SSC CGL 2025 (Held On: 25 Sept, 2025 Shift 3) (Previous Year Paper)

The logic followed here is:

Vowels are assigned even numbers as:

A = 2, E = 4, I = 6, O = 8, U = 10

Word: FUNDAMENTAL

Vowels in the word:

U, A, E, A

U = 10

A = 2

E = 4

A = 2

Sum of vowel values:

10 + 2 + 4 + 2 = 18

Hence, the correct option is Option 3.

Given Address:

Flat No. 101, Maple Residency, Koregaon Park, Pune, Maharashtra - 411001

Checking each option:

1. Flat No. 101, Maple Residency, Koregaon Park, Pune, Maharashtra - 411002 (Incorrect PIN Code)
   

2) Flat No. 101, Maple Residency, Koregaon Park, Pune, Maharashtra - 411001 (Exact Match)

3. Flat No. 101, Maple Residency, Koregaon Park, Pune, Maharashtra - 411011 (Incorrect PIN Code)
   

4. Flat No. 101, Maple Residency, Koregaon Park, Pune, Maharashtra - 410001 (Incorrect PIN Code)

Hence, the correct option is "Option 2".

Given:

Seminars were held daily from 11th Feb to 18th Feb 2022.

Raipur – 11th Feb 2022

Patna – 12th Feb 2022

Jamshedpur – 13th Feb 2022

Rourkela – 14th Feb 2022

Durgapur – 15th Feb 2022

Warangal – 16th Feb 2022

Tiruchirappalli – 17th Feb 2022

Calicut – 18th Feb 2022

Given that the seminar in Jamshedpur (13th Feb 2022) was held on Sunday.

So, 13th Feb 2022 = Sunday

Now counting forward:

14th Feb – Monday

15th Feb – Tuesday

16th Feb – Wednesday

17th Feb – Thursday

18th Feb – Friday

Hence, the correct option is Option 3.

Given:

3 @ 5 = 14

Finding the logic:

3 @ 5 = (3 × 3) + 5

= 9 + 5

= 14

Applying the same logic:

4 @ 100 = (4 × 4) + 100

= 16 + 100

= 116

Hence, the correct option is Option 2.

Given:

987 → 24

654 → 15

Finding the logic:

987 → 9 + 8 + 7 = 24

654 → 6 + 5 + 4 = 15

Applying the same logic:

321 → 3 + 2 + 1 = 6

Hence, the correct option is Option 1.

Given Equation:

72 ÷ 6 A 2 × 216 B 24 = 420

Solve the known operations first (BODMAS rule)

Now check the options:

A = − and B = +

12 − 432 + 24

= -420 + 24

= -396 (Not equal to 420)

A = + and B = ×

12 + 432 × 24

= 12 + 10368

= 10380 (Not equal to 420)

A = + and B = −

12 + 432 − 24

= 444 − 24

= 420

A = − and B = ×

12 − 432 × 24

= 12 − 10368

= -10356 (Not equal to 420)

Hence, the correct option is Option 3.

Given,

Kolkata, Darjeeling and Siliguri are cities in West Bengal, whereas Pune is in Maharashtra.

Pune is the odd one out.

Hence, the correct option is Option 4.

Given:

A is the son of C.

C is the wife of D.

Explanation

D is the father of A.

Hence, the correct answer is Option 2.

Given:

Increase in 1st year = 10%

Increase in 2nd year = 20%

Explanation:

Let the initial population be 100.

After 1st year:

100 + 10% of 100 = 100 + 10 = 110

After 2nd year:

110 + 20% of 110 = 110 + 22 = 132

Total increase = 132 − 100 = 32

Percentage increase = 32%

Hence, the correct answer is Option 2.

Logic: (1st number)³ - (1st number)² = 2nd number.

Given: 5 : 100

⇒ 5³ - 5² = 100

⇒ 125 - 25 = 100.

Similarly,

6 : ?

⇒ 63 - 62 = ?

⇒ 216 - 36 = 180

Hence, the correct option is Option 3.

The least possible Venn diagram for the given statements is as shown below

Conclusions:

I. All engineers are allowed → not follow (As being skilled is a necessary but not sufficient condition for being allowed).

II. No skilled person is unskilled → not follow (As this relies on external definitions not provided in the premises).

Thus, neither I nor II follows.

Hence, the correct answer is “Option 4”.

Statement:

All boxes are bags.

Some bags are blue.

No blue is a plastic

Conclusion I: Some boxes are not plastic.

Some bags are blue and no blue is plastic, but it is not given that those blue bags are boxes. Hence, we cannot definitely conclude that some boxes are not plastic.

Conclusion II: All plastics are bags.

There is no information that plastics are bags. So, this conclusion does not follow.

Hence, the correct answer is Option 4.

Given:

BHABHA = AGZAGZ

Finding the logic:

B → A (−1)

H → G (−1)

A → Z (−1, cyclic)

B → A (−1)

H → G (−1)

A → Z (−1, cyclic)

Each letter is replaced by the previous letter in the English alphabet.

Applying the same logic to RAMAN:

R → Q

A → Z

M → L

A → Z

N → M

RAMAN = QZLZM

Hence, the correct answer is Option 1.

Given Sequence:

B7, D9, F11, H13, J17

Observing the pattern:

Letters:

B, D, F, H, J → Each letter increases by 2 positions in the alphabet.

Numbers:

7, 9, 11, 13, 17

7 → 9 (+2)

9 → 11 (+2)

11 → 13 (+2)

13 → 15 (+2) should come, but given 17.

So, the number with J should be 15, not 17.

Hence, the pair that does NOT belong to the sequence is Option 4.

Given Series:

1, 8, 27, ?

Observing the pattern:

1 = 1³

8 = 2³

27 = 3³

The pattern is cubes of natural numbers.

Next term = 4³ = 64

Hence, the missing number is 64.

Given

Ganga, Brahmaputra and Yamuna are rivers of India, whereas Amazon is a river in South America.

Hence the correct option is Option 3.

Given Series:

AZ, CY, EX, GW, ?

Observing the pattern:

First letters:

A, C, E, G → Each increases by 2 positions.

Next letter after G = I

Second letters:

Z, Y, X, W → Each decreases by 1 position.

Next letter after W = V

So, the next term = IV

Hence, the correct answer is Option 4.

Given Series

13 + 3 = 16

16 × 2 = 32

32 + 3 = 35

35 × 2 = 70

The pattern is:

+3, ×2, +3, ×2 ...

Continuing the pattern

70 + 3 = 73

Hence, the correct option is Option 1.

Given:

Principal (P) = ₹10,000

Rate of Interest (R) = 8% per annum

Time (T) = 2 years

Formula for Compound Interest:

Amount = P (1 + R/100)^T

Calculation:

Amount = 10000 (1 + 8/100)²

= 10000 (1.08)²

= 10000 × 1.1664

= 11,664

Hence, the correct answer is Option 1.

Given:

FIGHT = EJHIS

Finding the pattern:

F → E (−1)

I → J (+1)

G → H (+1)

H → I (+1)

T → S (−1)

First and last letters −1, middle letters +1.

Applying the same logic to BRAVE:

B → A (−1)

R → S (+1)

A → B (+1)

V → W (+1)

E → D (−1)

BRAVE = ASBWD

Hence, the correct answer is Option 3.

Given:

A number is divisible by both 5 and 9.

Checking options:

35 → Not a multiple of 9

90 → multiple of  both 5, 9

47 → Not a multiple of 9

60 → Not a multiple of 9

Hence, the correct answer is Option 2.

Given:

WORLD = 23-15-18-12-4

Finding the logic:

W = 23

O = 15

R = 18

L = 12

D = 4

Each letter is coded according to its position in the English alphabet.

Applying the same logic to DORLW:

D = 4

O = 15

R = 18

L = 12

W = 23

DORLW = 4-15-18-12-23

Hence, the correct answer is Option 1.

Given statement: Pointing to a photograph, Rohan said, "He is the only son of my father's only daughter.

Case 1) If Rohan is a male member

Case 2) If Rohan is a female member

So, person in the photograph is Rohan's either nephew or son. As out of all the options "Nephew" is given. Therefore case 2 is eliminated.

Thus, person in the photograph is Rohan's nephew.

Hence, 'option 2' is the correct answer.

Given:

MARKET = NBSLFU

Finding the pattern:

M → N (+1)

A → B (+1)

R → S (+1)

K → L (+1)

E → F (+1)

T → U (+1)

Each letter is replaced by the next letter in the English alphabet.

Applying the same logic to NBSLFU:

N → O

B → C

S → T

L → M

F → G

U → V

Hence, the correct answer is Option 1.

Given Series:

11, 13, 17, 19, 23, ?

Observing the pattern:

11 → Prime number

13 → Prime number

17 → Prime number

19 → Prime number

23 → Prime number

All numbers are consecutive prime numbers.

The next prime number after 23 is 29.

Hence, the correct option is Option 2.

The correct answer is Guillotine.

Key Points

• The term Guillotine is used in parliamentary procedures to refer to the process where the discussion on demands for grants is curtailed or bypassed, and the demands are put to vote after the stipulated time limit has been reached.
• This practice ensures that the Parliament can meet its deadlines for financial legislation without unnecessary delays.
• It is primarily applied during the budget session of the Parliament when the discussion on the budget has to be completed within a specific timeframe.
• The Speaker or Chairperson of the House applies the guillotine to speed up the process of voting on the remaining demands for grants without detailed discussions.
• Once the guillotine is invoked, all remaining demands are voted upon in a single motion, saving valuable legislative time. This practice is essential for maintaining the fiscal schedule of the government.

Additional Information

• Cut Motions:
• A cut motion is a parliamentary procedure allowing members of the opposition to question specific allocations in the budget.
• There are three types of cut motions: Policy Cut, Economy Cut, and Token Cut.
• For example, a Policy Cut motion seeks to express disapproval of the policy underlying the budget, while an Economy Cut motion focuses on reducing expenditure by a specific amount.
• Motions:
• Motions are formal proposals made by members of Parliament to discuss, approve, or disapprove policies, issues, or specific matters.
• Examples include adjournment motions, no-confidence motions, and censure motions.
• Point of Order:
• A point of order is raised when a member believes that parliamentary rules or procedures are being violated during a debate.
• The Speaker or Chairperson decides whether the point of order is admissible or not, ensuring that the proceedings remain in order.
• Budget Process in India:
• The Union Budget is presented by the Finance Minister in Parliament, detailing the government’s revenue and expenditure for the fiscal year.
• The budget process includes stages such as the presentation of the budget, general discussion, scrutiny by departmental standing committees, and voting on demands for grants.
• The application of the guillotine marks the end of the voting process in the Lok Sabha.

Important Points

• Purpose of Guillotine:
• The guillotine ensures that the financial business of the government is not delayed, allowing funds to be released for government activities and programs.
• It is a time management tool for the legislature to handle its workload efficiently.
• Historical Context:
• The term "guillotine" originates from the French Revolution, where it was used for a swift and decisive execution process. In parliamentary terms, it symbolizes the swift closure of discussions.
• Constitutional Provisions:
• The budget process in India is governed by Articles 112 to 117 of the Constitution.
• Article 112 refers to the annual financial statement, while Article 113 deals with demands for grants.

The correct answer is Puducherry (Pondicherry).

Key Points

• Puducherry is renowned for its French colonial architecture, reflecting its history as a former French colony.
• The city features quaint streets with French names, lined with colonial-style villas, beautiful churches, and vibrant cultural buildings.
• Puducherry's architecture is influenced by the French design style, characterized by symmetrical layouts, pastel-colored facades, and ornate doorways.
• The French Quarter, also known as the "White Town," is a prominent area where colonial heritage is most evident. It contains numerous heritage buildings that are well-preserved and reflect French aesthetics.
• Puducherry is famous for landmarks like the Notre Dame des Anges Church, the French War Memorial, and Aurobindo Ashram, which display the city's unique blend of Indian and French cultural influences.

Additional Information

• Historical Background:
• Puducherry was under French rule from 1674 until 1954 when it was formally transferred to India.
• During their rule, the French developed Puducherry as a trading post and established urban planning that adhered to French architectural standards.
• The city is divided into French and Tamil quarters, showcasing a blend of cultural styles.
• Key Architectural Features:
• The French colonial villas in Puducherry are characterized by high ceilings, large windows, and spacious courtyards.
• The use of colorful facades and intricate ironwork highlights the artistic influence of French architecture.
• The streets of the French Quarter are laid out in a grid pattern, a hallmark of colonial urban planning.
• Tourism and Preservation:
• Puducherry attracts visitors from around the world due to its unique cultural and architectural heritage.
• The Indian government and local authorities have taken steps to preserve heritage buildings, ensuring the legacy of French architecture endures.
• Several buildings in Puducherry have been converted into heritage hotels and museums, allowing tourists to experience the colonial charm.
• Comparison to Other Indian Cities:
• Unlike cities like Goa, which reflects Portuguese colonial influences, Puducherry is distinctly influenced by French culture and architecture.
• Hyderabad and Kolkata, while rich in history, exhibit architectural styles rooted in Mughal and British colonial designs, respectively.

Important Points

• UNESCO World Heritage Status:
• Puducherry's French Quarter has been proposed for inclusion in the UNESCO World Heritage Site list due to its cultural and architectural significance.
• Efforts are underway to ensure the conservation of its unique heritage.
• Notable Events:
• The annual Bastille Day celebrations in Puducherry reflect its French connection, with locals and visitors participating in cultural events.
• The city also hosts the International Yoga Festival, showcasing its spiritual heritage alongside its colonial charm.

The correct answer is Annie Ernaux.

Key Points

• Annie Ernaux, a renowned French author, was awarded the Nobel Prize in Literature in 2022 for her exceptional contribution to literature.
• The Nobel Committee recognized her for her courage and clinical acuity in uncovering the roots, estrangements, and collective restraints of personal memory.
• Ernaux's works often explore the themes of memory, identity, and socio-political dynamics, making her one of the most influential voices in contemporary literature.
• Her autobiographical writings, such as "The Years" (2008), have been lauded for their innovative narrative style, blending personal and collective history.
• She is widely celebrated for her ability to transform personal experiences into universal narratives, which resonate with a global audience.

