SSC CGL 2025 (Held On: 25 Sept, 2025 Shift 1) (Previous Year Paper)

Given word: PHYSICS

The 2nd letter from the right is C, which has an alphabetical index of 3.

Hence, the correct answer is Option 1.

The address given in the problem is,

12-B, Sunrise Apartments, Sector 14, Gurugram, Haryana - 122001

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

So, the address that is exactly the same as the given address is in option (3).

Hence, the correct option is "option (3)".

Given:

Guntur is 2 days before Visakhapatnam.

Kakinada is 3 days after Visakhapatnam.

Kakinada = 20th Feb

Now,

Visakhapatnam = 20th Feb − 3 days = 17th Feb

Guntur = 17th Feb − 2 days = 15th Feb

Hence, the correct answer is Option 3.

The logic followed here is:

Logic: Last number = (1st number)^(2nd number)

So, 3 × 3 = 27

⇒ 3³ = 27

Similarly, 4 × 4 = ?

⇒ ? = 4⁴ = 256

Hence, the correct answer is 256.

The logic followed here is:

thus, "35" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given Statement:

F < B < E ≤ C = D ≥ A < G

From statement:

F < B < E ≤ C = D

and A < G

No direct relation between G and F

No direct relation between C and G

Conclusions:

I. G < F → False (no definite relation)

II. C > G → False (no definite relation)

Hence, the correct answer is Option 2.

The logic followed here is:

Logic: Identify the city belonging to a different state

Given:

Lucknow → Uttar Pradesh

Varanasi → Uttar Pradesh

Kanpur → Uttar Pradesh

Hyderabad → Telangana

So, three cities are in Uttar Pradesh, while one is in Telangana

Therefore, Hyderabad is different

Hence, the correct answer is Option 4.

Family Chart:

Given:

A is the wife of B.

C is the son of A.

Therefore, C is the Son  of B.

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: (Sum of the digits of the first number)³ = second number.

Now,

For 23 : 125

⇒ (2 + 3)³

⇒ 5³ = 125

Similarly, for 31 : ?

⇒ (3 + 1)³

⇒ 4³ = 64

Hence, the correct answer is "Option 4".

Given Statements:

Some men are honest.

All honest people are brave.

From statements:

Conclusions:

I. Some men are brave → True (since some men are honest and all honest are brave)

II. All men are brave → False (only some men are honest, not all)

Hence, the correct answer is Option 1.

Given Statements:

All keys are locks.

Some locks are tools.

No tool is a weapon.

From statements:

Conclusions:

I. Some keys are not weapons → False (no direct link between keys and tools/weapons)

II. All tools are locks → False (only some locks are tools, not all tools are locks)

Hence, the correct answer is Option 4.

The logic followed here is:

Logic: Each letter is shifted −3 in the English alphabetical order

Given: MANDELA → JXKABIX

Now, GANDHI → ?

So, GANDHI → DXKAEF

Hence, the correct answer is Option 4.

The logic followed here is:

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

Hence, the correct answer is "Option 2"

The logic followed here is:

thus, "100" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

All the words except RAZ are names of animals:

So, the odd one out is RAZ.

Hence, the correct answer is "Option 4".

The logic followed here is:

Logic: Identify the city that does not match the common pattern

Given:

Patna → State capital (Bihar)

Jaipur → State capital (Rajasthan)

Raipur → State capital (Chhattisgarh)

Chennai → State capital (Tamil Nadu)

Surat → Not a state capital (Gujarat’s capital is Gandhinagar)

So, four are state capitals, while one is not

Therefore, Surat is different

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "66" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given:

Distance = 360 km

Time taken = 4 hours

Speed of the train = Distance ÷ Time

= 360 ÷ 4

= 90 km/h

If speed increases by 10 km/h, new speed = 100 km/h

New time = Distance ÷ New speed

= 360 ÷ 100

= 3.6 hours

Hence, the correct answer is Option 3.

Logic: Same words have the same code in all sentences.

Given codes:

'The book is interesting' → mo, no, lo, po

'The book is boring' → mo, no, to, po

'The movie is interesting' → do, no, lo, po

Therefore,

'The movie is boring' → do, no, po, to

Hence, the correct answer is Option 2.

The logic followed here is:

Logic: The word is reversed

Given:

NORTH → HTRON

Now, ORTHN → ?

Reverse it → NHTRO

Hence, the correct answer is Option 2.

Given:

8 pens cost ₹80

Cost of 1 pen = 80 ÷ 8 = ₹10

Number of pens that can be bought for ₹200:

200 ÷ 10 = 20

Hence, the correct answer is Option 2.

BODMAS Table: 

Given equation: 5 * 3 # 2 @ 4 = ?

Now, if * = +, @ = -, # = ×, then:

5 + 3 × 2 - 4 = ?

5 + 6 - 4 = ?

11 - 4 = 7

Hence, the correct answer is Option 1.

Given:

First five natural numbers = 1, 2, 3, 4, 5

Multiply them:

1 × 2 × 3 × 4 × 5

= 2 × 3 × 4 × 5

= 6 × 4 × 5

= 24 × 5

= 120

Hence, the correct answer is Option 2.

The logic followed here is:

Logic: Each letter is shifted +1 in the English alphabetical order

Given: WINDOW → XJOEPX

Now, XJOEPX → ?

So, XJOEPX → YKPFQY

Hence, the correct answer is Option 1.

The logic followed here is:

Logic: Each letter is shifted −1 in the English alphabetical order

Given: COVERT → BNUFQS

Now,  ABSENT → ?

So, ABSENT → ZARFMS

Hence, the correct answer is Option 1.

The correct answer is Humayun's Tomb.

Key Points

• Humayun's Tomb, located in Delhi, India, served as a major inspiration for the design and architecture of the Taj Mahal.
• It was built in 1565-1572 by Empress Bega Begum, the wife of Mughal Emperor Humayun, and is considered the first garden-tomb in the Indian subcontinent.
• Humayun's Tomb is known for its distinctive Persian-style architecture, which features a symmetrical layout, a double dome, and intricate carvings.
• The architectural elements of Humayun's Tomb, including the use of white marble and red sandstone, influenced the construction of the Taj Mahal in Agra.
• The Taj Mahal, built by Emperor Shah Jahan in memory of his wife Mumtaz Mahal, incorporated the aesthetic principles seen in Humayun's Tomb, particularly the symmetry and Mughal garden layout.

Additional Information

• Humayun's Tomb:
• Located in Delhi, India, it is a UNESCO World Heritage Site.
• It was the first structure in India to use red sandstone on a large scale.
• The tomb was designed by Mirak Mirza Ghiyas, a Persian architect.
• Its design features a charbagh (four-part garden), a hallmark of Mughal architecture.
• Taj Mahal:
• Built in 1632-1653 in Agra, Uttar Pradesh, by Mughal Emperor Shah Jahan as a mausoleum for his wife Mumtaz Mahal.
• It is made of white marble and is recognized as one of the Seven Wonders of the World.
• The layout of the Taj Mahal is inspired by the Persian-style charbagh gardens seen in Humayun's Tomb.
• The Taj Mahal is also a UNESCO World Heritage Site and symbolizes the zenith of Mughal architecture.
• Other Options Explained:
• Jama Masjid: Built by Shah Jahan in Delhi, it is one of the largest mosques in India but does not share direct architectural influence with the Taj Mahal.
• Victoria Memorial: Located in Kolkata, this structure was built during British rule and has no connection to Mughal architecture or the Taj Mahal.
• Qutub Minar: A UNESCO World Heritage Site in Delhi, it is an ancient minaret from the Delhi Sultanate period and is unrelated to the architectural inspiration of the Taj Mahal.
• Important Points:
• The use of symmetry, domes, and gardens in Mughal architecture is a defining characteristic that connects Humayun's Tomb and the Taj Mahal.
• Both structures reflect the fusion of Persian, Islamic, and Indian architectural styles.

The correct answer is Stimulants.

Key Points

• Stimulants are a category of substances used to enhance alertness, reduce fatigue, and improve focus and energy levels.
• They are often used in sports to provide a competitive edge by boosting physical and mental performance.
• These substances work by increasing the activity of the central nervous system (CNS), leading to heightened concentration and reduced perception of exertion.
• Common stimulants include amphetamines, caffeine, cocaine, ephedrine, and modafinil, among others.
• The use of stimulants in sports is prohibited as they can provide an unfair advantage and pose serious health risks such as addiction, cardiovascular issues, and anxiety disorders.

Additional Information

• Categories of Prohibited Substances in Sports:
• Anabolic Agents: These substances promote muscle growth and are commonly used for performance enhancement. Examples include anabolic steroids like testosterone.
• Peptide Hormones: These are synthetic substances that mimic natural hormones to stimulate growth or increase endurance. Examples include erythropoietin (EPO) and human growth hormone (HGH).
• Beta-2 Agonists: These are typically used to treat asthma by relaxing airway muscles. However, some are misused in sports to enhance performance.
• Stimulants: This category includes substances that increase alertness and reduce fatigue, as explained above.
• Why Stimulants Are Prohibited:
• They provide an unfair advantage by enhancing mental and physical performance.
• Prolonged use can lead to severe health consequences, including increased heart rate, high blood pressure, and addiction.
• They can impair judgment and decision-making, leading to potential risks for athletes and others.
• Regulatory Measures:
• The World Anti-Doping Agency (WADA) monitors and updates the list of prohibited substances, including stimulants.
• Strict drug testing protocols are in place during competitions and training to ensure fair play.
• Athletes found guilty of using banned substances face sanctions, including suspensions or lifetime bans.
• Examples of Stimulants in Sports:
• Ephedrine: A common decongestant misused as a performance enhancer.
• Cocaine: A powerful stimulant with severe health risks.
• Caffeine: While naturally occurring and widely used, its use is monitored in high concentrations in sports.

