SSC CGL 2025 (Held On: 15 Sept, 2025 Shift 1) (Previous Year Paper)

The logic followed here is:

Given:

In the sentence "The quantum universe expands silently into infinity," the 2nd words are quantum, expands, into.

Then, The 3rd letters of these words are a, p, t.

Thus, Combining them gives the word apt.

Hence, the correct answer is Option 4) apt.

The logic followed here is:

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

Hence, the correct answer is "Option 1".

The logic followed here is:

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

Hence, the correct answer is "Option 1".

BODMAS Table: 

Given expression: 5 (7 S 3) D 3 (8 P 6) V 7 D 11 (3 S 13 S 2) P 2 (41 S 9) = ?

Now, if ‘P’ means ‘+’, ‘D’ means ‘-’, ‘S’ means ‘×’, and ‘V’ means ‘÷’, then:

⇒ 5 (7 × 3) - 3 (8 + 6) ÷ 7 - 11 (3 × 13 × 2) + 2 (41 × 9)

⇒ 5 (21) - 3 (14) ÷ 7 - 11 (78) + 2 (369)

⇒ 105 - 42 ÷ 7 - 858 + 738

⇒ 105 - 6 - 858 + 738

⇒ 99 - 858 + 738

⇒ -759 + 738

⇒ -21

Hence, the correct answer is "Option 3".

The logic followed here is:

thus, "445" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given Statement:

The advancement of biotechnology has the potential to revolutionize medicine by allowing for the development of personalized treatments, but it also raises ethical questions regarding genetic manipulation and the potential for inequality in access to these treatments.

Now,

Assumptions:

1.Biotechnology has the power to create personalized medical treatments - Implicit (The statement clearly mentions development of personalized treatments through biotechnology.)

2.The ethical implications of biotechnology need to be addressed to prevent misuse - Implicit (Since the statement says it raises ethical questions, it assumes ethical concerns are significant and must be considered.)

3.Biotechnology will be accessible to everyone, ensuring equality in treatment access - Not Implicit (The statement mentions potential inequality in access, not guaranteed equality.)

Here, Only assumptions I and II are implicit.

Hence, the correct answer is "Option 1".

Given Statement: The economy grew by 8% this year, the highest in a decade.

Now,

Conclusions:

(i) Economic reforms have been successful → Does not follow (The reason for the growth is not mentioned.)

(ii) Growth in previous years was below 8% → Follows (Since 8% is the highest in a decade, previous years’ growth must have been less than 8%.)

Here, Only conclusion II follows.

Hence, the correct answer is "Option 2".

Given word: CAMERA

The first four letters = C, A, M, E

We need to form meaningful four-letter English words using these letters, each letter only once.

Possible combinations:

CAME (meaningful)

ACME (meaningful)

MACE (meaningful)

Other combinations like MEAC, ECAM, etc., either don’t form valid words or are repetitions.

Total meaningful words = 3

Hence, the correct answer is Option 4.

The address given in the problem is:

Vill: Sangrur, PO: Sangrur, PS: Patiala, Dist: Punjab – 148001

Now, match this address with the given options:

So, only Option 1 is exactly identical to the given address.

Hence, the correct answer is Option 1 Only.

The logic followed here is:

Logic: Each letter is represented by its alphabetical position (A = 1, B = 2, … Z = 26).

For, BxD : 2x4

B → 2

D → 4

So, the pair is written as 2x4.

Similarly, for GxJ : ?

G → 7

J → 10

So, the pair is written as 7x10.

Hence, the correct answer is "Option 2".

The logic followed here is:

Given:

In a geometric progression (GP), the nth term = first term × (common ratio)^(n-1)

First term = 3.6

Common ratio = 1.5

Here, 3rd term = 3.6 × (1.5)² = 3.6 × 2.25 = 8.1

Hence, the correct answer is Option 2.

The logic followed here is:

Option 1) Lok Sabha → part of the Parliament

Option 2) Rajya Sabha → part of the Parliament

Option 3) Vidhan Sabha → state legislative assembly

Option 4) Supreme Court → judiciary

Here, All except the Supreme Court are legislative bodies.

Hence, the correct answer is Option 4.

The logic followed here is:

thus, "36" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

Family Chart:

Given: 

Mother’s son-in-law =

The man who is married to mother’s daughter.

Mother’s daughter = Rani (since Rani is speaking).

 

So, Man is the Husband of Rani.

Hence, the correct answer is "Option 1".

The logic followed here is:

Solve inside the brackets:

(2 + 3) = 5

(7 - 4) = 3

Now, Square the second bracket:

3² = 9

Finally, multiply:

5 × 9 = 45

Hence, the correct answer is Option 1.

The logic followed here is:

Logic: The operation “#” is defined as a # b = a × b + b

4 # 2 = 4 × 2 + 2 = 10

6 # 3 = 6 × 3 + 3 = 21

Similarly, 8 # 4 = 8 × 4 + 4 = 36

Hence, the correct answer is Option 1.

The logic followed here is:

Given:

Cost of 4 apples = 4 × 18 = 72

Cost of 2 melons = 2 × 35 = 70

Here, Total cost = 72 + 70 = 142

Hence, the correct answer is Option 1.

The logic followed here is:

Logic: The pattern is +1,-1,+1,+1,+2.

For, MANGO → NZOHQ

Applying same to PEACH:

So, PEACH → QDBDJ

Hence, the correct answer is Option 2.

The logic followed here is:

For, FLOWER is written as GNQXFS

Similarly, for GARDEN

So, GARDEN will be coded as HCTEFO.

Hence, the correct answer is "Option 3".

BODMAS Table: 

Given equation: 7 × 3 ÷ 4 + 20 - 5 = ?

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After changing the signs,

⇒ 7 - 3 + 4 ÷ 20 × 5 = ?

⇒ 7 - 3 + 1 = ?

⇒ 8 - 3 = ?

⇒ 5 = ?

Hence, the correct answer is "Option 3".

Family Chart:

Given: 

A is the brother of B.

C is the father of A.

D is the brother of C.

E is the son of D.  

So, E is the Cousin of B.

Hence, the correct answer is "Option 1".

BODMAS Table: 

Given equation: 10 × 2 − 4 + 8 ÷ 2 = ?

Now, if ‘×’ means ‘−’, ‘−’ means ‘+’, ‘+’ means ‘÷’, and ‘÷’ means ‘×’, then:

⇒ 10 − 2 + 4 ÷ 8 × 2 = ?

⇒ 10 − 2 + 0.5 × 2 = ?

⇒ 10 − 2 + 1 = ?

⇒ 11 − 2 = 9

Hence, the correct answer is "Option 4".

BODMAS Table: 

Given equation: 18 + 6 − 2 × 3 = ?

Now, if ‘+’ means ‘÷’, ‘−’ means ‘×’, ‘×’ means ‘+’, and ‘÷’ means ‘−’, then:

⇒ 18 ÷ 6 × 2 + 3 = ?

⇒ 3 × 2 + 3 = ?

⇒ 6 + 3 = ?

⇒ 9

Hence, the correct answer is "Option 1".

Given Statements:

No circles are squares.

Some squares are triangles.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some triangles are not circles - True (Since some squares are triangles and no square is a circle, those triangles which are squares cannot be circles.)

II. No triangle is a circle - False (Only triangles that are squares are not circles. Other triangles may or may not be circles.)

So, Only conclusion I follows.

Hence, the correct answer is "Option 1".

BODMAS Table: 

Given equation: (5 # 2) @ 3 % 4 = ?

Now, if ‘#’ = ‘+’, ‘@’ = ‘×’, ‘%’ = ‘−’, then:

⇒ (5 + 2) × 3 − 4 = ?

⇒ 7 × 3 − 4 = ?

⇒ 21 − 4 = ?

⇒ 17

Hence, the correct answer is "Option 2".

The correct answer is 12 pairs.

Key Points

• The Konark Sun Temple, located in Odisha, India, is a UNESCO World Heritage Site and is renowned for its architectural brilliance.
• The temple is designed as a colossal chariot of the Sun God, Surya, with twelve pairs of intricately carved stone wheels.
• Each pair of decorative wheels represents the months of the year and symbolizes the passage of time.
• The wheels are approximately 3 meters in diameter and are adorned with detailed carvings depicting various motifs, including animals and floral designs.
• The temple was constructed in the 13th century by King Narasimhadeva I of the Eastern Ganga dynasty.

Additional Information

• Nagara Style (North India)
• Features: Curvilinear towers (shikharas), square ground plans, often with a water tank and no boundary walls, with amalakas (grooved stone discs) at the top.
• Examples: Kandariya Mahadeva Temple (Khajuraho), Lingaraja Temple (Bhubaneswar), Sun Temple (Konark).
• Dravida Style (South India)
• Features: Pyramidical towers (vimanas), large gateways (gopurams), enclosed compounds with walls, water tanks, and pillared halls (mandapas).
• Examples: Brihadeeswarar Temple (Thanjavur), Meenakshi Temple (Madurai), Ranganathaswamy Temple (Srirangam).
• Vesara Style (Deccan/Central India)
• Features: A hybrid style blending Nagara and Dravida elements, often with complex layouts, a pyramidal but shorter tower, and sometimes gopurams.
• Examples: Kailasanatha Temple (Ellora), Virupaksha Temple (Pattadkal).

The correct answer is Lord Shreeji and Raghunathji's processions on Jaljhulani Gyaras attract traders from several states.

Key Points

• Baran's Dol Mela is one of the most prominent cultural fairs in Rajasthan, celebrated with significant enthusiasm and devotion.
• The highlight of the festival is the grand processions of Lord Shreeji and Raghunathji on the occasion of Jaljhulani Gyaras, a religious event that attracts devotees and traders from multiple states.
• Jaljhulani Gyaras is a sacred day in the Hindu calendar, observed during the month of Bhadrapada (August-September), celebrated for the ceremonial immersion of deities in water.
• Traders from several states converge at the fair, making it an important hub for cultural exchange and commerce.
• The Dol Mela showcases traditional Rajasthani culture through folk dances, music, and various rituals, symbolizing the region's rich heritage and devotion.

Additional Information

• Major Festivals & Fairs
• Pushkar Camel Fair: An iconic annual event known for livestock trading, holy dips in Pushkar Lake, and cultural performances.
• Bikaner Camel Festival: Celebrates camels with processions, local crafts, and delicacies in winter.
• Jaisalmer Desert Festival: A colorful event in the desert showcasing traditional Rajasthani arts, music, and crafts.
• Marwar Festival: Held in Jodhpur, it honors the legendary heroes of Rajasthan with folk music and dance.
• Nagaur Fair: A significant cattle fair, second only to Pushkar, known for trade and cultural events.

The correct answer is Head of the trading guilds representing various trading communities.

Key Points

• The term Sarthavaha in the Gupta dynasty referred to the head of the trading guilds who represented various trading communities and managed commercial activities.
• Sarthavahas were responsible for overseeing the functioning of trading guilds, ensuring smooth trade operations, and resolving disputes within the trading community.
• They played a key role in maintaining economic stability and fostering long-distance trade, which flourished during the Gupta period.
• Trading guilds under the leadership of Sarthavahas were significant in connecting the Gupta Empire with other regions, including Southeast Asia, the Roman Empire, and Central Asia, thereby enhancing economic prosperity.
• The Gupta period is often referred to as the "Golden Age" of India, marked by flourishing trade, urbanization, and cultural advancements, with Sarthavahas contributing significantly to economic growth.

Additional Information

• Gupta dynasty's administration
• King (Maharajadhiraja): Supreme head, responsible for war, peace, justice, and appointing officials, often seen as divine.
• Council of Ministers (Mantri Parishad): Assisted the king, included high officials like the Kumaramatya (minister) and Sandhivigrahika (peace/war minister).
• Judiciary: King was the highest court; Mahanandanayaka (Chief Justice) oversaw justice, with local judges handling regional cases.
• • Vishaya (District): Headed by a Vishayapati, assisted by various officials like Saulkika (toll collector), Gaulmika (forest/fort officer), and Talavataka (accountant).
• Village: Administered by a headman (Gramapati) and elders, settling local disputes.