Additional Information

• Annie Ernaux:
• Born on 1 September 1940, in Lillebonne, France, she grew up in a working-class environment.
• Her writings reflect themes of class struggle, feminism, and personal identity.
• She has authored several acclaimed books, including "Simple Passion" (1992) and "Happening" (2000), which delve into personal and societal issues.
• The Nobel Prize in Literature:
• This prestigious award is presented annually by the Swedish Academy to authors who have produced outstanding work in the field of literature.
• It was established in 1901 as per the will of Alfred Nobel, the inventor of dynamite.
• Winners receive a medal, a diploma, and a monetary award.
• Other Nominees:
• Leila Slimani: A Moroccan-French author known for "The Perfect Nanny" (2016).
• Marie NDiaye: A French novelist awarded the Goncourt Prize in 2009.
• Delphine de Vigan: A celebrated French writer, known for her psychological novels like "No and Me" (2007).
• Important Points:
• "The Years", one of Ernaux's most celebrated works, is considered a masterpiece of modern literature.
• Her writings often challenge societal norms and explore feminist and existentialist themes.
• The Nobel Prize in Literature has been awarded to 16 women, including Ernaux, as of 2022.

The correct answer is The best athlete.

Key Points

• The David Dixon Award is presented at the Commonwealth Games to recognize the most outstanding athlete of the Games. This award is not given to teams, host cities, or nations, but rather to an individual athlete who has exhibited exceptional performance and sportsmanship.
• The award was first introduced in the 2002 Commonwealth Games held in Manchester, UK, and has since been a significant honor at every edition of the Games.
• Winners of the David Dixon Award are chosen based on their sporting excellence, adherence to the values of the Commonwealth Games, and demonstration of good sportsmanship throughout the competition.
• This prestigious award is named after David Dixon, the former honorary secretary of the Commonwealth Games Federation (CGF), in recognition of his immense contributions to the Games and its development.
• The selection process involves a panel of experts who evaluate athletes' performances across all disciplines, ensuring that the recipient truly embodies the spirit of the Commonwealth Games.

Additional Information

• History of the David Dixon Award:
• The award was introduced to honor David Dixon, who played a key role in shaping the modern Commonwealth Games. He was instrumental in promoting the values of fairness, inclusivity, and athletic excellence within the Games.
• It serves as a legacy of his dedication to the Commonwealth sporting community and his vision of fostering goodwill and camaraderie through sports.
• Criteria for the Award:
• Selection is not solely based on the number of medals won but also on the athlete’s overall impact and conduct during the Games.
• Attributes such as leadership, respect for opponents, and an ability to inspire others are key considerations in the selection process.
• Notable Recipients:
• In 2002, the first-ever recipient of the award was Francis Hoenselaar, a Dutch table tennis player, who impressed the panel with her skills and sportsmanship.
• Other notable winners include track and field athletes, swimmers, and gymnasts, showcasing the diversity and talent across different sports in the Commonwealth Games.
• Significance of the Award:
• The David Dixon Award highlights the importance of integrity and ethical conduct in sports, reinforcing the core values of the Commonwealth Games Federation.
• It serves as an inspiration for young athletes to excel not only in their respective sports but also in their behavior and attitude towards competitors and the sporting community.

Important Points

• About the Commonwealth Games:
• The Commonwealth Games is a multi-sport event held every four years, featuring athletes from the member nations of the Commonwealth.
• The event is guided by the principles of equality, inclusivity, and respect, promoting unity among countries with diverse cultures and backgrounds.
• Role of the Commonwealth Games Federation (CGF):
• The CGF is responsible for the overall management and organization of the Games.
• It emphasizes values such as humanity, equality, and destiny, which align with the objectives of the David Dixon Award.

The correct answer is Create a class of loyal zamindars who would ensure regular land revenue collection..

Key Points

• The Permanent Settlement was introduced in 1793 by Lord Cornwallis during British rule in India, specifically in the provinces of Bengal, Bihar, and Odisha.
• The primary objective of the settlement was to create a class of loyal zamindars (landlords) who would act as intermediaries between the British administration and the peasants and ensure the regular collection of land revenue.
• Zamindars were made the hereditary owners of the land under their control and were required to pay a fixed amount of revenue to the British government, irrespective of the agricultural output or economic conditions.
• This system ensured a steady and predictable income for the British administration while transferring the risk of agricultural failure to the zamindars and peasants.
• The system was intended to make the zamindars economically and politically loyal to the British, as they were given significant social and economic privileges under this arrangement.

Additional Information

• Objective of the Permanent Settlement:
• The Permanent Settlement aimed to bring stability to land revenue collection and provide a consistent source of income for the British government.
• It sought to reduce administrative burdens by delegating revenue collection responsibilities to zamindars.
• Key Features of the Permanent Settlement:
• Revenue was fixed permanently and was not subject to revision, which often led to financial distress for zamindars during poor harvests.
• If zamindars failed to pay the fixed revenue, they could lose their rights to the land, which would then be auctioned to recover dues.
• The system made zamindars de facto landowners, often leading to exploitation of peasants, as they were not directly under British protection.
• Social and Economic Impacts:
• While the system benefited the British government and zamindars, it left peasants in a highly vulnerable position, subject to high rents and exploitation.
• It led to the emergence of a wealthy zamindar class but also increased inequality and poverty in rural areas.
• The rigidity of the system caused many zamindars to default on payments, leading to frequent auctions of land.
• Comparison with Other Revenue Systems:
• In contrast to the Permanent Settlement, the Ryotwari System, implemented in southern India, directly involved peasants (ryots) in revenue collection, bypassing intermediaries.
• The Mahalwari System, prevalent in northern India, involved village communities in revenue collection.

Important Points

• Option Analysis:
• Option 1: "Ensure the welfare of the peasant farmers" – This was not the primary aim of the Permanent Settlement, as it often led to the exploitation of peasants.
• Option 3: "Promote the cultivation of cash crops like indigo and cotton" – This was a goal of other British policies but not the main objective of the Permanent Settlement.
• Option 4: "Abolish the zamindari system altogether" – The Permanent Settlement aimed to strengthen the zamindari system, not abolish it.
• The policy was a significant step in the colonial restructuring of Indian society and economy, with long-term implications for land ownership and agrarian relations in India.

The correct answer is e-FIR, e-summons and e-warrants.

Key Points

• Bharatiya Nagarik Suraksha Sanhita (BNSS) is a proposed legislation aimed at overhauling India’s criminal justice system, bringing in significant technological advancements to improve efficiency and transparency.
• One of the primary technological shifts under BNSS is the introduction of e-FIR, e-summons, and e-warrants, ensuring faster and more efficient processing of cases and reducing delays in legal procedures.
• An e-FIR enables citizens to lodge First Information Reports (FIRs) online, eliminating the need to visit police stations physically, thus increasing accessibility and reducing inconvenience.
• e-summons and e-warrants allow for the electronic issuance and transmission of legal documents, ensuring quicker delivery and reducing chances of tampering or delays in communication.
• This technological integration is expected to streamline criminal justice procedures and enhance accountability across all stakeholders in the legal and enforcement system.

Additional Information

• e-FIR:
• An e-FIR facility allows individuals to file complaints online, particularly for non-cognizable offenses, without visiting a police station.
• The system ensures better documentation, minimizes human errors, and provides a transparent mechanism for complaint registration.
• For instance, states like Delhi, Maharashtra, and Karnataka have already implemented e-FIR systems for public convenience.
• e-Summons:
• e-Summons allows courts to issue summons electronically, ensuring quicker communication with parties involved in legal cases.
• This system reduces delays caused by manual handling and enhances accountability in judicial proceedings.
• e-Warrants:
• e-Warrants are electronically generated and transmitted warrants, ensuring their secure and tamper-proof delivery to enforcement agencies.
• It simplifies the process for law enforcement to act on judicial orders, reducing procedural delays significantly.
• Technological Integration in Criminal Justice:
• The proposed technological advancements under BNSS aim to modernize the criminal justice system, focusing on accessibility, transparency, and efficiency.
• Other features under BNSS may include the digitization of case records, AI-based analytics for case management, and centralized databases for real-time updates.

Important Points

• AI-based trial judgment generator: While AI-based tools can assist in legal research and case management, the use of AI to generate trial judgments is not currently part of BNSS reforms as it raises ethical and legal concerns.
• Drone-based crime scene scanning: Though innovative, this technology is not explicitly mentioned under BNSS but can be considered in future reforms for forensic investigations.
• Virtual trial-only courts: Virtual courts are being introduced selectively for specific cases like traffic violations but are not mandated for all offenses under BNSS.
• e-FIR, e-summons, and e-warrants: These are confirmed technological shifts under BNSS, focusing on improving procedural efficiency and reducing delays in the criminal justice system.

The correct answer is Chowk and Tribhanga.

Key Points

• Chowk:
• The "Chowk" posture is a square-like stance that symbolizes stability and strength. It is a fundamental posture in Odissi dance and represents the masculine aspect of the dance, often associated with the depiction of Lord Jagannath.
• It is performed with bent knees and thighs turned out, creating a symmetrical and grounded form. This symbolizes the earth and conveys strength and sturdiness.
• The Chowk posture forms the basis for many movements and is integral to the dance's rhythmic patterns and footwork.
• Tribhanga:
• The "Tribhanga" posture is a characteristic pose in Odissi dance that involves the division of the body into three bends: at the neck, torso, and knee.
• It represents the feminine grace and elegance of the dance form, contrasting with the masculine Chowk posture.
• The Tribhanga posture is often used to depict the sensuality and fluidity of the dance, symbolizing beauty and divinity, especially in depictions of female deities like Goddess Radha.
• It is highly intricate and requires precise control over the body to maintain the three bends while executing fluid movements.
• These two postures, Chowk and Tribhanga, form the foundation of Odissi dance and are used extensively throughout its performance to convey a wide range of emotions and stories.
• Odissi is one of the eight classical dance forms of India, originating in Odisha, and is known for its lyrical and graceful movements. The interplay between Chowk (masculine) and Tribhanga (feminine) postures gives the dance its unique identity.

Additional Information

• Importance of Postures in Odissi Dance:
• The postures in Odissi dance are inspired by the sculptures of temples like the Sun Temple at Konark. These postures are not only aesthetic but also have spiritual significance, often representing divine figures and cosmic energies.
• While Chowk represents the grounded, earthly nature, Tribhanga symbolizes the celestial and graceful aspects of the dance form.
• Other Classical Dance Forms of India:
• Bharatanatyam: Originates from Tamil Nadu, characterized by fixed upper torso, bent legs, and intricate footwork.
• Kathak: Originates from North India, known for its intricate foot movements, spins, and emotive expressions.
• Kathakali: Originates from Kerala, noted for its elaborate costumes, makeup, and storytelling.
• Kuchipudi: Originates from Andhra Pradesh, combines fast rhythms with fluid movements and includes theatrical elements like dialogue delivery.
• Manipuri: Originates from Manipur, characterized by its focus on graceful, flowing movements.
• Mohiniyattam: Originates from Kerala, known for its feminine grace and storytelling.
• Sattriya: Originates from Assam, introduced by the Vaishnavite saint Shankaradeva, and performed in monasteries.
• Specific Characteristics of the Options:
• Tandava and Lasya: These are foundational concepts in Indian classical dance that represent the masculine and feminine aspects of dance. Tandava depicts vigor and strength, while Lasya represents grace and beauty. However, these are not specific to Odissi dance but are overarching themes in Indian classical dances.
• Chakkar and Mudra: Chakkar refers to spins or circular movements, while Mudra denotes hand gestures used in storytelling. These are elements of various Indian dance forms but not foundational postures in Odissi.
• Natya and Nritya: These are Sanskrit terms where Natya refers to dramatic art (dance-drama), and Nritya refers to expressional dance. While these are components of Odissi, they are not postures.
• Odissi's Historical and Cultural Significance:
• Odissi has its roots in the temples of Odisha and was originally performed as a form of devotional offering to deities.
• It is one of the oldest classical dance forms in India, with references dating back to the Natyashastra, an ancient treatise on performing arts attributed to Bharata Muni.
• The dance form was revived during the 20th century after facing decline during the colonial period.

The correct answer is Mauryan Period.

Key Points

• The life-size standing image of Yakshini holding a chauri (flywhisk) is an example of the sculptural excellence of the Mauryan Period.
• It is believed to have been created during the reign of Emperor Ashoka (268–232 BCE), a period known for its promotion of art and architecture.
• The Yakshini sculptures are noted for their voluptuous forms, intricate detailing, and lively expressions, reflecting the Mauryan emphasis on naturalism in art.
• The Chaitya halls, stupas, and pillars of the Mauryan Empire were other prominent examples of their artistic achievements.
• The Yakshini figure, often associated with fertility and abundance, was a part of the religious and cultural motifs of the time, blending animistic traditions with early Buddhist art.

Additional Information

• Characteristics of Mauryan Art:
• The Mauryan art is primarily characterized by its polished stone sculptures, which were often monumental in size.
• The famous Ashokan Pillars with inscriptions and animal capitals (e.g., the Lion Capital at Sarnath) are iconic examples of Mauryan craftsmanship.
• The art form blended local traditions with influences from the Achaemenid and Hellenistic styles.
• Figures like Yakshas and Yakshinis were sculpted to represent fertility, abundance, and protection in early Indian religious contexts.
• Comparison with Other Periods:
• Gupta Period: Known for its classical refinement, the Gupta art emphasized religious themes, particularly Buddhist and Hindu iconography, and was more focused on temples and cave murals than on standalone sculptures.
• Kushan Period: This period saw the rise of Gandhara and Mathura schools of art, which combined Indian and Greco-Roman styles. It was more focused on Buddhist iconography, especially images of Buddha.
• Sunga Period: This era was marked by the continuation of Mauryan traditions but with a greater emphasis on terracotta and stone carvings, often depicting scenes from everyday life and Jataka tales.
• Religious Significance:
• The Yakshinis were considered guardian deities and were often placed near stupas and other religious monuments.
• They are associated with nature worship, which was prevalent in pre-Buddhist and early Buddhist traditions in India.
• Important Points:
• The Yakshini holding a chauri is housed in museums like the Patna Museum, showcasing the rich heritage of Mauryan art.
• This sculpture is an important example of how art during the Mauryan period served both religious and aesthetic purposes.
• The Mauryan period laid the foundation for future developments in Indian art and architecture.

