
• Advisory Board: A person cannot be detained for more than three months unless an Advisory Board, consisting of persons qualified as High Court Judges, finds sufficient cause for extended detention.
• 44th Amendment Act (1978): This act sought to reduce the detention period without board approval to two months, but this specific provision has not yet been notified.
• Rights of Detainee: Although the authority must disclose grounds, they can refuse to disclose facts they consider against the public interest.

The correct answer is The fellowship provides financial support for Indian-origin scientists to relocate permanently to India within a maximum of five years..

Key Points

• The Vaishvik Bhartiya Vaigyanik (VAIBHAV) Fellowship was launched to foster a collaborative ecosystem between the Indian STEMM diaspora and Indian academic or research institutions to boost the nation's S&T infrastructure. Hence, statement 1 is correct.
• The fellowship is designed for short-term collaborative assignments rather than permanent relocation; it requires the fellow to spend a minimum of one month and a maximum of two months per year in an Indian host institution for up to three years. Hence, statement 2 is incorrect.
• It focuses on facilitating joint research projects and technology transfer in critical areas like Quantum Technology, Artificial Intelligence, and Health, encouraging mentorship for Indian students. Hence, statement 3 is correct.
• The Department of Science and Technology (DST), which operates under the Ministry of Science and Technology, is the primary body responsible for the implementation and administration of this fellowship. Hence, statement 4 is correct.

 Additional Information

• Eligibility Criteria:
• Open to Non-Resident Indians (NRIs), Persons of Indian Origin (PIOs), or Overseas Citizens of India (OCIs).
• Applicants must possess a PhD, MD, or MS degree from a recognized university.
• The candidate must be an active researcher in a top-ranked global institute (within the top 500 QS World University Rankings).
• Financial Grant Details:
• Fellows receive a monthly grant of INR 4,00,000 for the duration of their stay in India.
• The fellowship covers international airfare (business class), accommodation, and local contingency expenses.
• Host institutions in India are provided with funds to support the research work and consumables required for the collaboration.
• Research Areas:
• The program identifies 18 broad verticals and 80 horizontals including Energy, Water, Agriculture, and Advanced Materials.

The correct answer is A high CD ratio may indicate over-aggressive lending relative to mobilized deposits, raising concerns about asset-liability mismatches, liquidity risk, and long-term financial stability..

Key Points

• A high Credit-Deposit (CD) ratio suggests that a bank has lent out a significant portion of its deposits, which could lead to liquidity crunches if depositors demand their funds simultaneously. Hence, statement 1 is incorrect.
• The CD ratio is a measure of loan-to-deposit utilization; a high ratio specifically means banks are lending more extensively rather than maintaining large liquidity buffers. Hence, statement 2 is incorrect.
• When credit growth significantly outpaces deposit growth, banks face Asset-Liability Mismatches (ALM) because long-term loans are being funded by short-term deposits, which directly threatens financial stability. Hence, statement 3 is correct.
• The CD ratio is a critical macro-prudential indicator monitored by the Reserve Bank of India (RBI) to ensure systemic health, debunking the idea that it has no macroeconomic consequence. Hence, statement 4 is incorrect.

 Additional Information

• Understanding CD Ratio:
• The Credit-Deposit Ratio indicates how much of each rupee of deposit is going toward credit markets. In the Indian context, a ratio exceeding 75% to 80% is generally considered a signal for banks to increase their deposit mobilization.
• Factors like Statutory Liquidity Ratio (SLR) and Cash Reserve Ratio (CRR) mean that banks must keep roughly 22% to 25% of their deposits with the RBI or in liquid assets, making a 100% CD ratio mathematically impossible without external borrowing.
• Risks and Implications:
• Liquidity Risk: High ratios increase reliance on inter-bank call money markets and Certificate of Deposits, which are more expensive and volatile than retail deposits.
• Interest Rate Impact: To correct a high CD ratio, banks often raise Fixed Deposit (FD) rates, leading to a general rise in the cost of capital across the economy.

The correct answer is Thiruvathira Kali - Claps and Ilathalam.

Key Points

• Thiruvathira Kali is a traditional folk dance from Kerala, performed exclusively by women during the auspicious month of Dhanu and on the occasion of Onam. Hence, Thiruvathira Kali matches with Claps and Ilathalam.
• The performance is characterized by rhythmic clapping of hands by the dancers who move in a circular pattern around a Nilavilakku (ceremonial lamp).
• The musical foundation of the dance is provided by Thiruvathirappattu (vocal music) and the Ilathalam, which is a pair of bronze cymbals used to accentuate the rhythmic cycle.

 Additional Information

• Rouf:
• A celebrated folk dance of Jammu and Kashmir, performed primarily by women during the spring season and Eid.
• It involves intricate footwork and vocal singing in a choral format; it does not traditionally use heavy percussion like Dhol and Nagara as its primary rhythm base.
• Saila:
• A popular stick dance performed in Chhattisgarh and Madhya Pradesh by the Gond and Baiga tribes after the harvest.
• The dance involves rhythmic striking of sticks, usually accompanied by the Mandhar (drum) rather than the Rabab or Harmonium.
• Sattriya:
• A Classical Dance of Assam, introduced by the reformer Sankaradeva in the 15th century.
• The key musical instruments are the Khol (a unique Assamese drum), Bhortal (large cymbals), and the flute, while the Mridangam is more central to Carnatic music.

The correct answer is It reveals 3000-year-old settlements with Satavahana.

Key Points

• The archaeological site of Chandravalli is located in the Chitradurga district of Karnataka, positioned in a scenic valley formed by three hills.
• Excavations at the site have provided evidence of continuous human habitation for nearly 3,000 years, dating back from the Iron Age (Megalithic culture) through the Satavahana and Kadamba periods into the medieval era.
• The site is most renowned for revealing layers of the Satavahana dynasty (spanning the 2nd century BCE to the 3rd century CE), characterized by the discovery of unique lead and copper coins bearing the dynasty's symbols.
• Significant archaeological findings include russet-coated painted pottery, which is a diagnostic feature of the Satavahana period in South India, helping historians date the transition of regional cultures.
• Early excavations were conducted by M.H. Krishna in 1929, followed by Sir Mortimer Wheeler in 1947, who applied scientific stratigraphy to identify the various cultural sequences of the settlement.
• The site also proved the existence of international maritime trade, as Roman silver denarii belonging to Emperor Augustus and Emperor Tiberius were unearthed, suggesting that Chandravalli was a major trade hub connected to global networks.

 Additional Information

• Satavahana Dynasty:
• Known as the Andhras in the Puranas, they were one of the first major empires to rule large parts of the Deccan and Peninsular India.
• They are famous for their patronage of Prakrit literature and the construction of massive Buddhist stupas like the one at Amaravati.
• Kadamba Inscriptions:
• Apart from Satavahana relics, the site contains a vital rock inscription of King Mayurasharma, the founder of the Kadamba dynasty, who established the first indigenous Kannada empire.
• This inscription, written in Sanskrit, records the construction of a water tank and mentions the king's victories over several rival clans.
• Geographic Significance:
• The site is naturally protected by the hills of Cholagudda and Kirubanakallu, which likely contributed to its longevity as a strategic military and residential settlement in ancient Karnataka.

The correct answer is Organization of village panchayats.

Key Points

• The Directive Principles of State Policy (DPSP) are enumerated in Part IV of the Indian Constitution from Article 36 to 51. These principles are inspired by the Irish Constitution and are fundamental in the governance of the country.
• DPSPs are broadly classified into three categories based on their ideological leanings: Socialistic Principles, Gandhian Principles, and Liberal-Intellectual Principles.
• Gandhian Principles represent the program of reconstruction envisioned by Mahatma Gandhi. They focus on the upliftment of rural society, decentralization of power, and moral development.
• Article 40 of the Constitution directs the state to organize village panchayats and endow them with such powers and authority as may be necessary to enable them to function as units of self-government. This aligns with Gandhi's vision of Gram Swaraj (village self-rule).
• To give a practical shape to this principle, the 73rd Constitutional Amendment Act, 1992, was passed, which institutionalized Panchayati Raj across India.

 Additional Information

• Equal pay for equal work
• This is a Socialistic Principle under Article 39(d). It aims at ensuring economic justice and gender equality in wages for both men and women.
• Uniform Civil Code
• This is a Liberal-Intellectual Principle under Article 44. It directs the state to endeavor to secure a uniform set of personal laws for all citizens regardless of religion.
• Protection of monuments
• This is a Liberal-Intellectual Principle under Article 49. It mandates the state to protect objects and places of artistic or historic interest that are declared to be of national importance.
• Other Gandhian Principles
• Article 43: Promotion of cottage industries in rural areas.
• Article 46: Promotion of educational and economic interests of Scheduled Castes (SCs) and Scheduled Tribes (STs).
• Article 47: Prohibition of the consumption of intoxicating drinks and drugs.
• Article 48: Prohibition of the slaughter of cows and calves.

The correct answer is h / 2π.

Key Points

• According to Bohr’s atomic model, specifically his second postulate, the angular momentum of an electron orbiting the nucleus is quantized.
• The theory dictates that an electron can only revolve in specific stable orbits where its orbital angular momentum (L) is an integral multiple of the reduced Planck constant, which is h / 2π.
• Mathematically, this condition is expressed as L = mvr = n(h / 2π), where m represents the mass of the electron, v is its orbital velocity, and r is the radius of the stable orbit.
• The variable n is known as the principal quantum number and can take any positive integer value (1, 2, 3, ...), corresponding to the first, second, and subsequent energy levels or shells.
• This quantization explains why electrons do not continuously lose energy and spiral into the nucleus, as classical electromagnetic theory would otherwise predict for a moving charged particle.

 Additional Information

• Planck’s Constant (h):
• It is a fundamental physical constant used to describe the sizes of quanta in quantum mechanics.
• Its value is approximately 6.626 × 10⁻³⁴ Joule-seconds (J·s).
• Stationary Orbits:
• In these specific orbits, an electron is in a stationary state, meaning it does not radiate energy while moving.
• Energy is only emitted or absorbed when an electron jumps from one allowed orbit to another.
• Force Balance:
• The stability of these orbits is maintained because the centripetal force required for the electron’s circular motion is provided by the Coulombic force of attraction between the positive nucleus and the negative electron.

The correct answer is Laxman Nayak.

Key Points

• Laxman Nayak was a prominent tribal civil rights activist belonging to the Bhumia tribe of Malkangiri district in Odisha.
• Deeply influenced by the ideals of Mahatma Gandhi, he spearheaded the local movement against the British by promoting non-violence, khadi, and the message of Swaraj.
• During the Quit India Movement in 1942, he led a peaceful demonstration toward the Matili Police Station to protest against the oppressive British administration and local landlords.
• The police resorted to indiscriminate firing to disperse the crowd, during which a forest guard named G. Ramayya died under mysterious or accidental circumstances in the chaos.
• Despite his adherence to peaceful protest, the British authorities falsely implicated Laxman Nayak for the murder of the forest guard to crush the tribal resistance in the Koraput region.
• He was eventually convicted in a biased trial and executed by hanging on March 29, 1943, in the Berhampur Jail, becoming a legendary martyr of the Indian freedom struggle.

 Additional Information

• Tileswari Koch
• She was a young 12-year-old martyr from Assam who was shot by the British police while attempting to hoist the Tricolour at the Dhekiajuli police station in 1942.
• Arun Chandra Guha
• A noted revolutionary and politician from Bengal associated with the Yugantar group; he later became a member of the Constituent Assembly.
• Lala Hansraj
• An eminent educator and follower of Swami Dayanand Saraswati, he was instrumental in founding the Dayanand Anglo-Vedic (DAV) educational institutions.

The correct answer is Longer crushing season and higher sugar content in southern sugarcane.