The correct answer is Drums like Timila and Maddalam are mentioned in Silappadikaram and still used today.

Key Points

• Timila and Maddalam are traditional percussion instruments integral to Kerala's musical heritage and are prominently used in temple rituals and classical performances like Kathakali and Koodiyattam.
• Both instruments are mentioned in the ancient Tamil epic Silappadikaram, which provides insights into South India's rich cultural history and traditions.
• The Maddalam is a barrel-shaped drum played horizontally, while the Timila is a high-pitched percussion instrument used in temple music, especially in Panchavadyam ensembles.
• Kerala is renowned for its percussion-based music traditions, where drums like Chenda, Idakka, and Maddalam play a central role in performances such as Theyyam and temple rituals.
• These instruments symbolize the continuity of Kerala's cultural and musical heritage, showcasing how ancient traditions are preserved in the modern era.

Additional Information

• Maharaja Swathi Thirunal Rama Varma (1813-1847): Kerala's most celebrated historical composer, known for numerous works in Sanskrit, Marathi, Telugu, and Malayalam.
• Irayimman Thampi (1782-1856): Recognized as the first composer of Malayalam Kirthanas, enriching the state's classical music traditions.
• K. J. Yesudas: A legendary Carnatic vocalist and playback singer, widely known across India.

The correct answer is Fibreglass is light, flexible, and safer on impact.

Key Points

• Fibreglass is lightweight: Fibreglass has a low density, making it significantly lighter than metal bars, which is essential for ease of handling and transportation.
• Flexibility: Fibreglass has excellent flexibility, allowing it to absorb impact without permanent deformation. This property makes it ideal for high jump bars as it can bend and return to its original shape.
• Safety on impact: Fibreglass is less rigid than metal, reducing the risk of injuries to athletes if they collide with the bar during a high jump attempt.
• Durable and reliable: Fibreglass resists weathering, corrosion, and wear over time, ensuring consistent performance in various environmental conditions.
• Preferred in sports equipment: The unique combination of lightweight, flexibility, and safety features makes fibreglass a standard material for high jump bars in professional and amateur athletics.

Additional Information

• Metals and non-metals are fundamental categories of elements with contrasting properties:
• metals (like iron, copper) are typically shiny, malleable, ductile, and excellent conductors of heat/electricity, while non-metals (like oxygen, sulfur) are often dull, brittle, poor conductors (except graphite), and can be solids, liquids, or gases.
• Key differences also lie in their electron behavior, density, and oxide formation, with metals tending to lose electrons and form basic oxides, whereas non-metals gain or share electrons and form acidic/neutral oxides.

The correct answer is 3 - 4 - 1 - 2.

Key Points

• Qutb al-Din Aibak (1206–1210 CE) was the first ruler of the Delhi Sultanate and established the Slave Dynasty. He became the Sultan after the death of Muhammad Ghori.
• Iltutmish (1211–1236 CE), the successor of Qutb al-Din Aibak, is credited with strengthening the Delhi Sultanate and introducing a centralized administrative system.
• Razia Sultan (1236–1240 CE) was the daughter of Iltutmish and the first and only female ruler of the Delhi Sultanate. Her reign was marked by opposition from nobles.
• Ghiyas ud din Balban (1266–1287 CE) was a powerful ruler of the Slave Dynasty known for his strong administration and the theory of kingship.
• The correct chronological order of their reigns is Qutb al-Din Aibak (3) → Iltutmish (4) → Razia Sultan (1) → Ghiyas ud din Balban (2).

Additional Information

• Qutb al-Din Aibak:
• He was originally a slave of Muhammad Ghori and played a crucial role in the conquest of northern India.
• Founded the Qutb Minar in Delhi and laid the foundation of the Slave Dynasty.
• His sudden death in a polo accident ended his brief reign.
• Iltutmish:
• He was the first Sultan to obtain the investiture of sovereignty from the Abbasid Caliph, enhancing the legitimacy of his rule.
• Introduced the silver tanka and copper jital, standardizing the coinage system in the Sultanate.
• Expanded the Delhi Sultanate by annexing several territories, including Bengal and Sindh.
• Razia Sultan:
• She was chosen as the successor by Iltutmish due to her administrative skills and capabilities.
• Faced opposition from the Turkish nobles (Chihalgani), who were reluctant to accept a female ruler.
• Her reign was marked by political instability, leading to her eventual defeat and death.
• Ghiyas ud din Balban:
• He introduced the theory of kingship, emphasizing the divine right of the Sultan to rule.
• Implemented strict laws to maintain law and order and crushed rebellions with an iron hand.
• Strengthened the military and expanded the administrative framework of the Sultanate.

The correct answer is Scoring two goals in a match.

Key Points

• The term "brace" in football refers to when a player scores two goals in a single match.
• The phrase originated from hunting terminology, where "brace" was used to signify a pair of something, such as two birds or animals caught.
• It is considered a significant achievement for a football player, highlighting their contribution to the team's performance in a match.
• Scoring a brace often indicates a player's ability to perform under pressure and capitalize on opportunities.
• Players who score two goals in a match are often celebrated for their skill and consistency in finishing.

Additional Information

• Two consecutive wins:
• This refers to when a team wins two matches in a row, showcasing consistency and strong performance over multiple games.
• However, this is not related to individual player performance and is different from the concept of scoring a brace.
• Two yellow cards:
• In football, two yellow cards in a single match lead to a red card, resulting in the player being sent off the field.
• A yellow card is a caution given to a player for offenses such as unsporting behavior or repeated fouls.
• Saving two penalties:
• This refers to a goalkeeper successfully stopping two penalty kicks in a match, which is a rare and commendable feat.
• While it demonstrates exceptional goalkeeping skills, it is unrelated to the term "brace," which pertains to scoring goals.

The correct answer is Startup knowledge repository.

Key Points

• The BRICS Startup Knowledge Hub aims to serve as a centralized repository for information, resources, and best practices related to startups across BRICS nations.
• This initiative was launched in 2025 to promote innovation, collaboration, and entrepreneurship among the member countries: Brazil, Russia, India, China, and South Africa.
• It focuses on sharing knowledge and fostering partnerships among startups, investors, and governments of the BRICS nations.
• The hub supports the exchange of startup-related policies, research, funding opportunities, and technological expertise to accelerate growth in the startup ecosystem.
• BRICS Startup Knowledge Hub contributes to economic development and the creation of a robust innovation-driven ecosystem in member countries.

Additional Information

• BRICS
• Origin & Acronym: Coined by Goldman Sachs economist Jim O'Neill in 2001 (BRIC), representing Brazil, Russia, India, & China; South Africa joined in 2010, adding the 'S'.
• Expansion: Recently expanded to include Iran, Egypt, Ethiopia, UAE, and Indonesia, broadening its global reach.
• Purpose: A platform for dialogue and cooperation in trade, finance, technology, and security, aiming to increase Global South influence in international affairs.
• Economic Significance: Members collectively hold significant shares of the world's population, GDP, and trade, acting as a major engine for global growth.
• Key Initiatives: Includes the New Development Bank (NDB) for infrastructure financing and promoting local currency settlements.

The correct answer is Mrinal Pande.

Key Points

• Mrinal Pande is a renowned Indian author, journalist, and television personality, known for her significant contributions to Hindi literature.
• She wrote the novel "Sahela Re," which explores the intricacies of human relationships, particularly focusing on women’s lives and struggles.
• Mrinal Pande has a deep understanding of Indian societal norms, and her works often reflect feminist perspectives and cultural nuances.
• Besides being an author, she has served as the editor of the Hindi daily "Hindustan" and has worked as the chairperson of Prasar Bharati.
• Her literary works have earned widespread recognition for their vivid portrayal of Indian social issues and have contributed significantly to modern Hindi literature.

Additional Information

• Vasanti Muzumdar
• Vasanti Muzumdar is a less widely known name in Indian literature, and there is limited information on her contributions to Hindi or regional literature.
• She may have contributed to cultural or literary works in regional languages, but she is not associated with the novel "Sahela Re."
• Priyanka Sarkar
• Priyanka Sarkar is primarily recognized as an Indian actress known for her work in Bengali cinema.
• She has not written the novel "Sahela Re" or contributed significantly to Hindi literature.
• Shubhangi Tiwari
• Shubhangi Tiwari is not a recognized name in the field of literature or Hindi novels.
• Her name does not appear in association with the novel "Sahela Re" or any notable literary contributions.

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

Key Points

• In the girls' doubles final of the Khelo India Youth Games (Table Tennis Doubles), Kavya Bhatt and Divyanshi Bhowmick defeated Subhankrita Datta and Avisha Karmakar with a clean score of 3-0.
• The Reason (R) that they won the final match with a 3-0 score is correct.
• However, the Assertion (A), which states that Kavya Bhatt and Divyanshi Bhowmick dropped only one game in the entire tournament, is false.
• The pair did not lose any game during the final, as they achieved a flawless victory in the finals.
• This discrepancy makes the assertion (A) incorrect, but the reason (R) remains valid as it accurately describes the final match result.

Additional Information

• Khelo India Youth Games
• The Khelo India Youth Games is an initiative by the Indian Government to promote sports culture in the country at the grassroots level.
• It provides a platform for young athletes to showcase their talent and serves as a stepping stone for representing India at international levels.
• The games include multiple sports, including table tennis, where young talents compete in various categories like singles, doubles, and mixed doubles.

The correct answer is Gokul Chandra Das.

Key Points

• Gokul Chandra Das was honoured with the Padma Shri in 2025 for his contributions to empowering women through the teaching of traditional Bengali folk percussion.
• He has dedicated years to preserving and promoting Bengali folk music, particularly percussion instruments such as the dhak, which hold cultural significance in Bengal.
• Das has trained numerous women in rural areas to master the art of Bengali folk percussion, enabling them to gain financial independence and recognition.
• His work has helped revive traditional art forms while simultaneously providing a platform for women to showcase their talents at national and international events.
• The Padma Shri award recognizes his efforts to preserve India’s intangible heritage and promote gender equality through the arts.

Additional Information

• Farooq Ahmad Mir
• Farooq Ahmad Mir is a notable poet and writer from Jammu and Kashmir known for his contributions to literature.
• His works often focus on themes of communal harmony, nature, and the socio-political landscape of Jammu and Kashmir.
• Shyam Bihari Agrawal
• Shyam Bihari Agrawal is a philanthropist and social worker known for his contributions to community service and welfare programs.
• He has worked extensively to improve education, healthcare, and rural development in India.
• Sheen Kaaf Nizam
• Sheen Kaaf Nizam is a renowned Urdu poet celebrated for his literary works and contributions to Urdu poetry.
• His poetry is known for its depth, emotional resonance, and exploration of themes such as love and human values.
• Despite his literary achievements, he is unrelated to traditional Bengali folk percussion or women empowerment through music.

The correct answer is 2.4%.

Key Points

• India’s land area constitutes approximately 2.4% of the world’s total geographical area.
• India has a total land area of about 3.287 million square kilometers, making it the 7th largest country in the world in terms of area.
• The world’s total land area is estimated to be around 149 million square kilometers, and India’s share in this is 0.024 (or 2.4%).
• Despite its relatively small share in the global landmass, India is home to 17.7% of the world's population, as of the latest census data.
• India’s geographical diversity spans mountains, plains, plateaus, deserts, and coastal areas, contributing to its significance despite the limited land area.

Additional Information

• Geographical Importance of India
• India is strategically located in South Asia, bordered by the Himalayas to the north and surrounded by oceans on three sides.
• Its geographical position makes it a significant player in global trade and geopolitics.
• India’s diverse topography includes the fertile Indo-Gangetic Plain, the Thar Desert, and the Deccan Plateau.

The correct answer is SSO.

Key Points

• The NISAR (NASA-ISRO Synthetic Aperture Radar) satellite will operate in a Sun-Synchronous Orbit (SSO), which ensures consistent solar illumination for its observations.
• The satellite is set to be launched into space aboard an ISRO GSLV (Geosynchronous Satellite Launch Vehicle).
• NISAR is a joint mission between NASA and ISRO, designed to map the Earth's surface with unprecedented detail using advanced radar imaging.
• SSO is particularly suited for Earth observation satellites as it allows for consistent observation under similar lighting conditions, crucial for accurate and comparable data collection.
• The primary focus of NISAR is to monitor natural disasters, melting ice sheets, and changes in the Earth's surface to better understand environmental changes and their impacts.