The correct answer is 1) Both statements are true, and Statement 2 is the correct explanation of Statement 1..

Key Points

• Statement 1: The Shore Temple at Mahabalipuram exemplifies Dravidian architecture built of carved granite blocks.
• The Shore Temple is located in Mahabalipuram, Tamil Nadu, a UNESCO World Heritage Site.
• It was constructed during the reign of the Pallava king Narasimhavarman II (Rajasimha) in the 8th century CE.
• The temple is made of granite blocks, showcasing the brilliance of Pallava-era craftsmanship.
• It is one of the earliest examples of structural stone temples in South India, representing the transition from rock-cut to structural architecture. Hence, Statement 1 is correct.
• Statement 2: This temple showcases architectural continuity from the Pallava-era rock-cut cave temples to fully structural stone constructions.
• The Pallava dynasty is renowned for its contributions to early South Indian temple architecture.
• The Shore Temple illustrates the evolution of temple design, transitioning from rock-cut cave temples like those at Mahabalipuram to fully structural temples.
• The carvings and sculptures reflect the artistic and architectural continuity from earlier Pallava creations.
• The temple's aesthetic and construction techniques influenced subsequent Dravidian temple architecture. Hence, Statement 2 is correct.
• Explanation: Statement 2 correctly explains Statement 1 because it provides the historical and architectural context for the Shore Temple's design and construction.
• Therefore, the correct answer is Option 1.

Additional Information

• Shore Temple:
• The Shore Temple complex consists of three shrines dedicated to Lord Shiva and Lord Vishnu.
• The temple faces the Bay of Bengal and served as a landmark for sailors, hence the name "Shore Temple."
• The intricate carvings depict mythological stories, including scenes from the Mahabharata and Ramayana.
• Pallava Dynasty’s Contribution to Architecture:
• The Pallavas pioneered rock-cut architecture, as seen in the Mandapas (cave temples) and monolithic Rathas (chariots) at Mahabalipuram.
• Examples include Arjuna's Penance, the Pancha Rathas, and the Mahishasuramardini Cave.
• The Shore Temple marked the beginning of structural temple construction, which became a hallmark of later South Indian dynasties like the Cholas.
• Dravidian Architecture:
• Dravidian temple architecture is characterized by pyramidal towers (Vimanas), pillared halls (Mandapas), and enclosed courtyards.
• Intricate carvings and sculptures depicting deities, animals, and mythological scenes are common features.
• Later examples of Dravidian architecture include the Brihadeshwara Temple (Chola dynasty) and the Meenakshi Temple (Nayaka dynasty).
• UNESCO World Heritage Status:
• The Shore Temple is part of the Group of Monuments at Mahabalipuram, designated as a UNESCO World Heritage Site in 1984.
• This recognition highlights its cultural and historical significance.

Important Points

• Other Notable Pallava Monuments:
• Pancha Rathas: Five monolithic temples, each carved from a single granite block, named after the Pandavas from the Mahabharata.
• Arjuna’s Penance: A massive open-air rock relief depicting scenes of devotion and mythology.
• Mahishasuramardini Cave: A cave temple featuring carvings of Goddess Durga slaying the demon Mahishasura.
• Architectural Influence:
• The Shore Temple and other Pallava structures laid the foundation for later South Indian dynasties' architectural styles, particularly the Cholas and Vijayanagara rulers.
• The use of granite and intricate carvings became defining features of Dravidian temples.

The correct answer is Garo tribe of Meghalaya.

Key Points

• The Wangala Festival is a major harvest festival celebrated by the Garo tribe, predominantly residing in the state of Meghalaya in northeast India.
• This festival is also referred to as the Festival of 100 Drums, symbolizing the culmination of the agricultural season and the offering of gratitude to their deity Misi Saljong, the Sun God.
• The Wangala Festival typically takes place in the months of October and November, coinciding with the harvest season.
• During the festival, traditional dances are performed to the rhythm of drums, accompanied by folk songs, marking the cultural heritage and vibrancy of the Garo community.
• The festival fosters community bonding and promotes the preservation of the traditional practices of the Garo tribe.

Additional Information

• Garo Tribe:
• The Garo tribe is one of the three major matrilineal tribes of Meghalaya, alongside the Khasi and Jaintia tribes.
• The Garos predominantly inhabit the Garo Hills region of Meghalaya, with smaller populations in Assam and Bangladesh.
• They practice shifting cultivation and rely heavily on agriculture for their livelihood, growing crops like rice, maize, and millet.
• Their language belongs to the Tibeto-Burman family, and their culture is deeply rooted in traditional beliefs and practices.
• Harvest Festivals in India:
• India celebrates various harvest festivals across its diverse regions, such as Pongal in Tamil Nadu, Baisakhi in Punjab, and Onam in Kerala.
• These festivals mark the end of the agricultural season and express gratitude to nature and deities for a bountiful harvest.
• Other Tribes:
• Bhotia tribes of Uttarakhand: They are semi-nomadic and primarily inhabit the Himalayan region of Uttarakhand, practicing transhumance and trading activities.
• Balti tribes of Jammu and Kashmir: Known for their rich cultural heritage, they reside in the Ladakh region, practicing agriculture and livestock rearing.
• Nyishi tribes of Arunachal Pradesh: They are primarily agriculturalists and are known for their unique headgear made of cane and feathers.

Important Points

• Significance of Wangala Festival:
• The Wangala Festival is an expression of gratitude to Misi Saljong, considered the provider of prosperity and blessings.
• It is a vibrant celebration that showcases the traditional attire, dances, and music of the Garo tribe.
• This festival plays a crucial role in preserving the cultural identity and heritage of the Garo community.
• Role of Traditional Festivals:
• Traditional festivals act as a medium to preserve indigenous practices and pass them on to future generations.
• They promote community bonding, cultural exchange, and awareness of the rich diversity of India's heritage.

The correct answer is A collection of hymns across ages.

Key Points

• The Rigveda is one of the oldest religious texts in the world and is a significant part of ancient Indian literature. It is the earliest of the four Vedas, which are the foundational texts of Hinduism.
• The Rigveda is primarily a collection of hymns (Sanskrit: "Riks") that praise various deities like Agni (fire), Indra (war and rain), and Soma (ritual drink). These hymns are composed in Vedic Sanskrit and arranged into 10 books known as Mandalas.
• It was composed over a long period, estimated to be between 1500 BCE and 1200 BCE, during the early Vedic period of Indian history. The hymns were transmitted orally for generations before being written down.
• The Rigveda reflects the socio-religious, political, and economic conditions of early Vedic society. It is not just a religious text but also a historical document that provides insight into the culture and worldview of the ancient Indo-Aryans.
• The Rigveda is divided into 1,028 hymns and 10 Mandalas, each of which contains hymns dedicated to specific deities or purposes. For example, Mandala 10 contains the famous Purusha Sukta, which describes the creation of the universe.

Additional Information

• Other Vedas and their Significance:
• The four Vedas are: Rigveda, Samaveda, Yajurveda, and Atharvaveda. While the Rigveda is primarily a collection of hymns, the others focus on rituals, chants, and spells.
• The Samaveda contains melodies and chants derived from the Rigveda and is associated with musical traditions.
• The Yajurveda focuses on rituals and sacrifices, providing instructions for priests performing yajnas (ritual offerings).
• The Atharvaveda includes hymns, spells, and incantations related to everyday life, such as healing and protection from evils.
• Structure of the Rigveda:
• The Rigveda is divided into 10 Mandalas, which are further subdivided into hymns (suktas) and verses (mantras).
• Each Mandala serves different purposes, and the hymns vary in length. For example, the second Mandala is the shortest, while the seventh Mandala is dedicated to the god Indra.
• Philosophical and Cultural Insights:
• The Rigveda provides glimpses into the early Vedic religion, which was polytheistic and centered around natural forces like fire, wind, and the sun.
• It contains philosophical hymns such as the Nasadiya Sukta, which questions the origin of the universe and reflects early metaphysical thought.
• The text also describes the concept of Rita (cosmic order), which later evolved into the idea of Dharma in Hindu philosophy.
• Historical Importance:
• The Rigveda provides valuable information about the early Indo-Aryan society, including its social structure, economy, and political organization.
• It mentions rivers like the Saraswati and Sindhu, highlighting the geographical knowledge of the time.
• The hymns reveal a pastoral and agricultural society that relied on cattle, horses, and barley cultivation.
• Key Features of Other Options:
• Compilation of Upanishads: The Upanishads are philosophical texts that explore the nature of reality, Brahman, and Atman. They are later additions to Vedic literature and are not part of the Rigveda.
• Collection of Classical Music: While the Samaveda is linked to music, the Rigveda is not primarily a musical text.
• Collection of Royal Chronicles: The Rigveda contains hymns about gods and nature but does not serve as a historical chronicle of kings or royal events.

The correct answer is Only 1 is Correct.

Key Points

• Statement 1: Pole height is 1.20–1.25 m with a fixed 9 cm diameter. Correct.
• According to standard Kho-Kho Federation specifications, each pole is installed at the end of the central lane.
• The height of the pole is maintained between 120 cm and 125 cm.
• The pole diameter is approximately 9 cm, ensuring uniformity in official competitions.

Hence, Statement 1 is correct.

• Statement 2: Total match duration, including intervals, is 48 minutes. Incorrect.
• A standard Kho-Kho match consists of four turns of 9 minutes each, making the total playing time 36 minutes.
• There is generally a 5-minute interval between innings and 2-minute breaks between turns.
• The total duration is therefore not exactly 48 minutes.

Hence, Statement 2 is incorrect.

• Statement 3: Sub-junior chasers' block distance is the same as juniors: 2.30 m. Incorrect.
• The dimensions of the Kho-Kho field vary according to the age category.
• The 2.30 m block distance is applicable to Junior and Senior categories.
• The Sub-Junior category uses modified dimensions and does not have the same block distance as Juniors.

Hence, Statement 3 is incorrect.

• Therefore, only Statement 1 is correct.

Additional Information

• Kho-Kho is one of India's oldest traditional outdoor sports and is governed internationally by the International Kho Kho Federation (IKKF).
• The game is played between two teams of 12 players, out of which 9 players take the field at a time for chasing.
• A standard Kho-Kho match consists of two innings, and each inning contains two turns for chasing and defending.
• The game emphasizes speed, agility, reflexes, teamwork, and strategic movement, making it one of the fastest tag sports in the world.
• The first edition of the Kho Kho World Cup was hosted by India, reflecting the growing international popularity of the sport.

Important Points

• The playing area of Kho-Kho consists of a central lane, cross lanes, and two poles placed at opposite ends.
• A chaser can obtain a directional advantage by touching the pole and changing direction, a technique known as pole diving or pole turning.
• The term "Kho" is used by a chaser to pass the chasing turn to a seated teammate.
• Kho-Kho is included in several national-level competitions such as the National Games of India and Khelo India Games.

The correct answer is A group of peasant-pastoralists.

Key Points

• The term Kunbis refers to a significant community in the Maratha society, primarily classified as peasant-pastoralists. They were traditionally engaged in agriculture and cattle rearing.
• Kunbis formed the backbone of the agricultural economy in the Deccan region of India and were closely associated with the rural agrarian structure.
• The Kunbis were considered a part of the Shudra varna in the traditional caste hierarchy. However, they played a vital role in the Maratha Empire's socio-economic and political framework.
• During the rise of the Maratha Empire under leaders like Shivaji Maharaj, Kunbis often joined the army and became an integral part of the Maratha warrior class. Over time, they were assimilated into the broader Maratha identity.
• Their contributions to agriculture and their service as soldiers in the Maratha army underline their dual role as both economic producers and defenders of their territories.
• In modern India, Kunbis are recognized as an Other Backward Class (OBC) in several states, which provides them with reservations and welfare benefits.

Additional Information

• Historical Significance of Kunbis:
• The Kunbis were primarily farmers in the Deccan plateau, which includes regions of present-day Maharashtra, Karnataka, and Madhya Pradesh. Their agricultural practices supported the regional economy.
• They were one of the communities that contributed to the success of the Maratha Confederacy, as they provided both agrarian resources and military manpower.
• With the rise of the Maratha power under Shivaji Maharaj, many Kunbis were integrated into the military structure, eventually forming a significant portion of the Maratha warrior class.
• Kunbis and the Maratha Identity:
• Over time, the distinction between Kunbis and Marathas blurred as many Kunbis adopted the Maratha identity, particularly those who gained prominence in the military and administration.
• Historical records suggest that the Maratha rulers encouraged this assimilation to strengthen their support base among the agrarian communities.
• The social mobility experienced by Kunbis during the Maratha rule allowed them to rise in social status, although they retained their agricultural roots.
• Cultural Practices:
• Kunbis follow a mix of agrarian and pastoral traditions, often celebrating festivals related to harvests such as Makar Sankranti and Ganesh Chaturthi.
• They have a rich cultural heritage, including folk songs and dances that reflect their agrarian lifestyle and community values.
• Other Caste Groups in the Context of the Question:
• Warrior caste of Rajput clans: Refers to the Rajputs, known for their martial traditions and ruling dynasties in northern and western India. Not related to the Kunbis.
• Class of temple priests: Generally refers to the Brahmins, who are traditionally associated with priesthood and religious duties.
• Community of merchant bankers: Refers to communities like the Banias or Vaishyas, known for their roles in trade and finance.

Important Points

• Kunbis in Modern India:
• In present-day Maharashtra, Kunbis are classified as OBCs and benefit from affirmative action policies in education, employment, and political representation.
• They continue to be an important agricultural community, contributing significantly to the rural economy of the state.
• Role in the Maratha Empire:
• Kunbis were instrumental in the military campaigns of the Marathas, serving as foot soldiers and cavalrymen.
• Their contribution to the agrarian economy ensured the sustenance of the Maratha state during periods of conflict and expansion.

The correct answer is Only 1 is Correct.

