SSC CGL 2025 (Held On: 12 Sept, 2025 Shift 2) (Previous Year Paper)

The logic follows here,

• Liberty : Autonomy
• Autonomy is the state or condition of liberty (they are closely related in meaning)

Similarly,

• Equality is closely related in meaning to Fairness.
• Equality → Fairness

Hence, the correct answer is "Option 3"

The pattern followed here is

Hence, the correct answer is "Option 3"

The pattern followed here is:

Hence, the correct answer is "Option 4"

The sequence is arranged in increasing order of structural complexity.

• Atom → basic unit of matter
• Molecule → combination of atoms
• Cell → basic unit of life (made of molecules)
• The next higher level of organization after a cell is a Tissue.
• Atom, Molecule, Cell, Tissue

Hence, the correct answer is "Option 1"

The logic follows here, 

Hence, the correct answer is "Option 1"

The logic follows here, 

Hence, the correct answer is "Option 1"

The logic follows here, 

Conclusion I: The region does not have proper irrigation infrastructure.

Does not follow — Dependence on monsoon rains does not necessarily mean irrigation infrastructure is absent or inadequate. Some infrastructure may exist, but farmers may still rely mainly on rainfall.

Conclusion II: A weak monsoon may reduce agricultural output.

Follows — If most farmers depend on monsoon rains, a weak monsoon can negatively affect irrigation, leading to lower agricultural output.

Hence, the correct answer is "Option 2"

The logic follows here, 

Assumption I: Exoplanets in habitable zones may have the potential to support life.

Otherwise, their discovery would not create “exciting prospects” for extraterrestrial life.

Assumption II: Current technology is insufficient to directly explore exoplanets in habitable zones.

This directly supports the claim about “technological limitations” posing obstacles.

Assumption III: Direct exploration will be unnecessary if we detect signs of life remotely.

The statement does not suggest abandoning direct exploration; it only says it is difficult.

Only I and II are implicit

Hence, the correct answer is "Option 1"

The letters identified from the word PROVIDE are:

• 1st: P
• 2nd: R
• 3rd: O
• 6th: D

Using the letters P, R, O, D once each, the only meaningful English word that can be formed is:

1. DROP (to fall or let fall)
2. PROD (to poke or rouse to action)
3. DORP (a small village or hamlet, commonly used in South African English and recognized in major dictionaries)

Because all three are recognized English words, the correct count is three.

Hence, the correct answer is "Option 2".

Addresses are identical to each other:

2. Rajeev Chauhan 3-5, Lotus Enclave, Kanpur, 208012
3. Rajeev Chauhan 3-5, Lotus Enclave, Kanpur, 208012

Option 2 and 3

Hence, the correct answer is "Option 2"

Given Word : STAMINA

Arrange letters in alphabetical order

 Left    A A I M N S T   Right

First from the left = A

Fourth from the righ = M

• A = 1
• M = 13

Letters between them: 13 − 1 − 1 = 11

Hence, the correct answer is "Option 1"

The pattern followed here is:

NPR : OQS

Similarly,

XYZ : ?

Hence, the correct answer is "Option 1"

Given:  In a row of 50 books, Book A is 15th from left and Book B is 20th from right

Position of B from left = 50 − 20 + 1 = 31

• A → 15th from left
• B → 31st from left

 Find number of books between A and B

31 − 15 −1 = 15

Hence, the correct answer is "Option 2"

The logic followed here is:

NUMERAL : UEALRMN

Similarly,

ALGEBRA : ?

Hence, the correct answer is "Option 4"

The logic follows here, 

• TDS → Tax Deducted at Source (tax-related)
• GST → Goods and Services Tax (tax-related)
• VAT → Value Added Tax (tax-related)
• DNA → Deoxyribonucleic Acid (biology-related)
• TDS, GST, VAT are all taxation terms
• DNA is a biological term

Hence, the correct answer is "Option 4"

Family Chart:

As per the given information

Since we don't know the gender of 'C'.

So, we cannot determine how 'C' is related to 'A'

Hence, the correct answer is "Option 4"

The logic follows here,

• 1331 = 113113 11^3
• 2197 = 133133 13^3
• 3375 = 153153 15^3
• 4096 = 163163 16^3
• 1331, 2197, 3375 are cubes of odd numbers
• 4096 is the cube of an even number
• Hence, the correct answer is "Option 3"

The logic follows here, 

Thus, it should be 41 instead of 40.

Hence, the correct answer is "Option 3"

The logic followed here is:

MOUSE is coded as 45291

And,

RAT as 786

Similarly,

STORE is coded as

Hence, the correct answer is "Option 1"

The logic followed here is,

Logic: x $ y = x² + y²

Now,

→ 7 $ 3:

7² + 3² = 49 + 9 = 58 (Matches)

→ 6 $ 2:

6² + 2² = 36 + 4 = 40

Similarly, for 5 $ 1

→ 5² + 1² = 25 + 1 = 26.

Hence, the correct answer is "Option 1".

Decoding symbols for Family Tree:

Step-by-step Relationship Analysis:

Breakdown the statement made by the woman: "She is the daughter of my mother's only daughter."

• "My mother's only daughter" – Since the speaker is a woman, her mother's only daughter is the woman herself.
• Now, replace that part in the sentence: "She is the daughter of "me" (the woman)."

Consolidated Family Structure:

Thus, the girl is the Daughter of the woman.

Hence, the correct answer is "Option 1"

Decoding symbols for Family Tree:

Step-by-step Relationship Analysis:

Breakdown the statement: "This man's son's sister is my daughter."

• "This man's son's sister" – A son's sister is simply the man's Daughter.
• If the same girl is the daughter of the man AND the daughter of the woman, then the man and the woman must be the parents of that girl.

Consolidated Family Structure:

Thus, the woman is the Wife of the man.

Hence, the correct answer is "Option 1"

Shortcut Trick

Total units in the ratio = 7 + 9 = 16 units.

Given Sum = 128, so 16 units = 128 → 1 unit = 128 ÷ 16 = 8.

First number = 7 × 8 = 56.

Second number = 9 × 8 = 72.

∴ The correct answer is 56, 72.

Alternate Method

Given: Ratio of two numbers = 7 : 9 and Sum of numbers = 128.

Formula Used: If a sum S is divided in the ratio a : b, then the parts are (a / (a + b)) × S and (b / (a + b)) × S.

⇒ Let the two numbers be 7x and 9x.

⇒ According to the question, 7x + 9x = 128

⇒ 16x = 128

⇒ x = 128 ÷ 16

⇒ x = 8

⇒ First Number = 7 × 8 = 56

⇒ Second Number = 9 × 8 = 72

∴ The correct answer is 56, 72.

Additional Information

Ratio Definition

A ratio is a comparison of two or more quantities of the same kind by division. For a : b, 'a' is called the antecedent and 'b' is the consequent.

Inverse Ratio

The inverse ratio of a : b is b : a. This is commonly used in speed-time and work-efficiency problems.

Compounded Ratio

For two ratios a : b and c : d, the compounded ratio is (a × c) : (b × d).

Proportion

When two ratios are equal, they are said to be in proportion (a : b :: c : d), implying the product of extremes equals the product of means (ad = bc).

Given:

If 16 + 11 = A and (1)/(9)=B, A % B = (A×B)-1, then find A % B.

Formula used:

A % B = (A×B) - 1

Calculations:

A = 16 + 11 = 27

B = 1/9

A % B = (A×B) - 1

⇒ A % B = (27×1/9) - 1

⇒ A % B = 3 - 1

⇒ A % B = 2

∴ The correct answer is option (3).

Given:

Sum of 28 and 22 = 28 + 22

Multiplier = 8

Formula used:

Total = (Sum) × Multiplier

Calculation:

⇒ Total = (28 + 22) × 8

⇒ Total = 50 × 8

⇒ Total = 400

∴ The correct answer is option (2).

The correct answer is 1) Both A and R are true, and R is the correct explanation of A..

Key Points

• Assertion (A): Humayun's Tomb was the first major Mughal garden tomb and directly inspired the design of the Taj Mahal.
• Humayun's Tomb, built in 1565-1572, was commissioned by Empress Bega Begum, Humayun's widow.
• It was the first garden tomb of the Mughal Empire in India and represents a significant milestone in Mughal architecture.
• The tomb's design influenced later Mughal structures, including the Taj Mahal, particularly in its symmetry and grandeur. Hence, statement A is correct.
• Reason (R): It introduced the charbagh garden layout and double-domed mausoleum style that became characteristic of later Mughal architecture.
• The tomb's charbagh layout, a Persian-inspired quadrilateral garden divided by walkways, became a hallmark of Mughal architectural design. While the charbagh concept existed earlier in Persian traditions, Humayun's Tomb popularized this layout in India.
• It also featured the innovative double-domed structure, ensuring proportionality between the interior and exterior. This style was emulated in subsequent Mughal buildings.
• These features were pivotal in shaping Mughal architectural identity and were reflected prominently in the Taj Mahal. Hence, statement R is correct.
• Since both A and R are true, and R explains A, the correct option is 1.

Additional Information

• Humayun’s Tomb:
• It is located in Delhi and was constructed by Persian and Indian artisans under the supervision of Mirak Mirza Ghiyas.
• The tomb is made of red sandstone, with marble inlays, showcasing Indo-Islamic architectural fusion.
• Its symmetrical design reflects the Persian influence on Mughal architecture.
• Charbagh Layout:
• The charbagh is a quadrilateral garden divided into four sections by pathways or water channels.
• It symbolizes the Garden of Paradise described in Islamic texts.
• This design was used in other Mughal monuments like the Taj Mahal and Shah Jahan’s Red Fort.
• Double-Domed Architecture:
• A double dome consists of two layers, with the outer dome creating the exterior grandeur and the inner dome maintaining proportionality inside.
• It was first seen in Humayun’s Tomb and later became a defining feature of Mughal structures such as the Jama Masjid and Taj Mahal.
• The Taj Mahal:
• Constructed by Emperor Shah Jahan between 1632 and 1653, it was inspired by the design and features of Humayun’s Tomb.
• The Taj Mahal also incorporates the charbagh, with water channels and lush greenery surrounding the mausoleum.
• It is a UNESCO World Heritage Site and a globally recognized symbol of Mughal architectural excellence.

Important Points

• Historical Significance: Humayun’s Tomb marked the beginning of the grand Mughal architectural tradition in India.
• Architectural Innovations: Features like the charbagh garden and double dome influenced subsequent structures.
• UNESCO Recognition: Humayun’s Tomb was designated a World Heritage Site in 1993.
• Connection to the Taj Mahal: The tomb’s design served as the blueprint for the Taj Mahal, highlighting its cultural and architectural impact.

The correct answer is Chandragupta Maurya.

Key Points

• Chandragupta Maurya was the founder of the Mauryan Empire, which was established in approximately 321 BCE. He unified India under a centralized authority for the first time.
• His administration was characterized by a highly centralized structure, with the emperor holding supreme authority over the state and decision-making.
• The Mauryan administration was divided into provinces, each governed by a royal prince or trusted administrator. These provinces were further subdivided into districts and villages.
• The famous advisor and prime minister Kautilya (Chanakya) played a crucial role in guiding Chandragupta’s administration. His work, Arthashastra, provides detailed insights into the governance, economy, military, and judicial systems of the Mauryan Empire.
• The Mauryan government was highly organized, with a strong emphasis on collecting taxes, maintaining a large standing army, and establishing a network of spies to ensure stability and control.
• Chandragupta Maurya's capital was at Pataliputra (modern-day Patna), which served as the center of administration and governance.

Additional Information

• Harshavardhana:
• Harshavardhana ruled from 606 CE to 647 CE and controlled northern India. His administration was decentralized, with local leaders and feudal lords having significant autonomy.
• He was known for promoting Buddhism and organizing the Kannauj Assembly to discuss religious matters.
• Unlike Chandragupta Maurya, Harsha’s administration lacked the strong centralization seen in the Mauryan Empire.
• Samudragupta:
• Samudragupta, a ruler of the Gupta Empire, reigned between 335 CE and 375 CE. He was known more for his military conquests and cultural patronage than administrative centralization.
• The Gupta administration was a mix of centralized and decentralized governance, with local rulers maintaining autonomy under the emperor's oversight.
• Samudragupta’s reign marked the beginning of India’s classical age, with advancements in art, literature, and science.
• Pulakeshin II:
• Pulakeshin II was a ruler of the Chalukya dynasty, reigning during the 6th century CE. He was renowned for his military campaigns and resistance against Harshavardhana.
• His administration was primarily focused on maintaining regional authority rather than centralizing power across a vast empire.
• The Chalukya administration was largely decentralized, with local chiefs enjoying significant autonomy.
• Important Points:
• The Arthashastra, authored by Kautilya, remains one of the most comprehensive treatises on ancient Indian administration, economy, and politics.
• The Mauryan Empire's centralized administration laid the foundation for future governance models in India.
• Chandragupta Maurya's legacy is remembered for creating a unified empire in India and establishing efficient governance systems.
• The centralized nature of the Mauryan Empire facilitated effective tax collection, law enforcement, and resource management across its vast territory.

The correct answer is Ragas evolved from jatis; 72-mela was by Venkatamakhi.

Key Points

• Ragas evolved from jatis: The concept of jatis (a classification of musical scales) formed the basis of Indian classical music, and over time, these evolved into the more structured and expressive ragas. This evolution can be traced back to ancient Indian music treatises such as the Natya Shastra, which describes jatis in detail. Hence, this statement is correct.
• 72-mela system by Venkatamakhi: The 72-mela system (or Melakarta system) was developed in the 17th century by the South Indian musicologist Venkatamakhi. It is a systematic arrangement of 72 parent ragas (melas) that form the foundation of Carnatic music. Hence, this statement is correct.
• Jatis evolved from ragas: This statement is incorrect because historical evidence and ancient texts confirm that jatis preceded ragas. Jatis were simpler and less structured, while ragas represent a more developed form of musical expression.
• Bharata and the 72-mela system: The statement that Bharata developed the 72-mela system is incorrect. Bharata, the author of the Natya Shastra, lived around the 2nd century BCE and is credited with describing the earliest concepts of music, drama, and dance. The 72-mela system was introduced much later by Venkatamakhi.
• Folk music and mela/grama scales: Folk music did not originally use the structured mela and grama scales. These are theoretical constructs of classical music and are not traditionally associated with folk music. Hence, this statement is incorrect.