Important Points

• Health Risks of Stimulants:
• Can lead to cardiac arrhythmias, heart attacks, or strokes due to increased heart rate and blood pressure.
• May cause psychological dependency and addiction over time.
• Prolonged use can result in severe mental health issues like anxiety, paranoia, or depression.
• WADA Guidelines:
• The World Anti-Doping Code specifies strict penalties for the use of banned substances.
• Athletes are advised to check the WADA Prohibited List before taking any medication or supplements.

The correct answer is The Montagu-Chelmsford Reforms of 1919.

Key Points

• The Montagu-Chelmsford Reforms of 1919 introduced the system of dyarchy (dual government) in the provinces of British India. This was a major constitutional reform aimed at increasing Indian participation in governance.
• Under dyarchy, the provincial subjects were divided into two categories: Transferred and Reserved.
• Transferred subjects: These included areas like education, public health, agriculture, and local government, which were administered by Indian ministers responsible to the legislative councils.
• Reserved subjects: These included areas like finance, police, and revenue, which were controlled by the British Governor and his executive council.
• The reforms were based on the Montagu-Chelmsford Report of 1918, authored by Edwin Montagu (Secretary of State for India) and Lord Chelmsford (Viceroy of India).
• The objective of the reforms was to gradually introduce a responsible government in India while retaining British control over vital areas of administration.
• The system of dyarchy, however, was heavily criticized due to its complexity and inefficiency. It failed to meet the aspirations of Indian leaders and was eventually replaced by provincial autonomy under the Government of India Act of 1935.

Additional Information

• The Indian Councils Act of 1861:
• This Act marked the beginning of legislative devolution by allowing Indians to be included in the law-making process as non-official members in the Viceroy's council.
• It laid the foundation for representative institutions in India but did not introduce dyarchy.
• The Morley-Minto Reforms of 1909:
• Also known as the Indian Councils Act of 1909, this reform introduced the concept of separate electorates for Muslims.
• It allowed for greater Indian representation in legislative councils but did not involve the division of subjects or dyarchy.
• The Montagu-Chelmsford Reforms of 1919:
• Introduced dyarchy in provinces, which was a significant shift towards self-governance.
• It also expanded the legislative councils and introduced direct elections for some seats.
• This reform emphasized the principle of increasing Indian participation in governance.
• The Government of India Act of 1935:
• This Act replaced dyarchy with provincial autonomy, granting full control of provincial administration to Indian ministers.
• It also proposed the establishment of a federation of British India and princely states, though this was never implemented.

Important Points

• Dyarchy:
• It was the first step towards decentralization of power in British India.
• The system was criticized because the reserved subjects, controlled by the British, were often the more critical and resource-intensive areas.
• Failures of the Montagu-Chelmsford Reforms:
• The division of subjects created confusion and inefficiency in governance.
• Indian leaders criticized the reforms for not providing full self-governance or addressing key issues like taxation and economic exploitation.
• Historical Context: The Montagu-Chelmsford Reforms were introduced in the aftermath of World War I when India contributed significantly to the war effort. The reforms were a response to increasing demands for political reforms and self-governance.
• Legacy: Though the dyarchy system was short-lived, it laid the groundwork for future reforms, eventually leading to India's independence in 1947.

The correct answer is Use of video conferencing and identity concealment for vulnerable witnesses.

Key Points

• Video conferencing and identity concealment are provisions that enhance the protection of vulnerable witnesses during trials. These measures ensure that witnesses can provide testimony without being physically present, reducing the risk of intimidation or harm.
• Identity concealment is a critical safeguard for witnesses who may be threatened or coerced due to their participation in legal proceedings. It helps maintain their anonymity, ensuring their safety and encouraging truthful testimony.
• Such provisions are particularly important in cases involving sensitive matters, organized crime, or instances where witnesses face threats from powerful entities.
• The system allows witnesses to testify from secure locations, ensuring that they are shielded from direct confrontation with the accused or their associates.
• These measures align with the broader goal of ensuring justice while protecting individuals who play an instrumental role in legal processes.

Additional Information

• Witness Protection Scheme, 2018:
• This scheme was introduced by the Ministry of Home Affairs, Government of India, to provide adequate protection to witnesses in sensitive cases.
• The scheme emphasizes the confidentiality of identity, relocation of witnesses, and other measures to ensure their safety.
• It categorizes witnesses into three categories based on the threat perception and provides necessary protection accordingly.
• Use of Technology in Courts:
• Video conferencing is increasingly used to conduct hearings, particularly in cases involving vulnerable witnesses or during emergencies like the COVID-19 pandemic.
• Technology minimizes logistical challenges and provides a secure environment for witnesses to testify.
• It also enhances judicial efficiency by reducing delays caused by witness unavailability.
• International Practices:
• Countries like the United States and United Kingdom have adopted similar measures for witness protection, including the use of video conferencing and voice modulation technologies.
• The United Nations Office on Drugs and Crime (UNODC) also advocates for witness protection mechanisms as part of its guidelines for criminal justice reform.
• Legal Provisions in India:
• The Indian Evidence Act, 1872, and the Criminal Procedure Code have been amended to incorporate provisions for witness protection.
• Section 161 of CrPC allows for recording witness statements through electronic means, ensuring flexibility and security.

Important Points

• Anonymous Jury System:
• This concept is not prevalent in India but is used in certain international jurisdictions to protect jury members in high-risk cases.
• It involves concealing the identities of jury members to safeguard them from external pressures.
• However, this option does not address witness protection directly, making it less relevant to the Indian context.
• SMS Statement Recording:
• This is not a recognized method for recording witness statements as per Indian legal standards.
• While it might seem convenient, it lacks the safeguards necessary to ensure authenticity and reliability.
• No Attendance Requirement:
• Witnesses are generally required to attend trials unless provisions like video conferencing are used.
• Absence of attendance without alternative arrangements can compromise the integrity of judicial proceedings.
• Role of Judiciary:
• The judiciary plays a pivotal role in implementing witness protection measures, ensuring a fair trial while safeguarding the rights and safety of witnesses.
• Landmark judgments like State of Maharashtra v. Dr. Praful B. Desai have upheld the use of video conferencing for witness testimonies.

The correct answer is First Buddhist Council - Rajagriha.

Key Points

• First Buddhist Council - Rajagriha:
  The First Buddhist Council was held in Rajagriha (modern-day Rajgir), Bihar, shortly after the death of Buddha in 483 BCE. It was presided over by Mahakassapa and convened by King Ajatashatru. The purpose was to preserve Buddha’s teachings (Sutta) and monastic discipline (Vinaya). Hence, this statement is correct.
• Second Buddhist Council - Lumbini:
  The Second Buddhist Council was held at Vaisali, not Lumbini, around 383 BCE. It was convened to address disputes over monastic rules. Hence, this statement is incorrect.
• Third Buddhist Council - Kashmir:
  The Third Buddhist Council was held in Pataliputra, not Kashmir, around 250 BCE, during the reign of Emperor Ashoka. It aimed to purify the Buddhist community of corruption and heretical views. Hence, this statement is incorrect.
• Fourth Buddhist Council - Pataliputra:
  The Fourth Buddhist Council was actually held in Kashmir, not Pataliputra, during the reign of King Kanishka in the 1st century CE. It focused on compiling the Sarvastivada Abhidharma texts. Hence, this statement is incorrect.

Additional Information

• Key Objectives of Buddhist Councils:
• The Buddhist Councils played a vital role in preserving and spreading the teachings of Buddha.
• They ensured the compilation, categorization, and interpretation of Buddhist scriptures such as the Tripitaka (Sutta, Vinaya, and Abhidhamma).
• Details of Major Buddhist Councils:
• First Buddhist Council:
  Held in Rajagriha, led to the compilation of Buddha’s teachings into the Sutta Pitaka and monastic code into the Vinaya Pitaka.
• Second Buddhist Council:
  Held in Vaisali, it dealt with disputes over monastic discipline and led to the first schism in Buddhism, resulting in the formation of the Sthaviravada and Mahasanghika schools.
• Third Buddhist Council:
  Held in Pataliputra, it focused on purifying Buddhism and included the codification of the Abhidhamma Pitaka.
• Fourth Buddhist Council:
  Held in Kashmir under King Kanishka’s patronage, it compiled commentaries on Buddhist texts and spread Buddhism to Central Asia.
• Significance of King Ashoka:
• King Ashoka played a crucial role in spreading Buddhism globally after the Third Buddhist Council.
• He sent Buddhist missionaries to regions such as Sri Lanka, Southeast Asia, and the Mediterranean.
• Key Terms and Scriptures:
• Tripitaka: The foundational Buddhist scriptures, comprising the Vinaya Pitaka (monastic code), Sutta Pitaka (Buddha’s teachings), and Abhidhamma Pitaka (philosophical analysis).
• Mahavibhasha: A significant commentary text compiled during the Fourth Buddhist Council.

The correct answer is Ghudlia.

Key Points

• Ghudlia refers to a decorated earthen pot used during the Gangaur festival in Madhya Pradesh.
• During Gangaur, unmarried girls place lamps inside the decorated Ghudlia and carry them while singing traditional songs dedicated to Gauri and Shiva.
• This ritual is performed to seek blessings for a good husband and a prosperous married life.
• Gangaur is a prominent festival celebrated in parts of Rajasthan and Madhya Pradesh, especially by women.
• The term "Ghudlia" is specific to Madhya Pradesh and reflects the cultural practices of the region during this festival.