Key Points

• Statement 1: Athletes must provide accurate whereabouts information and comply with all sample collection procedures.
• This statement is correct. Anti-doping regulations require athletes to provide accurate and up-to-date information about their locations (also referred to as "whereabouts").
• Compliance with sample collection procedures is mandatory to ensure fairness and transparency in sports.
• Failure to provide accurate whereabouts or refusal to comply with sample collection can result in sanctions, including disqualification or suspension.
• Statement 2: Athletes are allowed to delegate the sample collection process to a team official if they have a valid reason.
• This statement is incorrect. Athletes cannot delegate the sample collection process to anyone else under any circumstances.
• The anti-doping process is an individual responsibility, and the integrity of the sample collection process depends on the direct involvement of the athlete.
• Allowing delegation would compromise the reliability and credibility of anti-doping tests.

Additional Information

• Whereabouts Requirements:
• Athletes in the registered testing pool must provide detailed information about their location for each day, including times and places where they can be available for testing.
• Whereabouts information is critical for out-of-competition testing, which can occur without prior notice.
• Sample Collection Process:
• The sample collection process is conducted by trained doping control officers (DCOs) under the supervision of anti-doping organizations.
• Athletes are notified in person of their selection for testing and must report to the designated testing area promptly.
• The process includes the collection of urine and/or blood samples under strict protocols to prevent contamination or manipulation.
• WADA (World Anti-Doping Agency):
• WADA oversees global anti-doping efforts and ensures compliance with the World Anti-Doping Code.
• The code outlines the roles and responsibilities of athletes, support personnel, and anti-doping organizations.
• Consequences of Non-Compliance:
• Failure to comply with anti-doping regulations can lead to penalties such as ineligibility, fines, or a ban from competition.
• Three missed whereabouts filings within 12 months can result in sanctions similar to a positive doping test.

Important Points

• Strict Liability Principle:
• Under the strict liability principle, athletes are responsible for any prohibited substance found in their system, regardless of intent.
• This emphasizes the need for vigilance in ensuring compliance with anti-doping rules.
• Therapeutic Use Exemptions (TUEs):
• Athletes with legitimate medical conditions can apply for a TUE to use a prohibited substance or method.
• The application process is overseen by an independent TUE committee to ensure fair adjudication.
• Global Anti-Doping Code:
• The code is a set of universal standards aimed at harmonizing anti-doping efforts worldwide.
• It applies to all sports and countries, ensuring consistency in anti-doping practices.

The correct answer is James Robinson.

Key Points

• Why Nations Fail is a widely acclaimed book co-authored by Daron Acemoglu, an economist, and James Robinson, a political scientist.
• The book was published in 2012 and explores the fundamental reasons behind the economic success or failure of nations.
• The central argument revolves around the importance of inclusive political and economic institutions in fostering prosperity, while extractive institutions lead to stagnation and poverty.
• James Robinson, the co-author, is known for his work on development economics and political theory. He has held prestigious academic positions and collaborated extensively with Acemoglu on various research projects.
• The book is based on extensive historical examples and empirical analysis, making it a cornerstone in understanding the relationship between political systems, economic policies, and societal outcomes.

Additional Information

• About Daron Acemoglu:
• Daron Acemoglu is a Turkish-American economist and professor at the Massachusetts Institute of Technology (MIT).
• He is renowned for his research in development economics, political economy, and economic growth.
• His contributions include groundbreaking studies on the role of institutions in economic performance.
• About James Robinson:
• James Robinson is a British-American economist and political scientist.
• He has served as a professor at prestigious institutions, including Harvard University and the University of Chicago.
• His research focuses on the impact of political systems and governance on economic development.
• Key Themes of 'Why Nations Fail':
• Inclusive Institutions: These ensure broad participation in economic and political activities, fostering innovation and economic growth.
• Extractive Institutions: These concentrate power and wealth in the hands of a few, leading to exploitation and economic stagnation.
• Historical Case Studies: Examples include the success of the United States with inclusive institutions and the failure of nations like North Korea due to extractive institutions.
• Impact of the Book:
• 'Why Nations Fail' has influenced policymakers, academics, and researchers in understanding the importance of institutional frameworks.
• The book is widely regarded as a seminal work in the fields of economics and political science.

Important Points

• Other Works by Daron Acemoglu and James Robinson:
• The Narrow Corridor: Published in 2019, this book elaborates on the balance between state and society for freedom to thrive.
• Economic Origins of Dictatorship and Democracy: Co-authored by Acemoglu and Robinson, this work explores the causes of democratic transitions and the role of economic factors.
• Recognition:
• 'Why Nations Fail' has been translated into multiple languages and remains a highly recommended read for understanding global disparities.
• It has received praise from notable figures, including Bill Gates, who recommended it for its insightful analysis.

The correct answer is Option 3 (Both 1 and 2 are correct).

Key Points

• Statement 1: Indian Army topped the medal tally with 10 golds
• This statement is correct. The Indian Army emerged as the top-performing team in the Khelo India Winter Games 2024, securing the highest number of gold medals, which was 10 golds in the overall tally.
• The Indian Army has consistently participated in various sports initiatives and its dominance in the Winter Games is a reflection of its focus on physical fitness, discipline, and sports excellence.
• Statement 2: Jammu & Kashmir, former champions, finished second in 2024 with nine gold medals
• This statement is also correct. The region of Jammu & Kashmir, which had won the overall championship in previous editions of the Winter Games, secured the second position in 2024 with nine gold medals.
• Jammu & Kashmir has been a prominent participant in winter sports, given its geographical advantage of snow-capped mountains and infrastructure for activities like skiing, ice skating, and snowboarding.
• The Khelo India Winter Games 2024 was organized to promote winter sports in India and nurture young talent in disciplines like skiing, snowboarding, ice hockey, and other snow-based activities.
• The event fosters a competitive spirit among athletes from different parts of the country and serves as a platform for identifying talent for international competitions such as the Winter Olympics.

Additional Information

• Khelo India Winter Games Overview:
• The Khelo India Winter Games is an initiative under the Government of India’s Khelo India programme to promote sports culture and identify talent in various sports disciplines.
• The winter edition of the games focuses on snow-based sports and is typically held in regions like Jammu & Kashmir and Himachal Pradesh.
• The games have been instrumental in encouraging participation from athletes in high-altitude regions and creating awareness about winter sports in India.
• Notable Sports in Khelo India Winter Games:
• Skiing: Events include slalom and giant slalom, requiring athletes to navigate a downhill course marked by gates.
• Snowboarding: A popular winter sport that involves descending a snow-covered slope while standing on a board.
• Ice Hockey: This team sport is played on ice and has gained traction in regions like Ladakh and Himachal Pradesh.
• Figure Skating: A sport where individuals or teams perform spins, jumps, and dance moves on ice.
• Khelo India Programme:
• The Khelo India programme was launched by the Ministry of Youth Affairs and Sports in 2018 to revive India’s sports culture at the grassroots level.
• It aims to identify and nurture young talent by providing scholarships, world-class training facilities, and exposure to international competitions.
• The programme includes sports infrastructure development, coaching support, and organizing national-level competitions.
• Jammu & Kashmir’s Contribution to Winter Sports:
• Jammu & Kashmir is home to popular tourist destinations like Gulmarg, which has become a hub for skiing and snowboarding in India.
• The region hosts training camps and competitions for athletes aspiring to represent India in international winter sports events.
• Infrastructure development and government support have significantly enhanced the participation of local athletes in national and international games.

Important Points

• Medal Tally in Khelo India Winter Games 2024:
• Indian Army: 10 gold medals, topped the tally.
• Jammu & Kashmir: 9 gold medals, secured second position.
• Significance of Winter Sports:
• Winter sports are gaining popularity in India, particularly in the Himalayan regions.
• These sports help in promoting tourism and creating job opportunities in high-altitude areas.
• Future Prospects:
• The Khelo India Winter Games serve as a platform for identifying talent for international events like the Winter Olympics.
• Continued investment in infrastructure and training is essential for India’s success in winter sports on the global stage.

The correct answer is 73rd Amendment.

Key Points

• The 73rd Amendment Act of the Constitution, passed in 1992, made it mandatory for states to establish Panchayati Raj Institutions as units of local self-governance.
• It added Part IX to the Indian Constitution, titled "The Panchayats," and introduced Articles 243 to 243-O, laying down the framework for the functioning of Panchayati Raj Institutions.
• The Amendment also provided for the creation of a three-tier system of Panchayati Raj, consisting of the Gram Panchayat at the village level, the Panchayat Samiti at the block level, and the Zila Parishad at the district level.
• This Amendment made it mandatory to conduct regular elections every five years for Panchayati Raj Institutions and reserved seats for Scheduled Castes (SCs), Scheduled Tribes (STs), and women.
• It empowered the Panchayats with responsibilities related to economic development and social justice, as listed in the Eleventh Schedule of the Constitution.

Additional Information

• 42nd Amendment:
• Known as the "Mini Constitution," it was enacted in 1976 during the Emergency period.
• This Amendment introduced changes to strengthen the central government's powers and added the words "Socialist," "Secular," and "Integrity" to the Preamble.
• It did not address the establishment of Panchayati Raj Institutions.
• 74th Amendment:
• Passed in 1992, this Amendment focused on Urban Local Bodies and introduced Part IX-A to the Constitution.
• It mandated the establishment of Nagar Panchayats, Municipal Councils, and Municipal Corporations for urban governance.
• It complements the 73rd Amendment but does not directly apply to rural Panchayati Raj Institutions.
• 44th Amendment:
• Enacted in 1978, this Amendment aimed to restore democratic rights curtailed during the Emergency under the 42nd Amendment.
• It reduced the scope of the government's power to impose Emergency and protected civil liberties, but it did not deal with local self-governance.
• Eleventh Schedule:
• Introduced by the 73rd Amendment, it lists 29 subjects under the jurisdiction of Panchayati Raj Institutions.
• These subjects include agriculture, education, health, sanitation, rural development, and social welfare.
• The Eleventh Schedule empowers Panchayats to formulate and implement policies for these areas.

Important Points

• Key Features of the Panchayati Raj System:
• Democratic Decentralization: The system ensures grassroots participation in governance.
• Reservation: Seats are reserved for women (33%), SCs, and STs to promote inclusivity.
• State Election Commission: The Amendment mandates the establishment of a State Election Commission to oversee Panchayat elections.
• Gram Sabha: The foundation of the Panchayati Raj system, the Gram Sabha is the assembly of all registered voters in a village, responsible for approving plans and policies.
• Historical Context:
• The concept of Panchayati Raj was first recommended by the Balwant Rai Mehta Committee in 1957.
• It emphasized the need for a three-tier system and grassroots democracy.
• The Ashok Mehta Committee in 1977 further suggested reforms to strengthen the system.

The correct answer is Myopia (nearsightedness) and needs a diverging lens.

Key Points

• Myopia (Nearsightedness) is a vision condition in which a person can see nearby objects clearly but struggles to see distant objects clearly. This is caused by the excessive elongation of the eyeball or excessive curvature of the cornea.
• The power of the lens in spectacles is denoted by diopters (D). A negative lens power (-2.5 D in this case) indicates the use of a diverging (concave) lens, which is used to correct myopia.
• The diverging lens functions by spreading out light rays before they enter the eye so that the focus point shifts backward onto the retina, correcting the blurred vision for distant objects.
• The spectacle prescription (-2.5 D) signifies the degree of myopia, and the higher the negative value, the more severe the condition.
• Myopia can occur due to hereditary factors, excessive screen time, or prolonged near work, and it is commonly diagnosed in childhood or adolescence.

Additional Information

• Hypermetropia (Farsightedness):
• In this condition, a person can see distant objects clearly but struggles to see nearby objects clearly.
• It is corrected using a converging (convex) lens, which helps in focusing light rays onto the retina.
• This condition is often caused by a shorter-than-normal eyeball or a less curved cornea.
• Presbyopia:
• This is an age-related condition where the eye's lens loses flexibility, causing difficulty in focusing on nearby objects.
• It commonly occurs after the age of 40 years and is corrected using bifocal or progressive lenses.
• Astigmatism:
• Astigmatism is caused by an irregularly shaped cornea or lens, leading to distorted or blurred vision at all distances.
• This condition is corrected using a cylindrical lens, which compensates for the uneven curvature of the cornea or lens.
• Important Points:
• Corrective lenses (like concave or convex lenses) are prescribed after an eye examination by an optometrist or ophthalmologist.
• Regular eye check-ups are crucial to detect and manage vision-related issues early.
• Uncorrected refractive errors, such as myopia or hypermetropia, can lead to complications like eye strain, headaches, or progressive worsening of vision.

The correct answer is Currencies are not readily convertible into gold.

Key Points

• Modern currencies are fiat currencies: Modern currency systems are based on fiat money, which means that the currency is not backed by a physical commodity like gold or silver. Instead, it derives its value from the trust and confidence of the people using it.
• Non-convertibility into gold: Unlike in the past under the gold standard, modern currencies are not readily convertible into gold or any other commodity. This shift occurred after the abandonment of the gold standard, particularly with the collapse of the Bretton Woods system in 1971.
• Trust in central authorities: The value of modern currencies is maintained by the monetary policies of central banks, such as the Reserve Bank of India (RBI) or the Federal Reserve in the United States, and the overall economic stability of the issuing country.
• Legal tender status: Modern currencies are declared as legal tender by governments, meaning they must be accepted as a medium of exchange for goods and services within the country.
• Global practices: Today, most countries operate under a fiat currency system, and the exchange rates between currencies are determined by market forces or managed exchange rate systems.

Additional Information

• Gold Standard:
• The gold standard was a monetary system where a country's currency or paper money had a value directly linked to gold.
• Countries agreed to convert paper money into a fixed amount of gold. This system was widely used in the 19th and early 20th centuries.
• The system ended in 1971 when the United States, under President Richard Nixon, formally abandoned it, leading to the current fiat currency system.
• Bretton Woods System:
• Established in 1944, the Bretton Woods system was an international monetary framework where currencies were pegged to the U.S. dollar, which was convertible to gold.
• It collapsed in 1971 when the U.S. suspended the dollar's convertibility to gold.
• Fiat Currency:
• Fiat currency is government-issued money that is not backed by a physical commodity but rather by the trust and credit of the economy.
• Examples include the Indian Rupee (INR), U.S. Dollar (USD), and Euro (EUR).
• Commodity-Backed Currencies:
• Unlike fiat currencies, commodity-backed currencies derive their value from a physical commodity such as gold, silver, or oil.
• These systems are rare today due to the limitations and rigidity they impose on economic growth and monetary policy.