Additional Information

• Jatis:
• Jatis were the precursors to ragas and are mentioned extensively in the Natya Shastra by Bharata.
• They described specific tonal sequences and were categorized into shuddha (pure), bhinna (divergent), and gaudi (regional) types.
• The development from jatis to ragas marked the transition from ancient to medieval Indian music.
• Ragas:
• A raga is a melodic framework for improvisation and composition in Indian classical music.
• Ragas are structured around specific notes (swaras) and rules for their progression, emphasizing aesthetics and emotional expression.
• Each raga is associated with a particular time of day, season, or mood (rasa).
• 72-Melakarta System:
• The Melakarta system categorizes parent ragas into 72 groups based on their scale structure (ascending and descending note patterns).
• It is fundamental to Carnatic music and aids in the systematic study and composition of ragas.
• Venkatamakhi's work, Chaturdandi Prakasika, outlines this system in detail.
• Role of Bharata:
• Bharata is regarded as the father of Indian classical music and dance due to his seminal work, the Natya Shastra.
• He provided the first theoretical framework for Indian music, including swaras, gramas (scales), and talas (rhythmic patterns).
• Folk Music vs Classical Music:
• Folk music is informal and often specific to regions, communities, or traditions.
• It lacks the rigid theoretical frameworks of classical music, which are based on ragas, talas, and structured improvisations.

Important Points

• Natya Shastra: Written by Bharata, it is one of the earliest treatises on performing arts in India, covering music, drama, and dance.
• Sangeet Ratnakara: A 13th-century text by Sharangadeva that bridges the gap between ancient and medieval Indian music traditions.
• Rasa Theory: A key concept in Indian aesthetics, linking music and art with emotional expression.
• Venkatamakhi: A 17th-century musicologist who systematized Carnatic music through his 72-Melakarta framework.

The correct answer is Only 1 is Correct.

Key Points

• Statement 1: The athlete attains maximum momentum by running on the runway before take-off.
• In the long jump, the runway phase is crucial as the athlete builds maximum speed and momentum.
• The purpose of this phase is to ensure the athlete achieves the highest horizontal velocity before take-off.
• This momentum directly influences the distance covered during the jump. Hence, statement 1 is correct.
• Statement 2: During take-off, the athlete pushes off with the weaker foot to maintain balance.
• This statement is incorrect because the athlete typically uses their stronger or dominant foot for the take-off.
• The take-off foot provides the required explosive force to propel the body upward and forward.
• Using the weaker foot would compromise the efficiency and distance of the jump. Hence, statement 2 is incorrect.
• In conclusion, only the first statement is correct, making the correct answer Only 1 is Correct.

Additional Information

• Phases of the Long Jump:
• The long jump consists of four main phases: runway, take-off, flight, and landing.
• During the runway phase, athletes aim to reach maximum controllable speed.
• In the take-off phase, the athlete converts horizontal momentum into vertical lift using their stronger foot.
• The flight phase involves maintaining balance and posture mid-air to optimize distance.
• The landing phase is critical for maximizing the jump distance and ensuring a clean landing without loss of points.
• Factors Influencing Long Jump Performance:
• Speed: High runway speed is essential for generating momentum.
• Strength: Explosive leg power determines the quality of the take-off.
• Technique: Proper posture, arm movement, and body control during the flight and landing phases significantly affect performance.
• Consistency: Accurate placement of the take-off foot on the board ensures legality and efficiency of the jump.
• Training for Long Jump:
• Athletes focus on sprinting, strength training, and plyometric exercises to improve explosive power and speed.

Practice drills include runway repetitions, jump simulations, and landing techniques.

• Coaches emphasize proper biomechanics to ensure efficient energy transfer during take-off.

Important Points

• International Long Jump Records:
• The men's world record in the long jump is held by Mike Powell (USA) with a jump of 8.95 meters, achieved in 1991.
• The women's world record is held by Galina Chistyakova (USSR) with a jump of 7.52 meters, achieved in 1988.
• Olympic Long Jump:
• The long jump has been part of the Olympic Games since their modern inception in 1896.
• Notable athletes include Carl Lewis, who won four consecutive Olympic gold medals from 1984 to 1996.
• Biomechanics of Long Jump:
• The jump relies on the principles of kinematics and dynamics.
• Efficient energy transfer from horizontal to vertical motion is crucial for achieving maximum distance.
• Body posture during flight affects air resistance and overall performance.

The correct answer is President's Cup, Indonesia.

Key Points

• Mary Kom, a six-time world boxing champion, won her most recent international gold medal at the President's Cup held in Indonesia.
• The event took place in July 2019, and Mary Kom secured the gold medal in the 51 kg category.
• This victory was significant as Mary Kom defeated her opponent with a 5-0 unanimous decision in the final, showcasing her exceptional skill and experience.
• The President's Cup is an international boxing tournament that attracts elite boxers from various countries, providing a platform for showcasing their talent.
• Mary Kom's victory at this tournament further cemented her reputation as one of the greatest boxers of all time.

Additional Information

• Mary Kom's Achievements:
• Mary Kom is a six-time world champion in women's boxing and has also won an Olympic bronze medal in the 2012 London Olympics.
• She is the first Indian woman boxer to qualify for the Olympics and has inspired many young athletes in India.
• Her achievements include winning a gold medal at the 2018 Commonwealth Games and a bronze medal at the 2010 Asian Games.
• President's Cup, Indonesia:
• This is an annual international boxing event organized in Indonesia, attracting top-tier athletes from across the globe.
• The tournament offers a competitive platform for boxers to gain international exposure and improve their rankings.
• Mary Kom's participation and success in this event highlight her dedication to the sport and her ability to perform at the highest level.
• Other Options:
• Asian Games, Hangzhou: Mary Kom did not participate in the 2022 Asian Games held in Hangzhou, as she had moved on to mentoring roles by this time.
• Commonwealth Games, Birmingham: The 2022 Commonwealth Games in Birmingham did not feature Mary Kom, as she was not competing actively in international events during that period.
• Olympic Games, Tokyo: Mary Kom participated in the 2020 Tokyo Olympics but did not win a medal. She narrowly lost in the pre-quarterfinals.
• Impact of Mary Kom's Career:
• Mary Kom has been instrumental in promoting women's boxing in India and has inspired generations of athletes.
• She was awarded the Padma Bhushan (2013) and Padma Vibhushan (2020), among other honors, for her exceptional contributions to Indian sports.

The correct answer is Company's constitution at home and territorial control in India.

Key Points

• The Regulating Act of 1773 was the first significant legislative intervention by the British Parliament in the affairs of the East India Company, aimed at restructuring its administrative framework both in India and in Britain.
• This act brought changes to the Company’s constitution at home by subordinating its commercial activities to the British Parliament, thus ensuring better oversight. The Board of Control was established to supervise the Company’s operations.
• In India, the act introduced changes in the territorial control by centralizing administration under the Governor-General of Bengal. Warren Hastings was appointed as the first Governor-General, with supervisory power over the presidencies of Bombay and Madras.
• It aimed to address issues of corruption and inefficiency within the Company’s administration in India, with the establishment of the Supreme Court of Judicature at Calcutta to oversee judicial matters.
• The act marked the beginning of the British government’s intervention in Indian affairs, establishing the foundation for the future Parliamentary system of governance in India.

Additional Information

• Provisions under the Regulating Act of 1773:
• The Governor of Bengal was designated as the Governor-General of Bengal with a council comprising four members to assist in administrative matters.
• A Supreme Court was established at Calcutta in 1774, which was the first British court in India to administer justice.
• Direct control was established by the British Crown over the Company’s policies, marking the beginning of the dual system of governance.
• Key Features of the Act:
• It limited the autonomy of the East India Company by ensuring its accountability to the British Parliament.
• The act also prohibited Company officials from engaging in private trade or accepting gifts, reducing corruption.
• The revenue from India was to be disclosed to the British government.
• Impact on Territorial Control in India:
• By centralizing power, the act reduced administrative conflicts between the presidencies of Bombay, Madras, and Bengal.
• This centralization laid the groundwork for the British government’s consolidation of its hold over Indian territories.
• Significance of the Act:
• The act was the first step towards regulating the affairs of the East India Company and marked the beginning of British administrative control in India.
• It also reflected Britain’s emerging interest in Indian governance, which eventually culminated in the Government of India Act of 1858.

Important Points

• Distinction Between Options:
• Option 1: The act focused on governance rather than judicial system reforms or revenue collection, which were addressed in later acts like the Pitt’s India Act of 1784.
• Option 2: Military strategy and trade regulations were not the primary focus of the Regulating Act but were instead managed by the Company’s policies during that period.
• Option 4: Relations with princely states and foreign policy were not directly influenced by this act but were later addressed in the 19th century.
• Historical Context: The act was a response to reports of corruption and mismanagement by the East India Company, particularly after its financial crisis in the 1770s.

The correct answer is Only 1 is Correct.

Key Points

• Statement 1: The Mahakali Treaty signed between India and Nepal in 1996 includes provisions for the integrated development of the Mahakali River, as well as the construction and operation of the Pancheshwar Multipurpose Project. This treaty ensures equitable utilization of the river's resources for irrigation, hydropower, and other benefits. Hence, this statement is correct.
• Statement 2: The Ganga Water Treaty signed between India and Bangladesh in 1996 provides a framework for the sharing of Ganga waters at Farakka. However, it does not allow for water sharing throughout the year. Instead, it specifies seasonal sharing agreements, particularly during the dry season (January to May). Hence, this statement is incorrect.
• The correct answer is based on the accuracy of the statements. Only Statement 1 is correct, as it aligns with the provisions of the Mahakali Treaty.

Additional Information

• Mahakali Treaty (1996):
• It is a treaty between India and Nepal for the integrated development of the Mahakali River, which forms part of the border between the two nations.
• The treaty covers the Pancheshwar Multipurpose Project, which aims to generate hydropower, enhance irrigation, and ensure flood control in both countries.
• It ensures equitable utilization of the river's waters and establishes mechanisms for dispute resolution and mutual cooperation.
• Ganga Water Treaty (1996):
• It is an agreement between India and Bangladesh to share the waters of the Ganga River at the Farakka Barrage.
• The treaty specifies water sharing during the dry season (January to May), with provisions for minimum guaranteed flows.
• It aims to promote cooperation and resolve conflicts over water distribution.
• Importance of Water-Sharing Treaties:
• They ensure equitable distribution of water resources between nations sharing transboundary rivers.
• Such treaties help maintain peaceful bilateral relations and address challenges like water scarcity and flooding.
• They promote sustainable development by enabling joint projects for irrigation, hydropower, and flood management.
• Challenges in Water Sharing:
• Seasonal variations in river flows can create disputes over water allocation.
• Climate change impacts, such as reduced rainfall or increased flooding, complicate water sharing agreements.
• Effective monitoring and enforcement mechanisms are essential for ensuring compliance with treaty provisions.

Important Points

• India-Nepal Relations: The Mahakali Treaty is a significant example of bilateral cooperation in the water sector, highlighting the mutual benefits of shared resources.
• India-Bangladesh Relations: The Ganga Water Treaty underscores the importance of addressing transboundary water issues through dialogue and agreements.
• Transboundary Rivers: India shares several major rivers with its neighbors, including the Indus, Brahmaputra, and Teesta, necessitating robust water-sharing frameworks.

The correct answer is 4.

Key Points

• Under the State Level Khelo India Centres (SLKIC) vertical of the Khelo India Programme, 4 Kendriya Vidyalayas have been adopted in partnership with the Kendriya Vidyalaya Sangathan (KVS).
• The initiative aims to strengthen the sports ecosystem at the grassroots level and identify talented young athletes from schools.
• The adopted Kendriya Vidyalayas are provided support for developing sports infrastructure, training facilities, and athlete development programmes.
• The collaboration between Sports Authority of India (SAI) and KVS seeks to promote a culture of sports and fitness among school students.
• Hence, the correct answer is 4 Kendriya Vidyalayas.

Additional Information

• Khelo India Programme:
• Launched by the Ministry of Youth Affairs and Sports in 2018.
• Aims to revive the sports culture in India and build a strong framework for sports development.
• Sports Authority of India (SAI):
• Established in 1984.
• Functions under the Ministry of Youth Affairs and Sports.
• Responsible for implementing several sports development schemes.
• Kendriya Vidyalaya Sangathan (KVS):
• An autonomous organization under the Ministry of Education.
• Manages Kendriya Vidyalayas across India to provide quality education to children of transferable central government employees.
• State Level Khelo India Centres (SLKIC):
• Designed to utilize existing sports infrastructure and coaching resources available in states.
• Focuses on nurturing athletes in priority sports disciplines.

The correct answer is Only statement 1 is correct.

Key Points

• "The Adventures of Kakababu, Volume 1" by Sunil Gangopadhyay is a pioneering Bengali adventure novel that introduces the character Kakababu, a detective with a unique personality. The novel is known for its blend of mystery, adventure, and cultural themes. Hence, statement 1 is correct.
• The book does not contain elements of political satire on post-independence Indian governance. Kakababu’s stories revolve around adventures and mysteries rather than political commentary. Hence, statement 2 is incorrect.
• This novel is not an autobiography of Sunil Gangopadhyay’s freedom fighter experiences. It is purely fictional and focuses on the character of Kakababu and his adventures. Hence, statement 3 is incorrect.
• The series primarily highlights adventure and cultural exploration, often intertwining puzzles, historical secrets, and thrilling escapades that are distinct from political or autobiographical themes.
• Thus, the correct answer is "Only statement 1 is correct."