Key Points

• The sugar industry in India is gradually shifting from the North to the South because the tropical climate of Peninsular India results in a higher sucrose content in the sugarcane, leading to a higher sugar recovery rate.
• Southern states enjoy a significantly longer crushing season, often extending to 7 or 8 months, compared to only 4 or 5 months in Northern states like Uttar Pradesh and Bihar.
• The maritime influence in the south prevents the sugarcane from drying out quickly, whereas the extreme heat and dry winds in the north during summer reduce the juice content rapidly.
• The success of cooperative societies in states like Maharashtra and Karnataka has provided a more organized and efficient structure for sugar production and distribution.

 Additional Information

• Geographical Growth Requirements:
• Sugarcane requires a hot and humid climate with temperatures ranging between 21°C to 27°C.
• It thrives in areas receiving annual rainfall between 75 cm and 100 cm; irrigation is essential in drier regions.
• It can be grown on a variety of soils, but deep rich loamy soil or black cotton soil is ideal for high yields.
• Industry Characteristics:
• Sugarcane is a weight-losing crop, meaning the sugar content begins to decrease as soon as it is harvested due to the evaporation of moisture.
• To maximize recovery, sugar mills must be located in close proximity to the cane-producing areas to minimize transport time.
• India is the second-largest producer of sugar in the world after Brazil, and it also produces Gur and Khandsari.

The correct answer is Regional integration through trade, tech, and transport.

Key Points

• The Bay of Bengal Initiative for Multi-Sectoral Technical and Economic Cooperation (BIMSTEC) is a regional organization that acts as a bridge between South Asia and Southeast Asia, focusing on the Bay of Bengal littoral areas.
• The BIMSTEC Charter, adopted during the 5th Summit held in Colombo, Sri Lanka in 2022, formalized the group as an international legal entity with a defined set of foundational goals.
• A primary objective outlined in the charter is to promote regional integration by fostering multi-sectoral cooperation in areas such as trade, investment, technology, and transportation.
• The initiative aims to accelerate economic growth and social progress through mutual assistance and cooperation in matters of common interest, particularly connectivity and disaster management.
• The charter emphasizes the importance of multimodal connectivity to enhance people-to-people contact and intra-regional trade, which are critical for the collective prosperity of the member nations.

 Additional Information

• Member States:
• The organization comprises seven members: Bangladesh, Bhutan, India, Myanmar, Nepal, Sri Lanka, and Thailand.
• It was established on 6 June 1997 through the Bangkok Declaration.
• Strategic Pillars:
• Following the 2022 Charter, cooperation was streamlined into seven pillars, with India leading the Security pillar and Thailand leading Connectivity.
• Other pillars include Trade and Investment, Environment and Climate Change, Agriculture and Food Security, and Science, Technology, and Innovation.

The correct answer is Increased role of multinational corporations.

Key Points

• In July 1991, India launched the New Economic Policy (NEP) to tackle a severe Balance of Payments (BoP) crisis and dwindling foreign exchange reserves, marking a shift from an inward-looking economy to a globalized one.
• Globalization involved the integration of the national economy with the world economy by removing trade barriers, reducing import duties, and encouraging foreign capital inflows.
• A central feature of these reforms was the liberalization of Foreign Direct Investment (FDI), which paved the way for the increased role of multinational corporations (MNCs) in sectors like telecommunications, electronics, and automobiles.
• The government significantly relaxed the Monopolies and Restrictive Trade Practices (MRTP) Act, allowing MNCs to expand and invest without the rigorous and restrictive prior approvals that characterized the pre-reform era.
• The shift allowed MNCs to bring in advanced technology, global management practices, and competitive pricing, which fundamentally transformed the Indian industrial landscape.

 Additional Information

• LPG Framework:
• Liberalization: Focused on deregulation, it ended the License Raj for most industries and reformed the financial sector and tax structures.
• Privatization: Involved the disinvestment of government shares in Public Sector Undertakings (PSUs) to improve operational efficiency and reduce the fiscal deficit.
• Currency Reform: To boost exports, the Indian Rupee was devalued in 1991, and eventually, partial convertibility of the rupee was introduced on the current account.
• FDI Limits: Instead of restricting investment, the 1991 policy allowed automatic approval for FDI up to 51 percent in high-priority industries, which was a sharp increase from previous restrictive caps.
• Regulatory Transition: The restrictive Foreign Exchange Regulation Act (FERA) was eventually replaced by the more liberal Foreign Exchange Management Act (FEMA) in 1999 to facilitate external trade.

The correct answer is Eighth Five-Year Plan.

Key Points

• The Eighth Five-Year Plan, spanning from 1992 to 1997, was launched during the tenure of Prime Minister P.V. Narasimha Rao and Finance Minister Dr. Manmohan Singh.
• It served as the formal roadmap for the implementation of the New Economic Policy (1991), which aimed to tackle the severe Balance of Payments crisis and high inflation.
• The plan was instrumental in dismantling the License-Permit Raj by removing industrial licensing requirements for most sectors and significantly reducing price controls and rationing.
• It transitioned the Indian economy from a centrally planned model toward a market-oriented economy, emphasizing the role of the private sector in national growth.
• One of the primary goals was Human Resource Development, focusing on employment generation, primary education, and the eradication of illiteracy among the age group of 15 to 35 years.
• The plan was highly successful, achieving an annual GDP growth rate of 6.8 percent, which significantly exceeded the initial target of 5.6 percent.

 Additional Information

• Economic Liberalisation Concepts:
• LPG Model: Stands for Liberalisation, Privatisation, and Globalisation, which allowed for increased Foreign Direct Investment (FDI) and reduced trade barriers.
• Indicative Planning: During this period, the role of the Planning Commission shifted from a directive authority to an indicative one, focusing on long-term vision rather than micro-management.
• Other Plans Context:
• Fifth Five-Year Plan (1974–1978): Primarily focused on Garibi Hatao (poverty removal) and self-reliance in the agricultural sector.
• Sixth Five-Year Plan (1980–1985): Marked the beginning of economic liberalization in a very limited scope but remained largely focused on infrastructure and social welfare.
• Seventh Five-Year Plan (1985–1990): Concentrated on Food, Work, and Productivity, emphasizing technology upgrades and food grain production.

The correct answer is 1-S, 2-P, 3-R, 4-Q.

Key Points

• Article 249 provides that if the Rajya Sabha declares by a resolution supported by not less than two-thirds of the members present and voting that it is necessary or expedient in the national interest, Parliament may make laws for the whole or any part of India with respect to any matter in the State List. Hence, 1 matches with S.
• Article 250 empowers the Parliament to legislate on any item mentioned in the State List while a Proclamation of Emergency (under Article 352) is in operation; such laws cease to have effect on the expiration of six months after the proclamation has ceased to operate. Hence, 2 matches with P.
• Article 252 states that if the Legislatures of two or more States pass resolutions requesting the Parliament to enact laws on a matter in the State List, the Parliament can pass an act for regulating that matter, which applies only to those states and any other state that adopts it later. Hence, 3 matches with R.
• Article 253 enables the Parliament to make any law for the whole or any part of the territory of India for implementing any treaty, agreement, or convention with any other country or countries or any decision made at any international conference. Hence, 4 matches with Q.

 Additional Information

• Legislative Relations:
• Part XI of the Indian Constitution (Articles 245 to 255) defines the legislative relations between the Union and the States.
• The Seventh Schedule contains three lists: the Union List (100 subjects), the State List (61 subjects), and the Concurrent List (52 subjects).
• Under Article 252, a law passed by Parliament can only be amended or repealed by the Parliament itself and not by the legislature of the states concerned.
• Article 248 vests Residuary powers of legislation in the Parliament, covering matters not enumerated in any of the three lists.

The correct answer is Sikandar Lodi.

Key Points

• Sikandar Lodi (reigned 1489−1517), born as Nizam Khan, was the most powerful ruler of the Lodi Dynasty and the son of Bahlul Lodi.
• He founded the modern city of Agra in 1504 to maintain strategic control over the Doab region and the trade routes leading to Malwa and Rajasthan.
• In 1506, he officially transferred the capital of the Delhi Sultanate from Delhi to Agra, significantly enhancing the city’s political importance.
• He introduced a standardized land measurement system known as the Gazz-i-Sikandari (Sikandari Gaj), which consisted of 32 digits (anguls) and remained the official standard for land revenue until the era of Akbar.
• A great patron of literature and art, he was a scholar himself and wrote Persian poetry under the pen name Gulrukhi.
• He implemented rigorous administrative reforms, established an efficient espionage system, and ensured that the prices of essential commodities remained low for the public.

 Additional Information

• Bahlul Lodi:
• He was the founder of the Lodi Dynasty in 1451 and the first Afghan ruler of the Delhi Sultanate.
• He is credited with the conquest of the Sharqi Kingdom of Jaunpur, expanding the sultanate’s eastern boundaries.
• Ibrahim Lodi:
• He was the last Sultan of the Delhi Sultanate and the son of Sikandar Lodi.
• His reign ended after his defeat and death at the hands of Babur in the First Battle of Panipat in 1526.
• Daulat Khan Lodi:
• He served as the Governor of Lahore during the turbulent reign of Ibrahim Lodi.
• Due to his conflicts with the Sultan, he invited the Mughal prince Babur to invade India, which eventually led to the fall of the Lodi Dynasty.

The correct answer is Panchkula, Haryana.

Key Points

• The 63rd National Inter-State Senior Athletics Championships were held at the Tau Devi Lal Stadium in Panchkula, Haryana.
• The four-day event took place from June 27 to June 30, 2024, and was organized by the Athletics Federation of India (AFI).
• This competition served as the final domestic qualifying event for Indian athletes to secure their spots for the Paris 2024 Olympics.
• The meet featured several high-stakes events across track and field categories, attracting the nation's elite athletes seeking to meet the World Athletics qualification standards before the June 30 deadline.

 Additional Information

• Athletics Federation of India (AFI):
• It is the apex body for athletics in India, headquartered in New Delhi and founded in 1946.
• The AFI is responsible for organizing national championships and managing Indian athletics' participation in international events like the Asian Games and Olympics.
• Olympic Qualification Highlights:
• During the 2024 Panchkula meet, Kiran Pahal made headlines by qualifying for the women's 400m race with a time of 50.92 seconds.
• The event was critical for relay teams and individual athletes to improve their World Athletics Road to Paris rankings.
• Venue Significance:
• The Tau Devi Lal Stadium is a multi-sport complex that previously hosted the Khelo India Youth Games in 2022, establishing Panchkula as a prominent sports hub in North India.

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

Key Points

• The National Anti-Doping Agency (NADA) is an autonomous body established by the Government of India to ensure a dope-free sports environment. It functions under the administrative control of the Ministry of Youth Affairs and Sports, not the Ministry of Education. Hence, assertion A is incorrect.
• The primary objective of NADA is to implement anti-doping rules and policies that conform to the World Anti-Doping Agency (WADA) code. It is responsible for regulating drug testing, monitoring anti-doping measures, and promoting education and research regarding the prohibited use of performance-enhancing drugs in Indian sports. Hence, reason R is correct.

 Additional Information

• National Anti-Doping Agency (NADA):
• It was established on November 24, 2005, and is registered under the Societies Registration Act of 1860.
• The agency is tasked with sample collection (both blood and urine) from athletes during in-competition and out-of-competition periods.
• It works in coordination with the National Dope Testing Laboratory (NDTL), which is responsible for the analytical testing of the samples.
• The NADA India App was launched to provide athletes with a comprehensive list of prohibited substances to prevent accidental doping.
• World Anti-Doping Agency (WADA):
• Founded in 1999 and headquartered in Montreal, Canada, it sets the global standards for anti-doping regulations through the World Anti-Doping Code.

The correct answer is Chandragupta Maurya.