Additional Information

• GEO (Geostationary Earth Orbit)
• Satellites in GEO orbit at an altitude of approximately 35,786 km above the equator, maintaining a fixed position relative to the Earth's surface.
• GEO is ideal for telecommunications, weather monitoring, and broadcasting as satellites appear stationary from the Earth.
• HEO (Highly Elliptical Orbit)
• HEO satellites follow an elliptical orbit with a high apogee (farthest point) and a low perigee (closest point).
• This orbit is commonly used for communication satellites in high-latitude regions and scientific missions requiring prolonged observation of specific areas.
• LEO (Low Earth Orbit)
• LEO satellites orbit at altitudes ranging from 160 km to 2,000 km above the Earth's surface.
• This orbit is commonly used for Earth observation, communication, and scientific research as it allows for high-resolution imaging.
• However, LEO satellites require frequent repositioning to maintain orbit and have limited coverage compared to GEO or SSO satellites.

The correct answer is Singhbhum - Jharkhand.

Key Points

• Singhbhum, located in Jharkhand, is one of the richest districts in India in terms of iron ore reserves.
• The Singhbhum region is well-known for its iron ore belt, which stretches across Jharkhand and Odisha.
• Jharkhand accounts for a significant portion of India's iron ore production, with Singhbhum being a major contributor.
• Major iron ore mines in the Singhbhum district include Chiria, Gua, and Noamundi.
• The high-grade iron ore from Singhbhum is crucial for India's steel industry and is also exported to various countries.

Additional Information

• Mayurbhanj - Uttar Pradesh:
• This is an incorrect pair as Mayurbhanj is located in Odisha, not Uttar Pradesh.
• Mayurbhanj is known for its mineral resources, including iron ore, but it is not in Uttar Pradesh.
• Uttar Pradesh, on the other hand, is not a significant iron ore-producing state in India.
• Sundargarh - Meghalaya:
• This is an incorrect pair as Sundargarh is located in Odisha, not Meghalaya.
• Sundargarh is a major district for iron ore mining in Odisha and contributes significantly to India's total iron ore output.
• Meghalaya, however, is known for limestone, coal, and uranium deposits, not iron ore.
• Keonjhar - Rajasthan:
• This is an incorrect pair as Keonjhar is located in Odisha, not Rajasthan.
• Keonjhar is one of the leading districts for iron ore production in Odisha and India.
• Rajasthan, on the other hand, is known for its reserves of non-ferrous minerals like zinc, lead, and gypsum.
• Iron Ore Reserves in India:
• India ranks among the top producers of iron ore in the world, with significant reserves located in states such as Odisha, Jharkhand, Chhattisgarh, Karnataka, and Goa.
• Odisha is the largest producer of iron ore, followed by Chhattisgarh and Karnataka.
• Iron ore is a crucial raw material for the steel industry, which is a key sector for India's economic development.

The correct answer is Phospholipids.

Key Points

• Phospholipids are a major component of the plasma membrane, forming a bilayer that provides structure and acts as a selective barrier for the cell.
• Each phospholipid molecule has a hydrophilic (water-attracting) "head" and two hydrophobic (water-repelling) "tails," which arrange themselves into a bilayer to create the plasma membrane.
• The phospholipid bilayer is responsible for the fluidity of the membrane, allowing for the movement of proteins and lipids within it.
• It plays a critical role in maintaining cell integrity, enabling the transport of substances in and out of the cell, and facilitating cell signaling.
• Phospholipids also contribute to membrane flexibility, which is essential for processes like endocytosis, exocytosis, and cell division.

Additional Information

• Cellulose
• Cellulose is a complex carbohydrate (polysaccharide) composed of glucose units linked by beta-1,4-glycosidic bonds.
• It is the primary structural component of plant cell walls, providing rigidity and support to plants.
• Cellulose is not present in the plasma membrane of animal cells; it is exclusive to plants and some algae.
• DNA
• DNA (Deoxyribonucleic Acid) is the hereditary material in humans and most other organisms, carrying genetic instructions for development, functioning, growth, and reproduction.
• DNA is located within the nucleus of eukaryotic cells and is not part of the plasma membrane structure.
• It plays no direct role in forming or maintaining the plasma membrane but is crucial for coding the proteins that may integrate into the membrane.
• Glycogen
• Glycogen is a polysaccharide that serves as a form of energy storage in animals, fungi, and bacteria.
• It is stored primarily in the liver and muscle tissues and is broken down into glucose when energy is needed.
• Glycogen is not a structural component of the plasma membrane and serves a different biological function related to energy storage and metabolism.

The correct answer is 42nd Amendment Act, 1976.

Key Points

• The 42nd Amendment Act, 1976, is often referred to as the "Mini-Constitution" due to the extensive changes it introduced to the Indian Constitution.
• It was passed during the Emergency period under the leadership of Prime Minister Indira Gandhi.
• This amendment brought significant changes to the Preamble, Fundamental Rights, Directive Principles of State Policy, and the powers of the Parliament and judiciary.
• It introduced terms like "Socialist" and "Secular" in the Preamble of the Constitution.
• It curtailed the power of the judiciary and strengthened the authority of the Parliament, especially in the context of amending the Constitution.
• The 42nd Amendment also extended the term of the Lok Sabha and State Legislative Assemblies from 5 to 6 years, though this was later reversed by the 44th Amendment.

Additional Information

• 44th Amendment Act, 1978:
• This amendment was enacted to undo the excesses of the 42nd Amendment, restoring the balance between the executive, legislature, and judiciary.
• It restored the term of the Lok Sabha and State Legislative Assemblies to 5 years.
• It also ensured that Fundamental Rights like the Right to Property were no longer included as part of Fundamental Rights but rather as a legal right under Article 300A.
• Additionally, it introduced safeguards against the misuse of emergency provisions by requiring stricter conditions for the proclamation of national emergencies.
• 52nd Amendment Act, 1985:
• This amendment introduced the anti-defection law in India by adding the Tenth Schedule to the Constitution.
• The purpose was to curb defections by legislators to ensure the stability of the government and uphold the integrity of the democratic process.
• It laid down conditions under which a legislator could be disqualified on grounds of defection.
• 86th Amendment Act, 2002:
• This amendment made education a Fundamental Right under Article 21A of the Constitution.
• It mandated free and compulsory education for children aged 6 to 14 years.
• It also altered the Directive Principles of State Policy by adding a new clause under Article 45, directing the state to provide early childhood care and education for children below six years of age.

The correct answer is Steam contains more latent heat of vaporization.

Key Points

• Steam at 100°C contains additional energy in the form of latent heat of vaporization, which is the energy required to convert water into steam without changing its temperature.
• This latent heat of vaporization is released when steam condenses on the skin, transferring a large amount of heat energy, causing more severe burns compared to boiling water.
• Boiling water at 100°C does not have this extra latent heat, so the amount of energy transferred to the skin is comparatively lower.
• The process of condensation of steam releases significant heat energy into the skin, which intensifies the damage caused by burns.
• Thus, the severity of burns caused by steam is primarily due to the release of this latent heat of vaporization during condensation.

Additional Information

• Latent Heat of Vaporization
• Latent heat of vaporization is the heat required to convert a unit mass of a liquid into vapor without a change in temperature.
• For water, the latent heat of vaporization is approximately 2260 kJ/kg at 100°C.
• This significant amount of energy is what makes steam burns more severe compared to burns from boiling water.

The correct answer is option 3.

Key Points

• eSanjeevani OPD is a teleconsultation platform under the Ministry of Health and Family Welfare that enables citizens to connect with doctors remotely via online mode for medical consultations.
• eSanjeevani AB-HWC is another branch of the eSanjeevani initiative that connects citizens with Ayushman Bharat Health and Wellness Centers (AB-HWCs) for teleconsultation services.
• The initiative aims to bridge the rural-urban divide by making quality healthcare accessible, especially in remote and underserved areas.
• Both eSanjeevani OPD and eSanjeevani AB-HWC were launched to strengthen India’s digital health ecosystem and improve healthcare delivery across the country.
• It has emerged as one of the largest telemedicine platforms globally, with millions of consultations completed, contributing to the Government of India’s goal of achieving universal healthcare access.

Additional Information

• eSanjeevani OPD:
• It allows patients to consult with doctors from the comfort of their homes using audio-video communication technologies.
• The service is available free of cost and ensures privacy and confidentiality of patient data.
• It has been particularly beneficial during the COVID-19 pandemic by reducing the need for physical visits to hospitals.
• eSanjeevani AB-HWC:
• This service connects beneficiaries with doctors and specialists at Ayushman Bharat Health and Wellness Centers (AB-HWCs).
• AB-HWCs focus on providing comprehensive primary healthcare services, including maternal and child health, non-communicable diseases, and mental health services.
• It is designed to strengthen the grassroots healthcare system by leveraging digital technology.
• Ayushman Bharat Initiative:
• Ayushman Bharat is a flagship healthcare initiative launched by the Government of India to achieve Universal Health Coverage (UHC).
• It has two components: Health and Wellness Centers (HWCs) and the Pradhan Mantri Jan Arogya Yojana (PMJAY).
• Under PMJAY, financial protection of up to Rs. 5 lakh per family is provided annually for secondary and tertiary healthcare services.

The correct answer is Option 3: Both 1 & 2 correct.

Key Points

• Jhora is a traditional folk dance performed in the Kumaon region of Uttarakhand, India.
• It is primarily performed during spring fairs, celebrating the arrival of the new season with joy and unity.
• The dance is characterized by mixed circles of men and women, symbolizing harmony and togetherness.
• The Hurka drum is the primary musical instrument that accompanies the Jhora dance, providing rhythm and vibrancy to the performance.
• This dance is an integral part of Kumaoni culture and highlights the region's rich heritage and traditions.

Additional Information

• Jhora Dance:
• Jhora is a community dance that is performed on special occasions, such as weddings and festivals, to mark celebrations.
• The dancers form a circular formation and move in coordinated steps, reflecting the unity and strength of the community.
• It is considered one of the most famous folk dances of the Kumaon region and is a significant part of its cultural identity.
• Hurka Drum:
• The Hurka drum is a traditional musical instrument used in Uttarakhand, especially in folk dances such as Jhora and Hurka Baul.
• It is a small, handheld drum that produces rhythmic beats, adding energy and enthusiasm to the dance.
• The Hurka drum is often accompanied by other traditional instruments, enhancing the overall musical experience during performances.
• Traditional Kumaoni Culture:
• The Kumaon region of Uttarakhand is known for its unique cultural heritage, which includes traditional dances, music, and festivals.
• Other notable folk dances of Kumaon include Chholiya, a sword dance performed during weddings, and Ramola, a dance that narrates local folklore.
• Kumaoni culture is deeply rooted in nature worship and the celebration of seasons, which is reflected in its folk traditions.

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

Key Points

• Global Ease of Doing Business (EoDB) indicators measure how conducive a country's regulatory environment is for starting and operating businesses.
• These indicators primarily focus on formal sectors, excluding informal enterprises, as they lack consistent data across nations.
• Informal enterprises are challenging to measure due to their non-standardized structure and operations, which vary significantly across countries.
• The World Bank, which publishes the EoDB rankings, generally relies on data from registered businesses and government regulations, leaving out the informal sector.
• Reason (R) correctly explains the exclusion of informal enterprises, as they are difficult to assess uniformly across different nations due to inconsistencies in data collection methods and definitions of informal businesses.

Additional Information

• Ease of Doing Business Rankings and Informal Sector
• The Ease of Doing Business rankings are published by the World Bank to compare the business environments of different countries.
• The rankings primarily evaluate regulations affecting registered businesses, ignoring informal sectors due to challenges in uniform data collection.
• Informal businesses often operate outside formal regulatory frameworks, making them difficult to assess using standardized metrics.
• Including informal enterprises would require developing globally consistent definitions and measurement methods, which are currently lacking.

The correct answer is Electrify broad gauge & use renewables.