Additional Information

• About Kakababu:
• The character Kakababu, originally named Raja Roy Chowdhury, is a fictional detective created by Sunil Gangopadhyay. He is portrayed as a former director of the Archaeological Survey of India.
• Kakababu’s adventures often include thrilling mysteries, archaeological findings, and interactions with various cultures.
• The series is popular among readers of young adult fiction and is celebrated for its ability to blend intellectual challenges with entertaining narratives.
• The Kakababu novels are widely appreciated for their depiction of Bengal’s heritage, history, and culture.
• Sunil Gangopadhyay:
• Sunil Gangopadhyay is a renowned Indian writer and poet, primarily known for his works in Bengali literature.
• He has authored multiple novels, poetry collections, and essays, contributing significantly to modern Bengali literature.
• His stories often reflect themes of adventure, romance, history, and social issues.
• He received the prestigious Sahitya Akademi Award for his contribution to literature.
• Other Adventure Series:
• Besides Kakababu, Sunil Gangopadhyay created other fictional works that showcase his literary prowess, but Kakababu remains one of his most iconic characters.
• The books have been adapted into films and television series, further cementing their popularity among audiences.
• Characteristics of Bengali Adventure Novels:
• Bengali adventure novels often feature strong protagonists, gripping mysteries, and cultural insights.
• Other notable authors in this genre include Satyajit Ray (creator of Feluda) and Hemendra Kumar Roy.
• The novels are known for their ability to engage readers with a mix of intellectual puzzles and entertaining narratives.

Important Points

• Popularity of Kakababu: The Kakababu series has inspired generations of readers and is considered a classic in Bengali literature.
• Adaptations: Several novels from the Kakababu series have been adapted into films, such as "Mishawr Rawhoshyo", directed by Srijit Mukherji.
• Literary Significance: The series is an excellent example of how Bengali literature combines thrilling adventures with an exploration of history, archaeology, and culture.
• Target Audience: Although primarily aimed at young adults, the books are enjoyed by readers of all ages due to their engaging storytelling and intellectual depth.
• Legacy: Sunil Gangopadhyay’s works have left an indelible mark on Indian literature, inspiring future writers and filmmakers.

The correct answer is Narendra Modi.

Key Points

• Narendra Modi, the current Prime Minister of India, is known for his active participation in global discussions and efforts to address issues like climate change, environmental sustainability, and global health challenges.
• In 2025, he addressed a special session focused on "Environment, COP-30, and Global Health". This event highlighted India's leadership in international climate negotiations and its commitment to sustainable development.
• India's active engagement in the COP-30 discussions is crucial, especially in the context of its growing role in renewable energy transition and efforts to mitigate climate change.
• Modi has launched major initiatives like the International Solar Alliance (ISA), reaffirming India's dedication to promoting clean energy globally.
• His emphasis during the session was on fostering international collaboration to address global health challenges like pandemics and ensuring equitable access to healthcare resources globally.

Additional Information

• COP-30:
• The 30th Conference of Parties (COP-30) under the United Nations Framework Convention on Climate Change (UNFCCC) is a global platform for discussing climate-related issues.
• India has been a significant participant, advocating for climate justice and emphasizing the principle of Common but Differentiated Responsibilities (CBDR).
• Recent COP sessions have focused on the transition to renewable energy, sustainable practices, and funding mechanisms for developing nations.
• India's role in COP-30 reflects its increasing influence in shaping global climate policies.
• Global Health:
• Global health encompasses challenges like pandemics, healthcare access, and the equitable distribution of medical resources.
• India has played a vital role in global healthcare, particularly during the COVID-19 pandemic, by providing vaccines under the Vaccine Maitri initiative.
• Focus during the session included addressing emerging infectious diseases, strengthening healthcare infrastructure, and fostering global cooperation.
• India's pharmaceutical industry is pivotal in supplying affordable medicines worldwide.
• International Solar Alliance (ISA):
• Founded in 2015, the ISA is a collaborative initiative led by India and France to promote solar energy use globally.
• It aims to mobilize resources for deploying solar energy solutions in member countries, particularly in tropical regions.
• India has committed to significant investments and partnerships within ISA to boost renewable energy adoption.
• This alliance is central to achieving sustainable energy goals and combating climate change.
• Important Developments:
• India has set ambitious targets under its Nationally Determined Contributions (NDCs), including achieving net-zero emissions by 2070.
• The country is actively increasing its renewable energy capacity, with a focus on solar, wind, and hydropower.
• India's leadership in addressing global environmental and health issues is recognized internationally, as demonstrated by Modi's participation in the 2025 session.

The correct answer is Slump.

Key Points

• A slump is a type of landslide characterized by the backward rotation of the slipping material along a curved surface.
• This rotational movement often creates a concave-upward shape in the displaced material and leaves a crescent-shaped scarp at the top.
• Slumps occur when a slope becomes unstable due to water infiltration, weak soil, or geological disturbances.
• They are commonly found in areas with clay-rich soils, as clay can become saturated and lose its cohesive strength.
• Unlike other types of landslides, slumps usually involve large blocks of material moving as a coherent unit, rather than chaotic debris flow.

Additional Information

• Rockslide:
• A rockslide involves the rapid movement of large masses of bedrock downslope.
• It typically occurs on steep slopes where rock layers are weakened by fractures or weathering.
• Unlike slumps, rockslides do not rotate; instead, they slide along a planar surface.
• Example: The Frank Slide in Canada (1903).
• Rock fall:
• A rock fall is characterized by the free fall or bouncing of individual rocks from steep cliffs or slopes.
• It is often triggered by freeze-thaw cycles, erosion, or seismic activity.
• Rock falls are the fastest type of landslide and can cause significant damage.
• Example: Yosemite National Park has frequent rock falls due to its granite cliffs.
• Debris fall:
• Debris falls involve the rapid downward movement of unconsolidated material, such as soil, rocks, and organic matter.
• They are often triggered by heavy rainfall or earthquakes, which destabilize the slope.
• Debris falls are chaotic and lack the coherent rotational movement seen in slumps.
• Example: The 2014 Oso landslide in Washington, USA, was a debris fall.
• Characteristics of Slumps:
• Slumps are slower than rockslides and debris falls due to their rotational movement.
• They are common in areas with poorly consolidated sediments, especially in regions prone to water saturation.
• Preventive measures include slope stabilization, proper drainage systems, and vegetation cover to improve soil cohesion.

Important Points

• Types of Landslides:
• Falls: Free-falling movement of material from a cliff or steep slope.
• Slides: Coherent movement along a planar or curved surface (e.g., slumps).
• Flows: Chaotic movement of water-saturated debris and soil.
• Creep: Slow, imperceptible movement of soil or rock downslope over time.
• Causes of Landslides:
• Natural factors: Heavy rainfall, earthquakes, volcanic activity, and weathering.
• Human activities: Deforestation, mining, construction, and improper land use.
• Preventive Measures:
• Stabilizing slopes using retaining walls and terracing.
• Improving drainage to reduce water saturation in soil.
• Planting vegetation to enhance soil cohesion.
• Monitoring slopes for signs of instability.

The correct answer is People per square kilometre.

Key Points

• Population density refers to the number of people living per unit area of land. It is typically measured as the number of people per square kilometre.
• This metric is used in demographic studies to understand the distribution of populations across various geographical locations.
• Population density helps in assessing the pressure on resources such as land, water, and infrastructure in specific regions.
• It is calculated using the formula: Population Density = Total Population / Land Area (in square kilometres).
• The concept is essential for urban planning, environmental management, and policy-making, especially in densely populated countries like India, China, and others.

Additional Information

• Global Population Density Variations:
• The world average population density is approximately 59 people per square kilometre (as per 2023 data).
• Countries like Bangladesh and Monaco have some of the highest population densities globally, whereas countries like Mongolia and Australia exhibit very low population densities.
• Applications of Population Density:
• Helps in determining areas that require infrastructure development or resource allocation.
• Used in studies related to urbanization, migration patterns, and environmental impacts.
• Plays a critical role in the planning of transport networks, housing projects, and healthcare facilities.
• Factors Affecting Population Density:
• Geography: Regions with harsh climates or rugged terrain usually have lower population densities.
• Economic Opportunities: Areas with abundant job opportunities often attract higher population densities.
• Infrastructure: Well-developed areas with amenities and services experience higher population densities.
• Government Policies: Settlement schemes and urban planning affect population distribution and density.
• Examples of Population Density Studies:
• India: India’s population density is around 464 people per square kilometre (2021 census), with Bihar being the most densely populated state.
• Urban Centers: Cities like Tokyo, Delhi, and Shanghai exhibit extremely high population densities due to urbanization.

Important Points

• Overpopulation Challenges: High population densities can lead to overcrowding, increased pressure on natural resources, and challenges in waste management.
• Underpopulation Challenges: Low population densities may result in underutilized resources and difficulties in maintaining infrastructure and services.
• Population Density Index: An important metric for evaluating the socio-economic development and environmental sustainability of regions.
• Historical Perspective: Population density has changed over time due to industrialization, wars, migration, and technological advancements.

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

Key Points

• Assertion (A): Population explosion occurs when death rates decline rapidly but birth rates remain high.
• This statement is correct. A population explosion typically occurs due to advancements in healthcare, leading to a significant reduction in mortality rates (death rates), while birth rates remain consistently high.
• It is observed in the demographic transition model, particularly during the early stages of development when better medical facilities, sanitation, and food supply reduce deaths, but societal or cultural changes do not immediately reduce birth rates.
• Reason (R): Societal reproductive behavior changes instantly with improvements in healthcare.
• This statement is false. While healthcare improvements can reduce death rates, societal reproductive behavior does not change immediately.
• Cultural, social, and economic factors heavily influence birth rates, and these changes often take decades to manifest. For example, education, particularly of women, and access to family planning methods play critical roles in reducing birth rates.
• In the demographic transition theory, population explosion is often observed in the second stage, where death rates decline due to healthcare advancements, but birth rates remain high due to traditional societal norms.
• The assertion (A) is true because it explains the phenomenon of population explosion accurately. However, the reason (R) is false as societal reproductive behavior does not change instantly with healthcare improvements.

Additional Information

• Demographic Transition Model (DTM):
• The DTM explains population changes as societies progress economically and socially.
• It consists of four stages:

1. Stage 1: High birth and death rates lead to a stable population.
2. Stage 2: Declining death rates due to improved healthcare and sanitation, but high birth rates result in a population explosion.
3. Stage 3: Birth rates begin to decline, leading to a slow population growth.
4. Stage 4: Low birth and death rates stabilize the population.

• Factors Influencing Birth Rates:
• Education: Higher literacy levels, especially among women, lead to lower fertility rates.
• Economic Development: Economic progress often reduces dependency on large families for labor and security.
• Access to Contraceptives: Availability and usage of family planning tools play a key role in controlling birth rates.
• Population Explosion:
• It refers to a rapid increase in the population size within a short period.
• It often places immense pressure on resources like food, water, housing, healthcare, and education.
• Examples:
• The period of the Industrial Revolution in Europe saw improved healthcare and sanitation, leading to population growth in the 18th and 19th centuries.
• In India, significant population growth occurred post-independence due to healthcare improvements, while birth rates remained high.

Important Points

• Measures to Control Population Explosion:
• Family Planning Programs: Governments promote the use of contraceptives and provide incentives for smaller families.
• Education and Awareness: Educating people about the socio-economic benefits of smaller families helps in controlling population growth.
• Healthcare Access: Ensuring access to reproductive healthcare services reduces unintended pregnancies.
• Challenges Posed by Population Explosion:
• Overburdened resources like food, water, and energy.
• Increased unemployment and poverty.
• Environmental degradation due to overconsumption and waste generation.

The correct answer is Optimizing mechanical configurations for minimum mass and maximum structural resilience.

Key Points

• Generative design in aerospace engineering focuses on creating optimized mechanical configurations that minimize mass while maximizing structural resilience. This ensures the systems are lightweight yet durable enough to withstand extreme conditions.
• It leverages advanced algorithmic design tools and AI-driven simulations to explore multiple design options and choose the most efficient structure.
• Reducing mass is critical in aerospace systems as it directly impacts fuel efficiency, payload capacity, and overall mission success. Structural resilience ensures that components endure high stress and varying environmental conditions.
• The process integrates finite element analysis (FEA) and computational fluid dynamics (CFD) to test designs for their ability to handle pressure, temperature, and aerodynamic forces effectively.
• Generative design enables engineers to develop lightweight components such as fuselage structures, wing designs, and spacecraft bodies without compromising safety or performance.

Additional Information

• Facilitating multi-payload platforms: This statement refers to the integration of multiple payloads into shared structures, but it is not the primary objective of generative design. While generative design can aid in achieving this, its focus is more on structural optimization. Hence, this statement is incorrect.
• Enhancing life-support redundancies: This pertains to ensuring crew safety in deep-space missions, which involves systems engineering but does not directly relate to the purpose of generative design. Hence, this statement is incorrect.
• Designing adaptive thrust mechanisms: Adaptive thrust mechanisms are crucial for orbital corrections but are more related to propulsion engineering rather than generative design's core objective. Hence, this statement is incorrect.
• Applications of generative design: Generative design is widely used in aerospace for creating lightweight and optimized components such as engine mounts, brackets, and spacecraft structures. It supports innovations in material science, such as using carbon composites and additive manufacturing techniques.
• Advantages: Generative design reduces design time, improves component efficiency, and supports sustainability by minimizing material usage.