Key Points

• Chandragupta Maurya established the Maurya Empire in 322 BCE, which is historically recognized as the first major pan-Indian empire that unified most of the Indian subcontinent.
• With the strategic assistance of his mentor and prime minister, Chanakya (also known as Kautilya), he successfully overthrew the Nanda Dynasty by defeating its last ruler, Dhana Nanda.
• His military campaigns led to the expansion of his territory from the Indo-Gangetic Plain to the North-West frontier, covering regions up to the Hindu Kush mountains in modern-day Afghanistan.
• In 305 BCE, he successfully defeated Seleucus I Nicator, a former general of Alexander the Great, leading to a peace treaty that ceded territories like Arachosia (Kandahar) and Gedrosia (Balochistan) to the Mauryan state.
• The Greek diplomat Megasthenes was sent to his court at Pataliputra, where he authored the famous work Indica, providing a detailed account of the socio-political and administrative structure of the empire.
• According to historical and religious traditions, he eventually abdicated the throne in favor of his son Bindusara, embraced Jainism under the influence of the monk Bhadrabahu, and retired to Shravanabelagola.

 Additional Information

• Ashoka:
• He was the grandson of Chandragupta Maurya and the third ruler of the dynasty, famous for the Kalinga War in 261 BCE.
• Following the bloodshed of the war, he converted to Buddhism and popularized the concept of Dhamma through his numerous Rock Edicts and Pillar Inscriptions.
• Kanishka:
• He was the most prominent ruler of the Kushan Empire, reigning during the 2nd century CE.
• He is noted for convening the Fourth Buddhist Council in Kashmir and initiating the Saka Era in 78 CE.
• Samudragupta:
• A powerful ruler of the Gupta Empire, he is often termed the Napoleon of India by historian V.A. Smith.
• His conquests and personality are detailed in the Prayag Prashasti, an inscription composed by his court poet Harisena.

The correct answer is PM2.5.

Key Points

• PM2.5 refers to fine Particulate Matter with a diameter of less than 2.5 micrometers, which is small enough to enter the alveoli of the lungs and the bloodstream.
• In Delhi, particularly during the winter months like January 2025, PM2.5 is the dominant pollutant that dictates the Air Quality Index (AQI) category due to its high concentration.
• The high levels of PM2.5 in the National Capital Region (NCR) are attributed to vehicular emissions, industrial smoke, and the persistence of secondary inorganic aerosols under low-temperature conditions.
• Meteorological factors such as low wind speed and temperature inversion prevent the dispersion of these fine particles, leading to 'Severe' or 'Severe Plus' air quality ratings as defined by the Central Pollution Control Board (CPCB).
• While the AQI is calculated based on eight pollutants, the index value is usually determined by the lead pollutant, which in Delhi's winter is almost exclusively PM2.5.

 Additional Information

• AQI Pollutants:
• PM10: These are coarser particles up to 10 micrometers in size, often originating from construction dust and mechanical grinding.
• Sulphur Dioxide (SO2): Primarily emitted from thermal power plants and metal smelting; it is a precursor to acid rain but rarely the primary driver of Delhi's AQI.
• Ozone (O3): A secondary pollutant formed by the reaction of NOx and VOCs in the presence of sunlight; it is more problematic during the summer months.
• Regulatory Framework:
• National Air Quality Index: Launched in October 2014, it comprises six categories ranging from 'Good' to 'Severe'.
• GRAP: The Graded Response Action Plan is a set of emergency measures implemented by the Commission for Air Quality Management (CAQM) based on pollution severity.

The correct answer is Deemed permission.

Key Points

• The Bharatiya Nagarik Suraksha Sanhita (BNSS), 2023, which replaced the Code of Criminal Procedure (CrPC), was implemented across India on July 1, 2024.
• Under the new legal framework, specifically under Section 218 of the BNSS, a time-bound mechanism has been introduced for the sanction of prosecution of public servants and police officers.
• The law stipulates that the Competent Authority (either the Central or State Government) must decide on the request for prosecution within a period of 120 days from the date of receipt of the proposal.
• If the government fails to provide a decision or communicate its refusal within this stipulated 120-day window, the permission is legally presumed to have been granted. This legal provision is termed Deemed Permission.
• This reform aims to eliminate the long-standing issue of administrative delays that often protected corrupt or erring officials from legal accountability for years under the previous CrPC regime.

 Additional Information

• Bharatiya Nagarik Suraksha Sanhita (BNSS) Context:
• The BNSS was enacted to modernize the criminal justice system, focusing on victim-centric justice and the integration of technology in investigations.
• The Section 218 provision specifically targets Section 197 of the CrPC equivalent, ensuring that official immunity does not lead to impunity.
• This provision is particularly relevant for offenses committed by public servants while acting or purporting to act in the discharge of their official duties.
• The implementation of Deemed Permission aligns with various Supreme Court observations regarding the necessity of speedy trials and the removal of hurdles in the cognizance of offenses involving corruption or misuse of power.

Shortcut Trick

Total deer in the reserve = 200.

Since 10% of 200 = 20.

Then, 30% of 200 = 3 × 20 = 60.

∴ The correct answer is 60 deer.

Alternate Method

Given:

Total number of deer = 200

Percentage of deer tagged with GPS collars = 30%

Formula Used:

Value = (Percentage ÷ 100) × Total Value

Calculations:

⇒ Number of tagged deer = (30 ÷ 100) × 200

⇒ Number of tagged deer = 30 × (200 ÷ 100)

⇒ Number of tagged deer = 30 × 2

⇒ Number of tagged deer = 60

∴ The correct answer is 60 deer.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide by 100. Example: 30% = 30/100 = 3/10.

Finding Percentage of a Number

To find x% of y, the formula is (x × y) ÷ 100. This is a fundamental concept in data interpretation and commercial arithmetic.

Complementary Percentage

If 30% are tagged, then the remaining (100% − 30%) = 70% are untagged. Untagged deer = 0.70 × 200 = 140.

Shortcut Trick

Time taken by Van 1 = 4h and Van 2 = 3.5h. Let total distance = 28 units (LCM of 4, 3.5).

Speeds: V₁ = 28 ÷ 4 = 7 units/h, V₂ = 28 ÷ 3.5 = 8 units/h.

Distance covered by Van 1 by 7:00 a.m. (in 2 hours) = 7 × 2 = 14 units.

Remaining distance at 7:00 a.m. = 28 − 14 = 14 units. Relative speed = 7 + 8 = 15 units/h.

Meeting time = 7:00 a.m. + (14 ÷ 15) × 60 min = 7:00 a.m. + 56 min = 7:56 a.m.

∴ The correct answer is 7:56 a.m.

Alternate Method

Given:

Van 1 (Jaipur to Agra): Departure = 5:00 a.m., Arrival = 9:00 a.m.

Van 2 (Agra to Jaipur): Departure = 7:00 a.m., Arrival = 10:30 a.m.

Formula Used:

Distance = Speed × Time | Relative Speed (Opposite) = S₁ + S₂ | Time = Distance ÷ Speed

⇒ Time taken by Van 1 = 9:00 − 5:00 = 4 hours

⇒ Time taken by Van 2 = 10:30 − 7:00 = 3.5 hours = 7/2 hours

⇒ Let total distance (D) = LCM(4, 3.5) = 28 km

⇒ Speed of Van 1 (S₁) = 28 ÷ 4 = 7 km/h

⇒ Speed of Van 2 (S₂) = 28 ÷ 3.5 = 8 km/h

⇒ Distance covered by Van 1 from 5:00 a.m. to 7:00 a.m. = 7 × 2 = 14 km

⇒ Remaining distance between vans at 7:00 a.m. = 28 − 14 = 14 km

⇒ Relative speed (opposite direction) = 7 + 8 = 15 km/h

⇒ Time to meet after 7:00 a.m. = 14 ÷ 15 hours

⇒ Converting to minutes = (14/15) × 60 = 56 minutes

⇒ Meeting Time = 7:00 a.m. + 56 minutes = 7:56 a.m.

∴ The correct answer is 7:56 a.m.

Additional Information

Relative Speed Concept

When two objects move toward each other, their relative speed is the sum of their individual speeds (S₁ + S₂).

Meeting Time with Different Starts

Always calculate the lead distance covered by the first object until the second object starts moving before applying relative speed.

Time Conversion Shortcut

To convert hours into minutes, multiply by 60. For example, 1/15 hour = 4 minutes; 14/15 hour = 56 minutes.

Shortcut Trick

In a 30°−60°−90° triangle, the ratio of the sides opposite to the angles is 1 : √3 : 2.

Side opposite to 30° (Height) = 1 unit = 50 m.

Side opposite to 60° (Distance) = √3 units = 50√3 m.

∴ The correct answer is 50√3 m.

Alternate Method

Given: Height of the tower (AB) = 50 m, Angle of elevation (∠ACB) = 30°.

Formula: tan θ = Perpendicular ÷ Base

Let the distance of the man from the foot of the tower be x.

⇒ In right-angled triangle ABC, tan 30° = AB ÷ BC

⇒ 1 ÷ √3 = 50 ÷ x

⇒ x = 50 × √3

⇒ x = 50√3 m

∴ The correct answer is 50√3 m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when the object is above the horizontal level.

Trigonometric Ratios for Standard Angles

tan 30° = 1 ÷ √3 | tan 45° = 1 | tan 60° = √3.

Angle of Depression

The angle formed by the line of sight with the horizontal when the object is below the horizontal level. Numerically, Angle of Elevation = Angle of Depression.

Shortcut Trick

For a 30° angle in a right triangle, the ratio of Height : Hypotenuse is 1 : 2.

Total Tree Height = Standing Height + Broken (Hypotenuse) part = 1 + 2 = 3 units.

Given, 3 units = 20 m ⇒ 1 unit = 20/3 m.

Height of the broken point is 1 unit = 20/3 m.

∴ The correct answer is 20/3 m.

Alternate Method

Given:

Total length of the tree = 20 m

Angle of elevation (θ) = 30°

Let the height of the break point from the ground be 'h' meters.

Length of the broken part (Hypotenuse) = (20 − h) meters.

Formula Used:

sin θ = Perpendicular ÷ Hypotenuse

Calculations:

⇒ sin 30° = h ÷ (20 − h)

⇒ 1/2 = h ÷ (20 − h)

⇒ 20 − h = 2h

⇒ 3h = 20

⇒ h = 20/3 m

∴ The correct answer is 20/3 m.

Additional Information

Trigonometric Ratios

For standard angles: sin 30° = 1/2, sin 45° = 1/√2, and sin 60° = √3/2.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when looking up at an object.

Right Triangle Ratios

In a 30°−60°−90° triangle, the sides are always in the ratio 1 : √3 : 2.

Shortcut Trick

Convert successive discounts into a single multiplier ratio: 20% = 1/5 and 25% = 1/4.

The Marked Price (MP) to Selling Price (SP) ratio is: (5 × 4) : (4 × 3) = 20 : 12 = 5 : 3.

Given final Selling Price (3 units) = ₹5,40,000.

Value of 1 unit = 5,40,000 ÷ 3 = ₹1,80,000.

Marked Price (5 units) = 5 × 1,80,000 = ₹9,00,000.

∴ The correct answer is Rs. 900000.

Alternate Method

Given:

Final Selling Price of the car = Rs. 5,40,000

First Discount (D₁) = 20% | Second Discount (D₂) = 25%

Formula Used:

Selling Price (SP) = Marked Price (MP) × (100 − D₁)/100 × (100 − D₂)/100

Calculations:

⇒ 5,40,000 = MP × (100 − 20)/100 × (100 − 25)/100

⇒ 5,40,000 = MP × (80/100) × (75/100)

⇒ 5,40,000 = MP × (4/5) × (3/4)

⇒ 5,40,000 = MP × (3/5)

⇒ MP = (5,40,000 × 5) ÷ 3

⇒ MP = 1,80,000 × 5

⇒ MP = 9,00,000

∴ The correct answer is Rs. 900000.

Additional Information

Single Equivalent Discount

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Marked Price vs Selling Price

Marked Price is the list price before discount, while Selling Price = MP × (1 − d/100) where d is the discount percentage.

Effective Multiplier Method

If multiple discounts are applied, the final SP = MP × m₁ × m₂... where m is the fraction (100−d)/100.

Shortcut Trick

Use the direct ratio: CP : MP = (100 − Discount%) : (100 + Gain%).

CP : MP = (100 − 34) : (100 + 10) = 66 : 110.

Simplified Ratio: CP : MP = 3 : 5.

