SSC CGL 2025 (Held On: 14 Sept, 2025 Shift 3) (Previous Year Paper)

Given word: TREATMENT

Number of letters = 9

Equidistant from both ends means the middle letter (5th letter)

So, The 5th letter is T

Hence, the correct answer is Option 2.

The logic followed here is:

So, 'MOQ' will replace the question mark in the given series.

Hence, the correct answer is "Option 4".

The logic followed here is:

So, 'ASI' will replace the question mark in the given series.

Hence, the correct answer is "Option 1".

The logic followed here is:

Option 1) Rose → This is a flower.

Option 2) Guava → This is a fruit.

Option 3) Banana → This is a fruit.

Option 4) Papaya → This is a fruit.

Thus, Rose is different because it is a flower, while the others are all fruits.

Hence, the correct answer is "Option 1".

The logic followed here is:

Here, 24 = 4! = 4 × 3 × 2 × 1

120 = 5! = 5 × 4 × 3 × 2 × 1

720 = 6!

5040 = 7!

40320 = 8!

thus, "362880" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

The logic followed here is:

thus, "241" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given Statement: Smoking is prohibited in public places as per the new law.

Now,

Conclusions:

(i) People can smoke in private spaces → Does not follow (The statement only talks about prohibition in public places; nothing is mentioned about private spaces.)

(ii) Smoking harms public health → Does not follow (The reason for the law is not stated.)

Here, Neither conclusion I nor II follows.

Hence, the correct answer is "Option 4".

Given Statement:

The Himalayas act as a natural barrier that prevents the monsoon winds from reaching the northern plains, thereby creating a dry region in the rain shadow area.

Now,

Assumptions:

1.The Himalayas prevent monsoon winds from reaching certain areas - Implicit (The statement directly says the Himalayas act as a barrier preventing monsoon winds.)

2.The rain shadow effect leads to dry conditions in specific regions - Implicit (The statement links the rain shadow area with creation of a dry region.)

3.Monsoon winds are the sole cause of rainfall in the northern plains - Not Implicit (The statement does not say monsoon winds are the only cause of rainfall.)

Here, Only assumptions I and II are implicit.

Hence, the correct answer is "Option 1".

Given word: LAPTOP

Arrange the letters in alphabetical order:

A L O P P T

The fourth letter from the left = P

The first letter from the right = T

Now, letters between P and T in the English alphabet = Q, R, S → 3 letters

Hence, the correct answer is Option 3.

Given word: HARMONIOUS

Arrange the letters in alphabetical order:

A H I M N O O R S U

First letter from the left = A

Fourth letter from the right = O (from U, S, R, O)

Now, letters between A and O in the English alphabet:

B, C, D, E, F, G, H, I, J, K, L, M, N → 13 letters

Hence, the correct answer is Option 2.

The address given in the problem is,

Priya Yadav B-22, Rosewood Heights, Sector 15, Gurgaon, 122002

Let's match this address with the given options one by one to figure out which one is exactly the same.

So, the address that is exactly the same as the given address is in option (1).

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

The logic followed here is:

BEKQ : YVPJ

Similarly,

CFGL : ?

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: The first word is antonym of the second word.

Here,

Growth : Stagnation → Growth and Stagnation are opposites.

Similarly,

Success : Failure → Success and Failure are opposites.

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: All words except one denote quietness or calmness.

Silent → quiet → fits

Muted → soft, quiet → fits

Tranquil → peaceful → fits

Calm → peaceful → fits

Clamorous → noisy, loud → does not fit

Hence, the correct answer is "Option 4".

The logic followed here is:

EXCUSE : CXEESU

Similarly,

ERODES : ?

Hence, the correct answer is "Option 2".

Family Chart:

Given: 

1. A’s father’s wife = A’s mother.

2. A’s father’s wife’s only daughter = The only daughter of A’s mother = A’s sister.

So, A is the Brother of B.

Hence, the correct answer is "Option 3".

Given:

Numbers = 1, 8, 27, 64, 121, 216

Formula used:

Perfect cube = n × n × n

Calculation:

1 = 1 × 1 × 1 = 1³

8 = 2 × 2 × 2 = 2³

27 = 3 × 3 × 3 = 3³

64 = 4 × 4 × 4 = 4³

121 = 11 × 11 = 11²

216 = 6 × 6 × 6 = 6³

All numbers except 121 are perfect cubes

∴ The odd number is 121.

Family Chart:

Decoding the Symbols:

 

Given: A × B + C

It means:

A is the Mother of B

B is the Husband of C.

 

So,  A is the mother-in-law of C.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: Number - letter position = 1

(17, P) → 17 - 16 = 1 

(13, J) → 13 - 10 = 3 (odd one)

(19, R) → 19 - 18 = 1 

(23, V) → 23 - 22 = 1 

So, the odd one is (13, J, !).

Hence, the odd one out is Option 2.

The logic followed here is:

Logic: The operation “@” is defined as a @ b = (a + b)²

2 @ 3 = (2 + 3)² = 5² = 25 

3 @ 4 = (3 + 4)² = 7² = 49 

Similarly, 4 @ 5 = (4 + 5)² = 9² = 81 

Hence, the correct answer is Option 1.

The logic followed here is:

Logic: Each letter is shifted forward by 1 in the English alphabet.

For, READ → SFBE

Similarly, for BOOK → ?

So, BOOK → CPPL 

Hence, the correct answer is Option 1.

The logic followed here is:

Logic: Using the simple interest formula: Amount = Principal + Principal × Rate × Time

Given:

Amount becomes 2.5 times the principal in 10 years

Amount = 2.5 × Principal

Simple Interest = Amount - Principal = 2.5P - P = 1.5P

Using SI = Principal × Rate × Time:

1.5P = P × R × 10

R = 1.5 / 10 = 0.15

Hence, the correct answer is Option 3.

Family Chart:

Given: 

X and Y are brothers → They are sons of A.

Z is the wife of Y.

B is the daughter of Z → Since Z is Y’s wife, B is the daughter of Y and Z.

So, B is the granddaughter of A.

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: The operation "$" is defined as a $ b = a² - b

Given: a $ b = 2 → a² - b = 2 → b = a² - 2

Check each option:

Option 1) a = 2, b = 2 → 2² - 2 = 4 - 2 = 2 

Option 2) a = 8, b = 1 → 8² - 1 = 64 - 1 = 63 

Option 3) a = 4, b = 5 → 16 - 5 = 11 

Option 4) a = 9, b = 6 → 81 - 6 = 75 

Hence, the correct answer is Option 1.

The logic followed here is:

Find the sum of 42 and 42 → 42 + 42 = 84

Multiply the sum by 4 → 84 × 4 = 336

Hence, the correct answer is Option 3.

The correct answer is Statement I is true but Statement II is false.

Key Points

• The harmika of a stupa is a small, square railing located at the top of the dome (anda) and is considered the symbolic abode of the gods.
• It is an important architectural feature of stupas and signifies the connection between the earthly realm and the divine.
• The harmika is typically surmounted by a mast (yasti) with umbrellas or chhatras, symbolizing honor and protection.
• Statement II is incorrect because the relics of the Buddha or other revered figures are not kept in the harmika; they are enshrined in the anda (the hemispherical dome) of the stupa.
• The anda serves as the primary burial mound or relic chamber, which houses sacred relics and is the focal point of veneration.

Additional Information

• Components of a Stupa:
• Anda (Dome): The hemispherical dome represents the mound of dirt used to cover the Buddha's relics. It is the most important part of the stupa, symbolizing the universe.
• Harmika: A square platform at the top of the dome, representing the abode of gods. It does not contain relics but serves as a symbolic feature.
• Chhatra: The umbrella-like structure above the harmika symbolizes protection and honor. It often represents the Three Jewels of Buddhism: the Buddha, the Dharma, and the Sangha.
• Medhi: A circular terrace at the base of the dome, used by devotees for circumambulation (pradakshina).
• Torana: The gateway of the stupa, typically elaborately carved, representing the entry point for devotees.

The correct answer is Yajurveda.

Key Points

• Yajurveda is one of the four Vedas and is primarily concerned with ritualistic formulas and the proper procedures for performing sacrifices (Yajnas).
• It serves as a guide for priests and offers detailed instructions on how to conduct rituals and ceremonies, including the recitation of mantras.
• It is divided into two parts: the Shukla (White) Yajurveda, which contains clear and organized rituals, and the Krishna (Black) Yajurveda, which includes a mix of mantras and explanatory prose.
• The Yajurveda is written in Sanskrit and focuses on the performance of Vedic sacrificial rituals to invoke deities and ensure harmony in the universe.
• Its primary purpose is to provide a detailed framework for conducting ceremonial offerings, including fire sacrifices, to maintain cosmic order and promote spiritual well-being.

Additional Information

• Rigveda
• The Rigveda is the oldest of the four Vedas and primarily contains hymns (Suktas) dedicated to various deities such as Agni, Indra, and Soma.
• It is composed of 10 Mandalas (books) and focuses on philosophical and spiritual concepts.
• The hymns in the Rigveda were recited during rituals but are less focused on procedural details compared to the Yajurveda.
• Samaveda
• The Samaveda is known as the "Veda of Melodies" and primarily consists of hymns from the Rigveda set to music.
• It is meant to be sung during Vedic sacrifices and rituals, especially during the Soma sacrifice.
• The Samaveda highlights the importance of sound and music in invoking divine forces and creating a sacred atmosphere.
• Atharvaveda
• The Atharvaveda contains hymns, spells, and incantations that deal with everyday life, health, and healing, as well as spiritual practices.
• It includes knowledge of Ayurveda, focusing on remedies for diseases and methods to ward off negative influences.
• Unlike the other three Vedas, it is more practical and addresses the needs of common people.

The correct answer is Solid, non-tunable instruments primarily used for rhythm in folk music.

Key Points

• Ghana Vadya refers to solid, non-tunable instruments used to produce rhythm and percussion in Indian traditional and folk music.
• These instruments are primarily made of solid materials like wood, metal, or clay and are struck or shaken to create sound.
• Unlike other musical instruments, Ghana Vadya do not require tuning as their sound is determined by the material and shape.
• Some prominent examples of Ghana Vadya include kartal, manjira, ghunghroo, jal tarang, and chimta, which are widely used in folk and devotional music.
• Ghana Vadya instruments are among the oldest types of musical instruments in India and are considered precursors to more complex percussion instruments.

Additional Information

• The classification of musical instruments in Indian tradition is based on the Natyashastra by Bharat Muni, which categorizes them into four types:
• Ghana Vadya (solid instruments), Avanaddha Vadya (membrane-covered instruments), Tat Vadya (string instruments), and Sushir Vadya (wind instruments).
• Each category plays a significant role in Indian classical, folk, and devotional music, contributing to the rich and diverse musical heritage of India.

The correct answer is Sculling maintains balance by moving hands under water..

Key Points

• Sculling is a swimming technique where small, controlled hand movements are used to generate propulsion and maintain balance in the water.
• It is primarily performed under the water surface by moving the hands in a figure-eight or sweeping pattern.
• The primary goal of sculling is to stabilize the body and maintain proper position in the water, essential for efficient swimming.
• Sculling improves swimming technique, allowing swimmers to refine their control, coordination, and feel for the water.
• This technique is commonly used by advanced swimmers, synchronized swimmers, and water polo players for balance and positioning.

Additional Information

• Key Indian swimmers include:
• Srihari Nataraj, a record-holder in multiple events and Olympic qualifier;
• Sajan Prakash, the first Indian swimmer with a direct Olympic qualification; and
• Dhinidhi Desinghu, a young sensation and Olympic participant.
• Other prominent names are Virdhawal Khade, an Olympian and former national champion, and Shikha Tandon, a respected female swimmer, alongside Para-swimmer Vishwas K.S.

The correct answer is Sachin Nag.