Important Points

• Generative Design in Aerospace: The aerospace industry extensively uses generative design to create structures that balance lightweight design and structural durability.
• Historical Examples: NASA and SpaceX utilize generative design for optimizing spacecraft components like lander legs and fuselages. For instance, SpaceX's Falcon series employs lightweight designs to maximize payload efficiency.
• Material Science Integration: Generative design often incorporates advanced materials like titanium alloys and carbon fiber composites, which are lightweight and strong.
• Future Prospects: With advancements in AI and machine learning, generative design will continue to play a crucial role in developing next-generation aerospace systems.

The correct answer is Intracellular digestion.

Key Points

• Lysosomes are membrane-bound organelles found in animal cells that contain digestive enzymes. These enzymes help in breaking down cellular waste, debris, and foreign particles through intracellular digestion.
• Intracellular digestion occurs when lysosomes fuse with vesicles containing cellular waste or foreign particles. The digestive enzymes degrade these substances into simpler molecules that can be reused or expelled from the cell.
• Lysosomes play a vital role in maintaining cellular health by removing damaged organelles, pathogens, and cellular debris. This process is known as autophagy.
• Lysosomal enzymes function optimally at acidic pH levels, making lysosomes a specialized compartment for digestion within the cell.
• Disorders related to lysosomal function, such as Lysosomal Storage Diseases (e.g., Tay-Sachs disease), occur when these enzymes are defective, leading to the accumulation of undigested substances in cells.

Additional Information

• Protein synthesis:
• Protein synthesis primarily occurs in the ribosomes, which are present in the cytoplasm or attached to the rough endoplasmic reticulum.
• This process involves two major steps: transcription (DNA to mRNA) and translation (mRNA to protein).
• Lysosomes are not involved in protein synthesis, but they help degrade proteins that are damaged or no longer needed.
• Energy production:
• Energy production in cells is mainly carried out by the mitochondria, known as the "powerhouse of the cell."
• Mitochondria generate energy in the form of ATP through processes such as cellular respiration and oxidative phosphorylation.
• Lysosomes do not contribute to energy production; their primary role is in digestion and waste removal.
• Genetic information storage:
• Genetic information is stored in the form of DNA within the cell nucleus and mitochondria.
• The nucleus contains the majority of the cell’s genetic material and regulates gene expression and replication.
• Lysosomes do not have any role in storing or managing genetic information.
• Function of lysosomes:
• Lysosomes are involved in processes such as endocytosis (intake of substances), exocytosis (expulsion of waste), and autophagy (self-digestion of cellular components).
• They contain hydrolytic enzymes such as proteases, lipases, and nucleases, which break down various biomolecules.
• Lysosomes are often referred to as the "cell's garbage disposal" due to their role in waste degradation.

Important Points

• Discovery: Lysosomes were first discovered by Belgian biologist Christian de Duve in 1955.
• Structure: Lysosomes are spherical vesicles surrounded by a lipid bilayer membrane that maintains their acidic environment.
• Enzymes: The enzymes within lysosomes are synthesized in the rough endoplasmic reticulum and processed in the Golgi apparatus before being enclosed in lysosomes.
• Diseases: Mutations affecting lysosomal enzymes can lead to conditions like Gaucher disease and Niemann-Pick disease.

The correct answer is 3) 1 & 2 only.

Key Points

• Article 270 of the Indian Constitution governs the sharing of taxes between the Union and the States. After the 101st Constitutional Amendment Act, 2016, it was amended to include provisions for the sharing of revenue from Goods and Services Tax (GST). Hence, statement 1 is correct.
• The 101st Constitutional Amendment added provisions in Article 270 to ensure that the revenue collected through GST is apportioned between the Union and the States based on the recommendations of the Finance Commission. Hence, statement 2 is correct.
• The GST Compensation Cess, however, is not shared between the Union and the States under Article 270. It is governed by the GST (Compensation to States) Act, 2017, which ensures compensation to States for any revenue loss due to the implementation of GST. Hence, statement 3 is incorrect.
• The Finance Commission plays a key role in determining the sharing formula for taxes, including GST, under Article 270.
• It is important to note that the sharing mechanism under Article 270 does not include non-divisible taxes, such as surcharges and cesses, which remain with the Union Government.

Additional Information

• 101st Constitutional Amendment Act, 2016:
• This amendment introduced the Goods and Services Tax (GST) in India, effective from 1st July 2017.
• It subsumed multiple indirect taxes levied by the Centre and the States, such as excise duty, VAT, and service tax, under a single tax regime.
• It led to the addition of Article 246A, empowering both the Parliament and State Legislatures to make laws on GST.
• It also introduced Article 269A, which deals with the apportionment of GST collected on inter-State trade.
• GST Compensation Cess:
• It is levied under the GST (Compensation to States) Act, 2017 to compensate States for revenue losses arising due to GST implementation.
• The compensation cess is levied on specific luxury and sin goods, such as tobacco, aerated drinks, and automobiles.
• The funds collected through the cess are not part of the divisible pool under Article 270 and are kept in a separate compensation fund.
• Finance Commission:
• It is constituted under Article 280 of the Constitution to recommend the distribution of tax revenues between the Centre and the States.
• The Commission takes into account factors such as population, area, fiscal capacity, and governance when recommending the sharing formula.
• The 15th Finance Commission, chaired by N.K. Singh, has been tasked with making recommendations for the period 2021-26.
• Key Features of GST:
• GST is a destination-based tax, meaning it is collected at the point of consumption.
• It is levied at multiple rates, categorized into 5%, 18%, and 40%, depending on the goods or services.
• It aims to create a unified tax structure, reducing cascading effects and improving tax compliance.

Important Points

• Revenue Distribution: Taxes collected under GST are distributed between the Centre and the States in a 50:50 ratio. For inter-State supplies, the tax collected is divided between the Centre and the destination State.
• GST Council: The GST Council, established under Article 279A, is responsible for making recommendations on GST rates, exemptions, and other related matters.
• Impact of GST: GST has simplified the tax structure, enhanced transparency, and contributed to the formalization of the economy. However, challenges such as compliance costs and revenue shortfalls for States persist.

The correct answer is Increase by approximately 2%.

Key Points

• The formula for gravitational acceleration (g) at the surface of the Earth is g = G × (M / R²), where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.
• If the Earth's radius is reduced by 1%, the new radius becomes R' = 0.99 × R.
• Gravitational acceleration is inversely proportional to the square of the radius. Thus, a decrease in radius increases the value of g.
• The percentage change in g can be calculated using the formula: Percentage Change ≈ 2 × Percentage Change in Radius. Since the radius decreases by 1%, gravitational acceleration increases by approximately 2%.
• This phenomenon occurs because the mass remains constant while the radius decreases, concentrating the mass closer to the surface, thereby increasing the gravitational pull.

Additional Information

• Gravitational Constant (G): The universal gravitational constant is approximately 6.674 × 10⁻¹¹ N·m²/kg². It is a fundamental constant used to calculate gravitational forces.
• Mass of the Earth: The Earth's mass is approximately 5.972 × 10²⁴ kg. It remains unchanged regardless of changes in the radius.
• Effect of Radius: The radius of the Earth is approximately 6,371 km. If the radius is reduced, the gravitational pull at the surface increases due to the inverse square relationship.
• Applications: This principle is essential in understanding planetary physics, satellite mechanics, and designing space missions, where accurate calculations of gravity are required.
• Examples: Similar calculations are applied to study the gravitational acceleration on other planets or celestial bodies, such as the Moon or Mars, where the radius and mass differ significantly from Earth.

Important Points

• Inverse Square Law: Gravitational acceleration (g) is inversely proportional to the square of the radius. A smaller radius results in a stronger gravitational force at the surface.
• Percentage Calculation: When radius decreases by a percentage, the gravitational acceleration increases approximately by twice that percentage.
• Impact on Weight: If g increases, the weight of objects on the Earth's surface will also increase proportionally because weight is calculated as W = mg.
• Practical Implications: Understanding changes in g is crucial for designing infrastructure and equipment that interact with gravity, such as bridges, airplanes, and space vehicles.

The correct answer is 15 days (for initial remand, extendable).

Key Points

• Bharatiya Nagarik Suraksha Sanhita (BNSS) governs various provisions related to criminal proceedings in India, replacing the earlier Criminal Procedure Code (CrPC).
• Under BNSS, a person can be held in police custody for an initial period of 15 days. This period can be extended periodically if the magistrate approves such extensions.
• The magistrate has the authority to grant extensions beyond the initial 15 days, but the cumulative period of police custody cannot exceed the prescribed limits.
• As per BNSS, police custody is limited to 15 days at a time, ensuring judicial oversight to prevent misuse of custodial provisions.
• This provision ensures that the rights of individuals are protected while allowing the police sufficient time to conduct investigations. Hence, option "15 days (for initial remand, extendable)" is correct.

Additional Information

• Police Custody under Indian Law:
• The concept of police custody refers to the detention of an accused by the police for questioning and investigation purposes.
• During police custody, the accused is under the direct supervision of the police and can be interrogated to gather evidence.
• As per Indian law, police custody cannot exceed 15 days without the approval of a magistrate.
• Magistrates have the discretion to extend custody if the police can justify the need for additional time for investigation.
• Judicial Custody:
• After the expiration of police custody, the accused may be moved to judicial custody, where they are detained in a jail or prison.
• Judicial custody does not involve police interrogation, and the accused is under the supervision of judicial authorities.
• The maximum period of judicial custody varies depending on the nature of the crime but cannot exceed 90 days for serious offenses and 60 days for other offenses unless charges are filed.
• Role of the Magistrate:
• The magistrate plays a crucial role in ensuring that the rights of the accused are upheld during police and judicial custody.
• The magistrate evaluates the justification provided by the police for extending custody and ensures that the process adheres to legal standards.
• Approval from the magistrate is mandatory for extending police custody beyond the initial 15-day period.
• Human Rights Considerations:
• The provision of limiting police custody to 15 days reflects the Indian judicial system's commitment to protecting individual rights and preventing abuse.
• International human rights principles also emphasize the need for judicial oversight during detention to safeguard against arbitrary actions.

Important Points

• BNSS and CrPC Differences: BNSS replaces certain provisions of the Criminal Procedure Code (CrPC) to modernize criminal law and make the process more efficient.
• Custodial Rights: The accused has the right to consult a lawyer and be informed of the reasons for their detention.
• Preventing Misuse: The limitation on police custody helps prevent potential misuse of power by law enforcement agencies.
• Judicial Oversight: Regular intervention by magistrates ensures that the detention process remains lawful and fair.

The correct answer is 18-40 years.

Key Points

• The Pradhan Mantri Kisan Maandhan Yojana (PM-KMY) is a government-backed pension scheme aimed specifically at providing social security to small and marginal farmers.
• The scheme is applicable to farmers in the age group of 18 to 40 years. This means individuals within this age range can enroll in the scheme and start contributing towards their pension.
• Under the scheme, farmers are required to contribute a specific amount monthly based on their age at the time of joining. The Government of India matches the farmer's contribution equally.
• Subscribers to the scheme will receive a monthly pension of Rs. 3000 after attaining the age of 60 years.
• The scheme primarily targets small and marginal farmers with landholding up to 2 hectares, ensuring financial security in their old age.

Additional Information

• Eligibility Criteria:
• The farmer must be in the age group of 18-40 years at the time of enrollment.
• He/she should own landholding of up to 2 hectares as per land records of the concerned State/Union Territory.
• The farmer should not be covered under any other statutory social security schemes, such as the Employees' Provident Fund Organization (EPFO) or National Pension Scheme (NPS).
• Farmers paying income tax are not eligible for this scheme.
• Features of the Scheme:
• The scheme is a voluntary and contributory pension scheme.
• The monthly contribution ranges between Rs. 55 and Rs. 200, depending on the age of the subscriber at the time of joining.
• The Life Insurance Corporation of India (LIC) is the pension fund manager for this scheme.
• In case the farmer dies while receiving the pension, the spouse will receive 50% of the pension amount as a family pension.
• Key Benefits:
• Ensures financial stability for small and marginal farmers post-retirement.
• Provides economic support to the farming community, which constitutes a significant portion of India’s workforce.
• The government’s equal contribution ensures better returns and reduces the burden on farmers.
• Implementation:
• The scheme is implemented through Common Service Centers (CSCs), making it accessible to farmers in rural areas.
• Farmers can also enroll through the official PM-KMY portal by providing their Aadhaar card and bank account details.

Important Points

• The Pradhan Mantri Kisan Maandhan Yojana (PM-KMY) was launched by the Government of India in 2019.
• The scheme is part of the government’s broader efforts to improve the social and economic status of farmers, particularly those with limited resources.
• The Ministry of Agriculture and Farmers’ Welfare oversees the implementation of the scheme.
• The PM-KMY is a step towards addressing the challenges faced by farmers in their old age, ensuring financial security and dignity.

The correct answer is 1) Both A and R are true, and R is the correct explanation of A..

Key Points

• Assertion (A): UDAN Scheme improves regional air connectivity:
• The UDAN Scheme, known as Ude Desh Ka Aam Nagrik, is a flagship regional air connectivity scheme launched by the Government of India in 2016.
• Its primary aim is to make air travel affordable and accessible for common citizens, especially in tier-2 and tier-3 cities.
• By connecting underserved and remote regions of India, UDAN improves regional connectivity and fosters economic growth. Hence, the assertion is correct.
• Reason (R): It provides financial viability gap funding to airlines:
• Under the UDAN scheme, airlines operating on regional routes receive Viability Gap Funding (VGF) from the government.
• The VGF reduces the financial burden on airlines to operate flights on less profitable routes, ensuring sustained operations.
• This funding is critical to achieving the goals of UDAN by incentivizing airlines to cover underserved routes. Hence, the reason is correct.
• The reason (R) directly explains the assertion (A), as the financial VGF mechanism is integral to improving regional air connectivity through the UDAN scheme. Thus, both A and R are true, and R is the correct explanation of A.