According to the question, 3 units = Rs. 6000 → 1 unit = Rs. 2000.

Marked Price (5 units) = 5 × 2000 = Rs. 10,000.

∴ The correct answer is Rs. 10,000.

Alternate Method

Given:

Cost Price (CP) = Rs. 6000

Gain Percentage = 10%

Discount Percentage = 34%

Formula Used:

Selling Price (SP) = CP × (100 + Gain%)/100

Marked Price (MP) = SP × 100 / (100 − Discount%)

Calculations:

⇒ SP = 6000 × (110 / 100)

⇒ SP = 6000 × 1.1 = Rs. 6600

⇒ MP = 6600 × 100 / (100 − 34)

⇒ MP = 6600 × 100 / 66

⇒ MP = 100 × 100 = Rs. 10,000

∴ The correct answer is Rs. 10,000.

Additional Information

Direct CP and MP Relation

The formula MP/CP = (100 + Gain%)/(100 − Discount%) is extremely useful for exams to find any one unknown variable out of the four.

Discount and Profit Bases

Always remember that Discount is calculated on the Marked Price, while Profit or Loss is always calculated on the Cost Price.

Successive Discounts

If two successive discounts x% and y% are given, the single equivalent discount is [x + y − (x × y)/100]%.

Shortcut Trick

Total work remains constant, so use M₁D₁ = M₂D₂.

46 × 16 = 32 × Required Days

Required Days = (46 × 16) ÷ 32

Required Days = 46 ÷ 2 = 23 days.

∴ The correct answer is 23 days.

Alternate Method

Given:

Initial count of men (M₁) = 46

Initial time taken (D₁) = 16 days

New count of men (M₂) = 32

Formula Used:

When the amount of work is the same, we apply the inverse variation rule:

M₁ × D₁ = M₂ × D₂

Calculations:

⇒ 46 × 16 = 32 × D₂

⇒ 736 = 32 × D₂

⇒ D₂ = 736 ÷ 32

⇒ D₂ = 23

∴ The correct answer is 23 days.

Additional Information

General Chain Rule Formula

The complete relation for work is (M₁×D₁×H₁)/W₁ = (M₂×D₂×H₂)/W₂, where H is hours and W is units of work.

Concept of Man-Days

Total work is often measured in man-days, which is the product of the number of men and the number of days they work.

Inverse Proportion Rule

If Work is constant, Men and Days are inversely proportional. If the number of men decreases by a certain ratio, the days must increase by the reciprocal of that ratio.

Shortcut Trick

The perpendicular distance from the center to chord AB is OC = √(13² − 12²) = 5 cm.

Given CF = 5 + 4√5 cm and CF = OC + OF (for parallel chords on opposite sides), then OF = 4√5 cm.

Half the length of chord DE is FD = √(13² − (4√5)²) = √(169 − 80) = √89 cm.

Full length of chord DE = 2 × √89 = 2√89 cm.

∴ The correct answer is 2√89 cm.

Alternate Method

Given:

Radius (R) = 13 cm

Chord AB = 24 cm, OC ⊥ AB

Distance CF = (5 + 4√5) cm, OF ⊥ DE

Formula Used:

Length of perpendicular from center to chord = √(Radius² − (Chord/2)²)

Calculations:

⇒ AC = CB = AB ÷ 2 = 24 ÷ 2 = 12 cm

⇒ In right triangle OCB, OC² + CB² = OB²

⇒ OC² + 12² = 13²

⇒ OC² + 144 = 169

⇒ OC² = 25 ⇒ OC = 5 cm

⇒ Since OC and OF are perpendicular to parallel chords, CF = OC + OF

⇒ 5 + 4√5 = 5 + OF

⇒ OF = 4√5 cm

⇒ In right triangle OFD, OF² + FD² = OD² (where OD = Radius = 13)

⇒ (4√5)² + FD² = 13²

⇒ 80 + FD² = 169

⇒ FD² = 169 − 80 = 89

⇒ FD = √89 cm

⇒ Length of chord DE = 2 × FD = 2√89 cm

∴ The correct answer is 2√89 cm.

Additional Information

Perpendicular from the Center

A perpendicular drawn from the center of a circle to a chord bisects the chord into two equal segments.

Distance between Parallel Chords

The distance between two parallel chords on opposite sides of the center is the sum of their perpendicular distances from the center (d = d₁ + d₂).

Chord Length Formula

The length of a chord L at a distance d from the center of a circle with radius R is given by L = 2√(R² − d²).

Shortcut Trick

Identify the last digit of the area: 5, so the side must end in 5.

Estimate the range: 180² = 32,400 and 190² = 36,100.

Since 34,225 lies between 32,400 and 36,100, the side must be 185.

Verification: 18 × 19 = 342, then append 25 → 34,225.

∴ The correct answer is 185 units.

Alternate Method

Given: Area of the square = 34,225 square units.

Formula Used: Area of square = (side)²

⇒ side² = 34,225

⇒ side = √34,225

⇒ side = √(25 × 1369)

⇒ side = 5 × √1369

⇒ side = 5 × 37

⇒ side = 185 units

∴ The correct answer is 185 units.

Additional Information

Perimeter of a Square

The total boundary length of a square is given by the formula: Perimeter = 4 × side.

Diagonal of a Square

The straight line distance between opposite corners is: Diagonal = side × √2.

Area using Diagonal

If the diagonal (d) is known, the area can be calculated as: Area = d² ÷ 2.

Shortcut Trick

Length of tangents from an external point are equal ⇒ PR = PS = 14 cm.

Radii of the same circle are equal ⇒ OR = OS = 10 cm.

Perimeter of quadrilateral ORPS = Sum of all four sides.

Perimeter = 2 × (Radius + Tangent Length) = 2 × (10 + 14) = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Radius of circle (OR = OS) = 10 cm

Length of tangent PR = 14 cm

Formula Used:

Perimeter of a quadrilateral = Sum of the lengths of its four sides

Tangent property: Tangents drawn from an external point to a circle are equal in length.

Calculations:

⇒ In quadrilateral ORPS, the sides are OR, RP, PS, and SO.

⇒ PR and PS are tangents from point P ⇒ PR = PS = 14 cm.

⇒ OR and OS are radii of the circle ⇒ OR = OS = 10 cm.

⇒ Perimeter of ORPS = OR + RP + PS + SO

⇒ Perimeter = 10 + 14 + 14 + 10

⇒ Perimeter = 24 + 24 = 48 cm.

∴ The correct answer is 48 cm.

Additional Information

Radius and Tangent Relationship

The radius of a circle is always perpendicular to the tangent at the point of contact (∠ORP = ∠OSP = 90°).

Cyclic Quadrilateral Property

Quadrilateral ORPS is a cyclic quadrilateral because the sum of opposite angles ∠ORP + ∠OSP = 180°, which implies ∠ROS + ∠RPS = 180°.

Line Joining Center and External Point

The line OP bisects the angle between the two tangents (∠RPS) and the angle between the two radii (∠ROS).

Shortcut Trick

In melting and recasting, the total volume remains constant.

Volume of Hemisphere = n × Volume of Cone

(2/3)πr³ = n × (1/3)πr²h

By canceling (1/3), π, and r²: 2r = n × h

n = 2r / h

∴ The correct answer is 2r/h.

Alternate Method

Given:

Radius of hemisphere = r

Base radius of cone = r

Height of cone = h

Number of identical cones = n

Formula Used:

Volume of Hemisphere = (2/3)πr³

Volume of Cone = (1/3)πr²h

Calculations:

⇒ Total volume of melted hemisphere = Total volume of n cones formed

⇒ (2/3) × π × r³ = n × (1/3) × π × r² × h

⇒ 2 × r³ / 3 = n × r² × h / 3

⇒ 2 × r³ = n × r² × h

⇒ 2 × r = n × h

⇒ n = 2r / h

∴ The correct answer is 2r/h.

Additional Information

Volume of Sphere

The volume of a full sphere is exactly double that of a hemisphere: V = (4/3)πr³.

Surface Area of Hemisphere

The curved surface area (CSA) is 2πr² and the total surface area (TSA) is 3πr².

Volume of Cylinder

A cylinder with the same radius and height as a cone has three times the volume: V = πr²h.

Shortcut Trick

Given height (h) is 4 times the radius (r) ⇒ h = 4r.

Total Surface Area (TSA) formula becomes: 2πr(h + r) = 2πr(4r + r) = 2πr(5r) = 10πr².

For r = 7 cm, the base area (πr²) is always 154 cm².

TSA = 10 × 154 = 1540 cm².

∴ The correct answer is 1540 cm².

Alternate Method

Given:

Base radius (r) = 7 cm

Height (h) = 4 × radius = 4 × 7 = 28 cm

Formula Used:

Total Surface Area of Cylinder = 2πr(h + r)

Calculations:

⇒ Total Surface Area = 2 × (22/7) × 7 × (28 + 7)

⇒ Total Surface Area = 2 × 22 × 35

⇒ Total Surface Area = 44 × 35

⇒ Total Surface Area = 1540 cm²

∴ The correct answer is 1540 cm².

Additional Information

Curved Surface Area (CSA)

The area of the side surface excluding the top and bottom bases is given by 2πrh.

Volume of a Cylinder

The space occupied by the cylinder is calculated using the formula V = πr²h.

Base Area

Since the base of a cylinder is a circle, its area is πr². A cylinder has two such bases.

Shortcut Trick

Ratio of radius (r) to height (h) = 7 : 12. Let r = 7x and h = 12x.

Volume = 1/3 × π × r² × h ⇒ 77 = 1/3 × 22/7 × 49x² × 12x.

77 = 616x³ ⇒ x³ = 77/616 = 1/8.

x = √³(1/8) = 0.5. Hence, Radius (r) = 7 × 0.5 = 3.5 cm.

∴ The correct answer is 3.5 cm.

Alternate Method

Given:

• Ratio of radius (r) and height (h) = 7 : 12

• Volume of the cone (V) = 77 cm³

• Value of π = 22/7

Formula Used:

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

Calculations:

⇒ Let the radius (r) be 7x and the height (h) be 12x.

⇒ Volume (V) = (1/3) × (22/7) × (7x)² × (12x) = 77

⇒ (1/3) × (22/7) × 49x² × 12x = 77

⇒ 22 × 7 × 4 × x³ = 77

⇒ 616x³ = 77

⇒ x³ = 77 ÷ 616

⇒ x³ = 1/8

⇒ x = 1/2 = 0.5

⇒ Radius (r) = 7 × 0.5 = 3.5 cm.

∴ The correct answer is 3.5 cm.

Additional Information

Slant Height of a Cone

The slant height (l) is the distance from the apex to any point on the base circumference: l = √(r² + h²).

Curved Surface Area (CSA)

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

Total Surface Area (TSA)

The total area including the circular base: TSA = π × r × (l + r).

Shortcut Trick

Given: sin x = 4/5 and cos x = 3/5.

Expression: sin[2(x + π/8)] = sin(2x + π/4) = (sin 2x + cos 2x) / √2.

sin 2x = 2 × (4/5) × (3/5) = 24/25.

cos 2x = cos² x − sin² x = (3/5)² − (4/5)² = −7/25.

Value = (24/25 − 7/25) / √2 = 17 / (25√2).

∴ The correct answer is 17 / (25√2).

Alternate Method

Given: cosec x = 5/4 and sec x = 5/3.

Formula Used: sin(A + B) = sin A cos B + cos A sin B; sin 2x = 2 sin x cos x; cos 2x = cos² x − sin² x.

⇒ sin x = 1/cosec x = 4/5

⇒ cos x = 1/sec x = 3/5

⇒ sin 2x = 2 × (4/5) × (3/5) = 24/25

⇒ cos 2x = (3/5)² − (4/5)² = 9/25 − 16/25 = −7/25

⇒ sin[2(x + π/8)] = sin(2x + π/4)

⇒ sin(2x + π/4) = sin 2x cos(π/4) + cos 2x sin(π/4)

⇒ sin(2x + π/4) = (24/25) × (1/√2) + (−7/25) × (1/√2)

⇒ sin(2x + π/4) = (24 − 7) / (25√2) = 17 / (25√2)

∴ The correct answer is 17 / (25√2).