Key Points

• Indian Railways is prioritizing the electrification of its broad-gauge railway lines to replace diesel locomotives, reducing dependency on fossil fuels and significantly cutting carbon emissions.
• It aims to become a net-zero carbon emitter by 2030, aligning with India’s commitment to global climate change mitigation efforts under the Paris Agreement.
• The initiative involves integrating renewable energy sources like solar and wind power into its energy mix to cater to its electricity requirements.
• Indian Railways has set a target of installing 30 GW of renewable energy capacity by 2030, including solar panels and wind farms at its stations, tracks, and other facilities.
• This electrification and renewable energy adoption is expected to reduce the annual carbon dioxide emissions by millions of tons, making it a major contributor to India’s overall decarbonization goals.

Additional Information

• Decarbonizing Indian Railways
• Indian Railways is one of the largest rail networks in the world and a critical component of India’s transportation infrastructure.
• It is working towards decarbonization by adopting energy-efficient technologies, renewable energy, and electrification of rail networks to mitigate its environmental impact.

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

Key Points

• The Mahalanobis strategy was developed by Prasanta Chandra Mahalanobis, an Indian statistician and economist, and was adopted in India's Second Five-Year Plan (1956–1961).
• Heavy industries and capital goods were prioritized under this strategy as these sectors were deemed crucial for laying the foundation of industrialization and long-term economic growth.
• The strategy emphasized import substitution industrialization to reduce dependency on foreign goods and build a self-reliant economy.
• The rationale behind prioritizing heavy industries and capital goods was that these sectors would generate the necessary infrastructure and machinery for other industries, thus creating a ripple effect of economic development.
• The approach was based on the belief that capital-intensive industries would create a strong base for technological advancements and economic independence.

Additional Information

• India's Second Five-Year Plan (1956-1961)
• focused on rapid industrialization, particularly heavy industries, using the Nehru-Mahalanobis model, aiming for a 25% increase in national income and new jobs by boosting public sector investment in steel, coal, and energy, establishing major steel plants (Bhilai, Durgapur, Rourkela) with foreign help, but faced forex shortages and price rises.
• Key Features & Objectives:
• Focus: Rapid industrialization with emphasis on basic and heavy industries, shifting away from agriculture.
• Model: Based on the Nehru-Mahalanobis model, prioritizing investment in capital goods.
• Goals: Increase national income by 25%, create more jobs, and develop infrastructure.
• Public Sector: Significant expansion of the public sector.

The correct answer is Article 343.

Key Points

• Article 343 of the Indian Constitution declares Hindi written in the Devanagari script as the official language of the Union.
• It also specifies that the numerals used for official purposes must be in the international form of Indian numerals.
• The Constitution initially allowed English to continue as an official language of the Union for a period of 15 years from the commencement of the Constitution (1950), which was later extended through amendments.
• Article 343(2) empowers Parliament to provide for the continued use of English for official purposes, alongside Hindi, even after the initial 15-year period.
• The Official Languages Act of 1963 extended the use of English as an associate official language indefinitely, ensuring bilingual usage for administrative purposes.

Additional Information

• Article 348:
• It provides for the language to be used in the Supreme Court and High Courts and for Acts, Bills, and other official documents.
• According to this Article, English shall be used for all proceedings in the Supreme Court and High Courts, as well as for authoritative texts of laws and bills at the Union and State levels unless otherwise specified by Parliament or the State Legislature.
• Article 344:
• This Article deals with the Constitution and functions of the Official Language Commission.
• The Commission is tasked with making recommendations regarding the progressive use of Hindi for official purposes and the restrictions on using English.
• It also advises on measures to promote the spread and development of Hindi throughout the country.
• Article 351:
• It directs the Union Government to promote the spread of the Hindi language and develop it to serve as a medium of expression for all elements of India's composite culture.
• The Government is also instructed to enrich Hindi by assimilating its forms, style, and expressions from other Indian languages and Sanskrit.

The correct answer is Option 1: 1-Divergent, 2-Convergent, 3-Transform.

Key Points

• Mid-ocean ridge is associated with divergent plate boundaries, where tectonic plates move apart, allowing magma to rise and form new oceanic crust. An example is the Mid-Atlantic Ridge.
• Mariana Trench is formed at a convergent plate boundary, where one tectonic plate is subducted under another. This creates deep oceanic trenches like the Mariana Trench, the deepest part of the ocean.
• San Andreas Fault is a classic example of a transform plate boundary, where two tectonic plates slide past each other horizontally, leading to frequent earthquakes.
• Each type of plate boundary has distinct geological features and processes, including the formation of volcanoes, trenches, or faults.
• These geological processes significantly impact Earth's surface structure and contribute to seismic activity in regions near these boundaries.

Additional Information

• Divergent Plate Boundaries:
• At divergent boundaries, tectonic plates move away from each other.
• These boundaries are often found along mid-ocean ridges, such as the Mid-Atlantic Ridge.
• Magma rises from the mantle to fill the gap, creating new oceanic crust and causing sea-floor spreading.
• Associated geological features include volcanic ridges and rift valleys.
• Convergent Plate Boundaries:
• At convergent boundaries, tectonic plates move toward each other.
• Subduction zones occur when an oceanic plate is pushed beneath a continental or another oceanic plate, leading to trench formation.
• Examples of convergent boundaries include the Mariana Trench and the Andes Mountains.
• These areas are prone to intense geological activity, such as earthquakes and volcanic eruptions.
• Transform Plate Boundaries:
• At transform boundaries, plates slide past each other horizontally.
• The San Andreas Fault in California is a well-known example of this type of boundary.
• These regions are characterized by frequent earthquakes caused by the friction and stress between the sliding plates.
• Unlike divergent and convergent boundaries, transform boundaries do not create or destroy crust.
• Plate Tectonics:
• The theory of plate tectonics explains the movement of Earth's lithospheric plates.
• It provides insights into the formation of continents, mountain ranges, earthquakes, volcanoes, and oceanic trenches.
• Understanding plate tectonics is crucial for predicting natural disasters and studying Earth's geological history.

The correct answer is Option 1: Only 1.

Key Points

• Statement 1 is correct: Parliament has the authority to create new All India Services under Article 312 of the Constitution of India. However, this requires the approval of the Rajya Sabha by a two-thirds majority of the members present and voting.
• Statement 2 is incorrect: Article 312 of the Indian Constitution does not mention "Security Services" as part of All India Services. The three All India Services currently in existence are the Indian Administrative Service (IAS), Indian Police Service (IPS), and Indian Forest Service (IFS).
• The creation of new All India Services is intended to ensure uniformity and efficiency in the administration of the country.
• All India Services officers are recruited and trained by the central government but serve both the central and state governments, ensuring better coordination between the two levels of administration.

Additional Information

• About Article 312:
• Article 312 of the Indian Constitution provides for the creation of one or more All India Services, including an All India Judicial Service, common to the Union and the States.
• The services created under this article are meant to ensure uniform standards of administration across the country.
• The central government handles recruitment and training for these services through institutions like the Union Public Service Commission (UPSC).
• Existing All India Services:
• Indian Administrative Service (IAS): Responsible for policy implementation and general administration in the government.
• Indian Police Service (IPS): Maintains law and order and ensures internal security.
• Indian Forest Service (IFS): Manages forests, wildlife, and environmental conservation.
• Role of the Rajya Sabha in Creating All India Services:
• The Rajya Sabha must pass a resolution by a two-thirds majority to authorize the creation of a new All India Service.
• This provision is intended to ensure that states are consulted before the creation of services that will operate at both the central and state levels.

Shortcut Trick

To find the ratio of the first and last terms (A : D), simply multiply all the given ratios together.

A/D = (A/B) × (B/C) × (C/D)

A/D = (7/5) × (2/3) × (4/5)

A/D = (7 × 2 × 4) / (5 × 3 × 5)

A/D = 56 / 75

∴ The correct answer is 56 : 75.

Alternate Method

Given:

A : B = 7 : 5

B : C = 2 : 3

C : D = 4 : 5

Method Used:

We can use the shift method to find the combined ratio A : B : C : D by copying the adjacent numbers into the empty spaces of each row.

ABCD75552233444556406075

Calculations:

⇒ Multiply down the columns to get the final combined ratio components:

⇒ A = 7 × 2 × 4 = 56

⇒ B = 5 × 2 × 4 = 40

⇒ C = 5 × 3 × 4 = 60

⇒ D = 5 × 3 × 5 = 75

⇒ The combined ratio is A : B : C : D = 56 : 40 : 60 : 75

⇒ Therefore, the required ratio is A : D = 56 : 75

∴ The correct answer is 56 : 75.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is called a compound ratio. For a:b and c:d, the compound ratio is (a × c) : (b × d).

Inverse Ratio

For any ratio a : b, its inverse ratio is b : a. The inverse of a compound ratio a : b : c is evaluated as 1/a : 1/b : 1/c.

Duplicate & Triplicate Ratio

The duplicate ratio of a : b is given by a² : b². The triplicate ratio of a : b is given by a³ : b³.

Shortcut Trick

Convert division to multiplication by taking the reciprocal of the second fraction in each bracket.

⇒ (7/5 × 15/14) × (10/7 × 3/5)

⇒ (3/2) × (6/7)

⇒ 18/14 = 9/7

∴ The correct answer is 9/7.

Alternate Method

Given:

Expression: (7/5 ÷ 14/15) × (10/7 ÷ 5/3)

Formula Used:

BODMAS Rule: Solve Brackets first, then Division (by converting to multiplication with reciprocal), then Multiplication.

Fraction Division: a/b ÷ c/d = a/b × d/c

Calculation:

Solving the first bracket:

⇒ 7/5 ÷ 14/15 = 7/5 × 15/14

⇒ (7 × 15) / (5 × 14) = 105 / 70 = 3/2

Solving the second bracket:

⇒ 10/7 ÷ 5/3 = 10/7 × 3/5

⇒ (10 × 3) / (7 × 5) = 30 / 35 = 6/7

Multiplying the results of both brackets:

⇒ 3/2 × 6/7

⇒ (3 × 6) / (2 × 7)

⇒ 18 / 14 = 9/7

∴ The correct answer is 9/7.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. Expressions must be solved in this exact sequential order.

Fraction Division

To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction: a/b ÷ c/d = a/b × d/c.

Simplifying Fractions

Always cancel out common factors from the numerator and denominator before multiplying to make calculations faster and avoid large numbers.

Shortcut Trick

Convert terms to simple fractions for faster and error-free calculation:

(3/2 + 3/8)³ − (9/5 × 5/4)

⇒ (15/8)³ − 9/4

⇒ 3375/512 − 1152/512 = 2223/512

⇒ 2223 ÷ 512 ≈ 4.3418

∴ The correct answer is 4.3418.

Alternate Method

Given:

(1.5 + 3/8)³ − (9/5 × 1.25)

Calculations:

⇒ Convert decimals to fractions to easily simplify the expression:

⇒ 1.5 = 15/10 = 3/2

⇒ 1.25 = 125/100 = 5/4

⇒ Substitute the fraction values back into the expression:

⇒ (3/2 + 3/8)³ − (9/5 × 5/4)

⇒ Make denominators equal inside the bracket:

⇒ (12/8 + 3/8)³ − 9/4

⇒ Add the terms together inside the bracket:

⇒ (15/8)³ − 9/4

⇒ Calculate the cube of the fraction (15³ = 3375, 8³ = 512):

⇒ 3375/512 − 9/4

⇒ Make denominators equal to perform subtraction (multiply 9/4 by 128/128):

⇒ 3375/512 − (9 × 128)/(4 × 128)

⇒ 3375/512 − 1152/512

⇒ Subtract the numerators:

⇒ (3375 − 1152) / 512

⇒ 2223 / 512

⇒ Convert the final fraction back to a decimal:

⇒ 2223 ÷ 512 = 4.341796875

⇒ Rounding to four decimal places, we get 4.3418.

∴ The correct answer is 4.3418.

Additional Information

BODMAS Rule

Always simplify complex mathematical expressions in the exact order of Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

Memorizing common fraction values makes calculations significantly faster. For example, 1/8 = 0.125, 3/8 = 0.375, and 1/4 = 0.25.