Important Points

• Option 1: "Currencies are fully backed by gold" is incorrect because modern currencies are not tied to the gold standard anymore.
• Option 3: "Currencies can always be exchanged for barrels of oil" is incorrect as currencies are not universally convertible into commodities like oil.
• Option 4: "Currencies are backed by silver" is incorrect as modern currencies are fiat and not tied to any commodity, including silver.

The correct answer is Third Plan.

Key Points

• The Green Revolution in India was introduced during the Third Five-Year Plan (1961-1966). This plan emphasized agriculture and aimed to make India self-sufficient in food production.
• The Green Revolution involved the adoption of new agricultural practices, including the use of high-yield variety (HYV) seeds, chemical fertilizers, pesticides, and improved irrigation methods.
• The revolution was spearheaded by the efforts of M.S. Swaminathan, known as the "Father of the Green Revolution in India," and supported by the government under Prime Minister Lal Bahadur Shastri.
• It primarily targeted regions like Punjab, Haryana, and Western Uttar Pradesh, which had favorable conditions for intensive agriculture.
• The Green Revolution led to a significant increase in food grain production, particularly wheat and rice, making India self-sufficient in food within a short period.

Additional Information

• Key Features of the Green Revolution:
• The use of high-yield variety (HYV) seeds, which were developed to produce more crops per hectare.
• Increased use of chemical fertilizers and pesticides to improve crop yield and protect against pests.
• Development of irrigation infrastructure, such as canals and tube wells, to ensure water supply for crops.
• Introduction of modern farming equipment like tractors and harvesters to improve efficiency.
• Impacts of the Green Revolution:
• India became self-sufficient in food production and reduced its dependence on food imports.
• It led to regional disparities, with agricultural benefits concentrated in states like Punjab and Haryana.
• Increased production of food grains, especially wheat and rice.
• Environmental concerns emerged due to excessive use of fertilizers, pesticides, and water resources.
• Five-Year Plans in India:
• The Second Five-Year Plan (1956-1961) focused on industrialization and infrastructure development.
• The Fourth Five-Year Plan (1969-1974) emphasized economic stability and poverty alleviation.
• The Fifth Five-Year Plan (1974-1979) focused on growth with equity and alleviating poverty.
• Global Context:
• The Green Revolution was inspired by global efforts to increase agricultural production, particularly in the United States and Mexico.
• Norman Borlaug, known as the "Father of the Green Revolution," played a key role in developing high-yield crops globally.

Important Points

• The Green Revolution was a turning point in India's agricultural history, transforming the country from a food-deficient nation to one of the largest producers of food grains.
• 1966 marked the official beginning of the Green Revolution in India, though its foundations were laid during the Third Plan.
• It introduced the concept of intensive farming, which required significant investment in seeds, fertilizers, and irrigation.
• While it boosted food security, it also led to challenges like soil degradation, overuse of water resources, and increased dependence on chemical inputs.

The correct answer is Greater journalistic freedom.

Key Points

• The World Press Freedom Index is published annually by Reporters Without Borders (RSF), an independent international organization.
• A higher ranking in this index signifies a country with greater press freedom, where journalists can operate freely without significant interference or control by the government, corporations, or other entities.
• The ranking is determined based on factors such as media independence, legislative framework, safety of journalists, and the level of pluralism in media.
• Countries with low rankings often experience issues such as media censorship, government control, harassment of journalists, and restrictions on free speech.
• Hence, a higher ranking reflects a country’s commitment to democratic principles, transparency, and the protection of journalistic rights.

Additional Information

• Factors influencing rankings in the World Press Freedom Index:
• The pluralism score assesses the diversity of opinions represented in the media.
• The media independence score evaluates the degree of editorial independence from political, governmental, or corporate pressures.
• The legislative framework score examines laws and regulations that impact press freedom, including those that may restrict or enable it.
• The transparency score measures the openness of institutions toward journalists and the public.
• The abuse score accounts for attacks, harassment, or imprisonment of journalists.
• Top-ranked countries in recent years:
• Countries like Norway, Finland, and Sweden consistently rank at the top due to their strong legislative protections, independent journalism, and minimal restrictions on media.
• These nations demonstrate high levels of transparency, press pluralism, and safety for journalists.
• Countries with low rankings:
• Countries such as North Korea, China, and Turkmenistan often rank at the bottom of the index due to strict censorship, suppression of dissent, and state-controlled media.
• Journalists in these nations face harsh penalties, surveillance, and limited freedom to report.
• Importance of press freedom:
• A free press serves as a watchdog for democracy, holding governments and corporations accountable.
• It facilitates the free flow of information, which is essential for informed decision-making by the public.
• Press freedom also supports human rights, transparency, and anti-corruption efforts.

Important Points

• Reporters Without Borders (RSF):
• Founded in 1985 and headquartered in Paris, France, RSF is an organization dedicated to promoting and protecting freedom of information worldwide.
• It monitors violations of press freedom, assists journalists in danger, and advocates for the free flow of information globally.
• India’s performance in the index:
• India has faced challenges in recent years due to increasing threats to journalists, internet shutdowns, and restrictive policies.
• Efforts are being made by civil society and media organizations to improve the safety and independence of journalists in India.
• Global trends:
• While some countries have shown progress, others have seen a decline in press freedom due to increased censorship, propaganda, and the suppression of dissent.
• Technological advancements, such as the rise of social media, have created both opportunities and challenges for press freedom.

The correct answer is Jharkhand.

Key Points

• Tana Bhagat Movement was a tribal uprising led by Tana Bhagat in the early 20th century among the Oraon community residing in what is now the state of Jharkhand.
• The movement began as a response to the oppression faced by the Oraon community under the British colonial rule and the local landlords (zamindars).
• The movement was unique as it emphasized non-violence and sought reforms in the tribal society, such as abstinence from alcohol, adopting cleanliness, and rejecting superstitions.
• The Tana Bhagat movement also had a religious dimension, as followers adopted practices inspired by Hindu rituals and Mahatma Gandhi’s ideals, including wearing khadi and boycotting British goods.
• The movement gained momentum in the region that is today known as Jharkhand, which became a separate state in 2000 after being carved out of Bihar.

Additional Information

• Oraon Community:
• The Oraons are one of the major tribal communities in India, primarily inhabiting the Chotanagpur Plateau region, which includes present-day Jharkhand, Chhattisgarh, and Odisha.
• Their traditional livelihood is based on agriculture, and they have a rich cultural heritage with unique dance, music, and festivals.
• The Oraon community faced exploitation under the British revenue system and local zamindars, which led to several tribal uprisings, including the Tana Bhagat movement.
• Significance of the Tana Bhagat Movement:
• The movement was inspired by Mahatma Gandhi's philosophy of non-violence and self-reliance, making it one of the few peaceful tribal uprisings during British rule.
• It sought to bring social and religious reform within the tribal community, emphasizing cleanliness, sobriety, and a rejection of traditional practices considered regressive.
• The movement also had a political dimension, as it opposed the oppression of the British and their policies affecting the tribal population.
• Other Tribal Movements in Jharkhand:
• Santhal Rebellion (1855-56): The Santhals revolted against the British and zamindars due to economic exploitation and land alienation.
• Birsa Munda Uprising (1899-1900): Led by Birsa Munda, this movement aimed at establishing Munda Raj and resisting British policies that disrupted the tribal way of life.
• Kol Rebellion (1831-1832): The Kols revolted against the British and local landlords for their exploitative practices.
• Formation of Jharkhand:
• The state of Jharkhand was formed on 15th November 2000, coinciding with the birth anniversary of tribal leader Birsa Munda.
• It was carved out of southern Bihar and was primarily created to address the socio-economic and political aspirations of the tribal population in the region.

Important Points

• The Tana Bhagat movement was a significant chapter in India’s tribal history, showcasing the unique blend of tribal traditions and Gandhian ideals.
• It highlights the role of tribal communities in India’s freedom struggle, often overlooked in mainstream narratives.
• The movement also reflects the socio-economic challenges faced by tribal communities under colonial rule, including land alienation, economic exploitation, and cultural suppression.
• Understanding such movements is crucial for appreciating the diversity of India’s freedom struggle and the contributions of various marginalized communities.

The correct answer is Opposing the grouping of Assam with Bengal.

Key Points

• Lokpriya Gopinath Bordoloi, the first Chief Minister of Assam, played a pivotal role in ensuring that Assam remained a part of India during the turbulent period of the Cabinet Mission Plan's proposal in 1946.
• He opposed the British plan to group Assam with Bengal in a single administrative unit. This grouping was seen as a precursor to Assam potentially being included in East Pakistan during Partition.
• Bordoloi’s opposition was based on the belief that Assam had a distinct cultural identity and should remain independent of Bengal's influence. This move safeguarded Assam’s territorial and cultural integrity.
• His efforts included lobbying at the national level and engaging with Congress leaders like Jawaharlal Nehru and Vallabhbhai Patel to ensure Assam’s interests were protected.
• Ultimately, Bordoloi’s firm stance and negotiations resulted in Assam being excluded from the proposed grouping, ensuring its integration into India and avoiding its annexation into East Pakistan.

Additional Information

• Cabinet Mission Plan (1946):
• The Cabinet Mission Plan was proposed by the British government to discuss and plan the transfer of power to Indian leadership.
• It suggested a grouping of provinces into three sections, with Assam and Bengal forming Group C. This grouping was opposed by many as it diluted Assam's autonomy.
• The plan also laid the groundwork for India’s partition, with the creation of Pakistan becoming increasingly likely.
• Gopinath Bordoloi’s Leadership:
• As the leader of Assam, Bordoloi championed the cause of Assamese culture and opposed any attempt to merge Assam with Bengal or Pakistan.
• He mobilized public opinion and worked with both local and national leaders to safeguard Assam’s interests.
• Assam’s Strategic Importance:
• Assam, rich in natural resources and with a unique cultural heritage, was considered a vital region in India’s northeastern frontier.
• Its inclusion in East Pakistan would have significantly altered the political and geographical landscape of the region.
• Outcome:
• Due to Bordoloi’s efforts, Assam was kept out of the grouping with Bengal, ensuring its integration into India post-Independence.
• He was later awarded the Bharat Ratna posthumously in 1999 for his contributions to the nation.

Important Points

• Other Options Analysis:
• Launching the Quit India Movement in Assam: While the Quit India Movement was significant, it was not directly related to Assam’s exclusion from East Pakistan. Hence, this is incorrect.
• Signing the Assam Accord: The Assam Accord was signed in 1985, much after Independence, and was unrelated to the events of 1946. Hence, this is incorrect.
• Rejecting the Instrument of Accession: The Instrument of Accession was related to princely states joining India or Pakistan post-Independence, not the grouping under the Cabinet Mission Plan. Hence, this is incorrect.

The correct answer is Third Plan.

Key Points

• The Green Revolution in India was introduced during the Third Five-Year Plan, which spanned from 1961 to 1966.
• The Green Revolution focused on increasing agricultural productivity through the adoption of new technologies, including the use of high-yielding variety (HYV) seeds, enhanced irrigation methods, and the application of chemical fertilizers and pesticides.
• This initiative was spearheaded by renowned agricultural scientist Dr. M.S. Swaminathan, who is often referred to as the "Father of the Green Revolution in India."
• The Green Revolution primarily targeted staple crops such as wheat and rice, leading to a significant increase in food grain production, especially in states like Punjab, Haryana, and Uttar Pradesh.
• The success of the Green Revolution reduced India's dependency on food imports and helped achieve self-sufficiency in food production, addressing the issue of food scarcity prevalent during that era.

Additional Information

• High-Yielding Variety Seeds:
• HYV seeds were introduced as part of the Green Revolution and were specifically designed to produce more output per hectare.
• These seeds required intensive irrigation, fertilizers, and pesticides for optimal results.
• Examples include Sonora 64 and Lerma Rojo varieties of wheat.
• Impact of Green Revolution:
• India's food grain production increased from 82 million tonnes in 1960-61 to over 100 million tonnes by 1970.
• The Green Revolution contributed to rural prosperity, especially in agriculturally rich states like Punjab and Haryana.
• It also led to the modernization of farming practices and the development of agro-industries.
• Challenges of the Green Revolution:
• While the Green Revolution boosted food production, it also resulted in environmental degradation due to excessive use of fertilizers and pesticides.
• It led to regional disparities, as benefits were concentrated in select states with better irrigation infrastructure.
• Small and marginal farmers faced difficulties due to the high cost of inputs required for HYV seeds.
• Role of Five-Year Plans:
• The Third Five-Year Plan (1961-1966) emphasized agriculture and rural development as key components of India's economic strategy.
• The plan aimed to achieve self-sufficiency in food production and laid the groundwork for the Green Revolution.
• It also introduced policies for the development of irrigation projects and agricultural research centers.

Important Points

• Key Figures:
• Norman Borlaug, an American agricultural scientist, is credited with pioneering the Green Revolution globally.
• In India, Dr. M.S. Swaminathan played a crucial role in the implementation of Green Revolution technologies.
• Geographical Focus:
• The Green Revolution primarily benefited regions with adequate irrigation facilities, such as Punjab, Haryana, and western Uttar Pradesh.
• Other parts of India, especially rain-fed areas, remained largely unaffected.
• Economic Effects:
• The Green Revolution contributed to the growth of agricultural exports and reduced India's reliance on food imports.
• It also fostered the development of agro-based industries, such as fertilizer and pesticide manufacturing.
• Social Impact:
• While the Green Revolution increased agricultural productivity, it also led to socio-economic disparities between large landowners and small farmers.
• It prompted the migration of rural laborers to urban areas due to mechanization and reduced labor demand in agriculture.

The correct answer is Agriculture.

Key Points

• Reduction in fertilizer subsidies post-1991: During the economic reforms of 1991, the Government of India reduced subsidies on fertilizers to address fiscal imbalances. This move significantly impacted the Agriculture sector, as farmers had to bear higher costs for fertilizers.
• Impact on farmers: Fertilizers are essential for enhancing soil fertility and crop yield. The reduction in subsidies increased the cost of agricultural inputs, directly affecting the financial burden on small and marginal farmers.
• Decline in agricultural productivity: Many farmers reduced fertilizer usage due to higher costs, leading to a decline in agricultural productivity in several regions, especially those dependent on fertilizer-intensive farming methods.
• Economic challenges for the agricultural workforce: The agriculture sector employs a significant portion of India's workforce, particularly in rural areas. Increased input costs strained the livelihood of farmers and agricultural laborers.
• Government measures: While fertilizer subsidies were reduced, the government introduced other schemes like Minimum Support Prices (MSP) and rural development programs to alleviate the impact on farmers.