Additional Information

• Gangaur Festival:
• Gangaur is celebrated in honor of Goddess Gauri, the consort of Lord Shiva.
• It marks the celebration of marital fidelity and the arrival of spring.
• Women worship Goddess Gauri with clay idols and observe rituals to pray for the longevity and prosperity of their husbands.
• The festival is celebrated with great enthusiasm in states like Rajasthan, Madhya Pradesh, and Gujarat.
• Use of Ghudlia:
• In Madhya Pradesh, unmarried girls decorate the Ghudlia with vibrant colors and designs.
• Lamps are placed inside the Ghudlia, symbolizing light and hope.
• The girls carry these pots on their heads and sing folk songs to invoke the blessings of the divine couple.
• Other Cultural Practices:
• Similar rituals exist in Rajasthan, where Gangaur is celebrated with fairs, processions, and dance performances.
• In Gujarat, the festival coincides with Navratri and includes similar worship of Goddess Gauri.
• Option Analysis:
• Kalash: A sacred pot used in Hindu rituals, but it is not specific to Gangaur in Madhya Pradesh.
• Matki: An earthen pot used in general for carrying water or cooking, not specifically for Gangaur rituals.
• Deepdan: Refers to the offering of lamps, but it is a general term and not specific to the decorated pot used in this festival.

Important Points

• Gangaur Festival's Significance:
• It is celebrated for 16 days starting from the day after Holi.
• Both married and unmarried women take part in the festivities. Married women pray for their husband's health and happiness, while unmarried girls pray for a suitable match.
• Regional Variations:
• The festival holds unique significance in different parts of India, with distinct rituals and traditions.
• In Rajasthan, Gangaur processions feature beautifully adorned elephants and horses.
• In Madhya Pradesh, the use of Ghudlia adds a unique touch to the celebrations.

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

Key Points

• Assertion (A): Gandhara images frequently feature halo and curly hair.
  Gandhara art is renowned for its distinctive style, which blends Indian traditions with Graeco-Roman elements. Halo and curly hair are prominent features of Gandhara sculptures, especially in depictions of Buddha. These elements are visually striking and symbolic of divinity and enlightenment. Hence, the assertion is correct.

• Reasoning (R): These elements symbolize Graeco-Roman influence in North-West India.
  

The influence of Graeco-Roman art on Gandhara sculptures is evident in details such as curly hair, drapery resembling Roman togas, and the use of a halo to signify divinity. These features were introduced due to cultural exchanges following Alexander the Great's invasion and subsequent Indo-Greek rule in North-West India. Hence, the reasoning is correct.

• Connection between A and R:
  

The reasoning effectively explains the assertion, as the halo and curly hair are direct results of Graeco-Roman artistic influence. This cultural amalgamation shaped Gandhara art during the Kushan period. Therefore, both A and R are true, and R is the correct explanation of A.

Additional Information

• Gandhara Art:
  

Gandhara art flourished primarily in the regions of modern-day Pakistan and Afghanistan during the Kushan Empire (1st–5th century CE). It is notable for its synthesis of Indian, Greek, and Roman artistic traditions.

• Graeco-Roman Influence:
  

Greek and Roman styles significantly influenced Gandhara art due to cultural interactions from Alexander's campaigns and subsequent Indo-Greek and Kushan rulers. Elements such as realistic human forms, curly hair, drapery folds, and halos were adapted into Buddhist iconography.

• Features of Gandhara Sculptures:
  

Gandhara sculptures often depict Buddha with Greek-style curly hair, a urna (symbol of enlightenment), and a halo. The use of mathematical proportions and realistic anatomy also reflects Greek artistic principles.

• Kushan Empire:
  

The Kushan Empire (30–375 CE) played a pivotal role in promoting Gandhara art. Rulers such as Kanishka the Great patronized Buddhist art and architecture, facilitating the integration of Graeco-Roman styles into Indian traditions.

• Historical Significance:
  

Gandhara art represents a unique cultural fusion and serves as an important visual record of the spread of Buddhism and its adaptation to local traditions across regions.

Important Points

• Halo:
  

The halo in Gandhara art, used around the head of Buddha, symbolizes divinity, spiritual radiance, and enlightenment. This element was borrowed from Graeco-Roman religious art.

• Curly Hair:
  

The depiction of Buddha with curly hair is inspired by Greek artistic conventions, particularly those seen in sculptures of Greek gods like Apollo.

• Trade and Cultural Exchange:
  

Gandhara's position on the Silk Route facilitated extensive trade and cultural exchange, allowing Graeco-Roman styles to influence local art.

• Legacy:
  

Gandhara art significantly influenced later Buddhist art traditions in Central Asia, China, Korea, and Japan.

The correct answer is Suktas.

Key Points

• The Rigveda, one of the oldest texts in the world and a foundational scripture of Hinduism, is divided into ten books called Mandalas.
• Each Mandala is further subdivided into Suktas, which are hymns dedicated to deities such as Agni (fire), Indra (rain and thunder), and others.
• The term Sukta literally means "well said" or "well recited" and refers to collections of verses or hymns.
• There are a total of 1,028 Suktas in the Rigveda, organized across the ten Mandalas. These hymns are composed in praise of natural forces and cosmic principles.
• The Suktas are further composed of verses known as Riks, which are the smallest units of the Rigveda.
• Each Sukta typically has a specific rishi (sage) credited with its composition, and it is directed towards specific deities or natural elements.

Additional Information

• Rigveda:
• The Rigveda is one of the four Vedas, the others being the Yajurveda, Samaveda, and Atharvaveda.
• It is written in Vedic Sanskrit and is considered to be composed around 1500 BCE to 1200 BCE.
• The Rigveda is primarily a collection of hymns (Suktas) dedicated to various deities and natural forces.
• It provides insights into the religious, social, and cultural life of the early Vedic period.
• Structure of Rigveda:
• The Rigveda is divided into 10 Mandalas (books).
• The Mandalas are not of uniform length; some are longer, while others are shorter.
• The second to seventh Mandalas are known as the "Family Books" because they are attributed to specific families of sages.
• Other Divisions in Vedic Literature:
• Kandas: This term is used in the context of other Vedic texts, such as the Yajurveda, where the content is organized into sections called Kandas.
• Puranas: These are a separate category of ancient Indian texts that primarily contain mythological stories, genealogies, and religious instructions. They are not part of the Rigveda.
• Samhitas: This term refers to collections of hymns or mantras in Vedic literature. The Rigveda itself is a Samhita, but within it, the subdivisions are called Suktas, not Samhitas.
• Key Features of Suktas:
• Suktas are primarily composed in praise of deities, such as Agni (fire), Indra (war and rain), Varuna (cosmic order), and others.
• They often reflect the Vedic people's understanding of the cosmos, nature, and human existence.
• Several hymns also include philosophical inquiries, such as the famous Nasadiya Sukta, which explores the origin of the universe.

Important Points

• Kandas: Used in the Yajurveda; not applicable to the Rigveda, which uses Mandalas and Suktas.
• Puranas: These are post-Vedic texts and are not part of the Rigveda's structure.
• Samhitas: The Rigveda itself is a Samhita, but its internal divisions are called Suktas.
• The Rigveda is considered the most sacred text in Hinduism and is an invaluable resource for understanding ancient Indian civilization.

The correct answer is 1, 2 and 3 are Correct.

Key Points

• Field Goal Validity: A field goal in field hockey is valid only if the ball is struck from inside the shooting circle. If the ball is played from outside this circle, the goal will not be counted. Hence, statement 1 is correct.
• Penalty Corner Requirements: For a goal to be valid during a penalty corner, the ball must first exit the shooting circle and then re-enter it before being scored. This ensures proper execution of the penalty corner as per the rules. Hence, statement 2 is correct.
• Penalty Stroke Rules: During a penalty stroke, the ball may be lifted to any height as long as it is pushed or flicked. This allows players to strategically place the ball during the stroke. Hence, statement 3 is correct.
• All three statements provided in the question align with the official rules of field hockey, verifying their correctness.

Additional Information

• Field Hockey Shooting Circle:
• The shooting circle is a semicircular area with a radius of 16 yards (14.63 meters) from the goal post.
• Only goals scored from within this circle are considered valid unless the ball touches a defender's stick inside the circle.
• Penalty Corner Rules:
• A penalty corner is awarded for specific fouls committed within the shooting circle or for intentional fouls in the defensive 23-meter area.
• The ball must be stopped or controlled outside the circle before a shot can be made.
• Penalty corners are often critical in matches, as they provide a high chance of scoring.
• Penalty Stroke Rules:
• A penalty stroke is awarded for serious fouls committed inside the circle that prevent a probable goal.
• The stroke is taken from a spot 6.4 meters away from the goal line.
• The goalkeeper must remain on the goal line until the stroke is taken.
• Scoring in Field Hockey:
• Goals can be scored in three main ways: field goals, penalty corners, and penalty strokes.
• Field goals are the most common and require the ball to be played within the shooting circle.
• Penalty corners and strokes are awarded for specific fouls and provide additional scoring opportunities.

Important Points

• Field Hockey Rules:
• Field hockey is governed by the International Hockey Federation (FIH), which regularly updates the rules to ensure fair play and safety.
• Matches are played in two halves of 35 minutes each (or four quarters of 15 minutes in modern formats).
• A team consists of 11 players, including a goalkeeper.
• Role of Umpires:
• Two on-field umpires officiate the game, ensuring the rules are followed.
• In major tournaments, video referees assist in reviewing contentious decisions.
• Penalty Situations:
• Penalty corners and strokes are vital moments in a match and can often change the outcome of a game.
• Teams use specialized strategies and players for these situations to maximize scoring chances.

The correct answer is Chitpavan Brahmanas.

Key Points

• After the death of Shivaji Maharaj in 1680, the Maratha Empire faced a period of political instability due to succession disputes and external invasions, especially by the Mughals.
• During this period, the Chitpavan Brahmanas, a community of Brahmins primarily located in the coastal regions of Maharashtra, rose to prominence and became influential in the Maratha administrative and political system.
• The Peshwa system (chief ministers) was established, and the Chitpavan Brahmanas dominated this position. The Peshwas became the de facto rulers of the Maratha Empire from the early 18th century.
• Notable Peshwas like Baji Rao I and Nana Saheb were instrumental in expanding the Maratha Empire and consolidating its governance under Brahmanical leadership.
• Under the leadership of the Peshwas, the Chitpavan Brahmanas controlled key administrative, military, and financial aspects of the Maratha state, ensuring their dominance in the political sphere.