Key Points

• Sachin Nag became India's first Asian Games gold medallist by winning the 100m freestyle swimming event during the inaugural Asian Games held in New Delhi in 1951.
• He was a prominent swimmer and water polo player from India, known for his significant contribution to Indian aquatic sports.
• At the 1951 Asian Games, Sachin Nag’s victory was a historic moment as it marked India's maiden gold medal at the continental event.
• His achievement laid the foundation for India’s future success in swimming and other aquatic events.
• The 1951 Asian Games was the first edition of the multi-sport event conducted by the Asian Games Federation, and it was hosted by India in New Delhi.

Additional Information

• Johnny Weissmuller
• Johnny Weissmuller was an American swimmer and actor known for his role as Tarzan in Hollywood films.
• He won five Olympic gold medals in swimming during the 1924 and 1928 Olympic Games.
• Weissmuller is not associated with Indian swimming or the Asian Games.
• Arjun Singh
• There is no record of Arjun Singh being associated with swimming or winning any medal at the Asian Games.
• The name might be linked to other notable Indian personalities in politics or other fields but is unrelated to this context.
• Mihir Sen
• Mihir Sen was a renowned Indian long-distance swimmer, famous for crossing the English Channel in 1958.
• He was the first Indian and the first Asian to achieve this feat.
• Although a celebrated swimmer, Mihir Sen’s accomplishments came after the 1951 Asian Games and were focused on long-distance swimming rather than competitive events like the 100m freestyle.

The correct answer is He campaigned using newspapers and engaged in debates.

Key Points

• Raja Rammohan Roy played a pivotal role in abolishing the practice of Sati (widow immolation) during the early 19th century in Bengal.
• He used newspapers and journals such as ‘Sambad Kaumudi’ to educate the masses and highlight the evils of Sati.
• Rammohan Roy engaged in debates and dialogues with orthodox Hindu leaders and British officials, advocating for social reform and raising awareness about the cruelty and religious misinterpretation related to Sati.
• He presented reasoned arguments and petitions to the British government, which culminated in the Sati Regulation Act of 1829 under Governor-General Lord William Bentinck.
• His efforts were instrumental in fostering a progressive mindset in society, paving the way for the eradication of practices rooted in superstition.

Additional Information

• Abolition of Sati
• The practice of Sati involved the immolation of widows on the funeral pyres of their husbands, and it was deeply rooted in societal and religious traditions in India.
• Rammohan Roy’s efforts played a crucial role in the enactment of the Sati Regulation Act of 1829, which prohibited the practice and made it illegal.
• His work is considered a landmark in the history of social reform in India, symbolizing the shift towards modernity and human rights.

The correct answer is Umroi, Meghalaya.

Key Points

• The 11th edition of the India-Bangladesh joint military exercise SAMPRITI-XI was conducted in Umroi, Meghalaya, in 2023.
• SAMPRITI is an annual bilateral military exercise between the Indian Army and the Bangladesh Army, aimed at enhancing cooperation and mutual understanding.
• The exercise focuses on counter-terrorism operations, humanitarian assistance, and disaster relief in a simulated environment.
• It aims to strengthen the friendship and bilateral relations between the two countries while fostering interoperability and coordination.
• Troops from both nations participated in joint training sessions, including field exercises, cultural exchanges, and discussions on military strategies.

Additional Information

• India-Bangladesh Military Exercise (SAMPRITI)
• SAMPRITI is a joint military exercise conducted annually between the armed forces of India and Bangladesh.
• It serves as a platform for both nations to share their experiences, strategies, and best practices in conducting military operations.
• The exercise focuses on enhancing defense cooperation and maintaining regional peace and security in South Asia.
• SAMPRITI also highlights the commitment of both nations toward combating terrorism and addressing disaster management challenges collaboratively.

The correct answer is Sir Manmatha Nath Roy Chowdhury.

Key Points

• The Santosh Trophy, launched in 1941, is named after Sir Manmatha Nath Roy Chowdhury, the Maharaja of Santosh, a region in present-day Bangladesh.
• Sir Manmatha Nath Roy Chowdhury was an eminent personality known for his contributions to Indian football and his patronage of sports.
• The Santosh Trophy is an annual Indian football tournament contested by states and government institutions.
• It was originally organized to promote football at the grassroots level and is considered one of the most prestigious state-level competitions in India.
• Initially, the tournament featured teams representing various states in India, encouraging regional talent to compete and gain recognition.

Additional Information

• Sir Henry Mortimer Durand:
• He was a British diplomat and civil servant, best known for the Durand Line Agreement of 1893, which established the border between British India and Afghanistan.
• The Durand Line remains a significant geopolitical boundary between modern-day Pakistan and Afghanistan.
• He played no role in Indian football or sports development, making this option unrelated to the Santosh Trophy.
• Syed Abdul Rahim:
• Syed Abdul Rahim was an iconic Indian football coach and manager who led the Indian national team during its golden era in the 1950s and 1960s.
• Under his guidance, India won the 1951 and 1962 Asian Games gold medals in football.
• Although he made significant contributions to Indian football, the Santosh Trophy was not named after him.
• Sailen Manna:
• Sailen Manna was a legendary Indian footballer and captain of the national team in the 1950s.
• He is regarded as one of the finest defenders in Indian football history and was awarded the Padma Shri for his contributions to the sport.

The correct answer is 3) A is true, but R is false..

Key Points

• Assertion (A): "The Moor's Last Sigh" is indeed acclaimed for its intricate narrative that blends historical events with magical realism to explore Indian identity and politics.
• The novel was written by Salman Rushdie and published in 1995.
• It is a multi-generational family saga infused with elements of magical realism, addressing themes like politics, history, art, and identity.
• Reason (R): The statement claiming that the novel won the Booker Prize is false.
• Although Salman Rushdie is a Booker Prize-winning author (for "Midnight's Children" in 1981), "The Moor's Last Sigh" did not win the Booker Prize.
• Hence, while the assertion is true, the reason provided is false, making option 3 the correct answer.

Additional Information

• Key Novels of Salman Rushdie:
• Midnight's Children (1981): A landmark of magical realism, exploring post-colonial India through a child born at the moment of independence.
• Shame (1983): A satirical look at Pakistan and the nature of shame.
• The Satanic Verses (1988): His most famous and controversial work, leading to a fatwa and extensive international debate.
• Shalimar the Clown (2006): Focuses on terrorism and Kashmir.
• The Golden House (2017): Explores the immigrant experience in America.
• Quichotte (2019): A modern reimagining of Don Quixote.

The correct answer is Amit Shah.

Key Points

• The Bharatpol portal was launched by Amit Shah, the Union Minister of Home Affairs, to enhance the efficiency of police services across India.
• This portal aims to strengthen the coordination among police departments of different states and union territories.
• It is designed to provide a centralized database of criminals, missing persons, and stolen property, facilitating better communication and tracking.
• The initiative is part of the broader goal of modernizing and digitizing the police system under the Smart Policing concept.
• The portal integrates technology like artificial intelligence and machine learning for predictive analysis and crime prevention.

Additional Information

• Narendra Modi
• PM Narendra Modi is known for launching key national initiatives like Digital India, Swachh Bharat Abhiyan, and Ayushman Bharat.
• He has been instrumental in policy reforms aimed at economic growth and technological advancements in India.
• Many initiatives under his leadership have promoted e-governance and digital infrastructure across sectors.
• Droupadi Murmu
• Droupadi Murmu is the current President of India, serving as the first tribal woman to hold the office of President.
• She focuses on social welfare and upliftment of marginalized communities.
• Although she hasn’t launched the Bharatpol portal, her leadership symbolizes inclusivity and representation of tribal voices in governance.
• Ashwini Vaishnaw
• Ashwini Vaishnaw is the Minister for Railways, Communications, and Electronics & IT.
• He has spearheaded projects like 5G rollout in India and modernization of the Indian Railways.
• His work primarily focuses on technological advancements in infrastructure and communication but does not involve police reforms or portals like Bharatpol.

The correct answer is Heating of air that rises and cools.

Key Points

• Convectional rainfall occurs due to the intense heating of the Earth's surface, which causes the air above it to rise.
• As the warm air rises, it expands and cools due to lower atmospheric pressure at higher altitudes.
• The cooling of the rising air leads to the condensation of water vapor present in it, resulting in the formation of clouds.
• Once the clouds become saturated, they release moisture in the form of rainfall, which is termed convectional rainfall.
• This type of rainfall is most common in regions near the equator, such as tropical areas, where the sun's heat is intense year-round.

Additional Information

• Cold fronts meeting warm air
• This phenomenon typically leads to frontal rainfall, not convectional rainfall.
• Frontal rainfall occurs when a cold air mass meets a warm air mass, causing the warm air to rise over the cold air, leading to condensation and precipitation.
• This type of rainfall is common in temperate regions and is often associated with cyclonic weather systems.
• Mountains forcing air to rise
• This process results in orographic or relief rainfall, not convectional rainfall.
• When moist air is forced to ascend over mountain ranges, it cools and condenses, leading to rainfall on the windward side of the mountains.
• The leeward side, also known as the rain shadow region, typically experiences dry conditions.
• Moist air from polar regions
• Moist air from polar regions is not associated with convectional rainfall.
• Instead, it plays a role in polar climates, where precipitation is typically in the form of snow rather than rain.
• Polar regions experience low precipitation levels due to the cold temperatures and limited moisture in the air.

The correct answer is They attract industries and create jobs.

Key Points

• Mineral deposits are often found in specific geographic locations, which leads to industrial development in those areas to extract and process these resources.
• Industries that utilize mineral resources, such as mining, metallurgy, and manufacturing, create a significant number of job opportunities, attracting workers and their families to live nearby.
• The presence of mineral resources stimulates the local economy, leading to the development of infrastructure such as roads, schools, and healthcare facilities, which further encourages population growth.
• Regions rich in minerals often experience urbanization as industries and ancillary services grow, leading to higher population densities in these areas.
• Examples include cities like Jamshedpur in India, which developed around the steel industry due to the availability of iron ore in the region.

Additional Information

• Role of Mineral Deposits:
• Mineral deposits are crucial for economic development and the industrialization of regions.
• They attract investment, industries, and workforce, leading to urbanization and population growth around mining and industrial hubs.
• However, their extraction and processing must be managed sustainably to minimize environmental degradation and ensure long-term benefits for society.

The correct answer is The transition from high growth to low growth aligns with reaching replacement level.

Key Points

• Demographic transition refers to the process of change in a society's population from high fertility and mortality rates to low fertility and mortality rates, typically in four stages.
• In the demographic transition model, the shift from high population growth to low growth (Stage 3 to Stage 4) is associated with fertility rates reaching replacement level fertility.
• Replacement level fertility is the average number of children per woman needed to maintain a stable population, which is approximately 2.1 in most developed countries.
• During Stage 3, fertility rates decline significantly due to increased access to education, family planning, and economic changes, leading to population stabilization in Stage 4.
• The alignment of demographic transition with replacement level fertility indicates the balance between births and deaths, ensuring sustainable population growth.

Additional Information

• Socioeconomic factors influencing demographic transition:
• Economic Development & Industrialization: Shifts from agrarian to industrial/service economies increase costs of raising children, reducing family size, and create demand for educated labor.
• Education: Higher education, especially for women, correlates with later childbearing, increased family planning knowledge, and lower fertility rates.
• Urbanization: Migration to cities for opportunities changes family dynamics and exposes people to new norms, often leading to smaller families.
• Income Levels: Increased wealth and job opportunities often lead to delayed marriage and childbearing.

The correct answer is To achieve temporal resolution for global coverage and land deformation monitoring.

Key Points

• The NISAR mission is jointly developed by NASA and ISRO to provide high-resolution radar imaging that supports monitoring of Earth's surface changes, including land deformation, ice dynamics, and ecosystem disturbances.
• The Synthetic Aperture Radar (SAR) on NISAR allows precise measurements and monitoring of surface changes by achieving temporal resolution through revisiting the same location every 12 days.
• Revisiting the same location ensures consistent monitoring of dynamic phenomena such as earthquakes, landslides, and glacier movements, contributing to disaster preparedness and response.
• The 12-day revisit cycle enables global coverage while maintaining a balance between high spatial resolution and temporal frequency, critical for analyzing long-term environmental and climatic patterns.
• NISAR's ability to monitor land deformation and surface changes supports applications in agriculture, forestry, hydrology, and urban planning, making it a valuable tool for scientific and societal benefits.