Additional Information

Double Angle Identities

sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ.

Compound Angle Formulas

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

Trigonometric Values of π/4

At π/4 (45°), the values are sin(π/4) = 1/√2 and cos(π/4) = 1/√2.

Shortcut Trick

A quadrantal angle is an angle in standard position whose terminal side lies on any coordinate axis.

In radians, quadrantal angles are always integral multiples of π/2 (e.g., 0, π/2, π, 3π/2, 2π etc.).

Checking the options: π/6, π/3, and π/4 are not multiples of π/2.

Only π/2 satisfies this condition as it lies on the positive y-axis.

∴ The correct answer is π/2.

Alternate Method

Given: Angles in radians π/6, π/3, π/4, and π/2.

Formula: An angle θ is quadrantal if θ = n × 90° or n × (π/2) radians, where n is an integer.

Step-by-step Calculation:

⇒ Convert π/2 to degrees: (π/2) × (180/π) = 90° (Lies on positive Y-axis).

⇒ Convert π/3 to degrees: (π/3) × (180/π) = 60° (Lies in Quadrant I).

⇒ Convert π/4 to degrees: (π/4) × (180/π) = 45° (Lies in Quadrant I).

⇒ Convert π/6 to degrees: (π/6) × (180/π) = 30° (Lies in Quadrant I).

⇒ Since only 90° falls on an axis, π/2 is the quadrantal angle.

∴ The correct answer is π/2.

Additional Information

Quadrantal Angles

These are angles whose terminal side coincides with the x-axis or y-axis. Examples include 0, π/2, π, 3π/2, 2π, and their negative counterparts like −π/2.

Trigonometric Values of Quadrantal Angles

For π/2 (90°), sin(π/2) = 1, cos(π/2) = 0, and tan(π/2) is undefined because the x-coordinate on the unit circle is 0.

General Form

Any angle θ that can be written as θ = n × (π/2) for any integer n is a quadrantal angle.

Shortcut Trick Sum of two complementary angles is always 90°. Larger angle = (Sum + Difference) ÷ 2 = (90° + 52°) ÷ 2 = 142° ÷ 2 = 71°. Smaller angle = (Sum − Difference) ÷ 2 = (90° − 52°) ÷ 2 = 38° ÷ 2 = 19°. ∴ The correct answer is 19°, 71°.

Alternate Method Given: Difference between angles = 52° Sum of complementary angles = 90°

Let the two angles be x and y (where x > y). ⇒ x + y = 90° --- (Equation 1) ⇒ x − y = 52° --- (Equation 2) Adding Equation 1 and Equation 2: ⇒ (x + y) + (x − y) = 90° + 52° ⇒ 2x = 142° ⇒ x = 142° ÷ 2 = 71° Substitute x in Equation 1: ⇒ 71° + y = 90° ⇒ y = 90° − 71° = 19° ∴ The correct answer is 19°, 71°.

Additional Information Complementary Angles Two angles are called complementary if their sum is exactly 90°. Supplementary Angles Two angles are called supplementary if their sum is exactly 180°. Conjugate Angles Two angles are called conjugate or explementary if their sum is 360°. Linear Pair A pair of adjacent angles formed by intersecting lines that are supplementary.

Shortcut Trick

The standard algebraic identity for the sum of two cubes is a³ + b³ = (a + b)(a² − ab + b²).

Dividing the identity (a + b)(a² − ab + b²) by the factor (a + b) directly yields the remaining factor.

Result = (a² − ab + b²).

∴ The correct answer is a² − ab + b².

Alternate Method

Given: (a³ + b³) ÷ (a + b)

Formula Used: a³ + b³ = (a + b)(a² − ab + b²)

⇒ Expression = (a³ + b³) / (a + b)

⇒ Expression = [(a + b) × (a² − ab + b²)] / (a + b)

⇒ Expression = a² − ab + b²

∴ The correct answer is a² − ab + b².

Additional Information

Difference of Two Cubes

The expression for the difference of cubes is a³ − b³ = (a − b)(a² + ab + b²).

Square of Binomials

Note the middle term difference: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

Difference of Two Squares

The identity for the difference of squares is a² − b² = (a − b)(a + b).

Shortcut Trick

Break each term into perfect square factors: √50 = √(25 × 2) and √8 = √(4 × 2).

Simplify to 5√2 and 2√2 respectively.

Sum the coefficients of the like surds: 5 + 2 = 7.

∴ The correct answer is 7√2.

Alternate Method

Given Expression: √50 + √8

Method: Simplification of radical expressions

⇒ √(25 × 2) + √(4 × 2)

⇒ (5 × √2) + (2 × √2)

⇒ (5 + 2) × √2

⇒ 7√2

∴ The correct answer is 7√2.

Additional Information

Like Surds

Surds that have the same irrational factor (radicand) in their simplest form are called like surds and can be added or subtracted.

Simplifying Radicals

A surd is simplified by extracting the greatest perfect square factor from the radicand using the property √(a²b) = a√b.

Product Rule for Surds

For any two positive numbers a and b, the rule √a × √b = √(ab) is used to combine or split square roots.

Shortcut Trick The ratio of perimeters of two similar triangles is always equal to the ratio of their corresponding sides. Given Side Ratio = 5 : 7 (Smaller : Larger) 7 units (Larger Perimeter) → 35 cm 1 unit → 5 cm 5 units (Smaller Perimeter) → 5 × 5 = 25 cm ∴ The correct answer is 25 cm. Alternate Method Given: Ratio of sides of triangular logo = 5 : 7 Perimeter of the larger logo = 35 cm Formula Used: Perimeter of Smaller Logo ÷ Perimeter of Larger Logo = Smaller Side ÷ Larger Side

Calculations: ⇒ Let the perimeter of the smaller logo be P. ⇒ P ÷ 35 = 5 ÷ 7 ⇒ P = (5 ÷ 7) × 35 ⇒ P = 5 × 5 ⇒ P = 25 cm ∴ The correct answer is 25 cm. Additional Information Properties of Similar Figures When two shapes are similar, the ratio of any linear measurements (like medians, altitudes, or angle bisectors) is equal to the side ratio. Area Ratio Theorem The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area1 ÷ Area2 = (s1 ÷ s2)2. Scale Factor (k) The scale factor represents the ratio of new dimension to original dimension. If k < 1, the figure is reduced; if k > 1, the figure is enlarged.

Shortcut Trick

For a circumscribed circle (circle outside), the diameter of the circle is equal to the diagonal of the square.

Diagonal of square = Side × √2 = 14 × 1.414 ≈ 19.8 cm.

Radius = Diagonal ÷ 2 = 19.8 ÷ 2 = 9.9 cm.

Rounding to the nearest option, we get 10 cm.

∴ The correct answer is ≈ 10 cm.

Alternate Method

Given:

Side of the square (a) = 14 cm

Formula Used:

Diagonal of a square = a√2

Radius of circumcircle (R) = Diagonal ÷ 2

Calculations:

⇒ Diagonal of the square = 14 × √2 cm

⇒ Diameter of the circumscribed circle = Diagonal = 14√2 cm

⇒ Radius of the circle (R) = (14√2) ÷ 2

⇒ R = 7√2 cm

⇒ Taking √2 ≈ 1.414

⇒ R ≈ 7 × 1.414

⇒ R ≈ 9.898 cm

⇒ R ≈ 10 cm (approx.)

∴ The correct answer is ≈ 10 cm.

Additional Information

Incircle of a Square

When a circle is inscribed inside a square, its diameter is equal to the side of the square. Radius (r) = Side ÷ 2.

Area Ratio

The ratio of the area of the circumcircle to the area of the incircle of the same square is always 2 : 1.

Diagonal Property

The diagonal of a square also acts as the angle bisector and measures Side × √2.

Seven persons – P, Q, R, S, T, U, and V – are sitting in a row facing North.

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1. S sits 3rd to the left of T.
   

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2. Only one person sits between Q and R.
   

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3. U is to the immediate right of R.
   

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4. P is not at any end.

T is sitting exactly in the middle.

Hence, the correct answer is “Option 1”.

Given Word**: FUTURE**

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The new word formed is GTUTSD.

Hence, the correct answer is “Option 2”.

The pattern observed is: √(First number) = Second number.

For 16 : 4 →

 √16 = 4

Similarly, for the second pair

For 81 : ? →

√81 = 9

Hence, the correct answer is “Option 3”.

1. Ink is the primary material required for writing, which is contained within a Pen.
2. A Pen is the tool used to write or record information on Paper.
3. A collection of printed or written Paper sheets bound together forms a Book.
4. After the Book is created, it is intended for the consumption of the Reader.

The sequence follows:

Ink → Pen → Paper → Book → Reader

Hence, the correct answer is “Option 1”.

<br> <br>

Pattern followed here is,

Hence, the correct answer is “Option 1”.

Circles in the given figure,

There are 7 circles present in the given figure.

Hence, the correct answer is “Option 1”.

<br> <br>

Pattern followed here is,

Hence, the correct answer is “Option 3”.

Logic followed here is,

For 71

Sum of digits = 7 + 1 = 8

For 64

Sum of digits = 6 + 4 = 10

For 53

Sum of digits = 5 + 3 = 8

For 44

Sum of digits = 4 + 4 = 8

Digit sum of 71, 53, and 44 is 8. However, the sum of the digits for 64 is 10.

Therefore, 64 is the odd one out in the group.

Hence, the correct answer is “Option 2”.

From given Venn diagram,

Players, only play cricket = 120

Hence, the correct answer is “Option 1”.

Triangles in the given figure,

There are 17 triangles present in the given figure.

Hence, the correct answer is “Option 2”.

According to the given information,

He is finally facing East direction.

Hence, the correct answer is “Option 4”.

Logic followed here is,

48 + 39 = 87

 

87 + 78 = 165

 

165 + 39 = 204

 

204 + 78 = 282

 

282 + 39 = 321

 

321 + 78 = 399

Hence, the correct answer is “Option 4”.

For Flower : Petal

A Petal is one of the many individual parts or segments that collectively form a Flower.

Similarly, for Book : ?

A Book is composed of many individual sheets or units known as Pages.

Hence, the correct answer is “Option 3”.

Here,

Cricket, Hockey, and Polo are all typically outdoor sports.

Whereas,

Badminton is an indoor sport.

Therefore, the outlier is Badminton.

Hence, the correct answer is “Option 4”.

For P2+Q3 is coded as R4+S5 →

Similarly,

let's find the code for E6+F7:

Hence, the correct answer is “Option 3”.

BODMAS Table: 

Given equation: (40 ÷ 8 + 12) × 3 = ?

⇒ (40 ÷ 8 + 12) × 3 = ?

⇒ (5 + 12) × 3 = ?

⇒ 17 × 3 = ?

⇒ 17 × 3 = ?

⇒ 51.

Hence, the correct answer is "Option 3".

BODMAS Table: 

Given equation: 14 @ 7 # 3 % 4 * 2 = ?

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Now, replacing the symbols:

⇒ 14 - 7 + 3 ÷ 4 × 2 = ?

⇒ 14 - 7 + 3 ÷ 4 × 2 = ?

⇒ 14 - 7 + 0.75 × 2 = ?

⇒ 14 - 7 + 1.5 = ?

⇒ 14 - 7 + 1.5 = ?

⇒ 7 + 1.5 = ?

⇒ 8.5.

Hence, the correct answer is "Option 1".

Family chart:

Family tree,

That man is father of Arjun.

Hence, the correct answer is “Option 3”.

Cubes in the given figure,

Layer 1 (top) → 20

Layer 2 → 15

Layer 3 → 15

Layer 4 → 15

Layer 5 → 20

Total Cubes = 20 + 15 + 15 + 15 + 20 = 105 

Hence, the correct answer is “Option 3”.

Pattern followed here is,

Hence, the correct answer is “Option 2”.