Additional Information

• Chitpavan Brahmanas:
• The Chitpavans were a community of Brahmins located primarily in the Kokan region of Maharashtra.
• They were known for their administrative skills, strategic thinking, and intellectual capabilities, which made them suitable for leadership roles in the Maratha Empire.
• They played a pivotal role in shaping the administrative and military policies of the Maratha state during the Peshwa era.
• Prominent Chitpavan Brahmanas like Baji Rao I were instrumental in establishing Maratha dominance across India, challenging the Mughal Empire.
• Peshwa System:
• The Peshwa system was established by Shivaji Maharaj's successors and became the hallmark of Maratha governance.
• Under the Peshwas, the Maratha Empire expanded significantly, reaching as far as Delhi in the north and Orissa in the east.
• Peshwas like Baji Rao I and Madhav Rao are celebrated for their military campaigns and administrative reforms.
• Decline of Mughals:
• After Shivaji's death, the Mughals tried to reassert control over the Deccan region but faced resistance from the Marathas.
• The decline of the Mughal Empire post-1707, especially after the death of Aurangzeb, provided an opportunity for the Marathas to establish their dominance.
• Other groups in the Maratha Empire:
• Deshmukhs: They were local landholders and administrators but did not achieve central political dominance like the Peshwas.
• Kunbis: A farming community that supported the Maratha movement but remained primarily agrarian and did not hold significant political power.

Important Points

• Baji Rao I: He was one of the most celebrated Peshwas and is remembered for his military genius and successful campaigns against the Mughals and other regional powers.
• Madhav Rao: A Peshwa known for his administrative reforms and efforts to stabilize the Maratha Empire during a period of internal and external turmoil.
• Role of Chitpavan Brahmanas: Their dominance in the Peshwa system highlights the socio-political dynamics of the Maratha Empire, where Brahmanical leadership played a key role.
• Maratha Expansion: Under the Peshwa leadership, the Marathas emerged as one of the most powerful entities in India, challenging the declining Mughal Empire and other regional states.

The correct answer is Murlikant Petkar - Gold, men's 50m freestyle, Heidelberg 1972.

Key Points

• Murlikant Petkar was India's first Paralympic gold medallist. He won the gold medal in the men's 50m freestyle swimming event at the 1972 Paralympics held in Heidelberg, Germany.
• Petkar set a world record in the event with a timing of 37.33 seconds, showcasing his exceptional talent and determination.
• Before his Paralympic success, he was a soldier in the Indian Army and sustained injuries during the Indo-Pak war of 1965, which led to his disability.
• Despite his injuries, he showed immense resilience by transitioning to sports and excelling in swimming, table tennis, and javelin throw at national and international events.
• Murlikant Petkar's achievement was a landmark moment in Indian sports history, as it highlighted the capabilities of differently-abled athletes and paved the way for increased recognition and support for Paralympians in India.

Additional Information

• Devendra Jhajharia:
• Devendra Jhajharia is a celebrated Indian Paralympian who won the gold medal in javelin throw at the 2004 Athens Paralympics and repeated the feat at the 2016 Rio Paralympics.
• He is the first Indian Paralympian to win two gold medals and holds multiple world records in javelin throw.
• He lost his left arm due to an accident at the age of eight, which did not deter him from pursuing excellence in sports.
• Mariyappan Thangavelu:
• Mariyappan Thangavelu won the gold medal in high jump at the 2016 Rio Paralympics, becoming the first Indian Paralympian to achieve this feat in the event.
• He suffered a permanent disability in one leg following an accident at a young age.
• Thangavelu was awarded the Padma Shri and Arjuna Award for his contribution to sports.
• Avani Lekhara:
• Avani Lekhara won the gold medal in air rifle shooting at the 2020 Tokyo Paralympics, becoming the first Indian woman to win a Paralympic gold.
• She also secured a bronze medal in a separate shooting event at the same Paralympics.
• Avani's achievement marked a significant milestone for Indian women in Paralympic sports.
• Paralympics in India:
• The Paralympics have grown in importance in India over the years, with increasing participation and medal wins by Indian athletes.
• The government has introduced various schemes, such as the Target Olympic Podium Scheme (TOPS), to support Paralympic athletes financially and technically.
• India has produced several Paralympic champions, inspiring millions and promoting inclusivity in sports.

Important Points

• Challenges faced by Paralympic athletes:
• Limited access to training facilities and equipment tailored to differently-abled athletes.
• Lack of awareness and recognition for Paralympic sports in comparison to mainstream sports.
• Financial constraints and dependency on government schemes for support.
• Government Initiatives:
• The Paralympic Committee of India (PCI) oversees the development and promotion of Paralympic sports in India.
• Programs like Khelo India and TOPS aim to nurture talent among differently-abled athletes.
• Increased prize money and awards for Paralympic medal winners to motivate and recognize their efforts.
• Historical Milestones:
• India's first participation in the Paralympics was at the 1968 Tel Aviv Games.
• India has consistently improved its performance in Paralympic Games, winning a record-breaking 19 medals at the 2020 Tokyo Paralympics.

The correct answer is Shashi Tharoor.

Key Points

• The book 'Why I Am a Hindu' is authored by Shashi Tharoor, an Indian politician, diplomat, and acclaimed author.
• It was published in 2018 and explores the essence and interpretation of Hinduism as a religion and philosophy.
• Tharoor, in the book, discusses his own journey of being a Hindu and delves into the expansive nature of Hinduism, emphasizing its inherent pluralism and inclusivity.
• The book critiques the political misuse of Hinduism by certain groups and stresses the distinction between Hinduism as a way of life and the ideology of Hindutva.
• 'Why I Am a Hindu' also provides insights into the scriptures, philosophies, and practices of Hinduism, contrasting them with the narrow interpretations often propagated in the political sphere.

Additional Information

• Shashi Tharoor:
• Shashi Tharoor is a renowned Indian politician, diplomat, and former Under-Secretary-General of the United Nations.
• He is a Member of Parliament representing the Thiruvananthapuram constituency in Kerala.
• Tharoor is an accomplished author, having written several bestsellers, including 'The Great Indian Novel', 'Inglorious Empire', and 'The Paradoxical Prime Minister'.
• His works often focus on India's history, culture, and politics, and he is known for his eloquent writing and oratory skills.
• Hinduism:
• Hinduism is one of the world's oldest religions, with no single founder or central religious authority.
• It is characterized by a diversity of beliefs, practices, and scriptures, such as the Vedas, Upanishads, Bhagavad Gita, and Ramayana.
• The religion emphasizes concepts such as Dharma (righteousness), Karma (action and consequence), and Moksha (liberation).
• Hinduism is inherently pluralistic and accommodates a wide range of philosophical schools and spiritual paths.
• Hindutva vs. Hinduism:
• Hinduism refers to the cultural and spiritual practices and beliefs of the Hindu people.
• Hindutva, on the other hand, is a political ideology that seeks to define Indian culture based on the supremacy of Hindu values.
• Shashi Tharoor, in his book, critiques the conflation of Hindutva with Hinduism, arguing that it narrows the inclusive and philosophical nature of the religion.
• Other Options:
• Pavan K. Varma: A former Indian diplomat and author known for books like 'The Great Indian Middle Class' and 'Being Indian', but not the author of 'Why I Am a Hindu'.
• Aravind Adiga: An Indian author and journalist, best known for his Booker Prize-winning novel 'The White Tiger'.
• Gurcharan Das: An Indian author and commentator, known for works like 'India Unbound', but not associated with the book in question.

The correct answer is The idea of fraternity as a binding force in democracy.

Key Points

• Fraternity is a central theme in the book "Fraternity: Constitutional Norm and Human Need" authored by Prof. Ashwini Kumar. The book explores how the concept of fraternity serves as a moral and constitutional principle that binds citizens together in a democracy.
• The idea of fraternity emphasizes solidarity, mutual respect, and collective responsibility among citizens, ensuring that democracy is not just a system of governance but a culture of cooperation and shared values.
• Fraternity is one of the core principles enshrined in the Preamble of the Indian Constitution, alongside liberty, equality, and justice. The book delves into how this principle is integral to achieving social harmony and true democracy.
• The book addresses the challenges posed by economic inequality, social divisions, and political polarization, arguing that fraternity can act as a unifying force to bridge these gaps.
• Prof. Ashwini Kumar emphasizes that fraternity is not just a legal concept but also a moral necessity for fostering inclusion, dignity, and human dignity in a democratic society.

Additional Information

• Fraternity in the Indian Constitution:
• Fraternity is explicitly mentioned in the Preamble of the Indian Constitution, emphasizing the promotion of dignity of the individual and unity and integrity of the nation.
• The concept aligns with the vision of the founding fathers like Mahatma Gandhi, Dr. B.R. Ambedkar, and Jawaharlal Nehru, who envisioned a society free from caste, class, and communal divisions.
• Relation to Democracy:
• Fraternity is essential for the functioning of a vibrant democracy, as it fosters mutual respect and understanding among citizens, ensuring that democracy thrives on cooperation rather than conflict.
• It addresses issues such as social exclusion, discrimination, and inequality, which can undermine the democratic fabric of a nation.
• Challenges to Fraternity:
• Economic disparities, social divisions, and political polarization pose significant challenges to achieving fraternity in society.
• The book highlights the role of policies and governance in promoting fraternity by addressing these challenges through inclusivity and equitable distribution of resources.
• Examples from History:
• The principle of fraternity played a key role in the Indian freedom struggle, uniting people across diverse backgrounds for a common cause.
• Historical movements like the civil rights movement in the USA also emphasize the importance of fraternity in achieving social justice.