Additional Information

• UDAN Scheme:
• The UDAN Scheme is jointly funded by the Central Government and the State Governments.
• It aims to reduce the cost of air travel by capping fares at Rs. 2,500 for an hour-long flight.
• The scheme also promotes the development of regional airports and infrastructure in remote areas.
• As of 2023, the scheme has connected several underserved regions, including cities like Darbhanga, Kanpur, and Jharsuguda.
• Viability Gap Funding (VGF):
• VGF is a financial support mechanism provided by the government to make projects commercially viable.
• Under UDAN, VGF is funded through a levy collected from airlines operating on metro routes.
• It ensures that airlines can operate on regional routes without incurring significant losses.
• Impact of UDAN:
• Boosts tourism and trade in remote and underserved areas.
• Generates employment opportunities in aviation and related sectors.
• Strengthens socioeconomic development by connecting regions with the rest of the country.
• Other Regional Connectivity Efforts:
• Development of regional airports under Regional Connectivity Infrastructure Investment.
• Promotion of private sector investments in aviation infrastructure.
• Collaboration with state governments to enhance last-mile connectivity.

Important Points

• UDAN Achievements:
• As of October 2023, the UDAN scheme has connected over 70 airports and initiated 425 routes in India.
• It has made air travel accessible to millions of citizens, especially in remote regions.
• Future Goals:
• The scheme aims to connect all regions of India, including the North-Eastern States and hilly terrains, by 2030.
• Plans to expand infrastructure and increase the number of operational airports to 100+.

The correct answer is London and New York.

Key Points

• The Global Cities Index, published by Oxford Economics, ranks cities based on their global influence, economic power, and connectivity. It evaluates cities across various metrics such as business activity, human capital, cultural experience, and political engagement.
• In the 2025 Global Cities Index, London and New York retained their top positions due to their significant contributions to global finance, international trade, and cultural influence.
• London is consistently ranked at the top due to its status as a leading financial center, housing major global corporations, and being a hub for international diplomacy and education.
• New York, often referred to as the "capital of the world," remains a major player in areas such as finance, media, and culture, hosting institutions like Wall Street and the United Nations Headquarters.
• The rankings reflect the resilience of these cities in maintaining their dominance despite challenges such as economic fluctuations, geopolitical tensions, and the COVID-19 pandemic.

Additional Information

• London's Key Features:
• London is home to institutions like the London Stock Exchange and the Bank of England, making it a critical global financial hub.
• The city is also a center for education and innovation, hosting world-renowned universities such as University College London (UCL) and Imperial College London.
• London’s cultural diversity and historical landmarks such as the British Museum and Tower of London further contribute to its global appeal.
• New York's Key Features:
• New York is a global leader in finance, with Wall Street serving as a symbol of economic power.
• The city hosts the United Nations Headquarters, emphasizing its importance in global diplomacy and international relations.
• As a cultural hub, New York is home to iconic institutions like the Metropolitan Museum of Art and Broadway.
• Global Cities Index:
• The index evaluates cities based on business activity, human capital, information exchange, cultural experience, and political engagement.
• It provides insights into the economic and social dynamics shaping global urban centers.
• Other cities frequently appearing in the top rankings include Tokyo, Singapore, and Paris.
• Impact of Rankings:
• High rankings attract foreign investment, talent migration, and tourism.
• They also influence urban planning and policy decisions aimed at sustaining global competitiveness.

The correct answer is Use light materials to make consumer goods.

Key Points

• Light industries primarily utilize lightweight materials such as plastics, fabrics, and smaller metals to produce consumer goods like clothes, electronics, furniture, and toys. This statement is correct.
• They are generally involved in the production of goods with higher value per unit weight, making transportation and logistics easier and cost-effective.
• Unlike heavy industries, light industries do not process heavy ores and minerals. Instead, they focus on finished goods rather than raw material processing. Hence, statement 1 is incorrect.
• These industries are not involved in the creation of large-scale industrial machinery. Their primary focus is on small-scale, high-demand products. Hence, statement 2 is incorrect.
• Light industries do not require bulky and heavy raw materials for production. Their raw materials are often compact and lightweight. Hence, statement 3 is incorrect.

Additional Information

• Definition and Scope:
• Light industries are characterized by their focus on low capital investment and less resource-intensive production processes.
• They are typically located near urban areas to cater to consumer needs and to minimize transportation costs.
• Examples of Light Industries:
• Textile industry: Produces clothing and other fabric-based products.
• Electronics: Manufacturing of small consumer appliances like mobile phones, laptops, and televisions.
• Food processing: Packaging and production of processed foods, dairy products, and beverages.
• Furniture-making: Use of light materials like plywood and aluminum to create household furniture.
• Differences Between Light and Heavy Industries:
• Raw Materials: Heavy industries process large, bulky raw materials like coal and iron, while light industries use smaller, lightweight materials.
• Location: Heavy industries are often located near mines and resources, while light industries are near markets or urban centers.
• Pollution: Heavy industries are generally more pollution-intensive, whereas light industries have less environmental impact.
• Economic Importance:
• Light industries are significant contributors to employment generation and urban economic development.
• They play a pivotal role in the export economy by producing high-demand consumer goods.

Important Points

• Global Examples of Light Industries:
• Japan: Known for high-quality electronics and textiles.
• Italy: Renowned for its fashion and furniture industries.
• China: A global hub for consumer goods production, especially electronics and toys.
• Government Support:
• In India, the government promotes light industries through initiatives like Make in India and MSME (Micro, Small, and Medium Enterprises) schemes.
• These industries are offered tax benefits, subsidized loans, and infrastructure support to encourage growth.

The correct answer is Three legislative lists.

Key Points

• Schedule VII of the Constitution of India deals with the division of powers between the Union and State governments. It contains three legislative lists: Union List, State List, and Concurrent List.
• The Union List includes subjects of national importance, such as defense, foreign affairs, atomic energy, and banking. The Parliament of India has exclusive power to legislate on these matters.
• The State List contains subjects of state and local importance like police, public health, agriculture, and local government. Only the state legislatures can legislate on these matters.
• The Concurrent List covers subjects of common interest to both the Union and State governments, such as education, marriage, trade unions, and forests. Both the Parliament and state legislatures can make laws on these subjects, but in case of conflict, the law made by the Parliament prevails.
• Schedule VII ensures a clear demarcation of legislative powers to maintain harmony between the Union and State governments and avoid overlapping jurisdictions.

Additional Information

• Union List:
• It currently consists of 100 subjects (originally 97).
• Examples of subjects include defense, atomic energy, railways, and foreign affairs.
• Legislation on these matters is exclusively under the Parliament's jurisdiction, ensuring uniformity across the nation.
• State List:
• It currently contains 61 subjects (originally 66).
• Key subjects include police, public order, public health, and agriculture.
• The state legislatures enjoy exclusive authority to legislate on these matters, ensuring regional autonomy.
• Concurrent List:
• It comprises 52 subjects (originally 47).
• Examples include education, criminal law, forests, and marriage.
• Both the Union and State legislatures can legislate on these matters, but in case of conflict, the Union law prevails under Article 254 of the Constitution.
• Historical Context:
• The concept of the three lists was borrowed from the Government of India Act, 1935, which also categorized powers into federal, provincial, and concurrent lists.
• This division of powers was designed to ensure a balance between centralization and decentralization, catering to India's federal structure.
• Amendments and Changes:
• Over time, the distribution of subjects in these lists has been altered through constitutional amendments to reflect changing needs and priorities.
• For instance, the 42nd Amendment Act of 1976 transferred subjects like education and forests from the State List to the Concurrent List.

The correct answer is Climate Change Adaptation.

Key Points

• Climate Change Adaptation refers to measures taken to adjust to the current or expected effects of climate change in order to minimize harm or exploit beneficial opportunities.
• The construction of sea walls and elevation of buildings specifically addresses the impacts of rising sea levels, which are a direct consequence of climate change. These are preventive actions to safeguard infrastructure and human lives.
• Adaptation strategies are often localized and cater to the specific challenges posed by climate change, such as floods, droughts, or temperature increases.
• The goal of adaptation is to build resilience in communities, economies, and ecosystems, allowing them to cope with the adverse effects of climate change.
• This approach differs from mitigation, which focuses on reducing the causes of climate change, such as greenhouse gas emissions.

Additional Information

• Climate Change Mitigation:
• Mitigation involves efforts to reduce or prevent the emission of greenhouse gases (GHGs) that contribute to global warming.
• Examples include switching to renewable energy sources like solar or wind, increasing energy efficiency, and adopting sustainable transportation practices.
• This strategy aims to address the root cause of climate change rather than its effects.
• For instance, the Paris Agreement focuses on mitigation by setting targets for reducing global GHG emissions.
• Carbon Sequestration:
• It refers to the process of capturing and storing atmospheric carbon dioxide (CO2) to reduce its concentration in the atmosphere.
• Natural methods include afforestation, reforestation, and soil management, while technological solutions include carbon capture and storage (CCS).
• Carbon sequestration is a key component of climate change mitigation strategies, aimed at offsetting emissions from industrial processes.
• Geoengineering:
• Geoengineering involves large-scale interventions in Earth's natural systems to counteract climate change.
• Examples include solar radiation management (reflecting sunlight back into space) and ocean fertilization to enhance carbon capture by marine ecosystems.
• These methods are controversial due to their potential unintended consequences and ethical concerns.
• Examples of Climate Change Adaptation:
• Constructing flood barriers in coastal cities prone to rising sea levels.
• Implementing drought-resistant crops in regions experiencing water scarcity due to climate changes.
• Developing early warning systems for extreme weather events such as hurricanes or heatwaves.
• Retrofitting infrastructure to withstand extreme weather conditions, such as stronger buildings to resist cyclones.

Important Points

• Rising Sea Levels:
• Sea levels have been rising globally due to melting ice caps and glaciers, as well as the thermal expansion of seawater caused by global warming.
• According to the Intergovernmental Panel on Climate Change (IPCC), global sea levels could rise by 0.3 to 1 meter by 2100.
• This poses threats to low-lying and coastal areas, leading to increased flooding, erosion, and loss of habitats.
• Global Adaptation Efforts:
• The UN Framework Convention on Climate Change (UNFCCC) emphasizes the need for adaptation measures alongside mitigation strategies.
• Programs like the Global Environment Facility (GEF) and the Green Climate Fund (GCF) provide financial support for adaptation projects in developing countries.
• Several nations, including the Netherlands, have implemented large-scale adaptation projects such as flood management systems and sustainable urban planning.

The correct answer is Orbital velocity.

Key Points

• Orbital velocity is the speed required for a satellite to maintain its orbit around a celestial body, like Earth, without propulsion or falling back due to gravity.
• It is determined by the balance between the satellite's inertia (from its motion) and the gravitational pull of the planet it is orbiting. This ensures the satellite remains in a stable orbit.
• For a satellite to remain in orbit, its orbital velocity must match the gravitational force pulling it toward the planet. If the velocity is too high, the satellite will escape into space; if too low, it will fall back to Earth.
• The formula for orbital velocity is given by v = √(GM/r), where G is the gravitational constant, M is the mass of the celestial body, and r is the radius of the orbit.
• Atmospheric drag can influence orbital velocity in low Earth orbits, causing satellites to lose speed and eventually deorbit. However, in the absence of atmospheric drag in higher orbits, orbital velocity alone ensures stability.

Additional Information

• Mass: A satellite's mass does not directly determine its orbit. While mass affects gravitational attraction, the satellite's orbital velocity is independent of its weight.
• Gravitational pull: The gravitational pull of a planet provides the force that keeps the satellite in orbit. However, it needs to be balanced by the satellite's orbital velocity to ensure stable motion.
• Atmospheric drag: This factor is significant in low Earth orbits (LEO). Atmospheric particles can slow the satellite over time, causing it to lose altitude and eventually re-enter the atmosphere.
• Examples: Geostationary satellites, such as communication satellites, maintain their orbit by traveling at speeds matching the Earth's rotation, ensuring they appear stationary from the surface.
• Applications of Orbital Velocity: Orbital velocity is crucial for launching satellites, space exploration, and maintaining space stations like the International Space Station (ISS). It ensures that objects remain in their designated trajectories.

Important Points

• Low Earth Orbit (LEO): Satellites in LEO typically have orbital velocities of around 7.8 km/s. They are closer to Earth, making them more prone to atmospheric drag.
• Geostationary Orbit: Satellites in geostationary orbit (~36,000 km above Earth) travel at a speed of approximately 3.07 km/s. This ensures they remain fixed relative to Earth's surface.
• Escape Velocity: If a satellite's speed exceeds the escape velocity (~11.2 km/s for Earth), it will leave Earth's gravitational influence and move into interplanetary space.
• Space Missions: Missions like ISRO's Chandrayaan and NASA's Mars rovers calculate orbital and escape velocities accurately for successful launches and operations.
• Newton's Laws: Orbital velocity is derived using Newton's Laws of Motion and Universal Gravitation. These principles explain the relationship between force, mass, and motion in orbit.