Rounding Decimals

When rounding to 4 decimal places, check the 5th decimal digit. If it is 5 or more, round up the 4th digit by 1. Otherwise, leave it unchanged.

Shortcut Trick

Convert the given fractions directly to decimals for faster calculation:

⇒ 3/4 = 0.75 and 1/2 = 0.5

Substitute these values into the expression:

⇒ ((1.5 + 0.75) ÷ (2.25 − 0.5)) + 0.6

⇒ (2.25 ÷ 1.75) + 0.6 = (9 ÷ 7) + 0.6 ≈ 1.2857 + 0.6 = 1.8857

∴ The correct answer is 1.8857.

Alternate Method

Given:

Expression = ((1.5 + 3/4) ÷ (2.25 − 1/2)) + 0.6

Calculations:

To avoid decimal division, convert all values to fractions:

⇒ 1.5 = 3/2

⇒ 2.25 = 9/4

⇒ 0.6 = 3/5

Substitute the fractions back into the expression:

⇒ ((3/2 + 3/4) ÷ (9/4 − 1/2)) + 3/5

Make denominators equal to add and subtract the inner terms:

⇒ ((6/4 + 3/4) ÷ (9/4 − 2/4)) + 3/5

⇒ (9/4 ÷ 7/4) + 3/5

⇒ (9/4 × 4/7) + 3/5

⇒ 9/7 + 3/5

Take the LCM of 7 and 5, which is 35:

⇒ (45 + 21) / 35

⇒ 66 / 35 ≈ 1.885714...

⇒ Rounding to 4 decimal places, we get 1.8857.

∴ The correct answer is 1.8857.

Additional Information

BODMAS Rule

Always follow the sequence of Brackets, Orders, Division/Multiplication, and Addition/Subtraction while simplifying expressions.

Fraction to Decimal Conversion

Memorising common fraction-to-decimal values like 1/2 = 0.5, 3/4 = 0.75, and 1/5 = 0.2 helps in quick calculation without the need to take an LCM.

Shortcut Trick

Equating the initial and new ratios for liquids A and B: (2x + 6) ÷ (3x + 12) = 3 ÷ 5 ⇒ 10x + 30 = 9x + 36 ⇒ x = 6.

New quantity of A = 2(6) + 6 = 18 liters (which represents 3 parts of the new ratio).

Since 3 parts = 18 liters, 1 part = 6 liters, and new C (8 parts) = 8 × 6 = 48 liters.

Initial quantity of C = 5x = 5(6) = 30 liters.

Quantity of C added = Final C − Initial C = 48 − 30 = 18 liters.

∴ The correct answer is 18 liters.

Alternate Method

Given:

Initial ratio of A, B, and C = 2 : 3 : 5

Quantity added to A = 6 liters

Quantity added to B = 12 liters

New ratio of A, B, and C = 3 : 5 : 8

Formula Used:

New Quantity = Initial Quantity + Quantity Added

Liquid A2x → 12 L+ 6 L18 L (3 parts)Liquid B3x → 18 L+ 12 L30 L (5 parts)Liquid C5x → 30 L+ y L48 L (8 parts)Initial Ratio = 2 : 3 : 5y = 48 − 30 = 18 Liters

Calculations:

⇒ Let the initial quantities of A, B, and C be 2x, 3x, and 5x respectively.

⇒ After adding 6 liters to A and 12 liters to B, their respective new quantities are:

⇒ New A = 2x + 6

⇒ New B = 3x + 12

⇒ According to the new given ratio of A to B (3 : 5), we can set up the equation:

⇒ (2x + 6) ÷ (3x + 12) = 3 ÷ 5

⇒ 5 × (2x + 6) = 3 × (3x + 12)

⇒ 10x + 30 = 9x + 36

⇒ 10x − 9x = 36 − 30

⇒ x = 6

⇒ Initial quantity of C = 5x = 5 × 6 = 30 liters

⇒ New quantity of A = 2(6) + 6 = 12 + 6 = 18 liters

⇒ According to the new ratio (3 : 5 : 8), 3 parts correspond to 18 liters of liquid A.

⇒ 1 part = 18 ÷ 3 = 6 liters

⇒ The new quantity of C corresponds to 8 parts of this ratio.

⇒ New C = 8 × 6 = 48 liters

⇒ Quantity of C added = New quantity of C − Initial quantity of C

⇒ Quantity of C added = 48 − 30 = 18 liters

∴ The correct answer is 18 liters.

Additional Information

Ratio Multiplier Concept

When components of a mixture are given in a ratio a : b : c, their absolute quantities can be modeled as ax, bx, and cx where x acts as a common variable multiplier.

Handling Independent Alterations

When distinct amounts are added to specific mixture components, forming an equation using only the known ratio components will yield the multiplier first, before trying to evaluate the unknown variables.

Cross-Multiplication Method

For simple rational equations structured as a ÷ b = c ÷ d, you can quickly simplify and solve by using direct cross-multiplication: a × d = b × c.

Shortcut Trick

In a partnership, the profit ratio is directly proportional to the product of Investment and Time.

Gautam's equivalent investment = 60000 × 24 months

Jatin's equivalent investment = 35000 × 16 months

Profit Ratio = (60000 × 24) : (35000 × 16) = (60 × 3) : (35 × 2) = 180 : 70 = 18 : 7

∴ The correct answer is 18:7.

Alternate Method

Given:

Gautam's investment (C₁) = ₹ 60,000

Jatin's investment (C₂) = ₹ 35,000

Total business duration = 2 years = 24 months

Jatin joins 8 months later.

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Gautam (₹ 60,000)

24 months

Jatin (₹ 35,000)

16 months

Joined 8 months later

Profit Ratio = (60k × 24) : (35k × 16)

Calculations:

⇒ Gautam's time period (T₁) = 24 months

⇒ Jatin's time period (T₂) = 24 − 8 = 16 months

⇒ Profit Ratio = (60000 × 24) : (35000 × 16)

⇒ Simplifying the capitals by removing common zeros:

⇒ Profit Ratio = (60 × 24) : (35 × 16)

⇒ Profit Ratio = 1440 : 560

⇒ Dividing both sides by 80:

⇒ Profit Ratio = 18 : 7

∴ The correct answer is 18:7.

Additional Information

Compound Partnership

When the capitals of partners are invested for different time periods, it is called a compound partnership. The share of profit for each partner is calculated by multiplying their respective capital with the time period.

Joining Mid-way

If a partner joins the business t months after the business started, and the total duration to calculate profit is T months, their capital is actively invested for only (T − t) months.

Active vs Sleeping Partner

An active partner manages the operations and usually receives a fixed percentage or salary from the total profit before the remainder is distributed. A sleeping partner only provides capital and shares the remaining profit based entirely on the investment ratio.

Shortcut Trick

Weight of the left person = Old Average + (Remaining People × Decrease in Average)

Old Average = 65 kg, Remaining People = 11

Decrease in Average = 65 − 64.5 = 0.5 kg

Weight = 65 + (11 × 0.5)

Weight = 65 + 5.5 = 70.5 kg

∴ The correct answer is 70.5 kg.

Alternate Method

Given:

Initial average weight of 12 people = 65 kg

New average weight after 1 person leaves = 64.5 kg

12 People | Average: 65 kg (Total: 780 kg)

11 People | Avg: 64.5 kg

1 Left

Total: 709.5 kg ? kg

Calculations:

⇒ Total initial weight of 12 people = 12 × 65 = 780 kg

⇒ Total weight of remaining 11 people = 11 × 64.5 = 709.5 kg

⇒ Weight of the person who left = (Total initial weight) − (Total new weight)

⇒ Weight of the person who left = 780 − 709.5

⇒ Weight of the person who left = 70.5 kg

∴ The correct answer is 70.5 kg.

Additional Information

When a Person Joins

If a new person joins and the average increases, New Weight = Old Average + (New total people × Increase in average).

When a Person is Replaced

If one person is replaced by another and the average changes, Weight of New Person = Weight of Replaced Person ± (Total number of people × Change in average).

Weighted Average

When combining two groups, Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Let the assumed average be 65.

Deviations of Q's sales (65, 68, 62, 64) from 65 are 0, +3, −3, and −1.

Sum of deviations = 0 + 3 − 3 − 1 = −1

Actual Average = Assumed Average + (Sum of deviations ÷ 4)

Actual Average = 65 − (1 ÷ 4) = 64.75

∴ The correct answer is ₹64.75.

Alternate Method

Given:

Sales of salesperson Q in different months:

Jan = 65, Feb = 68, March = 62, April = 64

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Calculations:

⇒ Total Sales of Q = 65 + 68 + 62 + 64

⇒ Total Sales of Q = 259

⇒ Total number of months = 4

⇒ Average Sales = 259 ÷ 4

⇒ Average Sales = 64.75

∴ The correct answer is ₹64.75.

Additional Information

Average Formula

Average is calculated by dividing the sum of all values by the total number of values. Average = Sum of observations ÷ Total number of observations.

Net Deviation Method

By selecting an assumed average, calculate the positive or negative deviation of each term. Net average = Assumed average + (Sum of deviations ÷ Total number of items).

Combined Average

When combining two distinct groups with quantities n₁ and n₂ and averages A₁ and A₂ respectively, the combined average is: (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Shortcut Trick

Rate = 10% = 1/10 ⇒ Multiplying factor per year = 11/10

Ratio of Amount to Principal = 66550 ÷ 50000 = 1331 ÷ 1000

We know that (11/10)³ = 1331 / 1000

Since the cube corresponds to 3 periods, the time taken is 3 years.

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹50,000

Amount (A) = ₹66,550

Rate of Interest (R) = 10% per annum

Formula Used:

A = P × (1 + R/100)ⁿ

Step-by-Step Compound Interest Growth

₹50,000

₹55,000

₹60,500

₹66,550

+10% 1 Yr

+10% 1 Yr

+10% 1 Yr

Total Time = 3 Years

Calculations:

⇒ 66,550 = 50,000 × (1 + 10/100)ⁿ

⇒ 66,550 ÷ 50,000 = (1 + 1/10)ⁿ

⇒ 6,655 ÷ 5,000 = (11/10)ⁿ

⇒ 1,331 ÷ 1,000 = (11/10)ⁿ

⇒ (11/10)³ = (11/10)ⁿ

⇒ n = 3

∴ The correct answer is 3 years.

Additional Information

Effective Rate of CI

For a 10% annual interest rate, the effective CI rate for 2 years is 21%, and for 3 years it is 33.1%. Here, 33.1% of ₹50,000 exactly equals the interest of ₹16,550.

Difference between CI and SI (2 Years)

The difference between compound interest and simple interest for a period of 2 years is calculated directly as: Difference = P × (R ÷ 100)².

Difference between CI and SI (3 Years)

For a period of 3 years, the formula extends to: Difference = P × (R ÷ 100)² × (3 + R ÷ 100).

Shortcut Trick

Given the ratio of Profit to Cost Price is 2 : 7.

Let Profit = 2 units and Cost Price = 7 units.

Selling Price = Cost Price + Profit = 7 + 2 = 9 units.

Required Ratio (Selling Price : Profit) = 9 : 2.

∴ The correct answer is 9:2.

Alternate Method

Given:

Ratio of Profit to Cost Price (CP) = 2 : 7

Cost Price (CP) = 7x Profit = 2x

Selling Price (SP) = 7x + 2x = 9x Profit & Loss Visualisation

Formula Used:

Selling Price (SP) = Cost Price (CP) + Profit

Calculation:

⇒ Let the Profit be 2x and the Cost Price be 7x.

⇒ SP = 7x + 2x

⇒ SP = 9x

⇒ The required ratio of Selling Price to Profit is SP : Profit

⇒ Ratio = 9x : 2x

⇒ Ratio = 9 : 2

∴ The correct answer is 9:2.

Additional Information

Profit and Loss Percentage

Both profit percentage and loss percentage are always calculated on the Cost Price (CP) unless specified otherwise. Formula: Profit % = (Profit ÷ CP) × 100.

Selling Price under Loss

When a product is sold at a loss, the Selling Price is less than the Cost Price. Formula: SP = CP − Loss.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP) of the article. Selling Price = Marked Price − Discount.