Important Points

• Prof. Ashwini Kumar: The author of the book is a noted academician and political thinker who has extensively written on democracy and constitutional values.
• Preamble of Indian Constitution: The Preamble declares fraternity as one of its guiding principles to ensure national unity and integrity.
• Human Need: Fraternity is not just a constitutional norm but a basic human need that fosters belongingness and dignity.
• Global Perspectives: The concept of fraternity is relevant globally, as seen in the principles of Universal Declaration of Human Rights and other international charters promoting equality and solidarity.

The correct answer is The first ISL match in 2014 was held in Kolkata..

Key Points

• The Indian Super League (ISL) was launched in 2014 as a professional football league in India to promote football and improve its standard across the country.
• The inaugural match of the ISL was held on 12th October 2014, marking the beginning of a new era in Indian football.
• The first match of the ISL was conducted in Kolkata, at the Salt Lake Stadium (Vivekananda Yuba Bharati Krirangan), which is one of the largest football stadiums in the world.
• The match featured Atletico de Kolkata (now ATK Mohun Bagan) playing against Mumbai City FC. Atletico de Kolkata won the match with a scoreline of 3-0.
• The choice of Kolkata as the venue for the opening match highlighted the city's rich football culture and history, often referred to as the football capital of India.

Additional Information

• About the Indian Super League (ISL):
• The ISL was launched in 2014 as part of India's efforts to globalize football in the country.
• It is organized by the All India Football Federation (AIFF), in collaboration with IMG-Reliance and Star Sports.
• The league operates on a franchise-based system, similar to the Indian Premier League (IPL) for cricket, with teams representing various cities and states of India.
• ISL has significantly contributed to the development of football in India by bringing in international players and coaches, improving infrastructure, and increasing viewership.
• Salt Lake Stadium:
• The Vivekananda Yuba Bharati Krirangan, popularly known as Salt Lake Stadium, is located in Kolkata, West Bengal.
• It was established in 1984 and has a seating capacity of approximately 85,000, making it one of the largest stadiums in the world by capacity.
• Salt Lake Stadium has hosted numerous prestigious football events, including international matches, domestic leagues, and tournaments like the FIFA U-17 World Cup.
• Teams in the Inaugural ISL:
• The first season of ISL featured eight teams: Atletico de Kolkata, Chennaiyin FC, Delhi Dynamos, FC Goa, Kerala Blasters, Mumbai City FC, NorthEast United, and FC Pune City.
• These teams represented different regions of India and aimed to bring football closer to fans from all parts of the country.
• Legacy of the First Match:
• The opening match of the ISL in Kolkata set the tone for the league's success, with high viewership and a festive atmosphere in the stadium.
• It established ISL as a premier football league in India, attracting both national and international attention.

Important Points

• Atletico de Kolkata went on to win the inaugural ISL championship in 2014, defeating Kerala Blasters in the final.
• The ISL has played a crucial role in reviving Indian football, increasing youth participation, and creating opportunities for local talent.
• The league has also contributed to the development of grassroots football programs and the improvement of stadium infrastructure across India.
• ISL continues to grow in popularity, serving as a platform for Indian players to showcase their skills alongside international stars.

The correct answer is It is not mentioned in the Constitution and was granted by the National Development Council..

Key Points

• The status of Special Category State (SCS) in India is not a constitutional provision. It is an administrative classification introduced by the National Development Council (NDC) in 1969.
• The NDC granted this status to states that faced challenges such as hilly terrain, low population density, strategic location (bordering neighboring countries), economic backwardness, or non-viable nature of state finances.
• States with SCS status receive preferential treatment in the allocation of central funds and other financial assistance, particularly under the Gadgil-Mukherjee formula for central assistance.
• Originally, the SCS status was conferred to Assam, Nagaland, Jammu & Kashmir, and Himachal Pradesh. Over time, more states were added to the list, bringing the total to 11 states.
• The 14th Finance Commission (2015) recommended doing away with the distinction between SCS and other states. However, some benefits are still provided to SCS states through other mechanisms.

Additional Information

• Criteria for Special Category States:
• Hilly terrain: States with challenging geographical conditions, such as the northeastern states and Himachal Pradesh, were included.
• Low population density: Low-density areas often face higher costs in providing basic services.
• Strategic location: Many SCS states are located on international borders, requiring additional security and development measures.
• Economic backwardness: Poor infrastructure and lack of industrial development were key considerations.
• Non-viable nature of state finances: These states often had lower revenue generation capacity and higher dependency on central assistance.
• Benefits of Special Category Status:
• Central Assistance: SCS states received 90% of funds as grants and only 10% as loans under central schemes.
• Tax Concessions: Industries in SCS states were provided tax incentives to promote development.
• Preferential Allocation: SCS states were prioritized for funding under centrally sponsored schemes.
• States with Special Category Status:
• Initially, the status was granted to Assam, Nagaland, Jammu & Kashmir, and Himachal Pradesh.
• Later, states like Manipur, Meghalaya, Sikkim, Tripura, Mizoram, Arunachal Pradesh, and Uttarakhand were added.
• 14th Finance Commission Recommendations:
• The 14th Finance Commission recommended discontinuing the SCS distinction.
• Instead, it suggested transferring a greater share of central taxes to all states (42% from the divisible pool).
• However, some benefits of SCS status, like tax incentives, are still provided through central legislation.

Important Points

• Difference between Special Category Status and Special Status:
• Special Category Status: Administrative classification granted by the NDC for central assistance based on specific criteria.
• Special Status: Constitutional provision under Article 371 to provide certain privileges and autonomy to specific states.
• Recent Developments:
• The reorganization of Jammu & Kashmir in 2019 led to the abrogation of its SCS privileges.
• States like Andhra Pradesh continue to demand SCS status due to financial constraints post-bifurcation.

The correct answer is Real, inverted, and same size..

Key Points

• Concave mirror and its image formation: A concave mirror forms images based on the position of the object relative to its principal focus (F) and center of curvature (C).
• Object at the center of curvature (C): When an object is placed at the center of curvature of a concave mirror, the image is formed at the same location as the object, i.e., at C.
• Nature of the image: The image formed is real, inverted, and of the same size as the object.
• Reflection mechanics: In this position, light rays originating from the object strike the mirror and converge at the same distance, creating an image identical in size and orientation (except inverted).
• Ray diagram: A ray diagram of a concave mirror with the object at C clearly shows the reflected rays intersecting at the same point on the other side of the mirror, confirming the image characteristics.

Additional Information

• Concave Mirror Basics:
• A concave mirror is a spherical mirror with its reflective surface curving inward, resembling the interior of a sphere.
• It is also known as a converging mirror because it converges light rays that fall on its surface.
• Image Characteristics by Object Position:
• Beyond C: Image is real, inverted, and diminished.
• At C: Image is real, inverted, and of the same size.
• Between C and F: Image is real, inverted, and magnified.
• At F: Image is highly magnified and formed at infinity.
• Between F and Mirror: Image is virtual, erect, and magnified.
• Applications of Concave Mirrors:
• Shaving mirrors: Provide a magnified and upright image.
• Telescope: Used in reflecting telescopes to gather and focus light.
• Headlights: Used in car headlights to focus light into a beam.
• Real vs. Virtual Images:
• Real images: Formed when light rays converge and can be projected on a screen.
• Virtual images: Formed when light rays only appear to diverge and cannot be projected on a screen.

Important Points

• Mirror Equation: The mirror equation is given by 1/f = 1/v + 1/u, where:
• f: Focal length of the mirror.
• v: Image distance from the mirror.
• u: Object distance from the mirror.
• Magnification: The magnification (M) of a mirror is given by M = -v/u. For an object at C, magnification is -1 (image is of the same size but inverted).
• Principal Focus and Radius of Curvature:
• The distance between the mirror and the principal focus is called the focal length (f).
• The radius of curvature (R) is twice the focal length (R = 2f).

The correct answer is As price increases, demand decreases, ceteris paribus..

Key Points

• The Law of Demand: In microeconomics, the law of demand states that, ceteris paribus (all other factors being equal), as the price of a good or service increases, the quantity demanded of that good or service decreases, and vice versa. This reflects a fundamental inverse relationship between price and demand.
• Correct Statement: The correct option, "As price increases, demand decreases, ceteris paribus," accurately describes this concept. This statement aligns with the law of demand, which emphasizes the inverse relationship between price and quantity demanded.
• Behavior of Consumers: Higher prices typically deter consumers from purchasing more of a good or service, while lower prices encourage increased consumption, assuming no other factors change.
• Demand Curve Representation: Graphically, the law of demand is represented by a downward-sloping demand curve, where price is plotted on the vertical axis and quantity demanded on the horizontal axis.
• Incorrect Statements: Other options such as "As price increases, demand increases" or "Price has no effect on demand" are inconsistent with the law of demand and do not hold true under normal market conditions.

Additional Information

• Determinants of Demand: While the law of demand focuses on the price-quantity relationship, other factors that affect demand include income levels, tastes and preferences, prices of related goods (substitutes and complements), and expectations about future prices.
• Ceteris Paribus: This Latin term means "all other things being equal." It is a critical assumption in the law of demand, ensuring that the relationship between price and quantity demanded is not influenced by external factors.
• Exceptions to the Law of Demand: Certain goods, such as Giffen goods (inferior goods with upward-sloping demand curves) or Veblen goods (luxury goods where higher prices increase demand due to perceived status), may not follow the typical law of demand.
• Price Elasticity of Demand: This concept measures the responsiveness of quantity demanded to changes in price. Goods with high elasticity are significantly affected by price changes, while goods with low elasticity are less sensitive.
• Real-Life Examples: Everyday examples of the law of demand include seasonal discounts that increase sales, or price hikes during shortages that lead to reduced consumption.