Shortcut Trick Square both values to compare them directly without estimating decimal roots. x2 = (√3 + √5)2 = 8 + 2√15 y2 = (√8 + √2)2 = 10 + 2√16 = 18 Since √15 < 4, the expression 8 + 2√15 is strictly less than 16. Because 18 > 16, it follows that y2 > x2, which means y > x. ∴ The correct answer is y > x. Alternate Method Given: x = √3 + √5 y = √8 + √2 Formula Used: (a + b)2 = a2 + b2 + 2ab Calculations: ⇒ Square the value of x: ⇒ x2 = (√3 + √5)2 ⇒ x2 = (√3)2 + (√5)2 + 2 × √3 × √5 ⇒ x2 = 3 + 5 + 2√15 ⇒ x2 = 8 + 2√15 ⇒ Simplify and square the value of y: ⇒ y = 2√2 + √2 = 3√2 ⇒ y2 = (3√2)2 = 9 × 2 = 18 ⇒ Compare the squared values: ⇒ We know that √15 is strictly less than √16 (which evaluates to 4). ⇒ So, x2 = 8 + 2√15 < 8 + 2 × 4 ⇒ x2 < 16 ⇒ Since x2 < 16 and y2 = 18, it is completely clear that y2 > x2. ⇒ y > x ∴ The correct answer is y > x. Additional Information Squaring Method to Compare Surds When comparing irrational expressions involving square roots, squaring both sides removes the root operations and provides whole numbers or simplified surds for easier comparison. Simplifying Like Surds Surds that can be reduced to have the exact same irrational part can be added or subtracted directly like standard algebraic terms, for example, √8 + √2 = 2√2 + √2 = 3√2. Comparing Roots of Different Orders For comparing surds of different roots (such as cube roots and square roots), find the LCM of the orders of the roots and raise all surds to that LCM power to compare exact integer values.

Shortcut Trick

By applying the cross-multiplication method on Red and Blue marbles:

Initial Ratio → 3 : 4 | Final Ratio → 4 : 5 | Quantity Added → +20, +15

1 initial ratio unit = [(4 × 15) − (5 × 20)] ÷ [(3 × 5) − (4 × 4)] = (60 − 100) ÷ (15 − 16) = 40.

Initial Green marbles = 6 × 40 = 240.

New Red marbles = (3 × 40) + 20 = 140, which represents 4 parts in the final ratio ⇒ 1 final unit = 140 ÷ 4 = 35.

New Green marbles = 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

Marbles added = 20 red, 15 blue, and an unknown number of green marbles (y).

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

Initial RatioRed = 3xBlue = 4xGreen = 6xFinal RatioRed = 4 partsBlue = 5 partsGreen = 7 parts+20+15+y

Calculations:

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

⇒ After adding 20 red and 15 blue marbles, the new counts become (3x + 20) and (4x + 15).

⇒ Given the new ratio of red to blue is 4 : 5.

⇒ (3x + 20) ÷ (4x + 15) = 4 ÷ 5

⇒ 5 × (3x + 20) = 4 × (4x + 15)

⇒ 15x + 100 = 16x + 60

⇒ 16x − 15x = 100 − 60

⇒ x = 40

⇒ Initial red marbles = 3 × 40 = 120

⇒ Initial green marbles = 6 × 40 = 240

⇒ New red marbles = 120 + 20 = 140

⇒ Let the number of green marbles added be y. The new count of green marbles is (240 + y).

⇒ Given the new ratio of red to green is 4 : 7.

⇒ 140 ÷ (240 + y) = 4 ÷ 7

⇒ 240 + y = (140 × 7) ÷ 4

⇒ 240 + y = 35 × 7 = 245

⇒ y = 245 − 240 = 5

∴ The correct answer is 5.

Additional Information

Ratio and Proportion

A ratio compares two quantities, indicating how many times one number contains another. For quantities a and b, it is represented as a : b or the fraction a/b.

Cross-Multiplication Method in Ratios

When a given ratio a : b changes to c : d by adding quantities x and y respectively, the value of 1 initial ratio unit is directly obtained using the formula (cy − dx) ÷ (ad − bc).

Shortcut Trick

For continued fractions, solve directly from the bottommost denominator to the top.

Bottom fraction: 4 + 1/2 = 9/2

Middle fraction: 3 + 2/9 = 29/9

Top fraction: 2 + 9/29 = 67/29

∴ The correct answer is 67/29.

Alternate Method

Given:

x = 2 + 1 / [3 + 1 / (4 + 1/2)]

Calculations:

Solve the innermost fraction first (bottom-up approach):

⇒ 4 + 1/2 = (8 + 1) / 2 = 9/2

Substitute this back into the expression:

⇒ x = 2 + 1 / [3 + 1 / (9/2)]

⇒ x = 2 + 1 / (3 + 2/9)

Now, simplify the next denominator:

⇒ 3 + 2/9 = (27 + 2) / 9 = 29/9

Substitute this back into the expression:

⇒ x = 2 + 1 / (29/9)

⇒ x = 2 + 9/29

⇒ x = (58 + 9) / 29

⇒ x = 67/29

∴ The correct answer is 67/29.

Additional Information

BODMAS Rule in Fractions

Always follow the order of Brackets, Orders, Division, Multiplication, Addition, and Subtraction. In continued fractions, treat the bottommost denominator as an implicit bracket.

Reciprocal of a Fraction

Dividing by a fraction is the same as multiplying by its reciprocal. For example, 1 ÷ (a/b) = b/a.

Continued Fractions

A continued fraction is evaluated through a step-by-step upward process of simplifying the lowest mixed fraction and reciprocating it.

Shortcut Trick

Profit Ratio = Capital Ratio (since time is constant) = 40000 : 60000 = 2 : 3

Total profit ratio parts = 2 + 3 = 5 parts

5 parts = ₹50,000 ⇒ 1 part = ₹10,000

A's share (2 parts) = 2 × 10,000 = ₹20,000

∴ The correct answer is ₹20,000.

Alternate Method

Given:

A's investment = ₹40,000

B's investment = ₹60,000

Total Profit = ₹50,000

Formula Used:

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

Capital & Profit Distribution

Total Profit = ₹50,000 (5 units)

A ₹40,000

B ₹60,000

Ratio: 2 Units Ratio: 3 Units

A's Share = 2/5 × ₹50,000 = ₹20,000

Calculations:

⇒ Since the investment time is exactly the same for both A and B, their profit ratio will be directly equal to their investment ratio.

⇒ Profit Ratio = 40000 : 60000

⇒ Profit Ratio = 4 : 6 = 2 : 3

⇒ Total profit parts = 2 + 3 = 5

⇒ A's share = (2 ÷ 5) × Total Profit

⇒ A's share = (2 ÷ 5) × 50000

⇒ A's share = 2 × 10000 = 20000

∴ The correct answer is ₹20,000.

Additional Information

Simple Partnership

When partners invest their capital for the exact same time period, all final profits or losses are distributed in the direct ratio of their original investments.

Compound Partnership

When partners invest for varying time periods, their profit distribution ratio is calculated as (Capital₁ × Time₁) : (Capital₂ × Time₂).

Working vs Sleeping Partner

A working partner manages the business operations and generally receives a fixed salary or percentage of the profit before the remaining balance is divided among all partners based on their respective investments.

Shortcut Trick

Total share of A = (Salary as working partner) + (Investment share of remaining profit)

⇒ A's total share = 10% of 64000 + (3 ÷ 8) × 90% of 64000

⇒ A's total share = 6400 + (3 ÷ 8) × 57600

⇒ A's total share = 6400 + 21600 = 28000

∴ The correct answer is ₹28,000.

Alternate Method

Given:

Total profit = ₹64,000

Investment ratio of A and B = 3 : 5

A's extra share (working partner) = 10% of total profit

Total Profit: ₹64,000A's 10% Extra₹6,400Remaining 90%₹57,600A (3 parts)₹21,600B (5 parts)₹36,000A's Total = ₹28,000

Calculations:

⇒ A's extra profit = 10% of ₹64,000

⇒ A's extra profit = (10 ÷ 100) × 64,000 = ₹6,400

⇒ Remaining profit = 64,000 − 6,400 = ₹57,600

⇒ Total ratio parts = 3 + 5 = 8 parts

⇒ A's share from remaining profit = (3 ÷ 8) × 57,600

⇒ A's share from remaining profit = 3 × 7,200 = ₹21,600

⇒ A's total share = Extra profit + Investment share

⇒ A's total share = 6,400 + 21,600 = ₹28,000

∴ The correct answer is ₹28,000.

Additional Information

Active vs Sleeping Partner

An active partner contributes both capital and time to manage the business, often receiving an extra salary or fixed percentage of profit before the remainder is distributed. A sleeping partner only contributes capital.

Profit Distribution Formula

Profit ratio is directly proportional to the product of invested capital and time period. Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Equivalent Capital

When investments are made for unequal times, we calculate the equivalent capital for 1 unit of time. Equivalent capital = Investment × Time invested.

Shortcut Trick

Using the Net Deviation Method:

Ratio of Junior to Senior employees = 30 : 45 = 2 : 3

Junior group deviation from overall average = 1.2 − 1.8 = −0.6 lakh

Total deviation of Junior parts = 2 × (−0.6) = −1.2 lakh

To balance the mean, total deviation of 3 Senior parts must be +1.2 lakh ⇒ per part = +0.4 lakh

Average sales of Senior employees = Overall average + Senior deviation = 1.8 + 0.4 = 2.2 lakh

∴ The correct answer is ₹2.2 lakh.

Alternate Method

Given:

Total employees = 75

Overall average monthly sales = ₹1.8 lakh

Junior employees = 30

Average sales of junior employees = ₹1.2 lakh

Formula Used:

Total Value = Average × Number of Entities

1.2 Junior (₹ lakh)

x Senior (₹ lakh)

1.8 Overall Avg

x − 1.8 Emp: 30 (Ratio 2)

0.6 Emp: 45 (Ratio 3)

Calculations:

⇒ Number of senior employees = 75 − 30 = 45

⇒ Total sales of all employees = 75 × 1.8 = 135 lakh

⇒ Total sales of junior employees = 30 × 1.2 = 36 lakh

⇒ Total sales of senior employees = 135 − 36 = 99 lakh

⇒ Average sales of senior employees = 99 ÷ 45

⇒ Average sales of senior employees = 2.2 lakh

∴ The correct answer is ₹2.2 lakh.

Additional Information

Weighted Average

When combining two different groups, the overall average is calculated using the formula: A = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where 'n' represents the number of elements and 'x' represents their respective averages.

Method of Deviation

This method assumes a central mean value and calculates the difference (deviation) of all group components from it. For any perfectly balanced average, the net sum of all weighted deviations must always be zero.

Alligation Cross Rule

A visual shortcut applied when two differently priced or valued sets are mixed to get a final average. The ratio of the quantities mixed is always given by (Higher Average − Target Average) : (Target Average − Lower Average).

Shortcut Trick

Total fuel consumed = Total distance ÷ Mileage = 3045 ÷ 15 = 203 litres

Increase in price per litre = 5% of ₹82 = ₹4.1

Total increase in expenditure = 203 × 4.1 = ₹832.3 ≈ ₹832

∴ The correct answer is ₹832.

Alternate Method

Given:

Initial price of petrol = ₹82 per litre

Price hike = 5%

Monthly distance travelled = 3045 km

Car mileage = 15 km/l

Petrol Consumed 3045 km ÷ 15 km/l = 203 litres

Price Increase 5% of ₹82 = ₹4.1 / litre

Total Increase 203 × 4.1 = ₹832.3 ≈ ₹832

Calculations:

⇒ Total petrol consumed per month = Total Distance ÷ Mileage

⇒ Total petrol consumed = 3045 ÷ 15 = 203 litres

⇒ Increase in petrol price = 5% of 82 = (5 ÷ 100) × 82 = ₹4.1 per litre

⇒ Total increase in monthly expenditure = Total petrol consumed × Increase per litre

⇒ Total increase = 203 × 4.1 = ₹832.3

⇒ Rounding to the nearest rupee, the increase is ₹832.

∴ The correct answer is ₹832.

Additional Information

Percentage Change

The formula to calculate the absolute change is: Change = Original Value × (Percentage Change ÷ 100).

Fuel and Distance Relationship

Total fuel required = Total Distance ÷ Fuel Efficiency (Mileage). Total cost is evaluated by multiplying the fuel required by the cost per unit of fuel.

Rounding Values

If the decimal part is less than 0.5, round down to the nearest integer. If it is 0.5 or greater, round up to the next integer.

Shortcut Trick

Let desktop price = 1, laptop price = 5, and the number of desktops ordered = x.

Original bill = 20 × 5 + x × 1 = 100 + x

New interchanged bill = x × 5 + 20 × 1 = 5x + 20

Given the new bill is 80% more, so (5x + 20) = 1.8 × (100 + x)

⇒ 5x + 20 = 180 + 1.8x ⇒ 3.2x = 160 ⇒ x = 50

∴ The correct answer is 2:5.

Alternate Method

Original OrderLaptops = 20Desktops = xBill = 100 + x+80%New OrderLaptops = xDesktops = 20Bill = 5x + 20Required Ratio = 20 : x

Given:

⇒ Number of high-end laptops originally ordered = 20

⇒ Let the number of standard desktops originally ordered = x

⇒ Price of a high-end laptop = 5 × Price of a standard desktop

Formula Used:

⇒ Total Bill = (Quantity of Laptops × Price of Laptop) + (Quantity of Desktops × Price of Desktop)

⇒ New Bill = Original Bill × (100% + Increase%)

Calculations:

⇒ Let the price of a standard desktop be 1 unit. Then the price of a high-end laptop is 5 units.

⇒ Original Bill = (20 × 5) + (x × 1) = 100 + x

⇒ Due to the mistake, the new quantity of laptops = x, and desktops = 20.

⇒ New Bill = (x × 5) + (20 × 1) = 5x + 20

⇒ It is given that the new bill is 80% higher than the original bill. This means the new bill is 180% of the original bill.

⇒ 5x + 20 = (100 + x) × (180 ÷ 100)

⇒ 5x + 20 = (100 + x) × 1.8

⇒ 5x + 20 = 180 + 1.8x

⇒ 5x − 1.8x = 180 − 20

⇒ 3.2x = 160

⇒ x = 160 ÷ 3.2

⇒ x = 50

⇒ Number of standard desktops originally ordered = 50

⇒ Required Ratio = (Number of High-End Laptops) : (Number of Standard Desktops) = 20 : 50 = 2 : 5

∴ The correct answer is 2:5.