Additional Information

• Fertilizer subsidy system:
• Subsidies for fertilizers like urea, DAP, and MOP were introduced to ensure affordable access to fertilizers for farmers, especially after India's Green Revolution in the 1960s.
• The subsidy system aimed to promote agricultural production and food security in the nation.
• Economic reforms of 1991:
• The Indian Government initiated Liberalization, Privatization, and Globalization (LPG) reforms to stabilize the economy and reduce fiscal deficits.
• One of the measures included reducing subsidies in various sectors, including agriculture.
• Effect on other sectors:
• The Industrial sector saw benefits from economic reforms due to deregulation and increased foreign investment.
• The Banking sector was strengthened through reforms, including improved regulations and the introduction of private banks.
• The Service sector grew rapidly post-reforms, driven by technology and outsourcing opportunities.
• Government initiatives to support agriculture:
• Programs like Pradhan Mantri Krishi Sinchai Yojana (PMKSY) were launched to improve irrigation facilities and reduce dependence on expensive fertilizers.
• Soil Health Card Scheme: This initiative helps farmers monitor soil quality and optimize fertilizer usage.
• Subsidy restructuring: The government periodically revises fertilizer subsidies to balance farmer support and fiscal responsibility.

Important Points

• Green Revolution:
• India's Green Revolution in the 1960s-70s relied heavily on fertilizers to boost agricultural productivity.
• The government provided subsidies to promote fertilizer usage, ensuring food security for the growing population.
• Current agricultural challenges:
• Dependence on fertilizers and high input costs remain significant challenges for farmers.
• Climate change and soil degradation further complicate agricultural productivity.
• Data and statistics:
• India is one of the largest consumers of fertilizers globally, with over 50% of the population dependent on agriculture.
• As of 2020, the government spent approximately ₹70,000 crore annually on fertilizer subsidies.

Shortcut Trick

Effective rate for 3 years at 5% p.a. = 3R . 3R² | R³ = 15.7625%.

Total amount at maturity = Principal (100%) + CI (15.7625%) = 115.7625%.

Amount = 115.7625% of 6000

Amount = 1.157625 × 6000 = 6945.75

∴ The correct answer is Rs. 6945.75.

Alternate Method

Given:

Principal (P) = Rs. 6000

Rate (R) = 5% p.a.

Time (n) = 3 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Year 0 Year 1 Year 2 Year 3

₹6000 ₹6945.75

• 5%

• 5%

• 5%

Compound Interest Progression

Calculations:

⇒ A = 6000 × (1 + 5/100)³

⇒ A = 6000 × (1 + 1/20)³

⇒ A = 6000 × (21/20)³

⇒ A = 6000 × (9261 / 8000)

⇒ A = 6 × 9261 ÷ 8

⇒ A = 55566 ÷ 8

⇒ A = 6945.75

∴ The correct answer is Rs. 6945.75.

Additional Information

CI vs SI Difference (2 Years)

The difference between Compound Interest and Simple Interest for 2 years is calculated directly as D = P(R/100)².

CI vs SI Difference (3 Years)

The difference between Compound Interest and Simple Interest for 3 years is calculated directly as D = P(R/100)² × (3 + R/100).

Effective Rate of CI

If interest is compounded annually at R% per annum, the effective compound interest rate for 2 years is given by the successive percentage formula: [2R + (R²/100)]%.

Shortcut Trick

Natural numbers are strictly counting numbers starting from 1 (1, 2, 3, 4...).

Because we do not count in negatives, this set contains only positive values.

Integers, rational numbers, and real numbers all include negative values by their standard definition.

∴ The correct answer is Natural Numbers.

Alternate Method

Let's evaluate each given set mathematically:

⇒ Integers (Z): The set containing negative integers, zero, and positive integers. Example: {..., −3, −2, −1, 0, 1, 2, ...}

⇒ Natural Numbers (N): The set of positive integers used for counting. Example: {1, 2, 3, 4, ...}

⇒ Rational Numbers (Q): Any number expressed as a fraction p/q where q ≠ 0. Example: −1/2, −5, 3/4

⇒ Real Numbers (R): The overarching set containing all rational and irrational numbers. Example: −10.5, −√3, π

⇒ Comparing the sets, it is clearly evident that only Natural Numbers do not have any negative elements.

∴ The correct answer is Natural Numbers.

Additional Information

Whole Numbers

Whole numbers are simply natural numbers including zero: {0, 1, 2, 3, ...}. Like natural numbers, they also do not contain negative numbers.

Irrational Numbers

These are real numbers that cannot be written as a simple fraction, meaning their decimal expansions are non-terminating and non-repeating (e.g., √2, π).

Number System Hierarchy

The relationship between the primary number sets is defined as: Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real Numbers.

Shortcut Trick

When distance is constant, Distance = (S₁ × S₂) ÷ (S₁ − S₂) × Δt

⇒ Time difference Δt = 36 minutes = 36 ÷ 60 hours = 3 ÷ 5 hours

⇒ Distance = (10 × 12) ÷ (12 − 10) × (3 ÷ 5)

⇒ Distance = (120 ÷ 2) × (3 ÷ 5) = 60 × 0.6

∴ The correct answer is 36 km.

Alternate Method

Given:

Speed of Suman = 10 km/hr

Speed of Sujata = 12 km/hr

Difference in time = 36 minutes = 36 ÷ 60 = 3 ÷ 5 hours

Start End

Distance = D km

Suman: 10 km/hr (T + 36 min)

Sujata: 12 km/hr (T min)

Formula Used:

Time = Distance ÷ Speed

Calculation:

Let the distance traveled by each be 'D' km.

⇒ Time taken by Suman − Time taken by Sujata = 3 ÷ 5 hours

⇒ (D ÷ 10) − (D ÷ 12) = 3 ÷ 5

⇒ (6D − 5D) ÷ 60 = 3 ÷ 5

⇒ D ÷ 60 = 3 ÷ 5

⇒ D = (3 × 60) ÷ 5

⇒ D = 36

∴ The correct answer is 36 km.

Additional Information

Inverse Relation of Speed and Time

When the distance is constant, Speed and Time are inversely proportional. A ratio of speeds = a : b means the ratio of time taken will be = b : a.

Average Speed (Same Distance)

If equal distances are covered at two different speeds S₁ and S₂, then the Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂).

Relative Speed

If two bodies move in the same direction, Relative Speed = S₁ − S₂. In opposite directions, Relative Speed = S₁ + S₂.

Shortcut Trick

Surface Area ∝ r² and Volume ∝ r³

Ratio of Surface Area = 100 : 121

Ratio of Radius = √100 : √121 = 10 : 11

Ratio of Volume = 10³ : 11³ = 1000 : 1331

Percentage increase in volume = (1331 − 1000) ÷ 1000 × 100 = 33.1%

∴ The correct answer is 33.1%.

Alternate Method

Given:

Increase in surface area = 21%

Formula Used:

Surface Area of a sphere = 4πr²

Volume of a sphere = (4/3)πr³

r

1.1r

+21% Area +33.1% Vol

Original Sphere Area: 100% | Vol: 100%

New Sphere Area: 121% | Vol: 133.1%

Calculations:

⇒ Let the original radius of the sphere be r.

⇒ Original surface area, S₁ = 4πr²

⇒ New surface area, S₂ = S₁ + 21% of S₁ = 1.21 × 4πr²

⇒ We know, S₂ = 4π(r_new)²

⇒ 4π(r_new)² = 1.21 × 4πr²

⇒ (r_new)² = 1.21r² ⇒ r_new = 1.1r

⇒ Original volume, V₁ = (4/3)πr³

⇒ New volume, V₂ = (4/3)π(1.1r)³

⇒ V₂ = 1.331 × (4/3)πr³ = 1.331V₁

⇒ Increase in volume = V₂ − V₁ = 1.331V₁ − 1V₁ = 0.331V₁

⇒ Percentage increase in volume = (0.331V₁ ÷ V₁) × 100 = 33.1%

∴ The correct answer is 33.1%.

Additional Information

Successive Percentage Change Trick

If the radius increases by x%, the volume (which is a 3D measure) increases by successive changes of x%, applied 3 times. Formula for effective % change of three identical values: 3x + 3x²/100 + x³/10000.

Scaling of 2D and 3D Shapes

For any similar figures or solids, if the linear dimension (like radius, edge, or side) scales by a factor k, the Area scales by k² and the Volume scales by k³.

Hemisphere Relationships

For a hemisphere of radius r: Curved Surface Area = 2πr², Total Surface Area = 3πr², Volume = (2/3)πr³.

Shortcut Trick

Convert dimensions to cm to match the tiles: 4.5 m = 450 cm and 2.8 m = 280 cm.

Base number of tiles = (Wall Area) ÷ (Tile Area) = (450 × 280) ÷ (30 × 20) = 126000 ÷ 600 = 210 tiles.

Tiles required including 8% wastage = 210 × 1.08 = 226.8 tiles.

Since tiles must be purchased in whole numbers, we round up to the nearest integer, which is 227.

∴ The correct answer is 227 tiles.

Alternate Method

Given:

Length of the wall (L) = 4.5 m = 450 cm

Height of the wall (H) = 2.8 m = 280 cm

Dimensions of one tile = 30 cm × 20 cm

Extra tiles required for cuts and wastage = 8%

Formula Used:

Total Base Tiles = Total Area of Wall ÷ Area of One Tile

Area of Rectangle = Length × Width

Wall Area 450 cm × 280 cm L = 4.5 m (450 cm) H = 2.8 m

Tile Area 30 × 20

Base Tiles = 126000 ÷ 600 = 210 tiles

Calculation:

⇒ Area of the wall = 450 × 280 = 126,000 cm²

⇒ Area of one tile = 30 × 20 = 600 cm²

⇒ Base number of tiles = 126,000 ÷ 600

⇒ Base number of tiles = 210

⇒ Extra tiles required for wastage = 8% of 210

⇒ Extra tiles = (8 ÷ 100) × 210 = 16.8 tiles

⇒ Total tiles required = 210 + 16.8 = 226.8 tiles

⇒ Since you cannot purchase a fraction of a tile, you must round up to the next integer to ensure you have enough.

⇒ Total tiles to be purchased = 227

∴ The correct answer is 227 tiles.

Additional Information

Unit Conversions for Area

Since 1 m = 100 cm, the conversion for area is 1 m² = 10,000 cm². Always ensure both the surface area and item area are in the same units before dividing.

Tiling Problems Concept

When covering a larger 2D region with smaller identical rectangular shapes, Total Items = Total Area ÷ Area of 1 Item. This assumes optimal fitting without edge-case limitations.

Accounting for Wastage

Wastage or cuts signify a relative percentage increase in the requirement. Final amount = Base amount × (1 + Rate ÷ 100). For physical, indivisible items like tiles or bricks, the final calculated amount must always be rounded up (ceiling value).

Shortcut Trick

Profit ratio = Sum of (Capital × Time)

A : B : C = (3 × 6 + 6 × 6) : (2 × 6 + 1 × 6) : (4 × 12)

A : B : C = 54 : 18 : 48 = 9 : 3 : 8

Total profit ratio parts = 9 + 3 + 8 = 20

B's share = (3 ÷ 20) × 1,80,000 = Rs. 27,000

∴ The correct answer is Rs. 27,000.

Alternate Method

A's Effective Capital:

3 × 6m 6 × 6m = 54

B's Effective Capital:

2×6 1×6 = 18

C's Effective Capital:

4 × 12m = 48

Profit Ratio A : B : C = 54 : 18 : 48 = 9 : 3 : 8

Given:

Initial investment ratio of A : B : C = 3 : 2 : 4

Total profit after 1 year = Rs. 1,80,000

Formula Used:

Profit Share ∝ Effective Capital

Effective Capital = Investment × Time Period

Calculations:

⇒ Let the initial investments of A, B, and C be 3x, 2x, and 4x respectively.

⇒ A invests 3x for 6 months, then doubles it to 6x for the remaining 6 months.

⇒ A's effective capital = (3x × 6) + (6x × 6) = 18x + 36x = 54x

⇒ B invests 2x for 6 months, then halves it to 1x for the remaining 6 months.

⇒ B's effective capital = (2x × 6) + (1x × 6) = 12x + 6x = 18x

⇒ C maintains his investment of 4x for the entire 12 months.

⇒ C's effective capital = 4x × 12 = 48x

⇒ Ratio of profit sharing A : B : C = 54x : 18x : 48x

⇒ Dividing by 6x, we get A : B : C = 9 : 3 : 8

⇒ Total profit ratio parts = 9 + 3 + 8 = 20 parts

⇒ 20 parts = Rs. 1,80,000

⇒ 1 part = 1,80,000 ÷ 20 = Rs. 9,000

⇒ B's profit share = 3 parts = 3 × 9,000 = Rs. 27,000

∴ The correct answer is Rs. 27,000.

Additional Information

Compound Partnership

When partners invest their capital for different durations, the partnership is called a compound partnership. The profits are divided proportionally based on the product of capital and the time period.

Working (Active) Partner

A partner who actively manages the business is called a working partner. They usually receive a fixed salary or a specific percentage of the total profit for their management, after which the remaining profit is distributed among all partners based on their effective investments.

Sleeping Partner

A partner who only invests money and does not partake in the day-to-day business operations is known as a sleeping partner. Their share of the profit solely depends on their financial investment and duration.

Shortcut Trick

Let the initial investment be 100 units.

Multiplying factors for +25%, −20%, and +10% are 5/4, 4/5, and 11/10 respectively.

Final Value = 100 × (5/4) × (4/5) × (11/10) = 110 units.

Overall Change = 110 − 100 = +10 units (10% gain on base 100).

∴ The correct answer is 10% gain.

Alternate Method

Given:

First year change = +25%

Second year change = −20%

Third year change = +10%

Formula Used:

Final Value = Initial Value × (100 ± R₁)/100 × (100 ± R₂)/100 × (100 ± R₃)/100

Percentage Change = [(Final Value − Initial Value) ÷ Initial Value] × 100%

Calculations:

Let the initial investment be 100.