Additional Information

• Synthetic Aperture Radar (SAR) Technology
• SAR technology uses radar signals to create high-resolution images of Earth's surface, regardless of weather conditions or time of day.
• NISAR is equipped with dual-frequency SAR systems (L-band and S-band), enabling detailed imaging of natural and man-made features.
• This cutting-edge technology is crucial for monitoring Earth's dynamic processes, including environmental changes, disaster management, and resource planning.

The correct answer is Glycogen.

Key Points

• Glycogen is the primary energy storage molecule in animals, particularly in the liver and muscles.
• It is a polysaccharide made up of glucose monomers linked together by alpha-1,4-glycosidic bonds with branching via alpha-1,6-glycosidic bonds.
• Glycogen serves as a readily mobilizable source of glucose to meet energy needs during periods of fasting or intense physical activity.
• The process of glycogen synthesis is called glycogenesis, while its breakdown is known as glycogenolysis.
• In humans, the liver stores approximately 100 grams of glycogen, while muscles store about 400 grams, providing energy for metabolic processes.

Additional Information

• Starch
• Starch is the primary energy storage polysaccharide in plants, not animals.
• It is composed of two molecules: amylose (linear chains of glucose) and amylopectin (branched chains of glucose).
• Humans and animals can digest starch by breaking it down into glucose with the help of the enzyme amylase.
• Starch is found in foods like potatoes, rice, wheat, and corn.
• Cellulose
• Cellulose is a structural polysaccharide found in the cell walls of plants, providing strength and rigidity.
• It is composed of glucose monomers linked by beta-1,4-glycosidic bonds, making it indigestible for humans due to the lack of the enzyme cellulase.
• It is a major component of dietary fiber, which aids in digestion and maintains gut health.
• Sucrose
• Sucrose is a disaccharide composed of one glucose and one fructose molecule, commonly known as table sugar.
• It is used as a quick source of energy and is found naturally in fruits, vegetables, and sugarcane.
• Sucrose is not a storage molecule but rather a transport molecule in plants.

The correct answer is Set up tribunals for water disputes and bar SC review.

Key Points

• Under Article 262 of the Indian Constitution, Parliament has the authority to legislate for the adjudication of inter-State water disputes.
• The provision explicitly allows Parliament to bar the jurisdiction of the Supreme Court or any other court in such disputes by creating specialized mechanisms like tribunals.
• In accordance with Article 262, the Inter-State River Water Disputes Act (1956) was enacted, which empowers the Union Government to establish tribunals for resolving inter-State water disputes.
• Tribunals constituted under this act are the highest authority for adjudicating water disputes, with no provision for judicial review by the Supreme Court to ensure quicker resolution.
• The purpose of this constitutional arrangement is to avoid legal delays and ensure impartial resolution of disputes through a focused mechanism.

Additional Information

• Inter-State Water Disputes
• Inter-State water disputes arise when States disagree over sharing, usage, or control of river waters flowing across their boundaries.
• Article 262 of the Constitution provides a framework for resolving these disputes through legislation and tribunals.
• Such disputes are handled under the Inter-State River Water Disputes Act, ensuring legal and administrative mechanisms for resolution without judicial delays.

The correct answer is Doppler effect.

Key Points

• The Doppler effect is the change in the frequency or wavelength of a wave concerning an observer moving relative to the wave source.
• It occurs for all types of waves, including sound, light, and water waves, and is commonly experienced in the case of sound waves.
• In the given scenario, as the train approaches, the sound waves compress, causing an increase in frequency (higher pitch), and as it moves away, the sound waves spread out, causing a decrease in frequency (lower pitch).
• The mathematical expression for the Doppler effect in sound is given by: f' = f(v + vo) / (v + vs), where f' is the observed frequency, f is the source frequency, v is the speed of sound, vo is the observer’s velocity, and vs is the source velocity.
• This phenomenon has practical applications in various fields, such as astronomy (to study the motion of stars and galaxies), radar technology, and medical imaging (Doppler ultrasound).

Additional Information

• Refraction of Sound
• Refraction refers to the bending of sound waves as they pass from one medium to another, where the speed of sound differs.
• For instance, sound waves bend towards cooler regions where the speed of sound is slower.
• While refraction affects the propagation direction of sound, it does not explain the change in pitch observed in the given scenario.
• Diffraction of Sound
• Diffraction is the bending of sound waves around obstacles or through openings.
• It allows sound to be heard even when the source is not in the direct line of sight.
• This phenomenon is unrelated to the change in frequency or pitch caused by relative motion between the source and observer.
• Resonance
• Resonance occurs when an object vibrates at its natural frequency due to an external force of the same frequency.
• This results in a significant increase in amplitude, as seen in musical instruments or structures like bridges.
• Resonance does not explain the change in pitch when the train passes by, as it is unrelated to relative motion or frequency shifts.

The correct answer is Theft committed by a gang.

Key Points

• The Bharatiya Nyaya Samhita (BNS) explicitly defines 'organized crime' and 'petty organized crime' to address offenses committed systematically by groups.
• Petty organized crime refers to smaller-scale, less severe offenses carried out by a group, such as theft or burglary committed by a gang.
• The focus is on crimes involving collaboration between multiple individuals, even if the financial or physical damage is relatively minor.
• The inclusion of 'petty organized crime' under BNS ensures that such offenses are penalized and deterred effectively.
• This categorization aims to tackle the growing threat posed by organized criminal behavior, including minor offenses, thereby maintaining law and order.

Additional Information

• Bharatiya Nyaya Samhita (BNS)
• The BNS is the proposed replacement for the Indian Penal Code (IPC) to address contemporary challenges in criminal justice.
• It introduces modern definitions and classifications of crimes, including 'organized crime' and 'petty organized crime'.
• The framework aims to simplify and enhance the efficiency of the Indian criminal justice system.

The correct answer is Paramparagat Krishi Vikas Yojana (PKVY).

Key Points

• Paramparagat Krishi Vikas Yojana (PKVY) is a sub-component of the Soil Health Management (SHM) scheme under the National Mission of Sustainable Agriculture (NMSA).
• The scheme was launched in 2015 to promote organic farming and improve soil health by reducing the dependence on chemical fertilizers and pesticides.
• PKVY helps farmers adopt organic farming practices such as the use of bio-fertilizers, vermicomposting, and traditional techniques to improve crop productivity sustainably.
• Under the scheme, farmers are incentivized to form clusters of 20 hectares to take up organic farming collectively, thereby reducing input costs and improving yields.
• The scheme also focuses on creating market linkages for organic produce, promoting Participatory Guarantee Systems (PGS), and assisting farmers in getting organic certification.

Additional Information

• Pradhan Mantri Kisan Maandhan Yojana (PM-KMY)
• PM-KMY is a pension scheme introduced in 2019 for small and marginal farmers.
• Farmers aged between 18 and 40 years can enroll in this scheme, contributing a small monthly premium based on their age.
• On attaining the age of 60, enrolled farmers receive a minimum fixed pension of Rs. 3,000 per month.
• The scheme aims to provide social security to farmers after their retirement.
• Soil Health Card Scheme (SHC)
• The Soil Health Card Scheme was launched by the Government of India in 2015 to promote soil testing and balanced use of fertilizers.
• Under this scheme, farmers receive soil health cards containing detailed information about the nutrient content of their soil.
• The cards provide recommendations on the type and quantity of fertilizers required to improve soil health and crop productivity.
• The scheme aims to reduce overuse of fertilizers, increase crop yields, and improve soil fertility.
• Agricultural Technology Management Agency (ATMA)
• ATMA is a central sector scheme launched in 2005 as part of the National Mission on Agricultural Extension and Technology (NMAET).
• It aims to strengthen extension services by integrating research and extension activities at the district level.
• ATMA focuses on capacity-building, training, and dissemination of technology to farmers through Farmer Field Schools (FFS) and demonstrations.
• The scheme emphasizes participatory and bottom-up planning for agricultural development.

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

Key Points

• The National Single Window System (NSWS) is an initiative by the Government of India that facilitates investors in obtaining clearances and approvals from various departments and ministries.
• The NSWS is designed to reduce bureaucratic hurdles and enhance the ease of doing business in India by providing a centralized platform.
• It integrates multiple government systems and services across Ministries and States, providing a single interface for businesses.
• The system includes a common registration form, document repository, and a dashboard for real-time tracking of approvals and clearances.
• Contrary to the Reason (R) in the question, the NSWS is not a separate ministry. Instead, it operates under the Department for Promotion of Industry and Internal Trade (DPIIT) within the Ministry of Commerce and Industry.

Additional Information

• National Single Window System (NSWS)
• The NSWS is a digital platform that provides guidance for investors to identify and apply for approvals as per their business requirements.
• It aims to reduce the compliance burden on businesses and increase transparency and accountability in government processes.
• The platform is designed to cater to both domestic and global investors, promoting India as a business-friendly destination.
• The Department for Promotion of Industry and Internal Trade (DPIIT)
• The DPIIT is a central government department under the Ministry of Commerce and Industry.
• It is responsible for formulating and implementing promotional and developmental measures for industrial growth in India.
• The NSWS is one of the key initiatives under DPIIT to improve ease of doing business.
• Ease of Doing Business in India
• India has made significant improvements in the World Bank's Ease of Doing Business rankings over the years, thanks to initiatives like the NSWS.
• These reforms aim to simplify regulatory processes, digitize services, and provide a conducive environment for businesses to thrive.

The correct answer is Most of the top 30 TTDI scorers are high-income economies from Europe and Asia-Pacific.

Key Points

• The 2024 Travel and Tourism Development Index (TTDI) highlights the dominance of high-income economies from Europe and the Asia-Pacific region among the top 30 scorers.
• Developed nations such as Japan, the United States, and several European countries consistently rank high due to their superior infrastructure, safety, and sustainability measures.
• The TTDI assesses countries on various parameters, including business environment, safety, health, infrastructure, and natural resources.
• Europe remains a strong performer, with many of its high-income nations being global leaders in areas such as tourism infrastructure and cultural resources.
• The Asia-Pacific region, led by countries like Japan and Singapore, has been recognized for its high-quality infrastructure and focus on sustainable tourism practices.

Additional Information

• The Travel and Tourism Development Index (TTDI)
• The TTDI is a global benchmark developed by the World Economic Forum to assess travel and tourism development conditions across countries.
• The index evaluates factors such as enabling environment, travel and tourism policy, and infrastructure to measure how countries perform in fostering sustainable tourism development.
• The TTDI is an essential tool for policymakers and stakeholders in identifying strengths and areas for improvement in their tourism sectors.

The correct answer is Restoration of degraded forest patches in Aravalli range.

Key Points

• The Aravalli Green Wall Project was launched in 2023 by the Ministry of Environment, Forest, and Climate Change (MoEFCC) to address environmental degradation in the Aravalli range.
• The initiative focuses on restoring degraded forest lands in the Aravalli range, which spans across Haryana, Rajasthan, Gujarat, and Delhi.
• The project aims to increase green cover and biodiversity in the region, thereby mitigating the adverse effects of desertification and climate change.
• One of the key objectives is to create a continuous green barrier along the Aravalli range to prevent the spread of the Thar Desert and improve ecosystem services.
• The project involves community participation, afforestation drives, and soil conservation measures to restore the ecological balance of the region.