From the given Venn diagram,

Those who play all three games = D

Hence, the correct answer is “Option 4”.

Pattern followed here is,

Similarly,

Hence, the correct answer is “Option 4”.

Logic: The direction of the rotation of the curved arrow within the square determines which image does not belong to the group.

Here,

The arrows in images J, K, and M represent a counter-clockwise rotation.

Whereas the arrow in image L represents a clockwise rotation.

Therefore, the image that does NOT belong to the group is L.

Hence, the correct answer is “Option 3”.

Given:

Total number of pens = 150

Cost Price (CP) per pen = ₹12

Number of pens in Batch 1 = 60 (at 20% profit)

Number of pens in Batch 2 = 50 (at 25% profit)

Formula Used:

Profit = Number of units × CP per unit × Profit%

Calculations:

⇒ Remaining pens = 150 − (60 + 50) = 40 pens

⇒ Profit from Batch 1 = 60 × 12 × (20/100) = 60 × 2.4 = ₹144

⇒ Profit from Batch 2 = 50 × 12 × (25/100) = 50 × 3 = ₹150

⇒ Profit from Batch 3 = 40 × 12 × (30/100) = 40 × 3.6 = ₹144

⇒ Total Profit = 144 + 150 + 144

⇒ Total Profit = ₹438

Hence, the correct answer is “Option 4”.

The logic followed here is:

For ACE to CEG

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Applying the same logic for GPQ

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Hence, the correct answer is 'Option 1'.

The correct answer is: Option 4 i.e. mentioned indirectly.

Key Points

• The word "alluded" means to refer to something in a subtle or indirect way, without explicitly stating it. (संकेत करना)
• Example: She alluded to her childhood in the speech, but she didn’t provide many details.
• "Mentioned indirectly" is the correct synonym for "alluded." It means referring to something in a roundabout or non-explicit manner. (अप्रत्यक्ष रूप से उल्लेख करना)
• Example: The teacher mentioned indirectly that there might be a quiz tomorrow.
• Hence, the correct answer is "mentioned indirectly."

Therefore, the correct answer is: Option 4 i.e. mentioned indirectly.

Additional Information

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

• Referred constantly (निरंतर संदर्भित करना): Refers to mentioning something repeatedly or frequently.
• Example: He referred constantly to his previous achievements during the meeting.
• Spoke callously (निर्दयता से बात करना): Refers to speaking in an insensitive or unkind manner.
• Example: She spoke callously about the incident, showing no sympathy for the victims.
• Mentioned directly (सीधे तौर पर उल्लेख करना): Refers to explicitly stating something without any ambiguity.
• Example: The manager mentioned directly that the project deadline would not be extended.

The correct answer is: Option 2 i.e. to nurture.

Key Points

• The word "foster" means to encourage or promote the development of something, especially feelings, ideas, or growth. (प्रोत्साहित करना या पोषण करना)
• Example: Teachers play a crucial role in fostering creativity and critical thinking in students.
• "To nurture" refers to providing care, support, and encouragement for someone or something to grow or develop. (देखभाल करना या पोषण करना)
• Example: Parents nurture their children by providing love, education, and emotional support.
• Hence, the word "foster" is synonymous with "to nurture", as both involve promoting growth or development.

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

Additional Information

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

• To look after (देखभाल करना): To take care of someone or something.
• Example: She is looking after her younger brother while their parents are away.
• To contain (रोकना या समाहित करना): To hold or include something within.
• Example: The container can hold up to five liters of water.
• To bring (लाना): To carry or take something to a place.
• Example: Please bring your books to the class tomorrow.

The correct answer is: 3) Competent.

Key Points

• The word "inept" means lacking skill, ability, or competence to perform a task effectively. It refers to someone who is clumsy, awkward, or unfit for a particular role. (अयोग्य, अकुशल)
• Example: His inept handling of the situation made the problem worse instead of solving it.
• "Competent" is the opposite of "inept." It means having the necessary skill, knowledge, or ability to do something successfully. (योग्य, कुशल)
• Example: She is a highly competent engineer who always delivers quality work on time.
• The given question asks for the most opposite word (antonym) to "inept." Since "competent" directly contrasts with "inept," it is the correct answer.

Therefore, the correct answer is: 3) Competent.

Additional Information

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

• Clumsy (असावधान, अनाड़ी): Lacking grace or skill in movement or action, awkward.
• Example: His clumsy attempt to fix the broken chair resulted in more damage.
• Bungling (गड़बड़ करने वाला): Performing a task in an awkward or inefficient manner, often resulting in mistakes.
• Example: Due to his bungling, the entire project got delayed by weeks.
• Gauche (असभ्य, भद्दा): Lacking social grace, sensitivity, or tact; awkward or unsophisticated.
• Example: His gauche remarks during the meeting made everyone uncomfortable.

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

Key Points

• The word "plunged" means to fall or jump suddenly from a high place or into a deep area. It also conveys the idea of moving or being thrown with force. (गिर जाना, कूद जाना)
• Example: The car plunged into the river after the accident.
• The word "ascended" refers to the action of moving upward, rising, or climbing to a higher position. (ऊपर चढ़ना)
• Example: The balloon ascended into the sky as the children watched in awe.
• Since "ascended" is the opposite of "plunged," it is the correct answer here.

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

Additional Information

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

• Dived (गहरे पानी में गोता लगाना): To jump or plunge headfirst into water or another substance.
• Example: The swimmer dived into the pool with perfect form.
• Cruised (आराम से यात्रा करना): To move smoothly and steadily, especially at a moderate or leisurely pace.
• Example: The ship cruised through the calm waters of the ocean.
• Plateaued (स्थिर होना): To reach a level, period, or condition of stability or no change after a period of activity or progress.
• Example: His performance plateaued after several years of rapid improvement.

The correct answer is: 2) Acoustics.

Key Points

• The word "Acoustics" refers to the branch of science that deals with the study of sound, including its production, control, transmission, reception, and effects. (ध्वनि विज्ञान)
• Example: The architect ensured that the acoustics of the concert hall were perfect to enhance the listening experience.
• Acoustics is a specialized field that plays a critical role in designing spaces like theaters, auditoriums, and soundproof rooms. (ध्वनिकी का उपयोग ध्वनि को समझने और नियंत्रित करने के लिए किया जाता है।)

Therefore, the correct answer is: 2) Acoustics.

Additional Information

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

• Sound barrier (ध्वनि अवरोध): A phenomenon occurring when an object moves faster than the speed of sound, causing a sharp change in pressure and a sonic boom.
• Example: The jet broke the sound barrier, creating a loud sonic boom.
• Peripheral (परिधीय): Related to the outer edges or less important aspects of something.
• Example: Peripheral devices like keyboards and mice are essential for computer operation.
• Bystander (दर्शक): A person who is present at an event but does not take part in it.
• Example: The bystander watched the accident unfold but did not intervene.

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

Key Points

• The word "Explorer" refers to a person who travels to unknown or new places to discover something new. (अन्वेषक)
• Example: Christopher Columbus is one of the most famous explorers who discovered the Americas.
• Explorer is the correct word as it describes someone who ventures into unfamiliar territories to gain knowledge, often for scientific, geographical, or adventurous purposes. (अन्वेषक)
• Example: The explorer spent years mapping the uncharted rainforests of South America.

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

Additional Information

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

• Tourist (पर्यटक): A person who travels for pleasure, typically to visit attractions or for leisure purposes.
• Example: The tourist visited the Eiffel Tower during their trip to Paris.
• Voyager (यात्री): A person who goes on a long journey, especially by sea or in space.
• Example: The voyagers set sail across the Atlantic to explore new lands.
• Traveller (यात्रा करने वाला): A general term for someone who travels from one place to another.
• Example: The traveller shared fascinating stories about their adventures in Asia.

The correct answer is 'is, drank'.

 

Key Points

• The first blank requires a singular verb because 'milk' is an uncountable noun, which is treated as singular in English. The verb 'is' correctly describes the current state of the milk.
• The second blank requires the simple past tense because the action is associated with a specific past time marker, 'last night'.
• The simple past tense of the verb 'drink' is 'drank'.
• Options 1, 2, and 3 are grammatically incorrect as they violate subject-verb agreement or tense rules.

Therefore, the correct answer is Option 4.

 

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

 

Additional Information

• Verb Forms: Drink (V1) − Drank (V2) − Drunk (V3).
• Option 1: 'has' is typically used for possession or the present perfect tense and does not fit the context of describing the milk's remaining quantity.
• Option 2: 'have' is a plural verb, and 'have drank' is incorrect because the present perfect requires the V3 form ('have drunk').
• Option 3: 'are' is a plural verb and cannot be used with the singular noun 'milk'.

The correct answer is: Option 2

Key Points

Let's match the homonyms to their correct meanings:

So, the matching pairs are:

• 1. Reel - A) Cylinder for winding thread or film
• 2. Real - B) Genuine or true
• 3. Dual - C) Two parts
• 4. Duel - D) Combat between two people

Therefore, Option 2 correctly matches the homonyms to their meanings:

Option 2) 1-A, 2-B, 3-C, 4-D

The correct answer is Option 1.

 

Key Points

• The primary grammatical issue in the sentence is the use of the preposition among to refer to a relationship involving only two people.
• The word between is used when naming distinct, individual items (often two), while among is used for undifferentiated groups or when referring to more than two.
• Since parents typically refers to a pair (mother and father), between is the appropriate preposition to describe sitting in the middle of them.
• Thus, the underlined part must be replaced by between her parents.

Therefore, the correct answer is between her parents.

 

Correct Sentence: 'She sat between her parents to watch the play.'

 

Additional Information

• besides her parents: The word besides means in addition to or apart from (e.g., 'Who else was there besides her parents?'). It does not indicate physical position. Beside (without the 's') means next to.
• next her parents: This is grammatically incorrect. The word next requires the preposition to to function as a phrase indicating position (next to her parents).
• among: This would be used if the subject were sitting in a large group or crowd where individuals are not specifically named (e.g., 'She sat among the spectators').

The correct answer is Option 1.

 

Key Points

• The phrase account on is grammatically incorrect in this context.
• The phrasal verb account for means to give a satisfactory record or explanation of one's actions, or to justify something.
• In the given sentence, the politician is under pressure to explain or justify his actions, making account for the correct substitution.
• Thus, Option 1 is the most appropriate improvement for the underlined part.

Therefore, the correct answer is account for.

 

Correct Sentence: 'The politician was under pressure to account for his actions.'

 

Additional Information

• Account of: This usually refers to a report or description of an event (e.g., 'She gave a detailed account of the incident').
• Account to: This means to be answerable or responsible to a person or an authority (e.g., 'Managers must account to the board of directors').
• No substitution: This is incorrect because account on is not a standard idiomatic expression for providing an explanation.

The correct answer is Option 1.

 

Key Points

• Passive voice structure: The object of the active voice sentence becomes the subject of the passive voice sentence.
• In the active voice, the sentence is: "The overseer instructed the workers to lift the bridge by two feet."
• To convert to passive voice: Place the object "the workers" at the beginning.
• Change the active verb "instructed" (simple past) to "were instructed" (simple past passive).
• Add the preposition "by" followed by the original subject "the overseer."
• Retain the infinitive phrase "to lift the bridge by two feet."

Therefore, the correct answer is Option 1.

 

Correct Sentence: "The workers were instructed by the overseer to lift the bridge by two feet."

 

Additional Information

• Option 2: "The workers was instructed by the overseer to lift the bridge by two feet." – Incorrect subject-verb agreement ("workers" is plural, so it should be "were").
• Option 3: "The workers were instructed by the overseer to lifting the bridge by two feet." – Incorrect verb form ("to lifting" instead of "to lift").
• Option 4: "The workers had been instructed by the overseer to be lifted the bridge by two feet." – Incorrect tense (past perfect) and incorrect construction ("to be lifted").

The correct answer is Option 2.