Shortcut Trick

Let the Marked Price (MP) of 1 box = 100.

Cost Price (CP) of 1 box = (MP of 8 boxes) ÷ 10 = 800 ÷ 10 = 80.

Selling Price (SP) of 1 box = MP − 2% discount = 100 − 2 = 98.

Profit % = [(SP − CP) ÷ CP] × 100 = [(98 − 80) ÷ 80] × 100.

Profit % = (18 ÷ 80) × 100 = 22.5%.

∴ The correct answer is 22.5%.

Alternate Method

Given:

Total number of boxes bought by the shop owner = 10

Price paid for 10 boxes = Price of 8 boxes

Discount allowed on each box during sale = 2% on Marked Price

<br>

Formula Used:

Profit = Selling Price (SP) − Cost Price (CP)

Profit % = (Profit ÷ CP) × 100

<br>

Cost Price (CP) of 1 box = 80

Marked Price (MP) of 1 box = 100

Selling Price (SP) of 1 box = 98 (-2% discount)

Profit = 18

Profit % = (18 ÷ 80) × 100 = 22.5%

<br>

Calculations:

⇒ Let the Marked Price (MP) of 1 box be 100.

⇒ The shopkeeper buys 10 boxes for the price of 8 boxes.

⇒ Cost Price (CP) of 10 boxes = 8 × 100 = 800.

⇒ Cost Price (CP) of 1 box = 800 ÷ 10 = 80.

⇒ He sells each box at a 2% discount on its Marked Price.

⇒ Selling Price (SP) of 1 box = 100 − (2% of 100) = 98.

⇒ Profit earned on 1 box = SP − CP = 98 − 80 = 18.

⇒ Profit % = (18 ÷ 80) × 100

⇒ Profit % = (9 ÷ 40) × 100 = 22.5%.

∴ The correct answer is 22.5%.

Additional Information

Marked Price and Discount

Discount is always calculated strictly on the Marked Price (MP) unless specified otherwise. The Selling Price is derived as SP = MP − Discount.

Buy X Get Y Free Offers

In promotional schemes, the discount percentage is based on the total number of items taken. Discount % = [Free Items ÷ Total Items (X + Y)] × 100.

Successive Discounts

When two consecutive discounts of a% and b% are applied on an item, the equivalent single discount percentage is given by the formula: (a + b − (a × b) ÷ 100)%.

Dishonest Dealer Profit

When a dealer claims to sell at cost price but uses a false weight, the actual profit percentage is calculated as: [(True Weight − False Weight) ÷ False Weight] × 100.

Shortcut Trick

Let the Cover Price be 100 units.

Discounted Price = 100 − 30% = 70 units.

Total Cost Price = 70 + (5% of 70) = 73.5 units.

Profit = 40% of Cost Price = 0.40 × 73.5 = 29.4 units.

Since 29.4 units = ₹1620, Cover Price (100 units) = (1620 ÷ 29.4) × 100 ≈ ₹5510.20

∴ The correct answer is ₹5,510.20.

Alternate Method

Given:

Discount on Cover Price = 30%

Shipping charge on discounted price = 5%

Profit percentage = 40%

Profit amount (sold for ₹1620 more) = ₹1620

Cover Price(x)-30% disc+5% shipCost Price(0.735x)+40% Profit= ₹1620Selling Price(CP + 1620)Equation: 40% of 0.735x = 1620

Formula Used:

Cost Price (CP) = Discounted Price × (1 + Shipping Rate)

Profit = Profit % × CP

Calculations:

⇒ Let the original Cover Price of the novel be x.

⇒ Discounted Price = x − 30% of x = 0.70x

⇒ Cost Price (CP) = 0.70x + 5% of 0.70x

⇒ CP = 0.70x + 0.035x = 0.735x

⇒ Profit = 40% of CP = 0.40 × 0.735x = 0.294x

⇒ We are given that the Profit earned is ₹1620.

⇒ 0.294x = 1620

⇒ x = 1620 ÷ 0.294

⇒ x = 1620000 ÷ 294

⇒ x ≈ 5510.20

∴ The correct answer is ₹5,510.20.

Additional Information

Marked Price (Cover Price)

It is the original price printed on an item. Successive discounts or flat discounts are always calculated on the Marked Price before deciding the Selling Price or Purchase Cost.

Effective Cost Price with Overheads

When extra charges like shipping, repairs, or transportation are incurred, they are added to the initial discounted purchase price to evaluate the total effective Cost Price.

Profit and Loss Application

All Profit and Loss calculations and percentage margins are applied purely on the final Cost Price, not on the original marked price. Formula: Profit % = (Profit ÷ CP) × 100.

Shortcut Trick

Fraction of syrup replaced each time = 10 ÷ 100 = 1/10.

Fraction of syrup left after 2 replacements = (1 − 1/10)² = (9/10)² = 81/100.

Final syrup = 81/100 × 100 L = 81 L.

Final water = Total volume − Final syrup = 100 L − 81 L = 19 L.

Ratio of syrup to water = 81 : 19.

∴ The correct answer is 81:19.

Alternate Method

Given:

Initial volume of pure syrup (V) = 100 L

Volume drawn and replaced by water (x) = 10 L

Number of operations (n) = 2

Formula Used:

Final amount of pure liquid = V × (1 − x/V)ⁿ

Initial 100L Pure Syrup −10L Mixture +10L Water (2 Times) Final 81L Syrup 19L Water Ratio 81:19

Calculations:

⇒ Final amount of syrup = 100 × (1 − 10/100)²

⇒ Final amount of syrup = 100 × (1 − 1/10)²

⇒ Final amount of syrup = 100 × (9/10)²

⇒ Final amount of syrup = 100 × 81/100 = 81 L

⇒ Volume of water in the final mixture = Total Volume − Final amount of syrup

⇒ Volume of water = 100 − 81 = 19 L

⇒ Ratio of syrup to water = 81 : 19

∴ The correct answer is 81:19.

Additional Information

Successive Replacement Formula

When a constant volume 'x' is replaced 'n' times from an initial volume 'V', Final amount = V × (1 − x/V)ⁿ.

Multiple Different Replacements

If replaced amounts are different (x, y, z) from initial volume V, Final amount = V × (1 − x/V) × (1 − y/V) × (1 − z/V).

Alligation Cross Method

Used to find the ratio in which two mixtures should be mixed. The ratio of quantities is inversely proportional to the differences in their unit concentrations.

Shortcut Trick

Volume is proportional to the cube of the radius, while surface area is proportional to the square of the radius.

Since 1 large sphere is melted into 8 identical smaller spheres, the volume ratio of 1 Large : 1 Small = 8 : 1.

Therefore, the radius ratio of Large : Small = 8^1/3 : 1^1/3 = 2 : 1.

The surface area ratio of 1 Large : 1 Small = 2² : 1² = 4 : 1.

Ratio of the surface area of 1 Large sphere to 8 Small spheres = 4 : (8 × 1) = 4 : 8 = 1 : 2.

∴ The correct answer is 1:2.

Alternate Method

Given:

Radius of original metal sphere (R) = 10 cm

Number of identical smaller spheres (n) = 8

Formula Used:

Volume of a sphere = (4/3) × π × r³

Surface Area of a sphere = 4 × π × r²

R = 10 Original Sphere

Melted into 8

r × 8 Smaller Spheres

Calculations:

⇒ Total volume remains constant when melting and recasting.

⇒ Volume of original sphere = 8 × Volume of one smaller sphere

⇒ (4/3) × π × R³ = 8 × (4/3) × π × r³

⇒ R³ = 8r³

⇒ 10³ = 8r³

⇒ 1000 = 8r³

⇒ r³ = 125 ⇒ r = 5 cm

⇒ Surface Area of Original Sphere (S₁) = 4 × π × 10²

⇒ S₁ = 400π cm²

⇒ Total Surface Area of 8 Smaller Spheres (S₂) = 8 × (4 × π × 5²)

⇒ S₂ = 8 × 100π = 800π cm²

⇒ Required Ratio = S₁ : S₂ = 400π : 800π

⇒ Ratio = 1 : 2

∴ The correct answer is 1:2.

Additional Information

Melting and Recasting Law

When a solid object is melted and recast into new shapes, the total volume of the material is always conserved, though the total surface area will change.

Dimensional Scaling Factors

If the linear dimension (radius, side) of a 3D figure changes by a factor of k, then its surface area scales by k² and its volume scales by k³.

Sphere vs Hemisphere Properties

For a full sphere, Total Surface Area = 4πr². For a solid hemisphere, the Curved Surface Area is 2πr² while the Total Surface Area is 3πr² because of the flat circular base.

Shortcut Trick

When a large shape is melted into smaller identical shapes, the number of objects is the ratio of their volumes.

For identical shapes like hemispheres, constants cancel out, leaving only the ratio of the radii cubed.

Number of small hemispheres = (Radius of large ÷ Radius of small)³

⇒ Number = (12 ÷ 3)³ = 4³

⇒ Number = 64

∴ The correct answer is 64.

Alternate Method

12 cm Large Hemisphere

Melted

3 cm

...

n = ? n Small Hemispheres

Given:

Radius of large hemisphere (R) = 12 cm

Radius of small hemisphere (r) = 3 cm

Formula Used:

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

Calculation:

Let the total number of small hemispheres produced be n.

Since the large hemisphere is melted to form smaller ones, the total volume of the material remains constant.

⇒ Volume of large hemisphere = n × Volume of small hemisphere

⇒ (2 ÷ 3) × π × R³ = n × (2 ÷ 3) × π × r³

Canceling out (2 ÷ 3) × π from both sides:

⇒ R³ = n × r³

⇒ 12³ = n × 3³

⇒ n = 12³ ÷ 3³

⇒ n = (12 ÷ 3)³

⇒ n = 4³

⇒ n = 64

∴ The correct answer is 64.

Additional Information

Melting and Recasting

When a solid object is melted and recast into new shapes, the total volume remains constant regardless of the new shape. Number of shapes = Total initial volume ÷ Volume of one new shape.

Volume of a Hemisphere

The space occupied by a hemisphere is exactly half the volume of a full sphere. It is given by V = (2 ÷ 3) × π × r³, where r is the radius.

Surface Areas of a Hemisphere

The Curved Surface Area (CSA) of a hemisphere is 2 × π × r². The Total Surface Area (TSA) includes the flat circular top, making it 3 × π × r².

Shortcut Trick

Since the cylinder and hemisphere have equal volumes on the same base radius, their height ratio h : r = 2 : 3.

Total height in terms of ratio units = 2 + 3 = 5 units.

Given total combined height = 30 cm.

Radius of hemisphere (3 units) = (30 ÷ 5) × 3 = 18 cm.

∴ The correct answer is 18 cm.

Alternate Method

Given:

Total combined height = h + r = 30 cm

Volume of the Cylinder = Volume of the Hemisphere

Formula Used:

Volume of Cylinder = πr²h

Volume of Hemisphere = (2/3)πr³

30 cm

h

r

r

Cylinder and Hemisphere

Calculations:

⇒ πr²h = (2 ÷ 3) × πr³

⇒ h = (2 ÷ 3)r

⇒ Substituting the value of h in the combined height equation:

⇒ (2 ÷ 3)r + r = 30

⇒ (5 ÷ 3)r = 30

⇒ r = 30 × 3 ÷ 5

⇒ r = 18 cm

∴ The correct answer is 18 cm.

Additional Information

Volume and Surface Area of a Cylinder

The volume is given by V = πr²h. The total surface area is 2πr(r + h), where r is the base radius and h is the height.

Volume and Surface Area of a Hemisphere

The volume is given by V = (2/3)πr³. The curved surface area is 2πr², and the total surface area is 3πr².

Ratio-Unitary Method in Geometry

When two 3D shapes have equal volumes and identical base radii, equating their volume formulas directly provides the ratio of their heights, significantly speeding up complex fractional calculations.

Shortcut Trick

The volume of any pyramid is directly one-third of the product of its base area and height.

Volume = (1 ÷ 3) × Base Area × Height

Volume = (1 ÷ 3) × 60 × 9 = 180 cm³

∴ The correct answer is 180 cm³.