Additional Information

• Afforestation in Himalayan States
• Afforestation efforts in Himalayan states focus on increasing forest cover in ecologically sensitive areas, which are crucial for mitigating climate change and preserving biodiversity.
• Such initiatives aim to reduce soil erosion, improve water retention, and support local livelihoods through sustainable forest management.
• These efforts are not directly linked to the Aravalli Green Wall Project, which specifically targets the Aravalli range.
• Building Forest Corridors Across Indo-Gangetic Plains
• Forest corridors are designed to connect fragmented forest patches to facilitate wildlife movement and conserve biodiversity.
• While the Indo-Gangetic plains are an important ecological zone, this option does not pertain to the Aravalli Green Wall Project, which is limited to the Aravalli range.
• Soil Carbon Sequestration in Desert Zones
• Soil carbon sequestration refers to capturing atmospheric carbon dioxide and storing it in soil organic matter, which improves soil fertility and mitigates climate change.
• Although soil conservation is a component of the Aravalli Green Wall Project, its primary focus is on restoring forests rather than exclusively targeting desert zones for carbon sequestration.
• Aravalli Range and Desertification
• The Aravalli range acts as a natural barrier against the spread of the Thar Desert into the Indo-Gangetic plains.
• Deforestation and mining activities have degraded the region, increasing vulnerability to desertification and biodiversity loss.
• The Aravalli Green Wall Project aims to counter these challenges by restoring forest patches and enhancing ecosystem resilience.

The correct answer is They are jointly owned by the public and private sectors..

Key Points

• Joint sector industries are enterprises where ownership is shared between the government (public sector) and private entities.
• The concept of joint sector industries was introduced to combine the strengths of both public and private sectors, such as government control and private sector efficiency.
• These industries aim to promote economic growth, generate employment, and develop industries in underdeveloped regions.
• Examples of joint sector industries include companies like Oil India Ltd., where both government and private stakeholders share ownership and management responsibilities.
• These industries play a vital role in bridging the gap between large-scale private monopolies and complete state control.

Additional Information

• By Economic Activity (The 5-Sector Model)
• Primary: Extracts or harvests natural resources (e.g., agriculture, fishing, mining, forestry).
• Secondary: Transforms raw materials into finished products (e.g., manufacturing, construction, food processing).
• Tertiary: Provides intangible services (e.g., retail, healthcare, education, transportation, banking).
• Quaternary: Focuses on information, knowledge, and research (e.g., IT, R&D, consultancy).
• Quinary: Involves high-level decision-making and top executives (e.g., government, senior management, scientific research).

The correct answer is Speaker of Lok Sabha.

Key Points

• The Speaker of Lok Sabha has the exclusive authority to decide whether a Bill is a Money Bill as per Article 110 of the Indian Constitution.
• A Money Bill is primarily concerned with matters such as taxation, borrowing by the government, expenditure, and financial obligations of the state.
• The Speaker’s decision regarding the classification of a Bill as a Money Bill is considered final and cannot be challenged in any court.
• The Money Bill can only be introduced in the Lok Sabha on the recommendation of the President, but its categorization rests with the Speaker.
• The Rajya Sabha can only make recommendations on a Money Bill, and these recommendations may or may not be accepted by the Lok Sabha.

Additional Information

• Finance Minister
• The Finance Minister is responsible for presenting the Union Budget and managing the financial affairs of the country.
• However, the Finance Minister does not have the authority to decide whether a Bill is a Money Bill.
• The Finance Minister may propose the Bill, but its classification as a Money Bill is decided solely by the Speaker of Lok Sabha.
• President
• The President of India has the power to recommend the introduction of a Money Bill in the Lok Sabha.
• Before the introduction of a Money Bill, the recommendation of the President is mandatory under Article 110.
• However, the President does not determine whether a Bill qualifies as a Money Bill; this decision is made by the Speaker of Lok Sabha.
• Comptroller & Auditor General (CAG)
• The Comptroller & Auditor General (CAG) is responsible for auditing the accounts of the central and state governments and ensuring financial accountability.
• The CAG does not play any role in determining the classification of a Bill as a Money Bill.
• The CAG’s functions are limited to auditing and reporting financial irregularities, not legislative procedures.

Important Points

• Money Bill
• A Money Bill is defined under Article 110 of the Indian Constitution. It deals exclusively with matters specified in Article 110(1), such as taxation, public expenditure, and borrowing.
• It can only be introduced in Lok Sabha and requires the President's recommendation before introduction.
• The Rajya Sabha cannot amend or reject a Money Bill; it can only make recommendations, which the Lok Sabha may or may not accept.
• The final authority to determine whether a Bill is a Money Bill rests with the Speaker of Lok Sabha.

The correct answer is Deep-sky objects.

Key Points

• Deep-sky objects refer to astronomical entities such as galaxies, nebulae, star clusters, and distant celestial objects that are faint and require dark skies for observation.
• Urban light pollution significantly hampers the visibility of deep-sky objects by increasing the sky brightness, reducing contrast, and obscuring faint objects.
• Light pollution from artificial sources, such as streetlights and buildings, scatters in the atmosphere, overwhelming the faint light emitted by deep-sky objects.
• Observing deep-sky objects typically requires remote locations or observatories situated far from urban areas to minimize the effects of light pollution.
• Efforts to mitigate light pollution, including the use of shielded lighting and adherence to dark-sky standards, are crucial for preserving the visibility of deep-sky objects.

Additional Information

• Sunspots
• Sunspots are dark, cooler areas on the Sun's surface caused by magnetic activity that inhibits convection.
• Observing sunspots does not typically require dark skies, as they can be seen using solar filters or telescopes during the daytime.
• Urban light pollution does not significantly affect the observation of sunspots, as they are observed against the bright background of the Sun.
• Planetary Transit
• Planetary transit occurs when a planet passes in front of its host star, causing a temporary dip in the star's brightness.
• These events are usually studied using telescopes equipped with photometers or spectrometers, and they are not as affected by light pollution as deep-sky objects.
• However, precise measurements of brightness changes may require optimal conditions, which can be influenced by urban light pollution.
• Lunar Phases
• Lunar phases refer to the appearance of the Moon as it orbits Earth, changing its illuminated portion visible from Earth.
• Light pollution has minimal impact on observing lunar phases because the Moon is bright enough to be visible even in urban environments.
• Lunar observation is often used as an introductory activity for amateur astronomers due to its ease of visibility.
• Light Pollution and Astronomy
• Light pollution refers to the excessive and misdirected artificial light that disrupts natural darkness, impacting astronomical observations.
• It reduces the visibility of faint celestial objects and interferes with professional and amateur astronomy.
• Organizations like the International Dark-Sky Association (IDA) advocate for measures to reduce light pollution through responsible lighting practices.

Shortcut Trick We can use approximate square root values to quickly compare the two given terms. ⇒ a = √7 − √3 ≈ 2.64 − 1.73 ≈ 0.91 ⇒ b = √5 − √2 ≈ 2.23 − 1.41 ≈ 0.82 Comparing the final decimal values, 0.91 > 0.82, which clearly indicates that a > b. ∴ The correct answer is a > b.

Alternate Method Given: a = √7 − √3 b = √5 − √2 Calculation: We can accurately compare a and b by finding their reciprocals and rationalizing the denominators: ⇒ 1/a = 1 / (√7 − √3) ⇒ 1/a = (√7 + √3) / (7 − 3) = (√7 + √3) / 4 ⇒ 1/b = 1 / (√5 − √2) ⇒ 1/b = (√5 + √2) / (5 − 2) = (√5 + √2) / 3 Now, let us compare 1/a and 1/b by equating their denominators (LCM of 3 and 4 is 12): ⇒ 1/a = 3(√7 + √3) / 12 = (√63 + √27) / 12 ⇒ 1/b = 4(√5 + √2) / 12 = (√80 + √32) / 12 Now comparing the numerators of both fractions: ⇒ Since 63 < 80, we have √63 < √80 ⇒ Since 27 < 32, we have √27 < √32 Therefore, (√63 + √27) < (√80 + √32), which directly implies that 1/a < 1/b. Since both a and b are positive numbers, taking the reciprocal reverses the inequality sign: ⇒ a > b ∴ The correct answer is a > b.

Additional Information Rationalization of Surds To remove roots from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator using the formula (x − y)(x + y) = x2 − y2. Reciprocal Inequality Rule For any two positive numbers x and y, if x > y, then taking their reciprocal reverses the inequality resulting in 1/x < 1/y. Comparing Pure Surds When comparing mixed surds like a√b, move the outside coefficient inside by squaring it as √(a2b) to make comparisons entirely based on the inner radicands.

Shortcut Trick

First bracket: (1/4 ÷ 1/2) = 1/4 × 2 = 1/2

Second bracket: (1/15 + 1 − 9/10) = 2/30 + 30/30 − 27/30 = 5/30 = 1/6

Combine results: 1/2 ÷ 1/6 = 1/2 × 6 = 3

∴ The correct answer is 3.

Alternate Method

Given:

Expression: (1/4 ÷ 1/2) ÷ (1/15 + 1 − 9/10)

Formula Used:

BODMAS Rule: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Calculations:

⇒ Step 1: Solve the first bracket (1/4 ÷ 1/2)

⇒ 1/4 ÷ 1/2 = 1/4 × 2/1 = 2/4 = 1/2

⇒ Step 2: Solve the second bracket (1/15 + 1 − 9/10)

⇒ Find the Least Common Multiple (LCM) of the denominators 15 and 10, which is 30.

⇒ 1/15 = 2/30

⇒ 1 = 30/30

⇒ 9/10 = 27/30

⇒ (2/30 + 30/30 − 27/30) = (2 + 30 − 27) / 30 = 5/30 = 1/6

⇒ Step 3: Divide the result of the first bracket by the result of the second bracket

⇒ 1/2 ÷ 1/6 = 1/2 × 6/1

⇒ 6 / 2 = 3

∴ The correct answer is 3.

Additional Information

BODMAS Rule

It dictates the order of operations: Brackets first, then Orders (powers/roots), Division, Multiplication, Addition, and finally Subtraction.

Fraction Division

To divide by a fraction, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Fraction Addition & Subtraction

Always find the LCM of the denominators to make them identical before adding or subtracting the numerators directly.

Shortcut Trick

Initial R : B = 3 : 4 and final R : B = 4 : 5. Quantities added are +20 and +15.

Using cross-multiplication: 1 unit = (20 × 5 − 15 × 4) / (4 × 4 − 3 × 5) = (100 − 60) / 1 = 40.

Initial green marbles = 6 units = 6 × 40 = 240. New red marbles = (3 × 40) + 20 = 140.

New red (140) equals 4 parts in final ratio. Thus, 1 final part = 140 ÷ 4 = 35.

New green = 7 final parts = 7 × 35 = 245. Green marbles added = 245 − 240 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Initial ratio of red, blue, and green marbles = 3 : 4 : 6

Red marbles added = 20, Blue marbles added = 15

Final ratio of red, blue, and green marbles = 4 : 5 : 7

Ratio Change of MarblesInitial Ratio3y (R)4y (B)6y (G)Added+ 20+ 15+ xFinal Ratio4 parts5 parts7 parts(3y + 20) : (4y + 15) = 4 : 5

Calculations:

⇒ Let the initial number of red, blue, and green marbles be 3y, 4y, and 6y respectively.

⇒ After adding the marbles, new number of red marbles = 3y + 20

⇒ New number of blue marbles = 4y + 15

⇒ The new ratio of red to blue marbles is 4 : 5.

⇒ (3y + 20) ÷ (4y + 15) = 4 ÷ 5

⇒ 5 × (3y + 20) = 4 × (4y + 15)

⇒ 15y + 100 = 16y + 60

⇒ 16y − 15y = 100 − 60

⇒ y = 40

⇒ Initial number of green marbles = 6 × 40 = 240

⇒ New number of red marbles = (3 × 40) + 20 = 140

⇒ The final ratio of red to green marbles is 4 : 7.

⇒ Let the new number of green marbles be G.

⇒ 140 ÷ G = 4 ÷ 7

⇒ 4 × G = 140 × 7

⇒ G = 980 ÷ 4 = 245

⇒ Number of green marbles added = New green marbles − Initial green marbles

⇒ Green marbles added = 245 − 240 = 5

∴ The correct answer is 5.