 

Key Points

• Active voice structure: The subject performs the action on the object.
• In the passive voice, the sentence is: "The director is expected to approve the new proposal soon."
• Identify the implied subject/agent: "They" (general expectation).
• Change the passive verb "is expected" (present simple passive) to "expect" (present simple active).
• Retain the object "the director" and the infinitive phrase "to approve the new proposal soon."

Therefore, the correct answer is Option 2.

 

Correct Sentence: "They expect the director to approve the new proposal soon."

 

Additional Information

• Option 1: "It is expected the director approve the new proposal soon." – Incorrect structure; missing "will" or "should" after "director."
• Option 3: "The director expects to approve the new proposal soon." – Incorrect meaning; changes the subject to the director himself.
• Option 4: "The new proposal expects the director's approval." – Incorrect and illogical; the proposal cannot "expect."

The correct answer is Option 2.

 

Key Points

• Active voice structure: The subject performs the action on the object.
• In the passive voice, the sentence is: "The game was won by the underdog team."
• Identify the subject/agent: "The underdog team."
• Change the passive verb "was won" (past simple passive) to "won" (past simple active).
• Retain the object "the game."

Therefore, the correct answer is Option 2.

 

Correct Sentence: "The underdog team won the game."

 

Additional Information

• Option 1: "The underdog team wins the game." – Incorrect tense (present simple instead of past simple).
• Option 3: "The underdog team has won the game." – Incorrect tense (present perfect instead of past simple).
• Option 4: "The underdog team is winning the game." – Incorrect tense (present continuous instead of past simple).

The correct answer is Option 4.

 

Key Points

• Indirect Speech: Rakesh told Ranjana that that book had been purchased by his father when he had been young.
• Key Conversions:
• "This book" → "that book"
• Demonstrative pronoun changes in indirect speech.
• "was purchased" → "had been purchased"
• Past tense passive changes to past perfect passive in indirect speech.
• "my father" → "his father"
• Pronoun shift to match the subject Rakesh.
• "I was young" → "he had been young"
• Past tense changes to past perfect to maintain sequence of tenses.
• Punctuation and Format:
• No quotation marks in indirect speech.
• Sentence flows naturally with reporting verb + indirect clause.

Therefore, the correct answer is Option 4.

 

Correct Sentence: Rakesh told Ranjana that that book had been purchased by his father when he had been young.

 

Additional Information

• Option 1: "Rakesh told Ranjana that book was purchased by his father when he was young."
• Incorrect because "this" should change to "that".
• Tense not shifted correctly ("was purchased" → "had been purchased").
• Option 2: "Rakesh told Ranjana that that book was purchased by my father..."
• Incorrect pronoun: "my father" should be "his father".
• Tense not shifted.
• Option 3: "Rakesh told Ranjana that that book had purchased by his father..."
• Grammatical error: "had purchased" (active) instead of "had been purchased" (passive).
• "he had young" is incorrect; should be "he had been young".

The correct answer is 'among'.

 

Key Points

• The preposition 'among' is used when talking about being in the middle of or surrounded by a group of people or things (usually more than two).
• In the given sentence, the phrase 'so many strangers' refers to a collective group, making 'among' the grammatically correct choice to show she was lost within that crowd.
• 'Between' is typically used when referring to two distinct individuals or objects.

Therefore, the correct answer is Option 1.

 

Complete Sentence: 'She felt lost among so many strangers.'

 

Additional Information

• Option 2: 'between' is used for specific, distinct items (e.g., 'The house is between the woods and the river').
• Option 3: 'beside' means 'next to' or 'at the side of' (e.g., 'Come and sit beside me').
• Option 4: 'with' indicates accompaniment or being in the company of someone (e.g., 'I am going with my friends'), but it does not express being surrounded by a mass of people as accurately as 'among'.

The correct answer is 'to'.

 

Key Points

• The preposition to is used after the adjective relevant to indicate a connection or relation to a particular subject.
• The term relevant to is a standard idiomatic expression meaning having a significant and demonstrable bearing on the matter at hand.
• In the given sentence, the lecture is described as being directly related to the subject of climate change.

Therefore, the correct answer is Option 2.

 

Complete Sentence: 'The lecture was relevant to the topic of climate change.'

 

Additional Information

• Option 1: 'in' is typically used for location or a specific state (e.g., in the room) and does not follow relevant.
• Option 3: 'on' is used for topics (e.g., a book on science), but it is grammatically incorrect when paired with the adjective relevant.
• Option 4: 'with' usually implies accompaniment or tools (e.g., with a friend) and is not the correct preposition for this context.

The correct answer is Option 4.

 

Key Points

• Indirect Speech: He wondered how such an obvious error had escaped everyone's notice.
• Direct Speech Conversion: He wondered, "How has such an obvious error escaped everyone's notice?"
• Key Conversions:
• "had escaped" → "has escaped"
• Past perfect in indirect speech changes to present perfect in direct speech to reflect immediacy of the wonder.
• Question form:
• Direct speech requires interrogative structure: "How has such an obvious error escaped...?"
• Punctuation:
• Question mark (?) is essential to reflect the interrogative tone of "wondered".

Therefore, the correct answer is Option 4.

 

Correct Sentence: He wondered, "How has such an obvious error escaped everyone's notice?"

 

Additional Information

• Option 1: "He wondered, 'How did such an obvious error escaped...'"
• Grammatical error: "did escaped" is incorrect; should be "did escape".
• Option 2: "He wondered, 'how such an obvious error had escaped...'"
• Incorrect because direct speech should use interrogative form, not statement order.
• Option 3: "He wondered, 'How could such an obvious error escape...'"
• Changes tense and meaning: introduces modal "could" which alters the sense of immediacy.

The correct answer is option 1.

Key Points

• The word "aberration" is correctly spelled and means "a deviation from what is normal".
• It is a noun and is used to describe something that is unusual or out of the ordinary.
• Example: The sudden drop in temperature was an aberration in the otherwise warm weather.

Correct sentence: The correct spelling is "aberration."

Additional Information

Let’s analyze the other options:

• Option 2: "Abberration" - This is incorrect because it contains an extra 'b'.
• Option 3: "Aberation" - This is incorrect because it is missing one 'r'.
• Option 4: "Abberation" - This is incorrect due to both an extra 'b' and a missing 'r'.

The correct answer is option 4.

Key Points

The misspelled word is 'Magnanomous'.

• The correct spelling would be 'Magnanimous'.
• The word 'magnanimous' is an adjective which means 'generous or forgiving, especially towards a rival or less powerful person'. (उदार)
• For Example -
• He showed a magnanimous gesture by forgiving his opponent after the heated debate.

Therefore, the correct answer would be 'Magnanomous'.

Additional Information

Let's understand the meanings of the other words mentioned in the options -

• Abhorrent - inspiring disgust and loathing; repugnant (घृणास्पद)
• Vicissitude - a change of circumstances or fortune, typically one that is unwelcome or unpleasant (उतार-चढ़ाव)
• Egregious - outstandingly bad or shocking (अत्यंत खराब)

The correct answer is 'Steadfast and firm'.

 

Key Points

• The passage describes Gunjan Saxena operating in high-altitude regions under constant threat of enemy fire, requiring “unwavering focus and precision.”
• The word “unwavering” means not shaking, not hesitating, and remaining firm despite challenges.
• Thus, “Steadfast and firm” is the closest meaning, as it conveys determination and consistency under pressure.
• Other options are incorrect: “Unsteady” implies instability, “Hesitant” implies doubt, and “Indifferent” implies lack of concern — none fit the context of military precision and courage.

Therefore, the correct answer is 'Option 4'.

 

Complete Passage: Operating in high-altitude regions under constant threat of enemy fire required (1) unwavering focus and precision. Gunjan's missions involved transporting essential supplies, evacuating wounded soldiers, and providing critical intelligence to ground forces. Her ability to remain calm under pressure earned her immense respect from peers and superiors alike.

 

Additional Information

• Unsteady: Suggests instability, opposite of “unwavering.”
• Hesitant: Implies doubt or reluctance, not suitable here.
• Indifferent: Means unconcerned, which does not fit the context of focus and precision.

The correct answer is 'Exploration or information-gathering for military purposes'.

 

Key Points

• The passage mentions Gunjan Saxena flew numerous “reconnaissance and supply missions in hostile territory.”
• “Reconnaissance” is a military term that refers to surveying or exploring an area to collect intelligence about enemy positions, terrain, or movements.
• Thus, the closest meaning is “Exploration or information-gathering for military purposes.”
• Other options are incorrect: “Celebration” is unrelated, “Retreat” means withdrawal, and “Combat” refers to fighting, not intelligence-gathering.

Therefore, the correct answer is 'Option 3'.

 

Complete Passage: Gunjan Saxena, widely known as the "Kargil Girl", holds a special place in Indian military history as one of the first female pilots in the Indian Air Force (IAF). Joining the IAF in the mid-1990s, she entered a field that was predominantly male-dominated, facing both operational challenges and social skepticism. Her courage and determination became legendary during the Kargil War of 1999, when she flew numerous (2) reconnaissance and supply missions in hostile territory.

 

Additional Information

• Celebration: Refers to festivities, irrelevant in a military context.
• Retreat: Means withdrawal from battle, opposite of reconnaissance.
• Combat: Refers to direct fighting, not intelligence-gathering.

The correct answer is 'Composed'.

 

Key Points

• The sentence is: “Her ability to remain ______(3) under pressure earned her immense respect from peers and superiors alike.”
• The blank requires an adjective or past participle that describes her state under pressure.
• “Composed” means calm, self-controlled, and collected — perfectly fitting the context.
• Other options are incorrect: “Compose” (verb, base form), “Composing” (present participle, grammatically unsuitable here), “Composes” (present tense verb, does not match the sentence structure).

Therefore, the correct answer is 'Option 3'.

 

Complete Passage: Operating in high-altitude regions under constant threat of enemy fire required unwavering focus and precision. Gunjan's missions involved transporting essential supplies, evacuating wounded soldiers, and providing critical intelligence to ground forces. Her ability to remain (3) composed under pressure earned her immense respect from peers and superiors alike.

 

Additional Information

• Compose: Base verb form, does not fit grammatically.
• Composing: Present participle, unsuitable in this sentence structure.
• Composes: Present tense verb, incorrect for describing her state under pressure.

The correct answer is 'Of'.

 

Key Points

• The sentence is: “Her journey has inspired a generation ______(4) young women to pursue careers in the armed forces.”
• The correct preposition is “Of” because it indicates belonging or association — the generation consists of young women.
• Thus, “a generation of young women” is grammatically and contextually accurate.
• Other options are incorrect: “For” suggests purpose but alters the meaning, “To” implies direction but is not idiomatic here, and “With” implies accompaniment, which does not fit.

Therefore, the correct answer is 'Option 1'.

 

Complete Passage: Beyond her battlefield contributions, Gunjan Saxena's journey has inspired a generation (4) of young women to pursue careers in the armed forces. Her story underscores the importance of breaking barriers and challenging stereotypes, paving the way for greater gender inclusion in the military.

 

Additional Information

• For: Suggests purpose, but grammatically incorrect in this sentence.
• To: Indicates direction, not suitable for describing a group.
• With: Implies accompaniment, which does not fit the context of inspiration.

The correct answer is 'Has been celebrated'.

 

Key Points

• The sentence is: “In recent years, her legacy ______(5) in books, documentaries, and a popular biographical film.”
• The context refers to her legacy continuing to be honored and recognized over time.
• The correct verb form is “Has been celebrated” because it indicates an action that started in the past and continues into the present.
• Other options are incorrect: “Celebrated” (simple past, does not capture ongoing recognition), “Is celebrate” (grammatically wrong), “Was being celebrated” (past continuous, does not fit the present relevance).

Therefore, the correct answer is 'Option 3'.

 

Complete Passage: In recent years, her legacy (5) has been celebrated in books, documentaries, and a popular biographical film. She stands not only as a decorated officer but also as a symbol of perseverance, patriotism, and the changing face of the Indian Air Force.

 

Additional Information

• Celebrated: Simple past, does not convey ongoing recognition.
• Is celebrate: Grammatically incorrect form.
• Was being celebrated: Past continuous, suggests temporary recognition in the past, not continuing into the present.