Alternate Method

Given:

Base Area of the pyramid = 60 cm²

Height of the pyramid = 9 cm

Formula Used:

Volume of a Pyramid = (1 ÷ 3) × Base Area × Height

h = 9 cm Base Area = 60 sq. cm

Calculations:

⇒ Volume = (1 ÷ 3) × 60 × 9

⇒ Volume = 20 × 9

⇒ Volume = 180 cm³

∴ The correct answer is 180 cm³.

Additional Information

Prism vs Pyramid Volume

The volume of a prism is directly calculated as Base Area × Height, whereas a pyramid's volume is exactly one-third of the prism's volume having the same base and height.

Total Surface Area of a Pyramid

Total Surface Area = Lateral Surface Area + Base Area. For a regular pyramid, Lateral Surface Area = (1 ÷ 2) × Base Perimeter × Slant Height.

Cone as a Pyramid

A cone is a specific type of pyramid with a circular base. Its volume formula is exactly (1 ÷ 3) × πr²h, which follows the same foundational logic.

Shortcut Trick

For an equilateral triangle circumscribing a circle of radius r, the remaining area can be directly calculated as:

⇒ Remaining Area = r² × (3√3 − π)

⇒ Remaining Area = 7² × (3 × 1.732 − 3.142)

⇒ Remaining Area ≈ 49 × (5.196 − 3.142) = 49 × 2.054

⇒ Remaining Area ≈ 100.64 cm²

∴ The correct answer is 100.6 cm².

Alternate Method

Given:

Radius of the inscribed circular disc (r) = 7 cm

Formula Used:

Inradius of equilateral triangle = a ÷ 2√3 (where a is the side length)

Area of equilateral triangle = (√3 ÷ 4) × a²

Area of a circle = π × r²

r = 7 cm

a = 14√3 cm Remaining Area = Area of Triangle − Area of Circle

Calculations:

⇒ a ÷ 2√3 = 7

⇒ a = 14√3 cm

⇒ Area of the equilateral triangle = (√3 ÷ 4) × (14√3)²

⇒ Area of the equilateral triangle = (√3 ÷ 4) × 196 × 3 = 147√3 cm²

⇒ Taking √3 ≈ 1.732

⇒ Area of the triangle ≈ 147 × 1.732 = 254.604 cm²

⇒ Area of the circle = (22 ÷ 7) × 7² = 154 cm²

⇒ Remaining Area = Area of the triangle − Area of the circle

⇒ Remaining Area = 254.604 − 154 = 100.604 cm²

∴ The correct answer is 100.6 cm².

Additional Information

Circumcircle of an Equilateral Triangle

The circumradius (R) of an equilateral triangle is exactly double its inradius. Formula: R = a ÷ √3 = 2r.

Area Ratio of Incircle and Circumcircle

For any equilateral triangle, the ratio of the area of the incircle to the area of the circumcircle is always 1:4.

Height of an Equilateral Triangle

The height or median (h) divides the triangle symmetrically. Formula: h = (√3 ÷ 2) × a.

Shortcut Trick

According to the geometric properties of a triangle, the centroid divides the triangle into three smaller triangles of equal area.

These three triangles are ΔGAB, ΔGBC, and ΔGCA.

Since their areas are equal, the area of ΔGAB is exactly one-third of the total area of ΔABC.

Ratio = Area(ΔGAB) : Area(ΔABC) = 1 : 3.

∴ The correct answer is 1:3.

Alternate Method

A B C D E F G Area(ΔGAB) = 1/3 × Area(ΔABC)

Given:

⇒ AD, BE, and CF are the medians of ΔABC.

⇒ G is the centroid where all three medians intersect.

Property Used:

⇒ The medians of a triangle divide it into six smaller triangles of equal area.

⇒ Area(ΔAFG) = Area(ΔBFG) = Area(ΔBDG) = Area(ΔCDG) = Area(ΔCEG) = Area(ΔAEG)

Calculation:

⇒ Let the area of one small triangle, Area(ΔAFG) = 1 unit

⇒ Total Area(ΔABC) = 6 units

⇒ Area(ΔGAB) = Area(ΔAFG) + Area(ΔBFG)

⇒ Area(ΔGAB) = 1 + 1 = 2 units

⇒ Ratio = Area(ΔGAB) ÷ Area(ΔABC)

⇒ Ratio = 2 ÷ 6

⇒ Ratio = 1 ÷ 3

∴ The correct answer is 1:3.

Additional Information

Centroid Division Property

The centroid divides each median of the triangle in the ratio 2:1, with the longer segment always originating from the vertex (e.g., AG:GD = 2:1).

Apollonius Theorem

It establishes a relationship between the lengths of the sides of a triangle and its median. For median AD on side BC: AB² + AC² = 2 × (AD² + BD²).

Medians and Area Rule

A new triangle formed by taking the lengths of the three medians of an original triangle as its sides will have an area exactly equal to 3/4 of the area of the original triangle.

Shortcut Trick

Area of △ADE : Area of trapezoid DECB = 4 : 21

Total Area of △ABC = Area(△ADE) + Area(DECB) = 4 + 21 = 25 parts

Ratio of corresponding sides AD : AB = √(Area of △ADE) : √(Area of △ABC) = √4 : √25 = 2 : 5

Since point D is on AB, DB = AB − AD = 5 − 2 = 3 parts

∴ The correct answer is 2:3.

Alternate Method

A B C D E Area = 4 units Area = 21 units

Given:

DE is parallel to BC (DE || BC).

Area of △ADE : Area of trapezoid DECB = 4 : 21.

Formula Used:

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

Area(△ADE) ÷ Area(△ABC) = (AD ÷ AB)²

Calculations:

⇒ Area of △ABC = Area of △ADE + Area of trapezoid DECB

⇒ Area of △ABC = 4 + 21 = 25 units

⇒ In △ADE and △ABC, DE || BC

⇒ ∠ADE = ∠ABC (Corresponding angles)

⇒ ∠AED = ∠ACB (Corresponding angles)

⇒ ∴ △ADE ~ △ABC (by AA similarity criterion)

⇒ Area(△ADE) ÷ Area(△ABC) = (AD ÷ AB)²

⇒ 4 ÷ 25 = (AD ÷ AB)²

⇒ AD ÷ AB = √(4 ÷ 25) = 2 ÷ 5

⇒ Let AD = 2x and AB = 5x

⇒ Since point D lies on AB, AB = AD + DB

⇒ DB = AB − AD

⇒ DB = 5x − 2x = 3x

⇒ Ratio of AD to DB = 2x : 3x = 2 : 3

∴ The correct answer is 2:3.

Additional Information

Area of Similar Triangles

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

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. ⇒ AD ÷ DB = AE ÷ EC.

AA Similarity Criterion

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are structurally similar.

Shortcut Trick

Perimeter of larger triangle = 3 + 4 + 5 = 12.

Area of larger triangle = 1/2 × 3 × 4 = 6.

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

Area of smaller triangle = 6 × (6 ÷ 12)²

⇒ Area = 6 × (1/2)² = 6 ÷ 4 = 1.5.

∴ The correct answer is 1.5 sq. units.

Alternate Method

Given:

⇒ Sides of the larger right-angled triangle = 3, 4, and 5

⇒ Perimeter of the inscribed smaller similar triangle = 6

Formula Used:

⇒ Area of a right triangle = 1/2 × Base × Perpendicular Height

⇒ For two similar triangles: Area₁ / Area₂ = (Perimeter₁ / Perimeter₂)²

3 4 5

2 1.5 2.5

Calculations:

⇒ The perimeter of the larger triangle = 3 + 4 + 5 = 12

⇒ Area of the larger triangle = 1/2 × 3 × 4 = 6 sq. units

⇒ The ratio of their perimeters (Smaller : Larger) = 6 / 12 = 1/2

⇒ According to the similarity theorem, the ratio of their areas is the square of the scale factor.

⇒ Area Ratio = (1 / 2)² = 1 / 4

⇒ Let the area of the smaller triangle be A.

⇒ A / 6 = 1 / 4

⇒ A = 6 × (1 / 4) = 1.5 sq. units

∴ The correct answer is 1.5 sq. units.

Additional Information

Similarity Area Theorem

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides, altitudes, or perimeters: Area₁ / Area₂ = (P₁ / P₂)².

Medial Triangle Property

The triangle formed by connecting the midpoints of the sides of a given triangle is called the medial triangle. It perfectly represents a similar inscribed triangle whose area is exactly 1/4^th of the original area, and perimeter is half of the original.

Right Angled Triangle Basics

The area of a right-angled triangle is simply 1/2 × Base × Perpendicular Height. The three sides strictly follow the Pythagorean theorem relation: a² + b² = c².

Shortcut Trick For Direct Common Tangent (DCT) and distance between centers (d): d2 = DCT2 + (R − r)2 ⇒ d = √(152 + 82) ⇒ d = √(225 + 64) = √289 = 17 cm ∴ The correct answer is 17 cm. Alternate Method

C1 C2 R = 12 cm r = 4 cm DCT = 15 cm d = ? T1 T2

Given: ⇒ Radius of larger circle (R) = 12 cm ⇒ Radius of smaller circle (r) = 4 cm ⇒ Length of Direct Common Tangent (DCT) = 15 cm Formula Used: ⇒ DCT = √[d2 − (R − r)2] Where 'd' is the distance between the centers of the two circles. Calculations: ⇒ 15 = √[d2 − (12 − 4)2] ⇒ 15 = √[d2 − 82] Squaring both sides: ⇒ 225 = d2 − 64 ⇒ d2 = 225 + 64 ⇒ d2 = 289 ⇒ d = 17 cm ∴ The correct answer is 17 cm. Additional Information Transverse Common Tangent (TCT) A tangent that crosses the line joining the centers of the circles. Formula: TCT = √[d2 − (R + r)2] Touching Circles Externally When circles touch each other externally, the distance between centers is d = R + r, and the DCT length simplifies directly to 2√(R × r). Number of Common Tangents If circles are completely disjoint (d > R + r), they have exactly 4 common tangents (2 direct and 2 transverse).

Shortcut Trick

The standard identity relating the direct common tangent (L), transverse common tangent (M), distance between centers (d) and radii (r₁, r₂) is:

L² + M² = 2d² − 2(r₁² + r₂²)

Given: L² + M² = 200, and r₁² + r₂² = 100

⇒ 200 = 2d² − 2(100)

⇒ 2d² = 400 ⇒ d² = 200 ⇒ d = 10√2

∴ The correct answer is 10√2 units.

Alternate Method

Given:

Distance between centers = d

Direct common tangent = L

Transverse common tangent = M

Sum of squares of common tangents, L² + M² = 200

Sum of squares of radii, r₁² + r₂² = 100

L (Direct Tangent) M (Transverse Tangent) d C₁ C₂

Formula:

Length of direct common tangent (L) = √[d² − (r₁ − r₂)²]

Length of transverse common tangent (M) = √[d² − (r₁ + r₂)²]

Calculations:

Squaring both formulas to eliminate the square roots:

⇒ L² = d² − (r₁² + r₂² − 2r₁r₂)

⇒ M² = d² − (r₁² + r₂² + 2r₁r₂)

Adding both equations together:

⇒ L² + M² = d² − r₁² − r₂² + 2r₁r₂ + d² − r₁² − r₂² − 2r₁r₂

⇒ L² + M² = 2d² − 2(r₁² + r₂²)

Substitute the given values into the derived equation:

⇒ 200 = 2d² − 2(100)

⇒ 200 = 2d² − 200

⇒ 2d² = 400

⇒ d² = 200

⇒ d = √200 = 10√2

∴ The correct answer is 10√2 units.

Additional Information

Direct Common Tangent (DCT)

A common tangent is called a direct tangent if it does not intersect the line segment joining the centers of the circles. Its length is greater than the transverse common tangent.

Transverse Common Tangent (TCT)

A common tangent is called a transverse tangent if it intersects the line segment connecting the centers of the two circles.

Number of Common Tangents

Two disjoint circles have exactly 4 common tangents (2 DCT and 2 TCT). If the circles touch externally, they have 3 common tangents. Intersecting circles only have 2 direct common tangents.

Shortcut Trick

The half-lengths of the two chords are 16 ÷ 2 = 8 cm and 12 ÷ 2 = 6 cm.