Additional Information

Ratio Comparison Method

When individual components of a ratio are increased by specific quantities, establishing an equation with any two known components allows determining the exact multiplier of the ratio.

Cross-Multiplication Trick

For an initial ratio a:b becoming c:d after adding quantities p and q respectively, 1 ratio unit can be calculated directly as (pd − qc) ÷ (ad − bc).

Proportional Value Assignment

Once a new concrete value corresponding to a final ratio part is found (like 140 equivalent to 4 parts), the exact quantities of all other parts in the final state can be determined by direct proportion.

Shortcut Trick

Profit ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Profit ratio = (25000 × 12) : (50000 × 6) = 300000 : 300000 = 1 : 1

Total profit ratio parts = 1 + 1 = 2 parts.

2 parts = ₹30,000 ⇒ 1 part = ₹15,000.

A's share is 1 part = ₹15,000.

∴ The correct answer is ₹15,000.

Alternate Method

Given:

A's Investment = ₹25,000, Time = 12 months

B's Investment = ₹50,000, Time = 6 months

Total Profit = ₹30,000

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Partner ACap: ₹25,000Time: 12 monthsWeight: 3,00,000Partner BCap: ₹50,000Time: 6 monthsWeight: 3,00,000Profit Ratio (A : B) = 1 : 1

Calculations:

⇒ Profit ratio of A and B = (25000 × 12) : (50000 × 6)

⇒ Profit ratio = 300000 : 300000 = 1 : 1

⇒ Sum of ratio parts = 1 + 1 = 2

⇒ A's share = (1 ÷ 2) × Total Profit

⇒ A's share = (1 ÷ 2) × 30000

⇒ A's share = 15000

∴ The correct answer is ₹15,000.

Additional Information

Simple Partnership

When all partners invest their capital for the same duration of time, the profit is distributed strictly in the ratio of their original investments.

Compound Partnership

When partners invest for different periods, the equivalent capital for each is calculated as Investment × Time. Profit is distributed in the ratio of these final products.

Working Partner vs Sleeping Partner

A working partner actively manages the business and is typically paid a fixed salary or a certain profit percentage before the rest of the profit is divided among all active and sleeping partners.

Shortcut Trick

Initial capital ratio = 5 : 3. Since A's capital = ₹50,000, B's = ₹30,000.

Equivalent capital of A = (50k × 6) + (40k × 6) = 540k

Equivalent capital of B = (30k × 6) + (60k × 6) = 540k

Profit ratio = 540k : 540k = 1 : 1

Profit share of A = (1 ÷ 2) × ₹84,000 = ₹42,000

∴ The correct answer is ₹42,000.

Alternate Method

Given:

Initial capital ratio of A : B = 5 : 3

Initial capital of A = ₹50,000

Total profit after 1 year = ₹84,000

Formula Used:

Profit Ratio = (Capital_A × Time_A) : (Capital_B × Time_B)

Partner A's Timeline₹50,000 × 6m₹40,000 × 6mPartner B's Timeline₹30,000 × 6m₹60,000 × 6mTotal Equivalent Capitals: A = ₹5,40,000 | B = ₹5,40,000

Calculations:

⇒ Initial capital of A = 5 units = ₹50,000

⇒ 1 unit = ₹50,000 ÷ 5 = ₹10,000

⇒ Initial capital of B = 3 units = 3 × ₹10,000 = ₹30,000

⇒ After 6 months, A withdraws ₹10,000, so new capital = ₹50,000 − ₹10,000 = ₹40,000

⇒ After 6 months, B doubles his investment, so new capital = ₹30,000 × 2 = ₹60,000

⇒ Equivalent capital of A = (₹50,000 × 6) + (₹40,000 × 6) = ₹3,00,000 + ₹2,40,000 = ₹5,40,000

⇒ Equivalent capital of B = (₹30,000 × 6) + (₹60,000 × 6) = ₹1,80,000 + ₹3,60,000 = ₹5,40,000

⇒ Ratio of profit share = 5,40,000 : 5,40,000 = 1 : 1

⇒ Sum of profit ratios = 1 + 1 = 2 units

⇒ A's profit share = (1 ÷ 2) × ₹84,000 = ₹42,000

∴ The correct answer is ₹42,000.

Additional Information

Compound Partnership

When capitals are invested for unequal time periods, profits are shared in the ratio of the product of capital and time, known as equivalent capital.

Simple Partnership

When partners invest their capitals for the exact same period, the profit is purely distributed in the respective ratio of their original investments.

Working vs Sleeping Partner

A working active partner manages the business and usually receives a salary or fixed percentage of profit. A sleeping partner only provides funds and gets a share solely based on capital and time ratio.

Shortcut Trick

The sequence of multiples forms an Arithmetic Progression (A.P.).

Average of an A.P. = (First Term + Last Term) ÷ 2

First 3-digit multiple of 23 = 115

Last 3-digit multiple of 23 = 989

Average = (115 + 989) ÷ 2 = 552

∴ The correct answer is 552.

Alternate Method

Given:

We need to find the average of all three-digit numbers divisible by 23.

Formula Used:

For an Arithmetic Progression (A.P.), Average = (a + l) ÷ 2

Where 'a' is the first term and 'l' is the last term.

115 First Term

989 Last Term

552 Average (Midpoint)

Average = (First Term + Last Term) ÷ 2

Calculations:

⇒ Smallest three-digit number = 100

⇒ 100 ÷ 23 = 4 with a remainder of 8

⇒ First 3-digit multiple of 23 (a) = 23 × 5 = 115

⇒ Largest three-digit number = 999

⇒ 999 ÷ 23 = 43 with a remainder of 10

⇒ Last 3-digit multiple of 23 (l) = 23 × 43 = 989

⇒ The sequence of numbers forms an A.P.: 115, 138, 161, ..., 989

⇒ Average of the A.P. = (115 + 989) ÷ 2

⇒ Average = 1104 ÷ 2 = 552

∴ The correct answer is 552.

Additional Information

Arithmetic Progression (A.P.)

A sequence of numbers where the difference between consecutive terms is constant. General form: a, a+d, a+2d, ...

Average of an A.P.

The average (mean) of an A.P. is strictly equal to the average of its first and last terms: Average = (a + l) ÷ 2.

Number of Terms in an A.P.

The total number of terms (n) in a finite arithmetic sequence can be calculated using the formula: n = [(l − a) ÷ d] + 1.

Shortcut Trick

Let the volume of each container be 100 units.

Container 1 has 40 units water ⇒ Milk filled = 100 − 40 = 60 units.

Container 2 has 50 units water ⇒ Milk filled = 100 − 50 = 50 units.

Total milk in mixture = 60 + 50 = 110 units; Total volume = 100 + 100 = 200 units.

Fraction of milk in final mixture = 110 ÷ 200 = 0.55.

∴ The correct answer is 0.55 (or 55%).

Alternate Method

Given:

⇒ Volume of Container 1 = Volume of Container 2

⇒ Water in Container 1 = 40%

⇒ Water in Container 2 = 50%

⇒ Remaining capacity of both containers is completely filled with milk.

Formula Used:

⇒ Fraction of Milk = (Total Volume of Milk) ÷ (Total Volume of Mixture)

Milk: 60Water: 40Vol: 100+Milk: 50Water: 50Vol: 100Total Milk: 110(55%)Total Water: 90(45%)Final Vol: 200Mixture Process

Calculations:

⇒ Let the capacity of each container be 100 units.

⇒ Volume of water in Container 1 = 40% of 100 = 40 units

⇒ Volume of milk added in Container 1 = 100 − 40 = 60 units

⇒ Volume of water in Container 2 = 50% of 100 = 50 units

⇒ Volume of milk added in Container 2 = 100 − 50 = 50 units

⇒ Total volume of milk in final mixture = 60 + 50 = 110 units

⇒ Total volume of the final mixture = 100 + 100 = 200 units

⇒ Fraction of milk in final mixture = 110 ÷ 200

⇒ Fraction = 11 ÷ 20 = 0.55

∴ The correct answer is 0.55 (or 55%).

Additional Information

Mixtures & Alligations

When mixing two quantities, the ratio of their volumes directly determines the weighted average of any concentration present in them.

Weighted Average Formula

Final Concentration = (C₁V₁ + C₂V₂) ÷ (V₁ + V₂), where C is concentration and V is volume.

Successive Replacement

If a fraction of a mixture is removed and replaced with a pure component iteratively, the remaining original component is calculated as Final = Initial × (1 − Removed/Total)ⁿ.

Shortcut Trick

Assume the price of Blue pens = 1 unit, Red pens = 3 units, and let Blue pens ordered = B dozens.

Original Bill = (6 × 3) + (B × 1) = 18 + B

Interchanged Bill = (B × 3) + (6 × 1) = 3B + 6

The new bill is 40% more, meaning it is 1.4 times the original bill.

⇒ 3B + 6 = 1.4 × (18 + B) ⇒ 1.6B = 19.2 ⇒ B = 12

∴ The correct answer is 1:2.

Alternate Method

Original BillRed : 6 × 3 = 18Blue : B × 1 = BTotal = 18 + BNew BillRed : B × 3 = 3BBlue : 6 × 1 = 6Total = 3B + 6+ 40% Increase

Given:

Number of red pens originally ordered = 6 dozens.

Let the number of blue pens originally ordered be B dozens.

Let the price of 1 dozen blue pens be x. Then, the price of 1 dozen red pens = 3x.

Calculations:

⇒ Original Bill = (6 × 3x) + (B × x) = 18x + Bx = x(18 + B)

⇒ When the quantities of the two colours are interchanged, Red = B dozens and Blue = 6 dozens.

⇒ New Bill = (B × 3x) + (6 × x) = 3Bx + 6x = x(3B + 6)

⇒ According to the problem, the new bill is 40% more than the original bill.

⇒ New Bill = Original Bill + 40% of Original Bill

⇒ x(3B + 6) = x(18 + B) × 1.4

⇒ Since price x is a positive value, we can divide both sides by x:

⇒ 3B + 6 = 1.4 × (18 + B)

⇒ 3B + 6 = 25.2 + 1.4B

⇒ 3B − 1.4B = 25.2 − 6

⇒ 1.6B = 19.2

⇒ B = 19.2 ÷ 1.6 = 12

⇒ The original order had 6 dozens of red pens and 12 dozens of blue pens.

⇒ Required Ratio = Red pens : Blue pens = 6 : 12 = 1 : 2.

∴ The correct answer is 1:2.

Additional Information

Percentage Increase Formula

If a value is increased by p%, the final value can directly be computed as Original Value × (1 + p/100).

Total Value Relationship

Total value of any item is given by Quantity × Unit Price. Any changes in quantities or unit prices directly impact the proportionate bill linearly.

Cross-Multiplication in Equations

To simplify decimal coefficients quickly, you can multiply the entire equation by a power of 10. For instance, 1.6B = 19.2 transforms immediately into 16B = 192.

Shortcut Trick

Total Selling Price required for a 30% profit = 130% of ₹800 = ₹1040.

Total sellable flowers = 200 − 20 = 180 flowers.

Selling Price per flower = 1040 ÷ 180 = 52 ÷ 9 ≈ ₹5.78.

∴ The correct answer is ₹5.78.

Alternate Method

Given:

Total cost price of 200 flowers = ₹800

Number of withered flowers = 20

Desired profit percentage = 30%

Formula Used:

Selling Price (SP) = CP × (100 + Profit%) ÷ 100

SP per unit = Total SP ÷ Total units available for sale

CP = ₹800 (200 flowers)

+30% Profit

Total SP = ₹1040 (130% of CP)

÷ 180 flowers

(200 − 20 withered)

SP / flower = ₹5.78

Calculations:

⇒ Total Selling Price = 800 × (100 + 30) ÷ 100

⇒ Total Selling Price = 800 × 130 ÷ 100 = ₹1040

⇒ Number of flowers available to sell = 200 − 20 = 180

⇒ Selling Price per flower = Total Selling Price ÷ 180

⇒ Selling Price per flower = 1040 ÷ 180

⇒ Selling Price per flower = 52 ÷ 9

⇒ Selling Price per flower = 5.777... ≈ ₹5.78

∴ The correct answer is ₹5.78.