⇒ Value after 1st year = 100 + (25% of 100) = 100 + 25 = 125

⇒ Value after 2nd year = 125 − (20% of 125) = 125 − 25 = 100

⇒ Value after 3rd year = 100 + (10% of 100) = 100 + 10 = 110

100

+25%

125

−20%

100

+10%

110

Overall Gain: +10%

⇒ Overall gain = Final Value − Initial Value = 110 − 100 = +10

⇒ Overall gain percentage = (10 ÷ 100) × 100% = 10%

∴ The correct answer is 10% gain.

Additional Information

Successive Percentage Change Formula

When a value experiences two successive percentage changes x% and y%, the net effect can be found directly using: Net Change = x + y + (xy ÷ 100). Use a negative sign for any decrease.

Multiplying Factor (Base Change)

Instead of calculating the percentage value and adding it, you can multiply the base directly by its fractional equivalent. An increase of r% gives a multiplier of (1 + r/100), while a decrease gives (1 − r/100).

Compound Interest Relation

Successive percentage changes behave identically to compound interest (CI). If an investment grows at rate r% over multiple years, calculating successive increments yields the final maturity amount.

Shortcut Trick

Area of the square = 20 × 20 = 400 cm²

Since the areas are equal, Area of the rectangle = 400 cm²

Breadth = Area ÷ Length = 400 ÷ 25

Breadth = 16 cm

∴ The correct answer is 16 cm.

Alternate Method

20 cm 20 cm Area = 400

=

25 cm b = ? Area = 400

Given:

Side of the square (a) = 20 cm

Length of the rectangle (l) = 25 cm

Area of square = Area of rectangle

Formula Used:

Area of a square = a²

Area of a rectangle = l × b

Calculations:

⇒ Area of the square = 20 × 20 = 400 cm²

⇒ Area of the rectangle = 25 × b

⇒ Since both areas are equal: 25 × b = 400

⇒ b = 400 ÷ 25

⇒ b = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Perimeter of a Square

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

Perimeter of a Rectangle

The total length of the boundary of a rectangle is given by: Perimeter = 2 × (Length + Breadth).

Diagonal of a Square

The length of the diagonal connecting opposite corners of a square is: d = a√2.

Diagonal of a Rectangle

The length of the diagonal of a rectangle is calculated using the Pythagoras theorem: d = √(l² + b²).

Shortcut Trick

Net Area = Area of Rectangle − Area of Triangle

⇒ Net Area = (12 × 9) − (1/2 × 4 × 3) = 108 − 6 = 102 m²

⇒ Total Cost = 102 × 120 = ₹12,240

∴ The correct answer is ₹12,240.

Alternate Method

Given:

Length of the wall = 12 m

Breadth of the wall = 9 m

Base of the window = 4 m

Height of the window = 3 m

Painting rate = ₹120 per m²

Formula Used:

Area of Rectangle = length × breadth

Area of Triangle = 1/2 × base × height

    12 m 9 m 4 m 3 m

Calculations:

⇒ Area of the rectangular wall = 12 × 9 = 108 m²

⇒ Area of the triangular window = 1/2 × 4 × 3 = 6 m²

⇒ Area to be painted = 108 − 6 = 102 m²

⇒ Total cost of painting = 102 × 120 = ₹12,240

∴ The correct answer is ₹12,240.

Additional Information

Area of Composite Shapes

The total area of a composite figure is calculated by adding or subtracting the areas of the basic geometric shapes that constitute it.

Area of a Triangle

The area is given by the formula: Area = 1/2 × base × height, where the height is the perpendicular distance from the base to the opposite vertex.

Cost Calculation

Total Cost = Total Area × Rate per unit square. Both the area and the rate must have matching unit dimensions before multiplication.

Shortcut Trick

Area of a regular hexagon = 6 × (√3 ÷ 4) × (side)²

Total Area = 6 × (√3 ÷ 4) × 18² = 486√3 cm² ≈ 486 × 1.732 = 841.75 cm².

Since 12% is covered by the frame, visible glass area is 100% − 12% = 88% of the total area.

Visible Area = 0.88 × 841.75 = 740.74 cm².

∴ The correct answer is 740.7 cm².

Alternate Method

Given:

Side of the regular hexagonal mirror (a) = 18 cm

Percentage of mirror covered by the frame = 12%

Formula Used:

Area of regular hexagon = (3√3 ÷ 2) × a²

Visible Area88%Frame (12%)18 cm

Calculations:

⇒ Total Area of the mirror = (3√3 ÷ 2) × (18)²

⇒ Total Area = (3 × 1.732 ÷ 2) × 324

⇒ Total Area = 486 × 1.732 = 841.752 cm²

⇒ Frame area = 12%

⇒ Visible Area = (100 − 12)% = 88% of Total Area

⇒ Visible Area = (88 ÷ 100) × 841.752

⇒ Visible Area ≈ 740.74 cm²

∴ The correct answer is 740.7 cm².

Additional Information

Area of a Regular Hexagon

A regular hexagon is composed of 6 equilateral triangles. The general area formula is (3√3 ÷ 2) × a², where 'a' represents the side length.

Inradius and Circumradius

For a regular hexagon of side 'a', the circumradius (R) is perfectly equal to its side (R = a), and the perpendicular inradius (r) is (√3 ÷ 2) × a.

Percentage Reduction Technique

When an area or specific quantity is reduced by a certain value of x%, the remaining effective quantity can be calculated directly as (100 − x)% multiplied by the original base value.

Shortcut Trick

The base price of ₹500 is not needed to find the percentage error; use 100% directly.

Intended SP = 100% − 10% = 90%

Erroneous SP = 100% + 10% = 110%

Percentage Error = [(Erroneous SP − Intended SP) ÷ Intended SP] × 100

= [(110 − 90) ÷ 90] × 100 = 200 ÷ 9 = 22.22%

∴ The correct answer is 22.22%.

Alternate Method

Given:

Original Price = ₹500

Intended price change = Decrease by 10%

Erroneous price change = Increase by 10%

Formula Used:

Percentage Increase = (Increase in Value ÷ Original Correct Value) × 100

Price Error Comparison

Original

₹500

Intended SP

₹450 (−10%)

Erroneous SP

₹550 (+10%)

Gap = ₹100

Error % = (100 ÷ 450) × 100 = 22.22%

Calculations:

⇒ Intended correct price = 500 − (10% of 500) = 500 − 50 = ₹450

⇒ Erroneous actual price = 500 + (10% of 500) = 500 + 50 = ₹550

⇒ Price increase due to error = 550 − 450 = ₹100

⇒ Percentage increase = (100 ÷ 450) × 100

⇒ Percentage increase = 1000 ÷ 45 = 200 ÷ 9 ≈ 22.22%

∴ The correct answer is 22.22%.

Additional Information

Percentage Error Base Requirement

Percentage error must always be calculated with respect to the true (intended) value acting as the base denominator. Formula: (|True Value − Actual Value| ÷ True Value) × 100.

Base Independence Trick

When both the intended change and the erroneous change are provided as percentages, the final error percentage is entirely independent of the original numerical value (like ₹500). Solving logically with an assumed base of 100 prevents unnecessary calculations.

Percentage Increase / Decrease Concept

A numerical value increased by x% effectively becomes Value × (1 + x/100). Similarly, a value decreased by x% mathematically becomes Value × (1 − x/100).

Shortcut Trick

Total Cost Price of 8 shirts = Rs. 4000

Total Selling Price = SP of 6 shirts + SP of 2 discarded shirts

Total Selling Price = 3900 + 0 = Rs. 3900

Loss = Total CP − Total SP = 4000 − 3900 = Rs. 100

∴ The correct answer is Rs. 100 loss.

Alternate Method

Total CP of 8 shirts = Rs. 4000

SP of 6 shirts = Rs. 3900

2 Discarded

Loss = Rs. 100

Overall Loss = Rs. 4000 − Rs. 3900 = Rs. 100

Given:

Total Cost Price (CP) of 8 shirts = Rs. 4000

Selling Price (SP) of 6 shirts = Rs. 3900

Number of shirts discarded = 2

Formula Used:

Loss = Total Cost Price − Total Selling Price

Calculations:

⇒ Selling Price of the 2 discarded shirts = Rs. 0

⇒ Total Selling Price of all 8 shirts = Rs. 3900 + Rs. 0 = Rs. 3900

⇒ Since Total Cost Price is greater than Total Selling Price (4000 > 3900), the man incurs a loss.

⇒ Overall Loss = 4000 − 3900 = Rs. 100

∴ The correct answer is Rs. 100 loss.

Additional Information

Profit and Loss Basics

Profit occurs when Selling Price (SP) > Cost Price (CP), calculated as Profit = SP − CP. Loss occurs when CP > SP, calculated as Loss = CP − SP.

Impact of Discarded Goods

When items are damaged, lost, or discarded, their individual selling price effectively becomes zero. This directly reduces the total revenue, often resulting in an overall loss unless the remaining items are sold at a sufficiently high premium.

Profit/Loss Percentage

The percentage of profit or loss is always calculated on the initial total Cost Price using the formula: Loss % = (Loss ÷ CP) × 100.

Shortcut Trick

Initial average of 8 numbers is 56.

When 2 numbers are removed, the average of the remaining 6 numbers decreases by 2 (from 56 to 54).

The total decrease in the sum of the remaining 6 numbers = 6 × 2 = 12.

This excess value was originally carried by the removed 2 numbers.

Average of the removed numbers = Original Average + (Total Decrease ÷ 2) = 56 + (12 ÷ 2) = 62.

∴ The correct answer is 62.

Alternate Method

Given:

Total numbers = 8, Initial average = 56

Remaining numbers = 6, New average = 54

Formula Used:

Total Sum = Average × Count

Total 8 numbers → Overall Avg = 56Remaining 6 numbersAvg = 54Removed 2Avg = XTotal Sum = (6 × 54) + (2 × X) = 8 × 56X = 62

Calculations:

⇒ Total sum of all 8 numbers = 8 × 56 = 448

⇒ Total sum of remaining 6 numbers = 6 × 54 = 324

⇒ Sum of the 2 removed numbers = 448 − 324 = 124

⇒ Average of the 2 removed numbers = 124 ÷ 2 = 62

∴ The correct answer is 62.

Additional Information

Weighted Average Concept

When combining multiple groups with individual averages, the overall combined average is calculated using the formula: (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method in Averages

If removing items causes the remaining group's average to decrease, it indicates that the removed items had a value significantly greater than the original average.

Constant Addition/Subtraction Rule

If a constant value 'k' is added to or subtracted from every element in a dataset, the overall average of the entire dataset changes by exactly that same value 'k'.

Shortcut Trick

Volume in m³ = 500 L ÷ 1000 = 0.5 m³

Volume of right prism = Area of base × Height

Area of square base = 0.5 m³ ÷ 2 m = 0.25 m²

Side of base = √0.25 = 0.5 m = 50 cm

∴ The correct answer is 50 cm.

Alternate Method

Given:

Volume of the storage box (V) = 500 L

Height of the prism (h) = 2 m

Formula Used:

Volume of a prism = Base Area × Height

Volume (in m³) = Volume (in L) ÷ 1000

Base Area of a square = a² (where a is the side of the square)

a

a

h = 2 m

Volume = 500 L V = 0.5 m³

Calculations:

⇒ Volume V = 500 ÷ 1000 = 0.5 m³

⇒ a² × 2 = 0.5

⇒ a² = 0.5 ÷ 2

⇒ a² = 0.25 m²

⇒ a = √0.25 = 0.5 m

⇒ a = 0.5 × 100 = 50 cm

∴ The correct answer is 50 cm.

Additional Information

Volume Unit Conversion

1 cubic meter (m³) is equivalent to 1000 liters (L). To accurately convert liters to m³, always divide the given value by 1000.

Right Prism Property

A right prism is a 3D geometric solid where the top and bottom bases are identical polygons, and the lateral faces are vertical rectangles. Its total volume equals Base Area × Height.

Square Base Area

For a right prism having a square base, the formula for the base area simplifies directly to a², where 'a' represents the uniform side length of the given square base.

Shortcut Trick

Let Cost Price (CP) = 100 units ⇒ Marked Price (MP) = 100 + 60% = 160 units.

SP after 25% and 10% successive discounts = 160 × (75/100) × (90/100) = 108 units.

Given final SP = Rs. 2,430 ⇒ 108 units = 2430 ⇒ 1 unit = 22.5.

Cost Price = 100 units = 100 × 22.5 = Rs. 2,250.

∴ The correct answer is Rs. 2,250.

Alternate Method

Given:

Markup = 60%

Successive discounts = 25% and 10%

Final Selling Price (SP) = Rs. 2,430

Formula Used:

SP = CP × (1 + Markup/100) × (1 − D₁/100) × (1 − D₂/100)

CP = 100x

MP = 160x

SP = 108x

+60% Markup

Successive Disc. -25% & -10%

108x = Rs. 2430 x = 22.5

100x = ? Rs. 2250

Calculations:

⇒ Let the Cost Price (CP) be Rs. x.

⇒ Marked Price (MP) = x × (1 + 60/100) = 1.6x

⇒ Selling Price (SP) = 1.6x × (1 − 25/100) × (1 − 10/100)

⇒ SP = 1.6x × (75/100) × (90/100)

⇒ SP = 1.6x × (3/4) × (9/10) = 1.08x

⇒ Since SP is given as Rs. 2,430:

⇒ 1.08x = 2430

⇒ x = 2430 ÷ 1.08 = 2250

∴ The correct answer is Rs. 2,250.

Additional Information

Successive Discounts

When two discounts a% and b% are applied successively, the effective single discount percentage is given by (a + b − ab ÷ 100)%.

Marked Price and Cost Price Relationship

The direct relation between CP and MP is given by: CP × (100 + Profit%) = MP × (100 − Discount%). If there is a loss, use (100 − Loss%).

Markup Percentage

Markup% is always calculated with respect to the Cost Price. MP = CP + (CP × Markup% ÷ 100).