Important Points

• Historical Origin: The concept of the law of demand was formalized by economists like Alfred Marshall in the late 19th century.
• Implications for Businesses: Understanding the law of demand helps businesses set optimal prices for their goods and forecast sales trends.
• Relation to Supply: The law of demand works in conjunction with the law of supply to determine market equilibrium, where quantity demanded equals quantity supplied.
• Government Policies: Policies such as subsidies, price ceilings, and taxes can impact demand by altering the effective price consumers face.
• Demand Analysis Tools: Economists use tools like consumer surveys, market experiments, and econometric models to study demand patterns in real-world scenarios.

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

Key Points

• Assertion (A): The government's revenue receipts don't bring with them any burden of repayment or financial liability.
• Revenue receipts include earnings from sources like taxes, duties, and other non-tax revenues.
• These receipts are considered non-repayable because they are generated from the country's economic activities and are not loans or borrowings. Hence, the statement is true.
• Reason (R): These earnings aren't meant to be returned and generally come from either tax-related sources or other regular inflows.
• Tax-related earnings include income tax, corporate tax, customs duty, and GST, which are collected by the government as part of its fiscal operations.
• Non-tax revenue sources include fees, fines, dividends, and interest on investments, which also do not require repayment. Hence, this statement is correct and explains the Assertion.
• The relationship between A and R confirms that the government's revenue receipts are free from repayment obligations because they stem from the nation's tax and non-tax economic activities, not borrowings. Hence, both A and R are true, and R is the correct explanation of A.

Additional Information

• Revenue Receipts:
• Revenue receipts are part of the government's income generated through current operations, such as taxes and fees, without creating liabilities.
• They are classified into two categories: Tax Revenue and Non-Tax Revenue.
• Examples of tax revenue include corporate tax, income tax, customs duty, and GST.
• Examples of non-tax revenue include earnings from public sector enterprises, dividends, interest income, and fines.
• Difference Between Revenue Receipts and Capital Receipts:
• Revenue Receipts: Do not create liabilities or involve asset reduction. Examples include taxes and interest income.
• Capital Receipts: Create liabilities or involve asset reduction. Examples include borrowings, loans, and disinvestment proceeds.
• Tax Revenue:
• Taxes are a compulsory contribution levied by the government on individuals and businesses.
• Examples include direct taxes like income tax and corporate tax, and indirect taxes like GST and customs duty.
• Non-Tax Revenue:
• Non-tax revenue is earned from sources other than taxes, such as fees, fines, dividends, and interest earnings.
• It plays a supplementary role in financing the government's operations.

Important Points

• Sources of Revenue Receipts:
• Revenue receipts are primarily derived from taxation policies and earnings from government assets.
• They contribute to the government's ability to fund welfare programs, infrastructure projects, and administrative expenses.
• Impact on Fiscal Policy:
• Revenue receipts are crucial for implementing fiscal policies aimed at economic stability and growth.
• They help reduce dependence on borrowings and maintain a balanced budget.
• Examples of Revenue Receipts:
• Tax Revenue: GST collections reached ₹1.5 lakh crore in September 2023.
• Non-Tax Revenue: Dividends from public sector enterprises contributed ₹50,000 crore in FY 2022-23.

The correct answer is P.C. Mahalanobis.

Key Points

• P.C. Mahalanobis, also known as the "Father of Indian Statistics," played a key role in shaping India's economic planning and development.
• He was the chief architect of India's Second Five-Year Plan (1956–1961), which focused on the rapid industrialization of the country.
• The Mahalanobis Model, a pivotal economic model, emphasized the growth of the heavy industry sector, which was considered essential for long-term economic development.
• He established the Indian Statistical Institute (ISI) in 1931 in Kolkata, which became a hub for statistical research and national development planning.
• P.C. Mahalanobis was instrumental in introducing the concept of Planned Economy and utilizing statistical methods for economic and social surveys in India.

Additional Information

• About India's Five-Year Plans:
• The Five-Year Plans were a series of centralized economic and social development initiatives launched by the Government of India post-independence.
• The First Five-Year Plan (1951–1956) primarily focused on agriculture, irrigation, and energy sectors to ensure food security.
• The Second Five-Year Plan, spearheaded by P.C. Mahalanobis, emphasized industrialization and building a self-reliant economy.
• The Plans were modeled on the Soviet Union's planning system but adapted to meet India's unique needs and challenges.
• P.C. Mahalanobis and His Contributions:
• He introduced the Mahalanobis Distance, a statistical measure used in cluster analysis and classification problems.
• The ISI, under his leadership, became a center for statistical innovation and played a crucial role in national policy formation.
• He was a member of the Planning Commission of India and influenced decisions on resource allocation, industrial policy, and economic development.
• He advocated for the use of statistics and data-driven decision-making in governance and planning.
• Other Options Explained:
• Sardar Patel: Known as the "Iron Man of India," he played a pivotal role in integrating princely states into the Indian Union but was not associated with Five-Year Plans.
• Jawaharlal Nehru: As India's first Prime Minister, he laid the foundation for economic planning and supported the establishment of the Planning Commission, but he did not directly design the Plans.
• B.R. Ambedkar: A key architect of the Indian Constitution and a social reformer, his contributions were more in law, social justice, and policy, not in economic planning.

Important Points

• The Mahalanobis Model:
• It emphasized the development of the capital goods sector, such as machinery and heavy industries, over consumer goods.
• The model aimed to reduce dependency on imports and achieve economic self-reliance.
• It was based on a two-sector model, dividing the economy into capital goods and consumer goods sectors.
• Legacy of Five-Year Plans:
• The Five-Year Plans continued until 2017 when they were replaced by the NITI Aayog's 15-year vision and three-year action agenda.
• The Plans laid the foundation for India's current economic structure and helped address issues such as poverty, unemployment, and infrastructure development.

The correct answer is Germanwatch & NewClimate Institute.

Key Points

• The Climate Change Performance Index (CCPI) is an annual publication that evaluates and compares the climate protection performance of different countries worldwide.
• The index is published by Germanwatch, the NewClimate Institute, and the Climate Action Network (CAN).
• The CCPI aims to enhance transparency in international climate politics by ranking countries based on their efforts to combat climate change.
• The index evaluates countries across four categories: Greenhouse Gas Emissions (40%), Renewable Energy (20%), Energy Use (20%), and Climate Policy (20%).
• The Climate Change Performance Index 2025 will analyze the progress made by various countries in achieving their climate goals, focusing on targets such as the Paris Agreement’s aim to limit global warming to below 2°C.

Additional Information

• Germanwatch:
• It is a non-governmental organization based in Germany, founded in 1991.
• Germanwatch focuses on promoting global equity and the preservation of livelihoods by advocating for sustainable global development and climate policy.
• The organization is known for its work on climate policy, corporate accountability, and sustainable development.
• NewClimate Institute:
• Based in Germany, the NewClimate Institute is a research organization established in 2014.
• Its focus is on the advancement of climate change mitigation and the development of sustainable climate strategies.
• The institute collaborates with governments, non-governmental organizations, and the private sector to enhance climate action at all levels.
• Climate Action Network (CAN):
• A global network of over 1,500 environmental NGOs in more than 130 countries.
• CAN works to promote government action, public engagement, and effective policy implementation to address climate change challenges.
• It actively contributes to international negotiations such as the UNFCCC and advocates for ambitious climate policies.
• Paris Agreement:
• The Paris Agreement is a legally binding international treaty on climate change adopted at COP21 in December 2015.
• Its goal is to limit global warming to well below 2°C, preferably to 1.5°C, compared to pre-industrial levels.
• Countries commit to nationally determined contributions (NDCs) outlining their climate action plans and emission reduction targets.

Important Points

• UNEP (United Nations Environment Programme):
• It is the leading global environmental authority established in 1972.
• UNEP does not directly publish the CCPI but works to promote sustainable development and environmental protection globally.
• IPCC (Intergovernmental Panel on Climate Change):
• It was established in 1988 by the World Meteorological Organization (WMO) and UNEP.
• IPCC assesses scientific research on climate change but does not produce the CCPI.
• WWF (World Wide Fund for Nature):
• Founded in 1961, WWF focuses on conserving biodiversity and reducing human impact on the environment.
• While WWF supports climate action, it is not the publisher of the CCPI.

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

Key Points

• Assertion (A): Alluri Sitarama Raju led the Rampa Rebellion against the British in the early 1920s. This statement is correct.
• Reason (R): The rebellion was primarily a response to the forest laws imposed by the British, which severely restricted tribal access to their traditional resources such as forests and agricultural land. This statement is correct.
• The Rampa Rebellion, also known as the Manyam Rebellion, took place in the tribal areas of Andhra Pradesh, where Alluri Sitarama Raju emerged as a leader fighting for tribal rights.
• The forest laws imposed by the British disrupted the livelihood of the tribal people, who were heavily dependent on forest resources for their survival. This was one of the main causes of the rebellion.
• Hence, both A and R are true, and R is the correct explanation of A.

Additional Information

• Alluri Sitarama Raju:
• Born: 1897 in Andhra Pradesh, India.
• He was a revolutionary freedom fighter who fought against British colonial rule, particularly focusing on tribal rights.
• He advocated the use of guerrilla warfare tactics and led tribal people in armed resistance against British forces.
• Rampa Rebellion:
• The rebellion occurred between 1922 and 1924 in the Rampa region of present-day Andhra Pradesh.
• It was triggered by oppressive forest policies that deprived tribal communities of their traditional rights to use forest resources.
• The rebellion involved armed resistance against the British and was a significant event in the history of India's freedom struggle.
• Forest Laws:
• The British colonial government implemented forest laws to gain control over forest resources, restricting access for local communities.
• These laws included measures like prohibiting shifting cultivation, banning grazing, and limiting access to forest produce.
• These policies led to widespread discontent among tribal populations who depended on forests for their livelihood.
• Legacy of Alluri Sitarama Raju:
• He is remembered as a hero and martyr of India's freedom movement.
• In honor of his contributions, various institutions and places have been named after him, including statues and memorials.
• His efforts showcased the role of tribal communities in resisting colonial oppression.