The correct answer is option 4.

Key Points

• The word 'Indispensable' is correctly spelled.
• Indispensable is an adjective that means "absolutely necessary or essential." (अनिवार्य)
• Examples:
• Good communication skills are indispensable for success in any career.
• A healthy diet is indispensable for maintaining good health.
• The other options are incorrect due to spelling errors:
• Option 1: 'Indespansable' - Incorrect spelling.
• Option 2: 'Indispansible' - Incorrect spelling.
• Option 3: 'Indespensable' - Incorrect spelling.

Additional Information

• Origin: The word "indispensable" comes from the Latin word "indispensabilis," meaning "not subject to dispensation."
• Synonyms: Essential, necessary, vital, crucial.
• Antonyms: Dispensable, unnecessary, optional.

The correct answer is - Show/Hide formatting marks

Key Points

• Show/Hide formatting marks
• The Show/Hide button, often represented by the Pilcrow symbol (¶), is located in the Paragraph group of the Home tab.
• It reveals non-printing characters such as spaces (dots), paragraph breaks (¶), and tab characters (arrows).
• This tool is essential for visualizing the exact structure of a document and troubleshooting layout issues caused by extra spaces or line breaks.
• The keyboard shortcut to toggle this feature is Ctrl + Shift + 8 (or Ctrl + *).

Additional Information

• Change Case
• Found in the Font group, this feature allows users to modify capitalization (e.g., lowercase, UPPERCASE, or Sentence case) without retyping text.
• Text Highlight Color
• This tool applies a vibrant color behind the text to make it stand out, functioning like a physical highlighter pen.
• It is primarily used to mark important information for editing or reviewing purposes.
• Clear All Formatting
• This button removes all styles, colors, and font changes from the selected text, resetting it to the default document style.
• It is highly effective when cleaning up text copied from websites or other documents with inconsistent formatting.

The correct answer is - The browser takes the URL in the link, sends a request to the target website server

Key Points

• HTTP Request Process
• When a user clicks a link, the browser acts as a client and identifies the URL associated with that link.
• The browser initiates a request specifically to the target website server where the page content is hosted.
• This communication typically uses the HTTP (Hypertext Transfer Protocol) or HTTPS (Secure) protocol.
• The server processes this request and sends back the necessary data for the browser to display.

Additional Information

• DNS Lookup
• Before connecting to the server, the browser uses the Domain Name System (DNS) to translate the human-readable domain name into a numeric IP address.
• This IP address is essential for routing the request across the internet to the correct physical server.
• Client-Server Architecture
• Web browsing is a classic example of the Client-Server model, where the browser (client) requests resources and the web server provides them.
• The search engine acts only as a directory or index; it does not host the actual content of the external sites it lists.
• HTTP Response and Rendering
• The server’s HTTP response usually contains HTML, CSS, and JavaScript files.
• The browser’s rendering engine then interprets these files to construct and display the final webpage on the user's screen.

The correct answer is - Google Search

Key Points

• Google Search
• It is a web search engine developed by Google, designed to find information on the World Wide Web.
• The engine functions by crawling (discovering pages), indexing (storing and organizing content), and ranking (displaying the most relevant results).
• It uses complex algorithms to provide users with links to websites, images, videos, and news based on specific keywords or queries.
• Unlike a browser, it is a service accessed through a website or application to locate data across the internet.

Additional Information

• Web Browsers
• Google Chrome and Mozilla Firefox are examples of web browsers, which are software applications used to access and view websites.
• A browser acts as an interface between the user and the internet, whereas a search engine is a tool used within a browser.
• Other popular browsers include Apple Safari, Microsoft Edge, and Opera.
• Search Engines
• Apart from Google, other notable search engines include Microsoft Bing, Yahoo! Search, DuckDuckGo, and Baidu.
• Communication Apps
• WhatsApp is an Instant Messaging (IM) platform and social media application used for communication, not for searching the public web.
• It provides end-to-end encryption for private text messages, voice calls, and video calls.

The correct answer is - To view and manage several different mailboxes from one interface

Key Points

• Centralized Management
• Desktop email clients allow users to consolidate multiple email accounts (such as personal, work, and school) into a single software application.
• This eliminates the need to open multiple web browsers or tabs to check different email providers.
• Users can perform tasks like searching across all accounts or moving messages between different mailboxes seamlessly.
• It streamlines the workflow by providing a consistent interface for managing various communications.

Additional Information

• Email Protocols
• IMAP (Internet Message Access Protocol): Allows the client to "sync" with the server, ensuring that changes made in the program are reflected on all devices.
• POP3 (Post Office Protocol): Typically downloads the email to the local storage and may remove it from the server.
• SMTP (Simple Mail Transfer Protocol): The primary protocol used by email clients for sending outgoing messages.
• Unified Inbox
• Many modern desktop clients offer a Unified Inbox feature that merges incoming mail from all configured accounts into one view.
• This feature helps in prioritizing unread messages regardless of which account they were sent to.
• Offline Access
• Desktop clients download and store emails locally, enabling users to read, search, and draft replies even without an active internet connection.
• Synchronized updates occur once the device reconnects to the network.

The correct answer is - Info

Key Points

• Info
• The Info option is found within the File tab (also known as the Backstage view) of Microsoft Excel.
• It displays comprehensive workbook properties such as file size, author, last modified date, and the time the document was last printed.
• This section provides the Protect Workbook command, which allows users to encrypt with a password, restrict access, or mark the file as final.
• It includes the Inspect Workbook tool to check the file for hidden metadata, personal information, or accessibility issues before sharing.
• Users can also manage different versions of the file and recover unsaved workbooks through this interface.

Additional Information

• Open
• Used to retrieve existing files from Local Folders, OneDrive, or the Recent documents list.
• Save As
• Allows the user to save a copy of the current workbook in a specific location or under a new file name.
• It provides options to change the file extension (e.g., from .xlsx to .csv or .xlsm).
• Export
• Facilitates the conversion of the Excel workbook into different formats like PDF/XPS documents.
• It allows users to Change File Type quickly without using the full Save As dialog.
• File Tab
• Introduced in Office 2010 to replace the Microsoft Office Button from the 2007 version.
• Clicking this tab takes the user out of the main editing interface into the Backstage view.

The correct answer is - Protect Workbook Structure

Key Points

• Protect Workbook Structure
• This command is specifically designed to maintain the integrity of the workbook's organization by preventing users from adding, deleting, hiding, or renaming worksheets.
• It ensures that the order of sheets remains fixed, which is critical for complex financial models that rely on specific sheet references.
• Unlike protecting a sheet, protecting the workbook structure allows data entry and editing within existing cells, provided the individual worksheets are not themselves protected.
• The feature can be accessed through the Review tab within the "Protect" group in Microsoft Excel.

Additional Information

• Protect Sheet
• Used to prevent changes to individual cell contents, formatting, or formulas within a specific worksheet.
• It allows the administrator to lock specific cells while leaving others (like input fields) unlocked for user interaction.
• Track Changes
• A collaborative tool used to record and highlight modifications made to the workbook by different users.
• It is primarily used for auditing and reviewing history rather than restricting structural or content changes.
• Password Protection
• When applying protection, users can set an optional password to prevent others from unprotecting the structure or sheets.
• This adds a layer of security, ensuring only authorized users can modify the file's configuration.
• Share Workbook
• An older Excel feature (replaced largely by Co-authoring) that allowed multiple users to work on the same file simultaneously, though it often placed restrictions on certain structural changes.

The correct answer is - Formula bar

Key Points

• Formula bar
• The Formula bar is a toolbar located at the top of the Microsoft Excel spreadsheet window, just below the ribbon.
• It is used to enter or edit formulas, text, or numbers in the active cell.
• Even if a cell is narrow and cannot display its full content, the Formula bar provides a dedicated space to view the entire value or logic.
• While the cell grid displays the calculated result of a formula, the Formula bar reveals the actual formula itself.

Additional Information

• Status bar
• Found at the bottom of the Excel window.
• Displays information about the current state of the worksheet, such as Page Numbers, and quick calculations like Sum, Average, and Count for selected cells.
• Includes the Zoom slider and View shortcuts (Normal, Page Layout, Page Break Preview).
• Title bar
• Located at the very top of the window.
• Shows the filename of the current workbook and the application name (Microsoft Excel).
• Contains the Minimize, Maximize/Restore, and Close buttons.
• Scroll bar
• Includes both vertical and horizontal bars.
• Used to navigate through a large worksheet that extends beyond the visible screen area.

The correct answer is - AutoCorrect

Key Points

• AutoCorrect
• It is a software feature that automatically detects and corrects misspelled words, common typos, and grammatical errors while the user is typing.
• The correction occurs immediately after the user presses the spacebar, the Enter key, or enters punctuation.
• It relies on a pre-defined AutoCorrect list that maps common mistakes (like "teh") to their correct forms (like "the").
• In addition to spelling, it can automatically fix capitalization errors, such as capitalizing the first letter of a sentence.

Additional Information

• Spell Checker
• A tool that scans a document and compares words against a built-in dictionary.
• Unlike AutoCorrect, it typically flags errors using a red wavy underline and often requires the user to manually select a correction from a list of suggestions.
• Thesaurus
• An software tool used to look up synonyms (words with similar meanings) and antonyms (words with opposite meanings).
• It is primarily used to enhance vocabulary and avoid word repetition within a document.
• AutoSummarize
• A legacy feature in word processors like Microsoft Word that identifies the key points of a document.
• It analyzes the text and assigns scores to sentences based on the frequency of keywords to generate a summary.

The correct answer is - Use the Change Case command on the selected paragraph and choose the UPPERCASE option

Key Points

• Change Case command
• The Change Case feature in word processing software (like Microsoft Word) allows users to modify the capitalization of text without the need to retype any content.
• Selecting the UPPERCASE option specifically converts every letter in the highlighted selection into capital letters instantly.
• This approach is the most efficient and consistent method for applying emphasis to specific sections of a document.
• In most interfaces, this command is represented by an 'Aa' icon located within the Font group of the Home tab.

Additional Information

• Types of Case Options
• Sentence case: Capitalizes only the first letter of the first word in a sentence.
• lowercase: Removes all capital letters from the selected text.
• Capitalize Each Word: Also known as Title Case, it capitalizes the first letter of every word.
• tOGGLE cASE: Switches the current case of every character (capitals become lowercase and vice versa).
• Keyboard Shortcuts
• The Shift + F3 shortcut is a quick way to cycle through different case options (Sentence case, UPPERCASE, and lowercase) for selected text.
• AutoCorrect Feature
• AutoCorrect is designed to fix common typos and automatically capitalize specific words, such as days of the week and months of the year, based on built-in dictionaries.

The correct answer is - The formula is trying to divide a number by zero or by an empty cell

Key Points

• #DIV/0! Error
• The #DIV/0! error occurs specifically when a formula attempts to divide a number by zero (0).
• It is also triggered when a formula refers to a blank or empty cell as the divisor, as Excel treats an empty cell as zero in this context.
• In mathematics, division by zero is undefined, and Excel uses this error code to indicate that the calculation cannot be performed.
• This error is commonly resolved by ensuring the divisor is not zero or by using the IFERROR function to manage the output.

Additional Information

• #NAME? Error
• Occurs when Excel does not recognize text within a formula, such as a misspelled function name (e.g., =SUMM instead of =SUM).
• It can also appear if a named range is referenced incorrectly or if text is not enclosed in double quotation marks.
• #VALUE! Error
• Indicates that the formula contains the wrong type of argument or operand.
• This most frequently happens when mathematical operations are performed on cells containing text strings instead of numerical values.
• #REF! Error
• Stands for Reference error and occurs when a formula refers to a cell that is no longer valid.
• This usually happens when the cells, rows, or worksheets used in the formula have been deleted.
• #N/A Error
• Stands for Not Available, meaning the formula cannot find the value it is tasked to search for.
• It is very common in lookup functions such as VLOOKUP, HLOOKUP, and MATCH when no match is found.