Shortcut Trick

Average of new entrants = Old Average + [(Total Final Students × Increase in Average) ÷ Number of New Students]

⇒ Average = 75 + [(65 × 1) ÷ 5]

⇒ Average = 75 + (65 ÷ 5)

⇒ Average = 75 + 13 = 88

∴ The correct answer is 88 marks.

Alternate Method

Given:

Initial number of students = 60

Initial average score = 75

New students added = 5

Increase in average = 1

Formula Used:

Total Sum = Average × Number of items

60 Students Avg = 75 Sum = 4500

5 Students Avg = x Sum = 5x

=

65 Students Avg = 76 Sum = 4940

4500 + 5x = 4940 → x = 88

Calculations:

⇒ Total sum of scores of 60 students = 60 × 75 = 4500

⇒ Total number of students after 5 new join = 60 + 5 = 65

⇒ New class average = 75 + 1 = 76

⇒ Total sum of scores of 65 students = 65 × 76 = 4940

⇒ Total score of the 5 new students = 4940 − 4500 = 440

⇒ Average score of the 5 new students = 440 ÷ 5 = 88

∴ The correct answer is 88 marks.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have individual averages of A₁ and A₂ respectively, their combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Replacement in Averages

When a new person replaces an old person and the average of the group increases, the value of the new person is computed directly by: Value of New Person = Value of Replaced Person + (Total Members × Increase in Average).

Weighted Average

Used when items have different importance or frequencies. It is given by the sum of the products of weights and items divided by the total sum of the weights.

Shortcut Trick Let sinA = x, meaning cosecA = 1 ÷ x. The given equation becomes: (1 ÷ x) − x = 2 ⇒ x2 + 2x − 1 = 0 Using the quadratic formula, the roots are: x = [−2 ± √(4 + 4)] ÷ 2 = −1 ± √2 Since the value of sinA must strictly lie within [−1, 1], we reject −1 − √2. ∴ The correct answer is −1 + √2.

Alternate Method Given: cosecA − sinA = 2 Formula Used: cosecA = 1 ÷ sinA Roots of ax2 + bx + c = 0 are x = [−b ± √(b2 − 4ac)] ÷ 2a Calculations: ⇒ (1 ÷ sinA) − sinA = 2 ⇒ 1 − sin2A = 2sinA ⇒ sin2A + 2sinA − 1 = 0 ⇒ sinA = [−2 ± √(22 − 4 × 1 × (−1))] ÷ (2 × 1) ⇒ sinA = [−2 ± √(4 + 4)] ÷ 2 ⇒ sinA = [−2 ± √8] ÷ 2 ⇒ sinA = [−2 ± 2√2] ÷ 2 ⇒ sinA = −1 ± √2 The range of sinA is strictly bounded such that −1 ≤ sinA ≤ 1. ⇒ −1 − √2 ≈ −2.414, which falls outside the valid trigonometric range. ⇒ −1 + √2 ≈ 0.414, which falls within the valid trigonometric range. ∴ The correct answer is −1 + √2.

Additional Information Reciprocal Trigonometric Identities cosecA = 1 ÷ sinA, secA = 1 ÷ cosA, cotA = 1 ÷ tanA Range of Trigonometric Functions The value of sinθ and cosθ always lies in the closed interval [−1, 1], meaning −1 ≤ sinθ ≤ 1 for all real numbers. Nature of Roots in Quadratic Equation For a standard quadratic equation ax2 + bx + c = 0, if the Discriminant D = b2 − 4ac > 0, the roots are real and distinct.

Shortcut Trick

The numbers 8, 15, and 17 form a standard Pythagorean triplet.

Given sin θ = Perpendicular / Hypotenuse = 8 / 17.

This directly implies that the Base is 15.

In the 2nd quadrant, the cosine ratio (Base / Hypotenuse) is always negative.

Therefore, cos θ = −15 / 17.

∴ The correct answer is −15/17.

Alternate Method

Given:

sin θ = 8 / 17

θ lies in the 2nd quadrant.

Formula Used:

sin²θ + cos²θ = 1

X Y

θ r = 17 y = 8 x = −15

Calculations:

⇒ cos²θ = 1 − (8 / 17)²

⇒ cos²θ = 1 − 64 / 289

⇒ cos²θ = (289 − 64) / 289

⇒ cos²θ = 225 / 289

⇒ cos θ = ±√(225 / 289)

⇒ cos θ = ±15 / 17

Since θ lies in the 2nd quadrant, the value of cos θ must be negative.

⇒ cos θ = −15 / 17

∴ The correct answer is −15/17.

Additional Information

ASTC Rule (Quadrant Signs)

Quadrant I: All positive | Quadrant II: sin & cosec positive | Quadrant III: tan & cot positive | Quadrant IV: cos & sec positive.

Pythagorean Triplets

Common triplets like (3, 4, 5), (5, 12, 13), and (8, 15, 17) help find missing sides of a right-angled triangle directly without manual calculations.

Trigonometric Identities

The three core Pythagorean identities are sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Shortcut Trick

A full circle is equal to 2π radians.

Required Percentage = (Given angle ÷ Total angle) × 100

⇒ % = (2 ÷ 2π) × 100

⇒ % = 100 ÷ π

⇒ % ≈ 100 ÷ 3.1416 ≈ 31.83%

∴ The correct answer is 31.83%.

Alternate Method

Given:

Angle θ = 2 radians

2 rad 2π − 2 rad

Total Circle = 2π radians

% = ?

Formula Used:

Percentage of a full circle = (θ in radians ÷ 2π) × 100

Calculations:

⇒ Total radians in a complete circle = 2π

⇒ Fraction represented by the angle = 2 ÷ 2π

⇒ Simplifying the fraction = 1 ÷ π

⇒ Percentage = (1 ÷ π) × 100

⇒ We know that the value of π ≈ 3.14159

⇒ Percentage = 100 ÷ 3.14159

⇒ Percentage ≈ 31.83%

∴ The correct answer is 31.83%.

Additional Information

Degrees vs Radians

1 radian is the angle subtended by an arc equal in length to the radius of the circle. 1 radian ≈ 57.296°. To convert an angle from radians to degrees, multiply by 180 ÷ π.

Arc Length Formula

The length of an arc s of a circle with radius r subtending an angle θ (in radians) at the center is given by s = r × θ.

Area of a Sector

The area of a circle sector with a central angle θ (in radians) is given by the formula Area = 0.5 × r² × θ.

Shortcut Trick

By complementary angle properties, cot(90° − A) = tan A.

The given expression simplifies to: tan A × tan A = tan² A.

Given sin A = 3 / 5, using the standard 3-4-5 Pythagorean triplet, the base is 4.

Thus, tan A = Perpendicular / Base = 3 / 4.

Value of expression = (3 / 4)² = 9 / 16.

∴ The correct answer is 9/16.

Alternate Method

Given:

sin A = 3 / 5

A is an acute angle.

Formula Used:

cot(90° − θ) = tan θ

tan θ = Perpendicular / Base

sin θ = Perpendicular / Hypotenuse

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

A Base = 4k Perpendicular = 3k Hypotenuse = 5k

Calculations:

⇒ sin A = 3 / 5 = Perpendicular / Hypotenuse

⇒ Let Perpendicular = 3k and Hypotenuse = 5k.

⇒ Base² = Hypotenuse² − Perpendicular²

⇒ Base² = (5k)² − (3k)²

⇒ Base² = 25k² − 9k² = 16k²

⇒ Base = √(16k²) = 4k

⇒ tan A = Perpendicular / Base = 3k / 4k = 3 / 4

⇒ Expression to find: tan A × cot(90° − A)

⇒ Since cot(90° − A) = tan A, substitute this back into the expression.

⇒ Expression = tan A × tan A = tan² A

⇒ tan² A = (3 / 4)² = 9 / 16

∴ The correct answer is 9/16.

Additional Information

Complementary Angles

Trigonometric ratios of complementary angles are related as: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Pythagorean Triplets

Sets of three positive integers a, b, and c that satisfy a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Basic Trigonometric Identities

The fundamental mathematical identities are: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.

Shortcut Trick The total distance between the centers is d = 8 cm. The sum of the radii of the two circles is r1 + r2 = 6 cm + 2 cm = 8 cm. Since d = r1 + r2, the two circles touch each other externally. Whenever two circles touch externally, exactly 3 common tangents can be drawn (2 direct and 1 transverse). ∴ The correct answer is 3 tangents.

Alternate Method Given: ⇒ Radius of the first circle, r1 = 6 cm ⇒ Radius of the second circle, r2 = 2 cm ⇒ Distance between the centers, d = 8 cm

Concept Used: ⇒ If d > r1 + r2, the circles do not touch → 4 common tangents ⇒ If d = r1 + r2, the circles touch externally → 3 common tangents ⇒ If r1 − r2 < d < r1 + r2, the circles intersect → 2 common tangents ⇒ If d = r1 − r2, the circles touch internally → 1 common tangent ⇒ If d < r1 − r2, one circle lies inside the other → 0 common tangents

C₁ C₂ P

6 cm 2 cm

d = 8 cm

Direct Tangent 1 Direct Tangent 2 Transverse Tangent

Calculations: ⇒ Sum of the radii = r1 + r2 ⇒ Sum of the radii = 6 + 2 = 8 cm. ⇒ We are given that the distance between their centers, d = 8 cm. ⇒ Since d = r1 + r2, the condition for externally touching circles is satisfied. ⇒ Therefore, exactly 3 common tangents can be drawn to the two circles. ∴ The correct answer is 3 tangents.

Additional Information Length of Direct Common Tangent (DCT) Length of DCT = √(d2 − (r1 − r2)2), where d is the distance between the centers. Length of Transverse Common Tangent (TCT) Length of TCT = √(d2 − (r1 + r2)2). Valid only when d ≥ r1 + r2. Intersection Point of Direct Tangents The direct common tangents intersect at a common external point that divides the line joining the centers externally in the ratio r1 : r2.

Shortcut Trick

In a circle, a chord subtending 60° at the center forms an isosceles triangle with the two equal radii.

Since the central angle is 60°, the remaining two angles are also (180° − 60°) ÷ 2 = 60° each.

This forms an equilateral triangle where all three sides are perfectly equal.

∴ Length of chord = Radius of the circle = 10 cm.

∴ The correct answer is 10 cm.

Alternate Method

Given:

Radius of the circle (r) = 10 cm

Angle subtended by the chord at the center (θ) = 60°

Formula Used:

Length of a chord = 2 × r × sin(θ ÷ 2)

O

A

B

60° 10 cm 10 cm ? cm

Calculations:

⇒ Chord length = 2 × 10 × sin(60° ÷ 2)

⇒ Chord length = 20 × sin(30°)

⇒ Chord length = 20 × 0.5

⇒ Chord length = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Chord Length Formula

The exact length of any chord is calculated as L = 2 × r × sin(θ ÷ 2), where r is the radius and θ is the central angle subtended by the chord.

Equilateral Triangle Property

When a chord subtends exactly 60° at the center of the circle, the triangle formed by the chord and the two radii is always an equilateral triangle, rendering the chord identical in length to the radius.

Perpendicular Bisector Theorem

A perpendicular drawn directly from the center of a circle to any given chord will automatically bisect the chord into two perfectly equal halves.

Shortcut Trick

Angles subtended by the same arc at any point on the remaining part of the circle are always equal.

Here, the arc PS subtends ∠PQS and ∠PRS at the circumference.

∴ ∠PRS = ∠PQS.

Given ∠PQS = 40°, we directly get ∠PRS = 40° (The given ∠RPS = 30° is redundant information).

∴ The correct answer is 40°.

Alternate Method

Given:

∠RPS = 30°

∠PQS = 40°

Concept Used:

Angles in the same segment of a circle are equal.

     P Q R S 40° 30° ? ∴ ∠PRS = 40°

Calculations:

⇒ In the given circle, consider the chord PS.

⇒ The chord PS subtends two angles at the circumference in the major segment: ∠PQS and ∠PRS.

⇒ According to the circle theorem, angles in the same segment of a circle are equal.

⇒ ∴ ∠PRS = ∠PQS

⇒ Since ∠PQS = 40°, we directly get ∠PRS = 40°.

∴ The correct answer is 40°.

Additional Information

Angles in Same Segment

Angles subtended by the same arc or chord at any point on the remaining part of the circle are always equal.

Angle at Centre vs Circumference

The angle subtended by an arc at the centre is double the angle subtended by it at any remaining part of the circle.

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on a circle. The sum of its opposite angles is always 180°.

Angle in a Semicircle

The angle subtended by the diameter of a circle at any point on its circumference is always a right angle (90°).

Shortcut Trick

Angles subtended by the same arc at any point on the circumference of a circle are always equal.

Here, both ∠RPQ and ∠RSQ are angles at the circumference subtended by the same arc RQ.

Therefore, we can directly conclude that ∠RSQ = ∠RPQ.

Given that ∠RPQ = 50°, it must be true that ∠RSQ is also 50°.

∴ The correct answer is 50°.

Alternate Method

Given:

Chords PR and QS intersect at the center O, meaning they are both diameters.

∠RPQ = 50°

50°

?

P

Q

R

S

O

Angles subtended by the same arc RQ are equal

Calculations:

⇒ Since PR and QS are diameters of the circle, the quadrilateral PQRS is a rectangle.

⇒ In a rectangle, opposite sides are always parallel to each other, so PQ || RS.

⇒ Taking PR as a transversal line intersecting the parallel lines PQ and RS, the alternate interior angles formed will be equal.

⇒ Therefore, ∠PRS = ∠RPQ = 50°.

⇒ Now, consider the triangle ORS formed by the center O and the chord RS. The side OR = side OS (since both are radii of the same circle).

⇒ Because angles opposite to equal sides in a triangle are equal, we have ∠OSR = ∠ORS.

⇒ Notice that ∠OSR is the same exact angle as ∠RSQ. Thus, ∠RSQ = ∠PRS = 50°.

∴ The correct answer is 50°.

Additional Information

Angles in the Same Segment

Angles subtended by the same arc at any point on the remaining part of the circle are always equal in measure.

Angle in a Semicircle

An angle subtended by a diameter at any point on the circumference of a circle is always a perfect right angle (90°).

Cyclic Quadrilateral

A quadrilateral whose four vertices lie exactly on the boundary of a circle is known as a cyclic quadrilateral. The sum of its opposite angles is always 180°.