Important Points

• Guerrilla Warfare: Alluri Sitarama Raju adopted guerrilla tactics to fight against British forces, utilizing the dense forest terrain to his advantage.
• Impact of Forest Policies: The forest policies disrupted the economic and social fabric of tribal communities, making them central to the freedom struggle in Rampa.
• Arrest and Execution: In 1924, Alluri Sitarama Raju was captured and executed by the British, marking the end of the rebellion.

The correct answer is She hoisted the Indian flag during the Quit India Movement..

Key Points

• Aruna Asaf Ali played a significant role in India's freedom struggle during the Quit India Movement, which began in August 1942.
• She is famously known for hoisting the Indian National Flag at the Gowalia Tank Maidan in Bombay (now Mumbai) on 9 August 1942, marking the launch of the movement.
• This act of defiance turned her into a prominent symbol of resistance against British rule.
• Her actions inspired many Indians to join the movement, and she became known as the "Heroine of the Quit India Movement."
• Aruna Asaf Ali continued her efforts for India's independence and was also active in post-independence India, advocating for various social and political causes.

Additional Information

• Quit India Movement:
• Launched by Mahatma Gandhi on 8 August 1942 during the Bombay session of the All India Congress Committee.
• Its primary aim was to demand an end to British rule in India and achieve complete independence.
• The movement's slogan, "Do or Die," urged Indians to act decisively against colonial rule.
• Aruna Asaf Ali's Legacy:
• She was born on 16 July 1909 in Kalka, Haryana, and married Asaf Ali, a prominent lawyer and freedom fighter.
• She was awarded the Bharat Ratna, India's highest civilian honor, in 1997 for her contributions to the nation.
• She also received the International Lenin Peace Prize for her efforts towards world peace.
• Role of Women in Freedom Struggle:
• Women like Aruna Asaf Ali, Sarojini Naidu, and Kamala Nehru played crucial roles in mobilizing support and leading protests against British rule.
• They participated in activities such as picketing foreign goods, organizing rallies, and spreading nationalist ideas.
• Gowalia Tank Maidan:
• Located in Mumbai, this place is historically significant as the site where the Quit India Movement was officially launched.
• Today, it is known as the "August Kranti Maidan" in honor of the movement.

Important Points

• Other Contributions of Aruna Asaf Ali:
• She was the first mayor of Delhi post-independence in 1958.
• She co-founded the National Federation of Indian Women in 1954, advocating for women's rights and social justice.
• Her life and work remain a source of inspiration for generations of Indians.

The correct answer is Capital goods sector.

Key Points

• Mahalanobis Strategy: The Mahalanobis strategy was introduced in the Second Five-Year Plan (1956-61) in India, emphasizing the importance of heavy industries and the capital goods sector for long-term economic growth.
• The strategy aimed at developing a strong industrial base, particularly focusing on sectors that produce machinery, tools, and equipment required for other industries.
• The capital goods sector was considered vital for self-reliance, as it reduces dependency on imported machinery and enables domestic industries to flourish.
• This approach was inspired by the Soviet model of economic planning, which emphasized state-led industrialization and investment in heavy industries.
• The policy was formulated by Prasanta Chandra Mahalanobis, a renowned Indian statistician, and played a pivotal role in shaping India's economic strategy during the early years of independence.

Additional Information

• Capital Goods Sector:
• The capital goods sector includes industries that manufacture machinery, tools, and equipment used for production in other industries. Examples include steel plants, machine tools, heavy electrical equipment, and chemical plants.
• Investments in the capital goods sector lead to the creation of infrastructure and production capacity, which are essential for long-term economic growth.
• This sector is considered the backbone of industrial development, fostering technological advancements and innovation.
• Countries like the Soviet Union and China successfully adopted similar strategies to enhance their industrial base.
• Mahalanobis Model:
• The Mahalanobis model is a two-sector model focusing on the allocation of resources between the consumer goods sector and the capital goods sector.
• The model assumes that investment in the capital goods sector results in higher economic growth due to its ability to produce goods for other industries.
• The strategy emphasized state-led industrialization to overcome the limitations of private investment in heavy industries.
• Comparison of Options:
• Agricultural inputs: While agriculture is crucial for food security and livelihood, the Mahalanobis strategy prioritized industrialization over agriculture for faster economic growth.
• Consumer goods sector: The consumer goods sector focuses on products for immediate consumption, but the strategy argued that investment in the capital goods sector would indirectly support the growth of consumer industries.
• Import-led services sector: This option contrasts with the Mahalanobis model, which emphasized reducing import dependency by strengthening domestic industries.

Important Points

• Second Five-Year Plan:
• The Second Five-Year Plan (1956-61) was heavily influenced by the Mahalanobis strategy and aimed at achieving a growth rate of 4.5% annually.
• The plan allocated significant resources to heavy industries like steel, engineering, and chemicals.
• This approach led to the establishment of public-sector enterprises such as BHEL, SAIL, and HEC, which remain integral to India's industrial landscape.
• Criticism:
• The strategy faced criticism for neglecting sectors like agriculture and small-scale industries, which are crucial for employment and rural development.
• It contributed to income inequality and regional disparities, as urban areas benefited more from industrialization than rural areas.

The correct answer is Gujarat.

Key Points

• Tippani Dance is a traditional folk dance performed primarily by women in the Indian state of Gujarat.
• It is particularly popular in the Saurashtra region of Gujarat, showcasing the vibrant and colorful culture of the state.
• The dance involves women using a long wooden stick, referred to as "tippani," to strike the ground rhythmically while performing.
• Costumes worn during the Tippani dance are traditionally colorful and bright, reflecting the lively spirit of Gujarati folk culture.
• This dance form is often performed during festivals, ceremonies, and special occasions, symbolizing unity and celebration among the community.
• The dance not only serves as entertainment but also emphasizes cooperation and synchronization among the performers.

Additional Information

• Traditional Folk Dances of Gujarat:
• Gujarat is known for its rich folk culture, including dance forms like Garba, Dandiya Raas, Tippani, and Bhavai.
• Garba and Dandiya Raas are widely recognized worldwide, performed during the festival of Navratri.
• Tippani Dance is unique in its use of tools (wooden sticks) to create rhythm and is a less commercialized yet culturally significant dance form.
• These dances reflect the agricultural and communal lifestyle of the state, celebrating nature, festivals, and social gatherings.
• Specific Characteristics of Tippani Dance:
• The wooden stick used in the dance is often decorated, symbolizing the artistic traditions of the region.
• The dance steps are synchronized and require coordination among the performers, reflecting the communal spirit of rural Gujarat.
• It is performed to the accompaniment of traditional folk music, enhancing its cultural essence.
• Other Indian Folk Dances:
• Rajasthan: Known for Ghoomar, Kalbelia, and Chari Dance, showcasing the royal heritage and desert lifestyle.
• Maharashtra: Famous for Lavani, a vibrant dance performed to rhythmic beats, often reflecting social themes.
• Punjab: Renowned for Bhangra and Giddha, dances full of energy and traditionally performed during harvest festivals.
• Cultural Significance:
• Folk dances like Tippani preserve the cultural heritage of the community and serve as a medium for storytelling and celebration.
• They act as a bridge between generations, passing down traditions, values, and artistic skills.
• Such dances are a part of the intangible cultural heritage of India and contribute to the diversity of the nation.

The correct answer is New Economic Policy.

Key Points

• The economic policy reforms of 1991 in India are popularly known as the New Economic Policy, which marked a paradigm shift in India's economic approach.
• It was introduced under the leadership of the then Prime Minister P.V. Narasimha Rao and Finance Minister Dr. Manmohan Singh to address the economic crisis caused by fiscal deficits and foreign exchange shortages.
• The reforms focused on Liberalization, Privatization, and Globalization (commonly referred to as LPG reforms), aiming to integrate India’s economy with the global market.
• The Liberalization measures removed trade restrictions, reduced import tariffs, and deregulated industries to encourage private sector participation.
• Privatization involved disinvestment in public sector enterprises to reduce the government's role in business and bring efficiency.
• Globalization opened Indian markets to foreign investments, promoting international trade and enhancing economic growth.
• This policy was implemented to address the balance of payments crisis, stabilize the economy, and promote growth by attracting foreign investments.

Additional Information

• Economic Crisis of 1991:
• India faced a severe balance of payments crisis in 1991, with foreign exchange reserves depleting to a level that could barely finance two weeks of imports.
• The fiscal deficit reached 8.4% of GDP, and inflation was rising at an alarming rate of 13%.
• The country sought assistance from the International Monetary Fund (IMF), which required India to implement structural reforms as part of the bailout package.
• Liberalization:
• Involved reducing government regulations and restrictions on businesses and industries.
• Key measures included abolishing the License Raj, reducing import tariffs, and easing restrictions on foreign investment.
• Allowed private entities to operate freely, fostering competition and innovation.
• Privatization:
• Focused on transferring ownership of public sector enterprises to private entities.
• Disinvestment in government-owned companies was undertaken to improve efficiency and reduce fiscal burden.
• Encouraged private sector participation in industries previously dominated by the government.
• Globalization:
• Opened Indian markets to foreign investments and international trade.
• Allowed multinational corporations to establish operations in India, leading to technological advancements and job creation.
• Enhanced India's integration into the global economy, fostering economic growth and development.
• Impact of New Economic Policy:
• India’s GDP growth rate increased significantly, averaging around 6-7% per year in the post-reform period.
• Foreign Direct Investment (FDI) inflows rose, boosting industrial development.
• India became a major player in global IT and service sectors due to increased competitiveness and technological advancements.
• While the reforms led to economic growth, they also resulted in income inequality and concerns about job security in certain sectors.