Additional Information

Profit and Loss Percentage

Profit% or Loss% is strictly calculated based on the Total Cost Price (CP) of the initial quantity bought, unless stated otherwise. Formula: Profit% = (Profit ÷ CP) × 100.

Effective Cost Recovery

When a fraction of goods gets damaged or lost, the desired revenue (Total SP) must be fully recovered by selling only the remaining intact goods. This inherently increases the required Selling Price per unit.

Marked Price and Discount

If a vendor wants to offer discounts while maintaining a fixed profit rate, the Marked Price must be calculated above this calculated Selling Price (SP). Selling Price = Marked Price − Discount.

Shortcut Trick

Given the initial Selling Price is ₹Z and the profit is 25%, the Cost Price (CP) = Z ÷ 1.25 = 0.8Z.

During the festive season, the new Marked Price is ₹1.6Z.

With a 20% discount on the Marked Price, the new SP = 1.6Z × (1 − 0.20) = 1.28Z.

Profit amount = New SP − CP = 1.28Z − 0.8Z = 0.48Z.

Percentage profit (as a fraction) = 0.48Z ÷ 0.8Z = 0.6.

∴ The correct answer is 60% (0.6).

Alternate Method

Given:

Initial Selling Price = ₹Z

Initial Profit = 25%

New Marked Price (MP) = ₹1.6Z

Festive Discount = 20%

Formula Used:

Cost Price (CP) = SP ÷ (1 + Profit/100)

Selling Price (SP) = MP × (1 − Discount/100)

Percentage Profit = [(SP − CP) ÷ CP] × 100

CP = 0.8ZSP = 1.28ZMP = 1.6Z−20% DiscountProfit = +0.48Z

Calculations:

⇒ First, calculate the original Cost Price (CP):

⇒ CP × (1 + 0.25) = Z

⇒ 1.25 × CP = Z

⇒ CP = Z ÷ 1.25 = 0.8Z

⇒ Next, calculate the new Selling Price after applying the 20% discount on the new MP:

⇒ New SP = 1.6Z × (1 − 0.20)

⇒ New SP = 1.6Z × 0.80 = 1.28Z

⇒ Calculate the profit earned during the festive season:

⇒ Profit = New SP − CP = 1.28Z − 0.8Z = 0.48Z

⇒ Calculate the profit fraction / percentage:

⇒ Profit fraction = Profit ÷ CP = 0.48Z ÷ 0.8Z = 0.6

⇒ This translates to exactly 60% profit.

∴ The correct answer is 60% (0.6).

Additional Information

Marked Price and Discount

A discount is strictly calculated on the Marked Price (MP), not the Cost Price. The Selling Price is derived directly by deducting the discount: SP = MP × (1 − Discount%).

Cost Price from Profit Percentage

When the final Selling Price and Profit% are provided, the Cost Price can be efficiently found using the formula: CP = SP × [100 ÷ (100 + Profit%)].

Successive Price Changes

When an article's price is marked up and then followed by a discount, a chain calculation or a successsive percentage formula [a + b + (a × b ÷ 100)] can be used to track the overall profit or loss.

Shortcut Trick

Initial Ratio (Acid : Water) = 5 : 3

Final Ratio (Acid : Water) = 5 : 4

Since only water is added, the ratio of acid (5 parts) correctly remains unchanged.

Increase in water ratio = 4 − 3 = 1 part.

1 part = 4 litres ⇒ Original Acid = 5 parts = 5 × 4 = 20 litres.

∴ The correct answer is 20 litres.

Alternate Method

Initial Acid = 5u Water = 3u

Final Acid = 5u Water = 4u

• 4 L Water

Water Change: 4u − 3u = 1u = 4 Litres Original Acid: 5u = 5 × 4 = 20 Litres

Given:

Initial ratio of acid to water = 5 : 3

Water added = 4 litres

New ratio of acid to water = 5 : 4

Calculations:

⇒ Let initial acid = 5x and initial water = 3x.

⇒ After adding 4 litres of water, new quantity of water = 3x + 4.

⇒ New ratio = 5x / (3x + 4) = 5 / 4

⇒ 20x = 15x + 20

⇒ 20x − 15x = 20

⇒ 5x = 20 ⇒ x = 4

⇒ Original quantity of acid = 5x = 5 × 4 = 20 litres.

∴ The correct answer is 20 litres.

Additional Information

Constant Component Concept

When one substance is added to a mixture, the quantity of the other substance remains strictly constant. Equating the ratio parts of the undisturbed component simplifies calculations directly without full equations.

Cross-Multiplication Method

If the initial ratio is a:b and final ratio becomes c:d after adding x and y respectively, the relation is: (a × d) − (b × c) = (y × c) − (x × d). Here, y = 4 and x = 0.

Replacement in Mixtures

When a portion of a mixture is withdrawn without adding anything, the ratio of its components remains identical to the original mixture. The ratio alters only upon adding a new quantity.

Shortcut Trick

Total work = LCM(12, 18, 5) = 180 units.

Efficiency of A = 180 ÷ 12 = 15, B = 180 ÷ 18 = 10, A+B+C = 180 ÷ 5 = 36.

Efficiency of C = 36 − (15 + 10) = 11.

Wage ratio A : B : C = 15 : 10 : 11.

C's share = (11 ÷ 36) × 7200 = ₹2,200.

∴ The correct answer is ₹2,200.

Alternate Method

Given:

Time taken by A alone = 12 days

Time taken by B alone = 18 days

Time taken by A, B, and C together = 5 days

Total wages = ₹ 7,200

Formula Used:

Total Work = LCM of time taken

Efficiency = Total Work ÷ Time

Wages are distributed in the ratio of efficiencies when workers work for the same total time.

Calculations:

⇒ Total efficiency of A, B, and C = 36 units/day

⇒ Efficiency of C = Total efficiency − (Efficiency of A + Efficiency of B)

⇒ Efficiency of C = 36 − (15 + 10) = 11 units/day

⇒ Since they worked together for all 5 days, wages are shared directly in the ratio of their individual efficiencies.

⇒ Ratio of wages A : B : C = 15 : 10 : 11

⇒ Total ratio parts = 15 + 10 + 11 = 36 parts

⇒ 36 parts = ₹ 7,200

⇒ 1 part = 7200 ÷ 36 = ₹ 200

⇒ C's share = 11 parts = 11 × 200 = ₹ 2,200

∴ The correct answer is ₹2,200.

Additional Information

Work and Wages

Wages are strictly proportional to the actual amount of work done by each individual. If workers work together for the same duration, their wages are distributed directly in the ratio of their daily efficiencies.

Different Working Days

If workers work for different numbers of days, their respective wage shares are calculated by multiplying their efficiency with the number of days they worked: Share = Efficiency × Days Worked.

Pipes and Cisterns

This concept mirrors Time and Work exactly. Filling pipes have a positive efficiency (doing work), while emptying pipes or leaks have a negative efficiency (undoing work).

Shortcut Trick

Let remaining mixture components A, B, and C be 7k, 8k, and 5k (Total = 20k).

Final quantity of A = 7k + 15 L and Final quantity of B = 8k.

Given, Final B − Final A = 25 ⇒ 8k − (7k + 15) = 25 ⇒ k = 40.

Total remaining mixture = 20 × 40 = 800 L.

Initial mixture = 800 L + 40 L (drained) = 840 L.

∴ The correct answer is 840 litres.

Alternate Method

Given:

⇒ Initial ratio of A : B : C = 7 : 8 : 5

⇒ Quantity of mixture drained = 40 litres

⇒ Added quantities: 15 litres of A and 5 litres of C

Remaining Mixture Components

7k

• 15 L Solution A

8k Solution B

5k

• 5 L Solution C

Final B − Final A = 8k − (7k + 15) = 25 L

Calculations:

⇒ Let the quantities of A, B, and C in the remaining mixture be 7k, 8k, and 5k respectively.

⇒ Total remaining mixture = 7k + 8k + 5k = 20k

⇒ After addition, Final quantity of A = 7k + 15

⇒ After addition, Final quantity of B = 8k

⇒ After addition, Final quantity of C = 5k + 5

⇒ We are given that Final A is 25 litres less than Final B:

⇒ 8k − (7k + 15) = 25

⇒ 8k − 7k − 15 = 25

⇒ k − 15 = 25

⇒ k = 40

⇒ Total remaining mixture = 20k = 20 × 40 = 800 litres

⇒ Initial quantity of mixture = Remaining mixture + Drained mixture

⇒ Initial quantity of mixture = 800 + 40 = 840 litres

∴ The correct answer is 840 litres.

Additional Information

Mixture Replacement Concept

When a part of a mixture is drained, the remaining mixture still retains the exact same ratio of ingredients as the original mixture before any subsequent pure addition is made.

Successive Replacement

If V is the initial volume of pure liquid, and x volume is replaced by water n times sequentially, the final volume of pure liquid is given by V × (1 − x/V)ⁿ.

Rule of Alligation

It allows calculating the required ratio in which two components of given prices or concentrations must be mixed to obtain a mixture of a desired average price or concentration.

Shortcut Trick

Time ratio of A : B = 3 : 2, so efficiency ratio of A : B = 2 : 3.

Total work = Efficiency of B × Time taken by B = 3 × 18 = 54 units.

Combined efficiency of A and B = 2 + 3 = 5 units/hour.

Time taken together = Total work ÷ Combined efficiency = 54 ÷ 5 = 10.8 hours.

∴ The correct answer is 10.8 hours.

Alternate Method

Given:

Ratio of time taken by A and B = 3 : 2

Time taken by B = 18 hours

Calculations:

⇒ Let the time taken by A be 3x and by B be 2x.

⇒ 2x = 18 hours

⇒ x = 9 hours

⇒ Time taken by A = 3 × 9 = 27 hours.

⇒ Total efficiency of A and B together = 2 + 3 = 5 units/hour.

⇒ Time taken by A and B working together = Total Work ÷ Total Efficiency

⇒ Time taken = 54 ÷ 5 = 10.8 hours.

∴ The correct answer is 10.8 hours.

Additional Information

Time & Efficiency Relationship

Efficiency is inversely proportional to the time taken. If the ratio of time taken by A and B is x : y, then the ratio of their efficiencies is y : x.

Work and Wages

Wages are generally distributed in the ratio of the total work done by each individual. If workers work for the same number of days, the wages are distributed directly in the ratio of their efficiencies.

Pipes & Cisterns

The concept of time and work directly applies to pipes. Inlet pipes have positive efficiency (filling work) while outlet pipes or leaks have negative efficiency (emptying work).

Shortcut Trick

Time taken for one round = Distance ÷ Speed.

T_x = 2/10 = 1/5 h, T_y = 2/12 = 1/6 h, T_z = 2/15 h.

Time to meet at start = LCM(1/5, 1/6, 2/15) = 2/1 = 2 hours.

Total time from 9:00 a.m. to 1:00 p.m. = 4 hours.

Number of meetings = Total time ÷ Meeting interval = 4 ÷ 2 = 2 times.

∴ The correct answer is 2 times.

Alternate Method

Given:

Circumference of the track (D) = 2 km

Speeds: X = 10 km/h, Y = 12 km/h, Z = 15 km/h

Cycling duration = 9:00 a.m. to 1:00 p.m. = 4 hours

Formula Used:

Time = Distance ÷ Speed

Meeting time at start point = LCM of individual times taken for one round

LCM of fractions = LCM of Numerators ÷ HCF of Denominators

Start PointD = 2 kmSpeeds:X = 10 km/hY = 12 km/hZ = 15 km/hTime to meet = LCM of individual times

Calculations:

⇒ T_x = 2 ÷ 10 = 1/5 h

⇒ T_y = 2 ÷ 12 = 1/6 h

⇒ T_z = 2 ÷ 15 = 2/15 h

⇒ Meeting time = LCM(1/5, 1/6, 2/15)

⇒ LCM of numerators (1, 1, 2) = 2

⇒ HCF of denominators (5, 6, 15) = 1

⇒ LCM(1/5, 1/6, 2/15) = 2 ÷ 1 = 2 hours

⇒ They all meet at the starting point every 2 hours.