Additional Information

Interchange of Quantities Rule

When the quantities of two items are interchanged, the change in total value is equal to the product of the difference in their unit prices and the difference in their original quantities: Change in Value = (P₁ − P₂) × (Q₂ − Q₁).

Percentage Increase Formula

The percentage increase between two values is calculated as: % Increase = [(New Value − Original Value) ÷ Original Value] × 100.

Base Value in Percentages

Whenever a value increases by R%, the new value becomes Original Value × (1 + R/100). Identifying the correct base value (the original order bill) is crucial to avoid calculation errors.

Shortcut Trick

Using the multiplying factor method for Compound Interest:

Rate for 3 full years = 12% (Multiplier = 1.12)

Rate for next 4 months = 12% × (4/12) = 4% (Multiplier = 1.04)

Amount = P × (1.12)³ × (1.04)

Amount = 8000 × 1.404928 × 1.04 ≈ 11689

Compound Interest = 11689 − 8000 = 3689

∴ The correct answer is ₹3,689.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate (R) = 12% per annum

Time (T) = 3 years 4 months = 3 + 4/12 years = 10/3 years

Formula Used:

A = P × (1 + R/100)ⁿ × (1 + r/100)

Where n is the full years and r is the rate for the fractional year.

Start ₹8000

+12% (1Yr)

Year 1 ₹8960

+12% (1Yr)

Year 2 ₹10035.2

+12% (1Yr)

Year 3 ₹11239.4

+4% (4Mo)

3 Yr 4 Mo ₹11689

Total CI = ₹11689 − ₹8000 = ₹3689

Calculations:

⇒ Rate for the fractional 4 months (1/3 year) = 12% × 1/3 = 4%

⇒ Amount (A) = 8000 × (1 + 12/100)³ × (1 + 4/100)

⇒ A = 8000 × (112/100)³ × (104/100)

⇒ A = 8000 × 1.12 × 1.12 × 1.12 × 1.04

⇒ A = 8000 × 1.404928 × 1.04

⇒ A = 11239.424 × 1.04 = 11689.00

⇒ Compound Interest (CI) = A − P

⇒ CI = 11689 − 8000 = 3689

∴ The correct answer is ₹3,689.

Additional Information

Fractional Year Compounding

When time is in fractions (e.g., n years and m months), apply the full compounding rate for complete years and a proportionally reduced rate for the remaining fractional year.

Effective Compound Interest Rate

For two consecutive periods at rates x% and y%, the overall effective compound interest percentage can be calculated using the successive percentage formula: x + y + (x × y)/100.

Difference Between CI and SI

For 2 years, the difference between compound and simple interest is CI − SI = P(R/100)². For 3 years, the difference is CI − SI = P(R/100)²(3 + R/100).

Shortcut Trick

CP per candy: Red = 15 ÷ 3 = ₹5, Blue = 18 ÷ 4 = ₹4.5, Green = 22 ÷ 5 = ₹4.4

Total CP for 4 mixed candies (in ratio 1:1:2) = 5 + 4.5 + (2 × 4.4) = ₹18.3

SP for 4 mixed candies = 4 × (10 ÷ 2) = ₹20

Profit = 20 − 18.3 = ₹1.7

Profit % = (1.7 ÷ 18.3) × 100 ≈ 9.29%

∴ The correct answer is Profit of 9.29%.

Alternate Method

Candy Mixture (1 : 1 : 2 Ratio)

1 Red @ ₹5

1 Blue @ ₹4.5

2 Green @ ₹4.4

Total CP = ₹18.3

• Profit ₹1.7

Total SP = ₹20

Profit % = (1.7 ÷ 18.3) × 100 ≈ 9.29%

Given:

Red candies bought at 3 for ₹15

Blue candies bought at 4 for ₹18

Green candies bought at 5 for ₹22

Mixture ratio = 1 : 1 : 2

Selling price = 2 for ₹10

Calculations:

⇒ Let the number of Red, Blue, and Green candies mixed be 1, 1, and 2 respectively.

⇒ Total candies mixed = 1 + 1 + 2 = 4 candies

⇒ Cost Price (CP) of 1 Red candy = 15 ÷ 3 = ₹5

⇒ Cost Price (CP) of 1 Blue candy = 18 ÷ 4 = ₹4.5

⇒ Cost Price (CP) of 1 Green candy = 22 ÷ 5 = ₹4.4

⇒ Total Cost Price of 4 mixed candies = (1 × 5) + (1 × 4.5) + (2 × 4.4)

⇒ Total CP = 5 + 4.5 + 8.8 = ₹18.3

⇒ Selling Price (SP) of 1 candy = 10 ÷ 2 = ₹5

⇒ Total SP of 4 mixed candies = 4 × 5 = ₹20

⇒ Since SP is greater than CP, there is a profit.

⇒ Profit = SP − CP = 20 − 18.3 = ₹1.7

⇒ Profit % = (Profit ÷ Total CP) × 100

⇒ Profit % = (1.7 ÷ 18.3) × 100

⇒ Profit % = 170 ÷ 18.3 ≈ 9.289% ≈ 9.29%

∴ The correct answer is Profit of 9.29%.

Additional Information

Profit & Loss Formulas

Profit occurs when Selling Price (SP) > Cost Price (CP). Profit = SP − CP. Profit Percentage is calculated on the total Cost Price: (Profit ÷ CP) × 100.

Mixture Average Cost

When different quantities of items are mixed, the Average Cost Price is simply the total cost of all components combined, divided by the total physical quantity.

Ratio Multipliers Trick

Instead of finding a large common multiple (LCM) to equate quantities, directly taking sample quantities equal to the given ratio (such as 1, 1, and 2 units) makes percentage-based calculations perfectly proportional and much faster.

Shortcut Trick

Initial Ratio of Boys : Girls = 7 : 5 and Final Ratio = 7 : 6

Since the number of boys remains the same, the ratio part for boys (7) is unchanged.

Increase in girls' ratio = 6 − 5 = 1 unit.

This 1 unit increase corresponds to the 20 girls who joined.

Total initial students = 7 + 5 = 12 units = 12 × 20 = 240.

∴ The correct answer is 240 students.

Alternate Method

Given:

Initial ratio of Boys : Girls = 7 : 5

Number of girls added = 20

Final ratio of Boys : Girls = 7 : 6

Formula Used:

If a ratio is given as a:b, the actual values can be taken as ax and bx.

Initial (7:5) Final (7:6)

Boys (7 units)

Girls (5 units)

Boys (7 units)

Girls (5 units)

+1

• 20 Girls

Calculations:

⇒ Let the initial number of boys be 7x and girls be 5x.

⇒ Total initial students = 7x + 5x = 12x.

⇒ After 20 girls join, the new number of girls = 5x + 20.

⇒ According to the given condition, 7x ÷ (5x + 20) = 7 ÷ 6

⇒ Cross-multiplying: 6 × 7x = 7 × (5x + 20)

⇒ 42x = 35x + 140

⇒ 42x − 35x = 140

⇒ 7x = 140

⇒ x = 20

⇒ Total initial students = 12x = 12 × 20 = 240

∴ The correct answer is 240 students.

Additional Information

Ratio Balancing Method

When one quantity changes while the other remains constant, ensure the ratio parts of the constant quantity are made equal before directly equating the change in units.

Compound Ratio

The compound ratio of two independent ratios a:b and c:d is evaluated as (a × c):(b × d).

Basic Proportion Property

If four quantities are in proportion such that a:b = c:d, then their cross products are equal (a × d = b × c).

Shortcut Trick

Total work = LCM(30, 45, 90) = 90 units.

Efficiencies: A = 3, B = 2, C = 1.

Work done in a 5-day cycle = (3 × 5) + 2 + 1 = 18 units.

Number of cycles = Total work ÷ Work per cycle = 90 ÷ 18 = 5 cycles.

Total days = 5 cycles × 5 days/cycle = 25 days.

∴ The correct answer is 25 days.

Alternate Method

Given:

Time taken by A = 30 days

Time taken by B = 45 days

Time taken by C = 90 days

Calculations:

⇒ In 1 cycle of 5 days, A works for all 5 days, while B and C work only on the 5th day.

⇒ Work done by A in 5 days = 5 × 3 = 15 units

⇒ Work done by B and C on the 5th day = 2 + 1 = 3 units

⇒ Total work done in 5 days = 15 + 3 = 18 units

⇒ Number of 5-day cycles required to complete 90 units = 90 ÷ 18 = 5 cycles

⇒ Total time taken = 5 cycles × 5 days/cycle = 25 days

∴ The correct answer is 25 days.

Additional Information

Total Work Concept

Total work is considered as the Least Common Multiple (LCM) of the days taken by individuals. Efficiency is the work done per unit time (Total Work ÷ Days).

Alternate or Cyclic Work

Find the work done in one complete cycle. Divide the total work by the work per cycle to find the number of cycles, then multiply by days per cycle to find the total time required.

Pipes and Cisterns

Works exactly on the same principle as Time and Work, but pipes that empty the tank are assigned a negative efficiency value.

Shortcut Trick

Apply the alligation method on the water concentration of the mixtures.

Initial mixture water = 40%, Added pure acid water = 0%, Final mixture water = 30%.

Ratio of remaining mixture to added pure acid = (30 − 0) : (40 − 30) = 30 : 10 = 3 : 1.

This means 1 part out of a total of (3 + 1) = 4 parts was replaced.

Replaced quantity = 1 / 4 of total volume = (1 / 4) × 100 = 25 litres.

∴ The correct answer is 25 litres.

Alternate Method

Given:

Initial volume of the solution = 100 litres

Initial ratio of acid to water = 3 : 2

Final ratio of acid to water = 7 : 3

Calculations:

⇒ Quantity of water in the initial solution = (2 / 5) × 100 = 40 litres

⇒ Let x litres of the solution be removed and replaced with pure acid.

⇒ Quantity of water removed = (2 / 5) × x = 0.4x litres

⇒ Remaining water in the solution = 40 − 0.4x litres

⇒ Since only pure acid is added back, the overall quantity of water in the final mixture does not increase.

⇒ Final volume of the solution is maintained at 100 litres.

⇒ Quantity of water in the final solution = (3 / 10) × 100 = 30 litres

⇒ Equating the remaining water to final water:

⇒ 40 − 0.4x = 30

⇒ 0.4x = 10

⇒ x = 10 ÷ 0.4 = 25 litres

∴ The correct answer is 25 litres.

Additional Information

Alligation Rule

It is a quick method used to find the ratio in which two ingredients at their given prices or concentrations must be mixed to produce a mixture of a desired average concentration.

Tracking the Untouched Component

When solving replacement questions, always frame your equation using the component that is NOT being added from the outside. In this case, since pure acid is added back, it is much faster to track the changing quantity of water.

Replacement Formula

If a container initially holds V units of pure liquid and x units are taken out and replaced by water n times sequentially, the final quantity of pure liquid remaining is given by V × (1 − x/V)ⁿ.

Shortcut Trick

Time and efficiency are inversely proportional to each other.

Ratio of Time taken by P and Q = 2 : 5

Here, 2 parts correspond to 12 days.

So, 1 part = 12 ÷ 2 = 6 days.

Time taken by Q (5 parts) = 5 × 6 = 30 days.

∴ The correct answer is 30 days.

Alternate Method

Given:

Ratio of the efficiencies of P and Q = 5 : 2

Time taken by P to complete the project = 12 days

Formula Used:

Total Work = Efficiency × Time

Time = Total Work ÷ Efficiency

Calculations:

⇒ Let the efficiency of P be 5 units/day and Q be 2 units/day.

⇒ Total Work = Efficiency of P × Time taken by P

⇒ Total Work = 5 × 12 = 60 units

⇒ Time taken by Q = Total Work ÷ Efficiency of Q

⇒ Time taken by Q = 60 ÷ 2 = 30 days

∴ The correct answer is 30 days.

Additional Information

Time and Efficiency Relationship

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

Combined Work Rule

If A can do a piece of work in x days and B can do it in y days, their combined efficiency is (1/x + 1/y). Together they complete the work in (x × y) ÷ (x + y) days.

Work and Wages

Wages are distributed in proportion to the total work done by each person. If workers work for the exact same number of days, wages are distributed in the direct ratio of their efficiencies.

Shortcut Trick

Relative speed of Car with respect to SUV = 15 − 10 = 5 m/s.

Time required to close the initial gap of 600 m = 600 ÷ 5 = 120 seconds.

Distance traveled by car in this time = Speed × Time = 15 × 120 = 1800 m.

∴ The correct answer is 1800 m.

Alternate Method

Point P Car (15 m/s)

Point Q SUV (10 m/s)

Meeting

Gap = 600 m

SUV dist = 1200 m

Car distance = 1800 m

Given:

⇒ Speed of car (S_C) = 15 m/s

⇒ Speed of SUV (S_S) = 10 m/s

⇒ Initial distance between them in direction of motion (d) = 600 m

⇒ Track circumference = 1.5 km = 1500 m (Extra information)

Formula Used:

⇒ Time taken to overtake = Distance ÷ Relative Speed

⇒ Distance = Speed × Time

Calculation:

⇒ Since both vehicles are moving in the same direction, Relative Speed = S_C − S_S

⇒ Relative Speed = 15 − 10 = 5 m/s

⇒ Time taken to overtake = 600 ÷ 5 = 120 seconds

⇒ Distance traveled by car = S_C × Time taken

⇒ Distance = 15 × 120 = 1800 m

∴ The correct answer is 1800 m.

Additional Information

Relative Speed (Same Direction)

When two bodies move in the same direction, their relative speed is the difference of their individual speeds: S_R = S₁ − S₂. The distance required to overtake is determined by the initial gap between them.