In a circle, the radius, half-chord, and perpendicular distance from the center form a right-angled triangle.

Notice that the lengths 6, 8, and 10 form a standard Pythagorean triplet (6² + 8² = 10²).

The distances from the center to the chords are 6 cm and 8 cm, and their difference is 8 − 6 = 2 cm, matching the given spacing perfectly.

Therefore, the hypotenuse, which represents the circle's radius, must be 10 cm.

∴ The correct answer is 10 cm.

Alternate Method

Given:

Length of larger chord (AB) = 16 cm

Length of smaller chord (CD) = 12 cm

Space between the chords = 2 cm

O

C D

A B

P Q

8 6 2 x r r

Formula Used:

A perpendicular drawn from the center of a circle to a chord bisects the chord.

According to Pythagoras theorem: r² = (Perpendicular distance)² + (Half of chord)²

Calculations:

Let the perpendicular distance of the larger chord from the center be x cm.

Since both chords are on the same side of the center, the distance of the smaller chord from the center will be (x + 2) cm.

Half of larger chord = 16 ÷ 2 = 8 cm

Half of smaller chord = 12 ÷ 2 = 6 cm

Applying the Pythagoras theorem for the larger chord:

⇒ r² = x² + 8²

⇒ r² = x² + 64 --- (Equation 1)

Applying the Pythagoras theorem for the smaller chord:

⇒ r² = (x + 2)² + 6²

⇒ r² = x² + 4x + 4 + 36

⇒ r² = x² + 4x + 40 --- (Equation 2)

Equating (Equation 1) and (Equation 2):

⇒ x² + 64 = x² + 4x + 40

⇒ 4x = 64 − 40

⇒ 4x = 24

⇒ x = 6 cm

Substituting the value of x into Equation 1:

⇒ r² = 6² + 64

⇒ r² = 36 + 64 = 100

⇒ r = √100 = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Perpendicular Bisector Theorem

A perpendicular dropped from the center of a circle to a chord bisects the chord into two equal halves.

Parallel Chords on Opposite Sides

If two parallel chords lie on opposite sides of the center, the perpendicular distance between them is the sum of their individual distances from the center (d₁ + d₂).

Pythagorean Triplets

Many geometry problems can be solved significantly faster by recognizing common triplets like (3, 4, 5), (6, 8, 10), or (5, 12, 13) which perfectly satisfy the equation a² + b² = c².

Shortcut Trick

Direct formula for the Area of a Segment:

⇒ Area = r² × [(π × θ ÷ 360°) − (sin θ ÷ 2)]

⇒ Area = 12² × [(π × 150° ÷ 360°) − (sin 150° ÷ 2)]

⇒ Area = 144 × [(5π ÷ 12) − (1 ÷ 4)]

⇒ Area = 60π − 36 cm²

∴ The correct answer is 60π−36 cm².

Alternate Method

150° 12 cm 12 cm O A B Segment Area of Segment = Area of Sector − Area of Triangle

Given:

⇒ Radius of the circle (r) = 12 cm

⇒ Angle subtended at the center (θ) = 150°

Formula Used:

⇒ Area of Sector = (θ ÷ 360°) × πr²

⇒ Area of Triangle = (1 ÷ 2) × r² × sin θ

⇒ Area of Segment = Area of Sector − Area of Triangle

Calculations:

⇒ Area of Sector = (150° ÷ 360°) × π × 12²

⇒ Area of Sector = (5 ÷ 12) × π × 144 = 60π cm²

⇒ To find the area of the triangle, we use sin(150°) = sin(180° − 30°) = sin(30°) = 1 ÷ 2

⇒ Area of Triangle = (1 ÷ 2) × 12² × (1 ÷ 2)

⇒ Area of Triangle = (1 ÷ 2) × 144 × (1 ÷ 2) = 36 cm²

⇒ Area of Segment = 60π − 36 cm²

∴ The correct answer is 60π−36 cm².

Additional Information

Area of Major Segment

It is calculated by subtracting the area of the minor segment from the total area of the circle: Area = πr² − Area of Minor Segment.

Length of an Arc

The arc length corresponding to a central angle θ is a fraction of the circle's circumference: L = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The boundary length of a sector comprises the arc length and two radii: Perimeter = L + 2r.

The correct answer is Option 4: Carefree.

Key Points

• The word "Nonchalant" means feeling or appearing casually calm and relaxed; not displaying anxiety, interest, or enthusiasm. (बेपरवाह / उदासीन)
• Example: She gave a nonchalant shrug when asked about the upcoming exam.
• "Carefree" refers to being free from anxiety or worry; relaxed and unconcerned. (निश्चिंत / बेफिक्र)
• Example: The children enjoyed a carefree day at the park without any worries.
• Thus, "carefree" closely matches the meaning of "nonchalant" as both describe a relaxed and unconcerned attitude.

Therefore, the correct answer is- Carefree.

Additional Information

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

• Anxious (चिंतित): Feeling worried, nervous, or uneasy about something.
• Example: He felt anxious before the interview.
• Tense (तनावग्रस्त): Unable to relax because of nervousness or anxiety.
• Example: The atmosphere in the room became tense during the discussion.
• Worried (चिंतित): Feeling troubled or anxious about actual or potential problems.
• Example: She was worried about her exam results.

The correct answer is Option 2 :To keep away or at a safe distance.

Key Points

• The idiom "Keep at bay" means to prevent someone or something from approaching or causing harm; to maintain distance. (दूर रखना / पास न आने देना)
• Example: She used mosquito repellent to keep insects at bay.
• "To keep away or at a safe distance" means to prevent danger or trouble from coming near. (सुरक्षित दूरी पर रखना)
• Example: Regular exercise helps keep diseases at a safe distance.
• Hence, the meaning of "Keep at bay" closely matches "to keep away or at a safe distance."

Therefore, the correct answer is Option 2.

Additional Information

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

• To monitor closely (ध्यानपूर्वक निगरानी करना): To watch or observe carefully.
• Example: The teacher monitored the students closely during the exam.
• To defeat completely (पूरी तरह पराजित करना): To win over someone decisively.
• Example: The team defeated their opponents completely in the final match.
• To take advantage of (फायदा उठाना): To use a situation for personal benefit.
• Example: He took advantage of the opportunity to improve his skills.

The correct answer is Option 3: Pragmatic.

Key Points

• The word "Quixotic" means exceedingly idealistic, unrealistic, or impractical in pursuing noble ideas. (अव्यावहारिक / आदर्शवादी लेकिन अवास्तविक)
• Example: His quixotic plan to eliminate poverty overnight was admirable but unrealistic.
• "Pragmatic" refers to dealing with things realistically and practically rather than theoretically. (व्यावहारिक)
• Example: She took a pragmatic approach to solving the financial crisis.
• Since "quixotic" suggests impractical idealism and "pragmatic" suggests practical realism, they are opposites in meaning.

Therefore, the correct answer is Option 3.

Additional Information

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

• Romantic (रोमांटिक / भावुक): Characterized by idealized or emotional views.
• Example: He has a romantic notion of living in the countryside.
• Visionary (दूरदर्शी): Having original and imaginative ideas about the future.
• Example: The visionary leader introduced revolutionary reforms.
• Idealistic (आदर्शवादी): Guided by ideals or high principles, often unrealistic.
• Example: Her idealistic views sometimes clash with reality.

The correct answer is Option 3: Onomatopiea.

Key Points

• The word "Onomatopiea" is incorrectly spelt. The correct spelling is "Onomatopoeia". (ध्वन्यात्मक शब्द)
• Example: Words like "buzz" and "hiss" are examples of onomatopoeia.
• "Onomatopoeia" refers to the formation of a word from a sound associated with what is named. (ध्वन्यनुकरण)
• Example: The word "clang" imitates the sound it represents, making it an example of onomatopoeia.
• Hence, "Onomatopiea" is the misspelt word among the given options.

Therefore, the correct answer is- Onomatopiea.

Additional Information

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

• Apothegm (सूक्ति): A short, witty, and instructive saying.
• Example: "Know thyself" is a famous apothegm.
• Esoterica (गूढ़ विषय): Things that are understood by only a small number of people with specialized knowledge.
• Example: The lecture was filled with philosophical esoterica.
• Sesquipedalian (दीर्घ शब्दों का प्रयोग करने वाला): Given to using long words.
• Example: His sesquipedalian style made the speech difficult to follow.

The correct answer is Option 3: Myocardial.

Key Points

• The adjective "Myocardial" refers to anything relating to the muscular tissue of the heart (myocardium). (हृदय की पेशीय ऊतक से संबंधित)
• Example: The patient was admitted after suffering a myocardial infarction.
• The word is derived from "myo" (muscle) and "cardial" (relating to the heart).
• Hence, "Myocardial" is the correct spelling that accurately describes the muscular tissue of the heart.

Therefore, the correct answer is- Myocardial.

Additional Information

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

• Myocardiac (गलत रूप): Incorrect formation; not the accepted medical adjective.
• Myocordial (गलत वर्तनी): Incorrect spelling; not used in medical terminology.
• Myocardiol (अशुद्ध शब्द): Not a valid medical term.

The correct answer is Option 3: Arson.

Key Points

• The term "Arson" refers to the criminal act of deliberately setting fire to property. (आगजनी)
• Example: The accused was arrested on charges of arson after the warehouse was set ablaze.
• It is considered a serious crime because it endangers life and causes heavy property damage.
• Thus, "Arson" precisely matches the given definition.

Therefore, the correct answer is Option 3.

Additional Information

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

• Burglary (सेंधमारी): Illegal entry into a building with intent to commit a crime, especially theft.
• Example: The police investigated the burglary at the jewellery shop.
• Vandalism (तोड़फोड़): Deliberate destruction or damage to public or private property.
• Example: The monument was damaged due to vandalism.
• Sabotage (तोड़फोड़ द्वारा नुकसान पहुँचाना): Deliberate action aimed at weakening or damaging something, often secretly.
• Example: The workers were accused of sabotage during the strike.

The correct answer is Option 1: Glossophobia.

Key Points

• The term "Glossophobia" refers to the fear of public speaking. (सार्वजनिक भाषण का भय)
• Example: Due to glossophobia, he avoided giving presentations in front of large audiences.
• The word is derived from the Greek word "glossa" meaning tongue and "phobia" meaning fear.
• Hence, "Glossophobia" correctly describes the fear of speaking in public.

Therefore, the correct answer is- Glossophobia.

Additional Information

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

• Sociophobia (सामाजिक भय): Fear of social interactions or social situations.
• Example: Sociophobia can make attending social gatherings stressful.
• Anthophobia (फूलों का भय): Fear of flowers.
• Example: People with anthophobia may feel anxious around gardens.
• Ergophobia (काम का भय): Fear of work or the workplace.
• Example: Ergophobia can interfere with a person’s professional life.

The correct answer is Option 2: was it.

Key Points

• The sentence begins with "Not until", which is a negative adverbial expression.
• When a sentence starts with a negative expression like "Not until", it requires subject-auxiliary inversion.
• The correct structure becomes: Auxiliary verb + subject + main verb.
• Therefore, "was it accepted" follows the correct inverted structure.

Correct sentence: Not until the data was independently verified was it accepted by the journal.

Additional Information

• Option 1: "it was" – Incorrect because inversion is required after "Not until."
• Option 3: "did it" – Incorrect as the sentence is in passive voice and requires "was," not "did."
• Option 4: "had it" – Incorrect because past perfect is not required in this context.

The correct answer is Option 1: whose

Key Points

• The blank introduces a non-restrictive relative clause giving additional information about "the chairperson".
• The word required must show possession (authority belonging to the chairperson).
• "Whose" is a possessive relative pronoun used for both people and things.
• Thus, the correct structure is: "the chairperson, whose authority was rarely contested,"

Correct sentence: Though the board remained divided on procedural issues, the chairperson, whose authority was rarely contested, pushed the resolution through.

Additional Information

• Option 2: "whom" – Used as an object pronoun, not to show possession.
• Option 3: "who's" – Contraction of "who is" or "who has"; incorrect in possessive context.
• Option 4: "which" – Used for things, not appropriate for referring to a person in this context.