⇒ Total duration from 9:00 a.m. to 1:00 p.m. = 4 hours.

⇒ Number of meetings = 4 ÷ 2 = 2 times

∴ The correct answer is 2 times.

Additional Information

Meeting at the Starting Point

When multiple runners or objects move around a circular path starting at the same time, the time required for all of them to meet at the starting point simultaneously is always the LCM of the times taken by each to complete exactly one round.

First Meeting on a Circular Track

If two bodies move in the same direction, they will meet for the very first time anywhere on the track after a time = Distance ÷ Relative Speed (Difference of their speeds). If moving in opposite directions, relative speed is the sum of their speeds.

LCM & HCF of Fractions

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.
HCF of fractions = HCF of Numerators ÷ LCM of Denominators.

Shortcut Trick

Area of a circular path = π × w × (2r + w)

Where w = 3 m (width of path) and r = 8 m (inner radius)

Area = π × 3 × (2 × 8 + 3) = π × 3 × 19

Area = 57π ≈ 57 × 3.14 ≈ 179 m²

∴ The correct answer is 179 m².

Alternate Method

Given:

Inner radius of the park (r) = 8 m

Width of the path (w) = 3 m

Outer radius of the park with path (R) = 8 m + 3 m = 11 m

8 m

3 m

11 m

Circular Park with Path

Formula Used:

Area of circular path = Area of outer circle − Area of inner circle

Area = πR² − πr² = π(R² − r²)

Calculations:

⇒ Area = π × (11² − 8²)

⇒ Area = π × (121 − 64)

⇒ Area = 57π

⇒ Area ≈ 57 × 3.14159

⇒ Area ≈ 179.07 m²

By comparing the options, the nearest value is 179 m².

∴ The correct answer is 179 m².

Additional Information

Area of a Circle

The total region enclosed by a circle is given by Area = πr², where r is the radius.

Area of a Circular Path

For a circular path of width w around a circle of radius r, the area is calculated as π(R² − r²) or πw(2r + w), where the outer radius R = r + w.

Circumference

The perimeter or length of the outer boundary of a circle is calculated as C = 2πr.

Shortcut Trick

The slope of the given line in the form y = mx + c is m = 1/4.

The slope of a perpendicular line is always the negative reciprocal of the original line's slope.

Perpendicular slope = −1 ÷ (1/4) = −4.

∴ The correct answer is -4.

Alternate Method

Given:

Equation of the line: y = 1/4x + 7

Formula Used:

For two perpendicular lines with slopes m₁ and m₂: m₁ × m₂ = −1

y = 1/4x + 7 m₁ = 1/4

Perpendicular Line m₂ = -4

Calculations:

⇒ The given line is structured in the standard form y = mx + c, where m represents the slope of the line.

⇒ By comparing the equations directly, the slope of the given line (m₁) = 1/4

⇒ Let the slope of the required perpendicular line be defined as m₂.

⇒ Since the properties of perpendicular lines state that their product is −1, we have m₁ × m₂ = −1

⇒ (1/4) × m₂ = −1

⇒ m₂ = −1 × 4

⇒ m₂ = −4

∴ The correct answer is -4.

Additional Information

Slope-Intercept Form

The equation of a straight line is typically expressed as y = mx + c, where m is the slope defining the steepness and c is the y-intercept.

Parallel Lines

If two straight lines are distinctly parallel to each other, their slopes will always be exactly equal (m₁ = m₂).

Perpendicular Lines

When two non-vertical straight lines intersect perfectly at a right angle, the resulting product of their slopes evaluates to −1 (m₁ × m₂ = −1).

Shortcut Trick

We know the standard conversion where 1 radian is approximately equal to 57.3°.

To convert 2.5 radians to degrees, we simply multiply the value by 57.3°.

⇒ 2.5 × 57.3° = 143.25°

Looking at the given options, the closest exact value is 143.24°.

∴ The correct answer is 143.24°.

Alternate Method

Given:

Angle = 2.5 radians

Formula Used:

Angle in Degrees = Angle in Radians × (180° ÷ π)

2.5 rad 143.24°

Calculation:

⇒ Angle in Degrees = 2.5 × (180° ÷ π)

⇒ Angle in Degrees = 450° ÷ π

Using the precise value of π ≈ 3.14159:

⇒ Angle in Degrees = 450° ÷ 3.14159

⇒ Angle in Degrees ≈ 143.2395°

⇒ Rounding off to two decimal places, we get 143.24°.

∴ The correct answer is 143.24°.

Additional Information

Radian to Degree Conversion

To convert any given angle from radians to degrees, multiply the radian value by 180° ÷ π.

Degree to Radian Conversion

To convert an angle from degrees back to radians, multiply the degree value by π ÷ 180°.

Standard Conversions

Some common unit circle angles include: π radians = 180°, π ÷ 2 radians = 90°, and π ÷ 4 radians = 45°.

Shortcut Trick

Direct formula for the area of a sector in radians: Area = ½ × r² × θ

Substitute the given values: r = 10 cm and θ = π/6.

Area = ½ × (10)² × (π/6) = ½ × 100 × π/6 = 25π / 3

Using π ≈ 3.14, Area = (25 × 3.14) ÷ 3 = 78.5 ÷ 3 ≈ 26.17 cm².

∴ The correct answer is 26.17 cm².

Alternate Method

Given:

Radius of the sector (r) = 10 cm

Central angle (θ) = π/6 radians

O π/6 10 cm 10 cm Radius (r) = 10 cm Angle (θ) = π/6 rad Area = ?

Formula Used:

Area of a sector = ½ × r² × θ (where θ is in radians)

Calculations:

⇒ Area = ½ × (10)² × (π / 6)

⇒ Area = ½ × 100 × (π / 6)

⇒ Area = 50 × π / 6

⇒ Area = 25π / 3

Using π ≈ 3.14:

⇒ Area = (25 × 3.14) / 3

⇒ Area = 78.5 / 3

⇒ Area = 26.166... ≈ 26.17 cm²

∴ The correct answer is 26.17 cm².

Additional Information

Area of a Sector (in degrees)

If the central angle θ is given in degrees instead of radians, the formula to calculate the area becomes Area = (θ ÷ 360°) × πr².

Length of an Arc

The length of the arc (s) is calculated directly as s = rθ (if θ is in radians) or s = (θ ÷ 360°) × 2πr (if θ is in degrees).

Perimeter of a Sector

The total perimeter bounds the sector, including the curved arc length and the two straight radii. It is found using Perimeter = s + 2r.

Shortcut Trick

If tan x = cot y, then their angles are complementary: x + y = 90°

Given: tan A = cot(2A − 30°)

⇒ A + (2A − 30°) = 90°

⇒ 3A = 120°

⇒ A = 40°

∴ The correct answer is 40°.

Alternate Method

Given:

tan A = cot(2A − 30°)

Formula Used:

cot θ = tan(90° − θ)

A

2A − 30° Adjacent to A Opposite to A If tan x = cot y, then x + y = 90°

Calculations:

⇒ tan A = tan(90° − (2A − 30°))

Comparing the angles on both sides:

⇒ A = 90° − (2A − 30°)

⇒ A = 90° − 2A + 30°

⇒ A + 2A = 120°

⇒ 3A = 120°

⇒ A = 40°

∴ The correct answer is 40°.

Additional Information

Complementary Angle Identities

sin(90° − θ) = cos θ | tan(90° − θ) = cot θ | sec(90° − θ) = cosec θ

Supplementary Angle Identities

sin(180° − θ) = sin θ | cos(180° − θ) = −cos θ | tan(180° − θ) = −tan θ

Negative Angle Identities

sin(−θ) = −sin θ | cos(−θ) = cos θ | tan(−θ) = −tan θ

Shortcut Trick

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding altitudes.

Ratio of areas = (Ratio of altitudes)²

⇒ Area Ratio = (2 ÷ 5)² = 4 : 25

∴ The correct answer is 4:25.

Alternate Method

Given:

Ratio of corresponding altitudes of two similar triangles (h₁ : h₂) = 2 : 5

Formula Used:

Area of Δ₁ ÷ Area of Δ₂ = (h₁ ÷ h₂)²

h1 = 2 Area = A1

h2 = 5 Area = A2

A1 : A2 = h1² : h2²

Calculations:

⇒ Area of Δ₁ ÷ Area of Δ₂ = (2 ÷ 5)²

⇒ Area of Δ₁ ÷ Area of Δ₂ = 4 ÷ 25

⇒ Ratio of their areas = 4 : 25

∴ The correct answer is 4:25.

Additional Information

Area of Similar Triangles

The ratio of the areas of two similar triangles is precisely equal to the square of the ratio of their corresponding sides, altitudes, or medians.

Perimeter of Similar Triangles

The ratio of the perimeters of two similar triangles is directly equal to the ratio of their corresponding sides, altitudes, or medians.

Inradius & Circumradius Ratio

For two similar triangles, the ratio of their corresponding inradii or circumradii follows the exact same ratio as their corresponding sides.

Shortcut Trick

Directly apply the algebraic identity: (x + y)² + (x − y)² = 2(x² + y²)

Here, replacing x with a and y with 1:

⇒ (a + 1)² + (a − 1)² = 2(a² + 1²)

Substitute a = √5 ⇒ a² = 5.

⇒ 2(5 + 1) = 2 × 6 = 12

∴ The correct answer is 12.

Alternate Method

Given:

a = √5

Formula Used:

(x + y)² = x² + 2xy + y²

(x − y)² = x² − 2xy + y²

Calculation:

⇒ (a + 1)² = a² + 2a + 1

⇒ (a − 1)² = a² − 2a + 1

Adding both expanded expressions together:

⇒ (a² + 2a + 1) + (a² − 2a + 1)

⇒ 2a² + 2

Substitute the given value a = √5:

⇒ 2(√5)² + 2

⇒ 2 × 5 + 2

⇒ 10 + 2 = 12

∴ The correct answer is 12.

Additional Information

Important Algebraic Identities

Another useful identity for subtraction is: (a + b)² − (a − b)² = 4ab.

Difference of Squares

The product of sum and difference is: a² − b² = (a − b)(a + b).

Square Root Properties

The square of a square root returns the original non-negative value: (√x)² = x. Also, roots can be multiplied directly: √x × √y = √(xy).

Shortcut Trick

Directly extract perfect squares from the square roots:

⇒ √50 + √18 − √8

⇒ √(25 × 2) + √(9 × 2) − √(4 × 2)

⇒ 5√2 + 3√2 − 2√2 = 6√2

∴ The correct answer is 6√2.

Alternate Method

Given:

Expression: √50 + √18 − √8

Method Used:

Prime factorization of the numbers inside the roots to extract terms outside the square root.

Calculations:

⇒ √50 = √(2 × 5 × 5) = 5√2

⇒ √18 = √(2 × 3 × 3) = 3√2

⇒ √8 = √(2 × 2 × 2) = 2√2

Substitute these values back into the expression:

⇒ 5√2 + 3√2 − 2√2

Take √2 as common:

⇒ (5 + 3 − 2) × √2

⇒ 6√2

∴ The correct answer is 6√2.

Additional Information

Similar Surds

Surds which have the same irrational factor are called similar surds. Only similar surds can be added or subtracted. Example: 3√2 and 5√2 are similar surds.

Multiplication and Division of Surds

Unlike addition and subtraction, surds with the same order can be multiplied or divided regardless of their base. √a × √b = √(ab) and √a ÷ √b = √(a/b).

Simplifying Surds

To simplify a square root, find the largest perfect square factor of the number inside the root, and bring its square root outside. Example: √(x²y) = x√y.