Relative Speed (Opposite Direction)

When two bodies move in opposite directions (e.g., towards each other), their relative speed is the sum of their individual speeds: S_R = S₁ + S₂.

Circular Track Properties

The circumference of the track is only needed when finding the time taken for the vehicles to meet again at the starting point, or for a subsequent overtake cycle (which requires covering an extra 1 full lap).

Shortcut Trick

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

Area = (22 ÷ 7) × 1.5 × (2 × 14 + 1.5)

Area = (33 ÷ 7) × 29.5 = 973.5 ÷ 7 ≈ 139.07 m²

∴ The correct answer is 139 m².

Alternate Method

O

14 m

1.5 m

R = 15.5 m

Given:

⇒ Radius of the circular park, r = 14 m

⇒ Width of the path built around it, w = 1.5 m

Formula Used:

⇒ Area of the path = Area of the outer circle − Area of the inner circle

⇒ Area of a circle = πR²

Calculations:

⇒ Outer radius, R = r + w

⇒ R = 14 + 1.5 = 15.5 m

⇒ Area of the path = πR² − πr²

⇒ Area = π × (15.5² − 14²)

⇒ Area = (22 ÷ 7) × (15.5 + 14) × (15.5 − 14)

⇒ Area = (22 ÷ 7) × 29.5 × 1.5

⇒ Area = (33 ÷ 7) × 29.5

⇒ Area = 973.5 ÷ 7 ≈ 139.07 m²

⇒ The approximate area of the path is 139 m².

∴ The correct answer is 139 m².

Additional Information

Area of Circular Path Outside

When a path of width 'w' is constructed outside a circle of radius 'r', the Area is calculated as πw(2r + w).

Area of Circular Path Inside

When a path of width 'w' is constructed inside a circle of radius 'r', the Area is calculated as πw(2r − w).

Area of a Circle

The total space occupied by a circular region is denoted by πr².

Shortcut Trick

To convert an angle from radians to degrees quickly, directly substitute π = 180°.

Given angle = 5π ÷ 4

⇒ Angle = (5 × 180°) ÷ 4

⇒ Angle = 5 × 45°

∴ The correct answer is 225°.

Alternate Method

Given:

Angle in radians = 5π ÷ 4

Formula Used:

Angle in degrees = Angle in radians × (180° ÷ π)

0°90°180°270°5π/4 = 225°225°

Calculations:

⇒ Angle in degrees = (5π ÷ 4) × (180° ÷ π)

⇒ Angle in degrees = (5 × 180°) ÷ 4

⇒ Angle in degrees = 900° ÷ 4

⇒ Angle in degrees = 225°

∴ The correct answer is 225°.

Additional Information

Degrees to Radians Conversion

To convert an angle from degrees to radians, multiply the given angle by (π ÷ 180°).

Radian Measure

A radian is the angle subtended at the center of a circle by an arc whose length is exactly equal to the radius of the circle.

Common Angle Conversions

π ÷ 2 = 90°, π ÷ 3 = 60°, π ÷ 4 = 45°, and π ÷ 6 = 30°.

Shortcut Trick

Rewrite the equation y = -x + 4 as x + y = 4.

For a point to lie on the line, the sum of its coordinates must be 4.

Check the options: (5, 4) → 9; (1, 2) → 3; (3, 6) → 9; (2, 2) → 4.

Only the point (2, 2) gives a sum of 4.

∴ The correct answer is (2, 2).

Alternate Method

Given:

Equation of the line: y = -x + 4

X Y O

y = -x + 4

(2, 2)

(5, 4)

(1, 2)

(3, 6)

Calculations:

⇒ Let us substitute the given coordinates (x, y) into the equation x + y = 4.

⇒ Option 1: For (2, 2), LHS = 2 + 2 = 4. This satisfies the equation.

⇒ Option 2: For (5, 4), LHS = 5 + 4 = 9 ≠ 4. This does not satisfy.

⇒ Option 3: For (1, 2), LHS = 1 + 2 = 3 ≠ 4. This does not satisfy.

⇒ Option 4: For (3, 6), LHS = 3 + 6 = 9 ≠ 4. This does not satisfy.

∴ The correct answer is (2, 2).

Additional Information

Slope-Intercept Form

The equation of a straight line is often written as y = mx + c, where m is the slope of the line and c is the y-intercept. For y = -x + 4, the slope is -1 and the y-intercept is 4.

Checking a Point on a Line

A coordinate point (x₁, y₁) lies on a specific line if and only if its coordinates satisfy the equation of that line.

Intercepts of a Line

The x-intercept is calculated by setting y = 0, giving (4, 0). The y-intercept is calculated by setting x = 0, giving (0, 4). These are the points where the line crosses the respective axes.

Shortcut Trick

The relationship between arc length (l), radius (r), and central angle (θ) in radians is directly calculated using the formula:

θ = l ÷ r

⇒ θ = 10 ÷ 5 = 2 radians

∴ The correct answer is 2 radians.

Alternate Method

θ = ? r = 5 cm r = 5 cm Arc length (l) = 10 cm

Center O

Given:

Arc length of the sector (l) = 10 cm

Radius of the circle (r) = 5 cm

Formula Used:

The central angle in radians subtended by an arc at the center of a circle is the ratio of the arc length to its radius.

Angle (θ) = Arc length (l) ÷ Radius (r)

Calculations:

⇒ θ = 10 cm ÷ 5 cm

⇒ θ = 2

Since the formula directly yields the angle in radians, the central angle is 2 radians.

∴ The correct answer is 2 radians.

Additional Information

Conversion between Radians and Degrees

To convert an angle from radians to degrees, multiply by (180° ÷ π). Similarly, to convert degrees to radians, multiply by (π ÷ 180°). Thus, π radians = 180°.

Area of a Sector

The area of a sector with radius r and central angle θ (in radians) is determined by the formula A = (1 ÷ 2) × r² × θ. It can also be found using the arc length as A = (1 ÷ 2) × l × r.

Full Circle Arc Length

The total arc length encompassing a complete circle is its circumference, calculated as C = 2 × π × r. This aligns with a full rotation having a central angle of 2π radians.

Shortcut Trick

Given: tan(90° − A) = √3

We know from standard trigonometric values that tan 60° = √3.

Comparing both sides, we get: 90° − A = 60°

⇒ A = 30°

Therefore, sin A = sin 30° = 1/2.

∴ The correct answer is 1/2.

Alternate Method

A Base = √3 Perp = 1 Hyp = 2

Given:

tan(90° − A) = √3

Formula Used:

tan(90° − θ) = cot θ

cot θ = Base ÷ Perpendicular

sin θ = Perpendicular ÷ Hypotenuse

Calculations:

⇒ cot A = √3

⇒ Base ÷ Perpendicular = √3 ÷ 1

⇒ Let Base = √3 and Perpendicular = 1

⇒ By Pythagoras Theorem, Hypotenuse = √(Base² + Perpendicular²)

⇒ Hypotenuse = √((√3)² + 1²) = √(3 + 1) = √4 = 2

⇒ sin A = Perpendicular ÷ Hypotenuse = 1/2

∴ The correct answer is 1/2.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Trigonometric ratios for complementary angles satisfy relations like sin(90°−θ) = cos θ and tan(90°−θ) = cot θ.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of its base and perpendicular: H² = P² + B².

Standard Trigonometric Values

Common values frequently used in exams include sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and tan 60° = √3.

Shortcut Trick

Substitute x = √7 directly into the expression x + 1/x.

Calculate 1/x = 1/√7 and rationalize it by multiplying the numerator and denominator by √7 to get √7/7.

Add the terms: √7 + √7/7 = (7√7 + √7) / 7.

Simplify the numerator to get 8√7/7.

∴ The correct answer is 8√7/7.

Alternate Method

Given:

x = √7

Formula Used:

For rationalization: 1/√a = √a / a

Calculations:

⇒ Substitute the given value of x in the expression x + 1/x

⇒ x + 1/x = √7 + 1/√7

⇒ Taking the common denominator as √7 for addition:

⇒ x + 1/x = (√7 × √7 + 1) / √7

⇒ x + 1/x = (7 + 1) / √7

⇒ x + 1/x = 8 / √7

⇒ Multiply the numerator and denominator by √7 to rationalize the denominator:

⇒ x + 1/x = (8 × √7) / (√7 × √7)

⇒ x + 1/x = 8√7 / 7

∴ The correct answer is 8√7/7.

Additional Information

Rationalizing the Denominator

When a fraction has an irrational number like √a in the denominator, multiplying both the numerator and denominator by √a removes the radical from the denominator. Formula: 1/√a = √a / a.

Operations with Surds

When multiplying square roots of the same base number, the square root symbol is eliminated: √a × √a = a. This basic property simplifies addition and multiplication of irrational fractions.

Standard Algebraic Identities

If x + 1/x = k, then x² + 1/x² = k² − 2. This rule is highly useful for finding higher powers of algebraic expressions quickly in quantitative aptitude exams.

Shortcut Trick

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Ratio of areas = (3 ÷ 5)²

Ratio of areas = 9 / 25

∴ The correct answer is 9:25.

Alternate Method

Side = 3 units Area 1

Side = 5 units Area 2

Given:

Ratio of corresponding sides of two similar triangles = 3:5

Formula Used:

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Area₁ ÷ Area₂ = (Side₁ ÷ Side₂)²

Calculations:

⇒ Area₁ ÷ Area₂ = (3 ÷ 5)²

⇒ Area₁ ÷ Area₂ = (3² ÷ 5²)

⇒ Area₁ ÷ Area₂ = 9 ÷ 25

⇒ The ratio of their areas = 9:25

∴ The correct answer is 9:25.

Additional Information

Similar Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. Similarity preserves the shape but not necessarily the size of the figure.

Ratio of Perimeters

The ratio of the perimeters of two similar triangles is identical to the ratio of their corresponding sides.

Ratio of Volumes

For similar three-dimensional objects, the ratio of their volumes is equal to the cube of the ratio of their corresponding sides (V₁ ÷ V₂ = s₁³ ÷ s₂³).

Shortcut Trick

Rationalize the denominator inside the root to find x.

x = √[(2 + √3)² ÷ (4 − 3)] = 2 + √3

Substitute x into the expression: x² + x − 9

⇒ (2 + √3)² + (2 + √3) − 9

⇒ 7 + 4√3 + 2 + √3 − 9 = 5√3

∴ The correct answer is 5√3.

Alternate Method

Given:

x = √[(2 + √3) ÷ (2 − √3)]

Calculations:

Rationalizing the denominator inside the square root:

⇒ x = √[((2 + √3) × (2 + √3)) ÷ ((2 − √3) × (2 + √3))]

⇒ x = √[(2 + √3)² ÷ (2² − (√3)²)]

⇒ x = √[(2 + √3)² ÷ (4 − 3)]

⇒ x = √[(2 + √3)² ÷ 1]

⇒ x = 2 + √3

Now, substitute the value of x in the given expression:

⇒ x² + x − 9

⇒ (2 + √3)² + (2 + √3) − 9

Using the algebraic formula (a + b)² = a² + 2ab + b²:

⇒ (2² + 2 × 2 × √3 + (√3)²) + 2 + √3 − 9

⇒ (4 + 4√3 + 3) + 2 + √3 − 9

⇒ 7 + 4√3 + 2 + √3 − 9

⇒ 9 + 5√3 − 9

⇒ 5√3

∴ The correct answer is 5√3.

Additional Information

Rationalization of Surds

To eliminate a square root from the denominator, multiply both the numerator and the denominator by its conjugate. The conjugate of a − √b is a + √b.

Algebraic Identity for Squares

The square of a binomial sum is calculated using the identity (a + b)² = a² + 2ab + b².

Difference of Squares

The product of the sum and difference of the same two terms yields the difference of their squares: (a − b)(a + b) = a² − b².

Shortcut Trick

The sum of opposite angles in a cyclic quadrilateral is exactly 180°.

The given ratio of angles is 1 : 2 : 3 : 4. Let the angles be x, 2x, 3x, and 4x.

Since the sum of opposite angle pairs must be equal, we must pair them as (x, 4x) and (2x, 3x) because x + 4x = 5x and 2x + 3x = 5x.

Equating the sum to 180°: 5x = 180° ⇒ x = 36°.

The smallest angle in the ratio is x, which is 36°.

∴ The correct answer is 36°.

Alternate Method

Given:

The angles of a cyclic quadrilateral are in the ratio of 1 : 2 : 3 : 4.

Formula Used:

The sum of the opposite angles of a cyclic quadrilateral is always 180°.

O x 2x 3x 4x

Calculations:

⇒ Let the four angles of the cyclic quadrilateral be x, 2x, 3x, and 4x.

⇒ In a valid cyclic quadrilateral, the pairs of opposite angles must sum to the same value (180°).

⇒ Grouping the angles to find equal sums: (x + 4x) = 5x and (2x + 3x) = 5x.

⇒ Therefore, the opposite angle pairs are (x, 4x) and (2x, 3x).

⇒ Summing either opposite pair gives: 5x = 180°

⇒ x = 180° ÷ 5 = 36°

⇒ The measures of the four angles are 36°, 72°, 108°, and 144°.

⇒ Smallest angle = x = 36°

∴ The correct answer is 36°.

Additional Information

Cyclic Quadrilateral

A quadrilateral is called cyclic if all four of its vertices lie on the circumference of a circle. The sum of either pair of opposite angles is exactly 180°.

Exterior Angle Property

If any side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the interior opposite angle.

Ptolemy's Theorem

For any cyclic quadrilateral with side lengths a, b, c, d and diagonals p, q, the product of the diagonals equals the sum of the products of opposite sides: p × q = (a × c) + (b × d).

Brahmagupta's Formula

The area of a cyclic quadrilateral is given by Area = √(s − a)(s − b)(s − c)(s − d), where s is the semi-perimeter calculated as s = (a + b + c + d) ÷ 2.