Shortcut Trick

The longest chord of a circle is its diameter.

Given the diameter of the circle is 25 cm.

Maximum length of the chord = Diameter = 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Diameter of the circle = 25 cm

Concept Used:

A chord is a line segment whose endpoints lie on the circumference of the circle.

As a chord moves closer to the center of the circle, its length continuously increases.

The chord that passes exactly through the center is the longest possible chord, which is known as the diameter.

Shorter Chord

Longest Chord (Diameter) = 25 cm Center O

Calculation:

⇒ Maximum length of chord = Length of diameter

⇒ Maximum length of chord = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Perpendicular from Center to Chord

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

Equal Chords and Subtended Angles

Equal chords of a circle subtend equal angles at the center of the circle.

Distance of Chords from Center

Equal chords of a circle are always equidistant from the center. Conversely, chords that are equidistant from the center are exactly equal in length.

Shortcut Trick

⇒ sin A + cos A = √2 sin A

⇒ cos A = sin A(√2 − 1) ⇒ cot A = √2 − 1

⇒ tan A = 1 ÷ cot A = 1 ÷ (√2 − 1)

⇒ Rationalizing the denominator gives tan A = √2 + 1

∴ The correct answer is √2 + 1.

Alternate Method

Given:

sin A + cos A = √2 sin A

Formula Used:

tan A = sin A ÷ cos A

(a − b)(a + b) = a² − b²

Calculations:

⇒ Divide the entire given equation by cos A:

⇒ (sin A ÷ cos A) + (cos A ÷ cos A) = √2(sin A ÷ cos A)

⇒ tan A + 1 = √2 tan A

⇒ 1 = √2 tan A − tan A

⇒ 1 = tan A(√2 − 1)

⇒ tan A = 1 ÷ (√2 − 1)

⇒ Multiply numerator and denominator by (√2 + 1):

⇒ tan A = (√2 + 1) ÷ [(√2 − 1)(√2 + 1)]

⇒ tan A = (√2 + 1) ÷ [(√2)² − (1)²]

⇒ tan A = (√2 + 1) ÷ (2 − 1)

⇒ tan A = √2 + 1

∴ The correct answer is √2 + 1.

Additional Information

Basic Trigonometric Ratios

tan A = sin A ÷ cos A and cot A = cos A ÷ sin A.

Pythagorean Identities

sin² A + cos² A = 1 | 1 + tan² A = sec² A | 1 + cot² A = cosec² A.

Algebraic Rationalization

To rationalize a denominator of the form (a − b), multiply numerator and denominator by its conjugate (a + b) using the identity (a − b)(a + b) = a² − b².

The correct answer is '2' i.e. Peak .

Key Points

• The word APOGEE refers to the highest point or the peak of something.
• Among the given options:
• Start: Refers to the beginning of something.
• Peak: Refers to the highest point or summit, which is synonymous with "apogee."
• Nadir: Refers to the lowest point, which is the opposite of "apogee."
• Base: Refers to the bottom or foundation, which is unrelated to "apogee."
• Thus, the synonym for "APOGEE" is Peak.

Correct sentence: The apogee of her career was when she won the prestigious award, reaching the peak of her success.

Additional Information

• Opposite Words: The opposite of "apogee" is "nadir," which denotes the lowest point.
• Usage in Context: "Apogee" is often used metaphorically to describe the pinnacle of achievement or success.

The correct answer is '2' i.e. To oppose or discourage.

Key Points

• The idiom "Throw cold water on" means to oppose or discourage an idea, plan, or enthusiasm.
• It is generally used in situations where someone is critical of or undermines excitement or positivity about something.

Correct usage example: She threw cold water on his idea of starting a new business by pointing out all the risks involved.

Additional Information

• Idioms:
• Idioms are phrases or expressions that have figurative meanings different from their literal meanings.
• Examples of idioms include "Break the ice," "Burn the midnight oil," and "Hit the nail on the head."
• Context:
• The meaning of an idiom often depends on the context it is used in.
• Understanding idioms can enhance communication and comprehension skills.

The correct answer is '3' i.e. Truthful .

Key Points

• The word Mendacious means dishonest or lying.
• To find the antonym, we need to look for a word that conveys the opposite meaning, i.e., being honest or truthful.
• Among the options, Truthful is the most appropriate antonym as it directly signifies honesty.
• The other options like Dishonest, Fraudulent, and Deceitful are synonyms of Mendacious, not antonyms.

Correct answer: Truthful is the antonym of Mendacious.

Additional Information

• Mendacious: Comes from the Latin word "mendax," meaning lying or deceitful.
• Synonyms: Dishonest, Fraudulent, Deceitful.
• Antonym: Truthful (meaning honest or sincere).

The correct answer is '4' i.e. Obfescate .

Key Points

• The word Obfescate is misspelled. The correct spelling is Obfuscate, which means to make something unclear or difficult to understand.
• The other options, Antediluvian, Infinitesimal, and Persiflage, are correctly spelled.
• This question tests your ability to identify spelling errors in vocabulary.

Correct sentence: The correct spelling is Obfuscate, which refers to making something obscure or unclear..

Additional Information

• Tips for identifying spelling errors:
• Pay attention to common misspellings or letter swaps in complex words.
• Use mnemonic devices to remember tricky spellings, such as “Obfuscate” (think of “obscure”).
• Learning new vocabulary:
• Read regularly to familiarize yourself with uncommon words.
• Use a dictionary to verify the meanings and spellings of unfamiliar terms.

The correct answer is '1' i.e. Photolithography .

Key Points

• The word "Photolithography" is a microchip printing method and is the correct spelling.
• The other options provided are incorrect spellings:
• Option 2: "Photolithographe" is incorrect because of the added "e" at the end.
• Option 3: "Photolitography" is incorrect due to the missing "h" in "litho".
• Option 4: "Potolithography" is incorrect as the "Ph" is misspelled as "Po".

Correct spelling: Photolithography

Additional Information

• Photolithography is a process used in semiconductor manufacturing to transfer patterns onto a substrate using light. It is a crucial step in creating microchips.
• The word comes from "photo" (light) and "lithography" (stone printing), which is derived from Greek words.

The correct answer is '2' i.e. Encomium .

Key Points

• The word Encomium refers to a speech or piece of writing that praises someone or something highly.
• Elegy: A poem of serious reflection, typically a lament for the dead.
• Sermon: A talk on a religious or moral subject, especially one given during a church service.
• Soliloquy: An act of speaking one's thoughts aloud when by oneself, especially by a character in a play.

Correct sentence: A speech or piece of writing that praises someone highly is called an Encomium.

Additional Information

• Ensure to differentiate among the terms provided in the options to select the most appropriate one-word substitute.
• Understand the context of the question to eliminate incorrect options effectively.

The correct answer is Option 1.

 

Key Points

• The phrasal verb "give out" can mean to reveal or disclose information, especially something that was intended to be kept secret.
• In the sentence, the journalist refused to reveal the identities or information concerning her sources despite legal threats.
• Therefore, "give out" is the most appropriate choice.

Therefore, the correct answer is Option 1: give out.

 

Correct Sentence: "The journalist refused to give out her sources despite repeated legal threats."

 

Additional Information

• Give out: To reveal, disclose, or distribute something. ✓
• Give over: To hand something over to someone or to stop doing something; it does not fit the context.
• Give away: To give something to someone freely or unintentionally reveal a secret. Although possible in some contexts, "give out" is the intended expression here.
• Give up: To stop trying, surrender, or relinquish something.

The correct answer is '2' i.e. with.

Key Points

• The preposition with is commonly used after the verb comply to indicate adherence or obedience to rules, regulations, or standards.
• The sentence refers to compliance with the revised code of ethics, so with is the appropriate choice.
• Other options such as to, by, or in are grammatically incorrect in this context.

Correct sentence: The memorandum clarified that employees must comply with the revised code of ethics.

Additional Information

• Usage of prepositions:
• Comply with: To follow rules or adhere to standards. Example: "You must comply with the safety regulations."
• Comply to: Incorrect usage; avoid using "to" after comply.
• Comply by: Rarely used and usually incorrect in this context.
• Comply in: Incorrect usage; "in" does not fit grammatically here.

The correct answer is '1' i.e. opposed .

Key Points

• The sentence discusses how the scientist's hypothesis was not just radical but also completely contrary to the foundational assumptions of her discipline.
• The correct word to fill the blank should mean "completely contrary or in disagreement with."
• Opposed is the most appropriate choice as it correctly conveys the meaning of being "diametrically contrary" or "against."
• Other options like opposing and opposite are not suitable in this context. Averse is incorrect as it refers to a strong dislike or disinclination, which doesn't fit the sentence.

Correct sentence: The scientist's hypothesis was not only radical but also diametrically opposed to the foundational assumptions of her discipline.

Additional Information

• Opposed: Means "completely against or in disagreement with something."
• Opposing: Refers to something that is in opposition, often used in contexts like teams or forces.
• Opposite: Indicates something that is entirely different, often used in physical or conceptual contrasts.
• Averse: Means "having a strong dislike for something," which doesn't fit the context here.

The correct answer is '3' i.e. Part 3 .

Key Points

• The Error: The error lies in the usage of the subjective pronoun 'they' at the end of Part 3.
• Grammatical Rule (Pronoun Case): In English, we use Subjective Pronouns (I, he, she, we, they) when the pronoun is the subject of a verb. We use Objective Pronouns (me, him, her, us, them) when the pronoun is the object of a verb or preposition.
• Analysis: In the phrase "who report they," the pronoun 'they' is intended to be the subject of the clause that follows ("they were never granted..."). However, the way the sentence is structured, the reporting act requires a conjunction (that) to lead into the next clause, or the pronoun must clearly act as the subject of the subsequent verb.
• Correction: The sentence should read "...auditors who report that they were never granted..." or simply ensure the pronoun 'they' is properly placed as the subject of the passive verb "were granted" in Part 4. The current transition in Part 3 is abrupt and grammatically incomplete.

Additional Information

• Not just... but also: This is a correlative conjunction. The sentence correctly uses this pair to link "from the agencies" and "from the auditors." (Parallelism).
• Concerned: When placed after a noun (agencies concerned), it means "involved" or "relevant." When placed before (concerned agencies), it often means "worried." Both are grammatically correct depending on meaning.
• Subject-Verb Agreement: The main subject "findings" (plural) correctly agrees with the verb "prompted" (past tense).

The correct answer is '2' i.e. to suppress dissent does not only reflect legal opportunism .

Key Points

• The phrase "does not only reflect" is incorrect. The correct phrase should be "not only reflects" to maintain proper grammatical structure and parallelism.
• Using "does not only reflect" introduces unnecessary complexity and breaks the flow of the sentence.
• The corrected portion ensures that the sentence is grammatically accurate and stylistically coherent.

Correct sentence: That certain state governments have invoked archaic laws to suppress dissent not only reflects legal opportunism but also undermines the constitutional ethos that such provisions were meant to uphold.

Additional Information

• Parallelism: When using phrases like "not only... but also," ensure that the structure of the sentence is consistent and grammatically correct.
• Examples:
• Incorrect: He does not only sing, but also dances.
• Correct: He not only sings but also dances.

The correct answer is '1' i.e. A unanimous verdict has been delivered by the jury. .

Key Points

• To change a sentence from active to passive voice, the subject becomes the receiver of the action, and the object becomes the doer.
• The active sentence here is: The jury has delivered a unanimous verdict.
• In passive voice, the object ("a unanimous verdict") comes first, followed by the verb in the passive form. The subject ("the jury") follows, preceded by "by".
• The correct passive construction is: A unanimous verdict has been delivered by the jury.

Correct sentence: A unanimous verdict has been delivered by the jury.

Additional Information

• Active voice: The subject performs the action expressed by the verb.
• Example: The chef prepared the meal.
• Passive voice: The subject receives the action expressed by the verb.
• Example: The meal was prepared by the chef.
• Verb tense in passive voice is formed by using the correct form of "to be" + the past participle of the main verb.