Important Points

• License Raj:
• Before 1991, businesses in India required multiple licenses and approvals for starting and operating industries.
• The New Economic Policy abolished many of these restrictions, fostering entrepreneurship and industrial growth.
• IMF Bailout:
• India pledged 67 tonnes of gold as collateral to secure financial assistance from the IMF.
• This marked a turning point in India's economic history, leading to the adoption of structural reforms.
• Key Figures:
• P.V. Narasimha Rao: Played a crucial role as the Prime Minister in implementing the reforms.
• Dr. Manmohan Singh: As Finance Minister, he spearheaded the policy changes and is considered the architect of the reforms.
• Foreign Investment:
• FDI inflows increased, particularly in sectors like telecommunications, automobiles, and IT.
• India became one of the fastest-growing economies globally, attracting significant foreign capital.

Shortcut Trick

For half-yearly compounding, new rate = 20% ÷ 2 = 10% and cycles = 2 × 2 = 4.

The effective compound interest rate for 4 cycles at 10% is widely known to be 46.41%.

Total CI = 46.41% of Principal

Total CI = 46.41% × 5000 = 46.41 × 50 = 2320.5

∴ The correct answer is Rs. 2320.5.

Alternate Method

Given:

Principal (P) = Rs. 5000

Annual Rate (R) = 20%

Time (T) = 2 years

Formula Used:

A = P × (1 + r/100)ⁿ

Where r = rate per period, n = number of periods, and CI = A − P

0m 5000

6m

12m

18m

24m 7320.5

+10%

+10%

+10%

+10%

Total CI = 7320.5 − 5000 = Rs. 2320.5

Calculations:

⇒ Compounded half-yearly, so rate per period (r) = 20% ÷ 2 = 10%

⇒ Number of half-years (n) = 2 × 2 = 4

⇒ Amount (A) = 5000 × (1 + 10/100)⁴

⇒ A = 5000 × (11/10) × (11/10) × (11/10) × (11/10)

⇒ A = 5000 × 1.4641

⇒ A = 7320.5

⇒ Compound Interest (CI) = A − P

⇒ CI = 7320.5 − 5000 = 2320.5

∴ The correct answer is Rs. 2320.5.

Additional Information

Quarterly Compounding

When interest is compounded quarterly, the rate is divided by 4 (R/4) and the time periods are multiplied by 4 (4T).

Effective Rate for 2 Cycles

The effective compound interest rate for 2 continuous cycles at R% can be found using the successive percentage formula: R + R + (R×R)/100.

Difference between CI and SI

For exactly 2 years, the difference is given by CI − SI = P(R/100)². For exactly 3 years, it is CI − SI = P(R/100)²(3 + R/100).

Shortcut Trick

Integers represent the complete set of positive counting numbers, their negative counterparts, and zero, without any fractional or decimal parts.

Since the given values -5, 0, and 6 exactly match these conditions, they belong to the set of integers.

∴ The correct answer is Integers.

Alternate Method

Given:

A set of numbers: -5, 0, and 6

Classification Analysis:

⇒ Natural Numbers: Includes only positive counting numbers (1, 2, 3...). It excludes 0 and -5.

⇒ Whole Numbers: Includes all natural numbers along with 0 (0, 1, 2, 3...). It excludes negative numbers like -5.

⇒ Irrational Numbers: Includes non-terminating, non-repeating decimals like π or √2.

⇒ Integers: Includes all positive whole numbers, their negative counterparts, and 0 (..., -3, -2, -1, 0, 1, 2, 3...).

Conclusion:

⇒ -5 is a negative integer.

⇒ 0 is a neutral integer (neither positive nor negative).

⇒ 6 is a positive integer.

⇒ Therefore, all three numbers completely satisfy the properties of Integers.

∴ The correct answer is Integers.

Additional Information

Rational Numbers

Any number that can be expressed as a fraction p/q where p and q are integers and q is not equal to 0. Every integer is a rational number because it can be written as p/1.

Real Numbers

The universal set combining all rational and irrational numbers. All integers, whole numbers, and natural numbers fall under the category of real numbers.

Even and Odd Integers

An integer is even if it is completely divisible by 2 (e.g., -4, 0, 6). It is odd if it leaves a remainder of 1 when divided by 2 (e.g., -5, 3, 7).

Shortcut Trick

When the distance is constant, Speed and Time are inversely proportional.

Ratio of Speeds (S₁ : S₂) = 50 : 40 = 5 : 4

Ratio of Times (T₁ : T₂) = 4 : 5

Given 4 units = 8 hours ⇒ 1 unit = 2 hours.

Required Time (5 units) = 5 × 2 = 10 h.

∴ The correct answer is 10 h.

Alternate Method

Given:

Speed of the car (S₁) = 50 km/h

Time taken (T₁) = 8 h

New speed (S₂) = 40 km/h

Total Distance = 400 km

Case 1: Speed = 50 km/h | Time = 8 h

Case 2: Speed = 40 km/h | Time = ?

Formula Used:

Distance = Speed × Time

Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = 50 × 8 = 400 km

⇒ Time taken at 40 km/h = 400 ÷ 40

⇒ Time taken = 10 h

∴ The correct answer is 10 h.

Additional Information

Average Speed

When equal distances are covered at speeds x and y, the Average Speed for the entire journey is given by (2 × x × y) ÷ (x + y).

Relative Speed

If two objects are moving in opposite directions with speeds x and y, their relative speed is (x + y). If they move in the same direction, it is (x − y).

Unit Conversion

To convert km/h to m/s, multiply the speed by 5/18. To convert m/s to km/h, multiply by 18/5.

Shortcut Trick

The volume of a sphere is directly proportional to the cube of its radius (V ∝ r³).

If the radius is doubled, the new volume becomes 2³ = 8 times the original volume.

New Volume = 8 × 288π

New Volume = 2304π cm³

∴ The correct answer is 2304π cm³.

Alternate Method

Given:

Volume of the original sphere = 288π cm³

Formula Used:

Volume of a sphere = (4/3)πr³

r V = 288π cm³

r × 2

2r New V = ?

Calculations:

⇒ (4/3)πr³ = 288π

⇒ (4/3)r³ = 288

⇒ r³ = 288 × 3 ÷ 4

⇒ r³ = 72 × 3 = 216

⇒ r = 6 cm

⇒ New radius (R) = 2 × 6 = 12 cm

⇒ New Volume = (4/3)πR³

⇒ New Volume = (4/3)π(12)³

⇒ New Volume = (4/3)π × 1728

⇒ New Volume = 4 × 576π

⇒ New Volume = 2304π cm³

∴ The correct answer is 2304π cm³.

Additional Information

Surface Area of a Sphere

The total surface area of a complete sphere is given by the formula 4πr².

Volume of a Hemisphere

A hemisphere is exactly half of a sphere. Its volume is derived directly as (2/3)πr³.

Scaling Effect on 3D Shapes

If all dimensions of a 3D figure are scaled by a factor of k, its surface area changes by a factor of k² and its volume changes by a factor of k³.

Shortcut Trick

Total exact tiles = (800 × 600) ÷ (40 × 40)

Total exact tiles = 480,000 ÷ 1,600 = 300

Total required with 10% extra = 300 × 1.1 = 330

∴ The correct answer is 330 tiles.

Alternate Method

Given:

Length of room (L) = 8 m = 800 cm

Width of room (W) = 6 m = 600 cm

Side of square tile = 40 cm

Extra tiles required = 10%

Length = 800 cm Width = 600 cm Area = 480,000 cm²

40 cm 40 cm Tile

Required = (Floor Area ÷ Tile Area) + 10% Extra

Calculations:

⇒ Area of the floor = 800 × 600 = 480,000 cm²

⇒ Area of one square tile = 40 × 40 = 1,600 cm²

⇒ Exact number of tiles needed = 480,000 ÷ 1,600

⇒ Exact number of tiles = 300

⇒ Extra tiles for cutting and breakage = 10% of 300

⇒ Extra tiles = (10 ÷ 100) × 300 = 30

⇒ Total tiles needed = 300 + 30 = 330

∴ The correct answer is 330 tiles.

Additional Information

Area and Perimeter of a Rectangle

Area = Length × Width | Perimeter = 2 × (Length + Width)

Area and Perimeter of a Square

Area = Side² | Perimeter = 4 × Side

Unit Conversions

1 meter (m) = 100 centimeters (cm) | 1 square meter (m²) = 10,000 square centimeters (cm²)

Percentage Increase

Final Value = Initial Value × (1 + Rate ÷ 100)

Shortcut Trick

Let B's initial salary = 100. Then A's initial salary = 140.

B's new salary after 20% increment = 120.

A's new salary is 25% more than B's new salary ⇒ 120 + (25% of 120) = 150.

Increment in A's salary = 150 − 140 = 10.

x = (10 ÷ 140) × 100 = 7.14%.

∴ The correct answer is 7.14%.

Alternate Method

Given:

A's salary is 40% more than B's.

B's salary increases by 20%.

A's new salary becomes 25% more than B's new salary.

B = 100

A = 140

B' = 120

A' = 150

• 20%

• x% = 7.14%

• 25%

• 40%

Calculations:

⇒ Let the initial salary of B be 100.

⇒ Initial salary of A = 100 + (40% of 100) = 140

⇒ New salary of B = 100 + (20% of 100) = 120

⇒ New salary of A = 120 + (25% of 120) = 120 + 30 = 150

⇒ Increase in A's salary = 150 − 140 = 10

The value of x is the percentage increase calculated on A's initial salary.

⇒ x = (Increase ÷ Initial Salary of A) × 100

⇒ x = (10 ÷ 140) × 100

⇒ x = 100 ÷ 14 = 7.14%

∴ The correct answer is 7.14%.

Additional Information

Percentage Change

Percentage change is calculated as [(Final Value − Initial Value) ÷ Initial Value] × 100. It determines the relative growth or decline over the original base value.

Successive Percentage

If a value is successively increased by A% and B%, the net effective percentage change can be calculated directly using the formula: (A + B + AB ÷ 100)%.

Base Shift Rule

If value A is R% more than B, then B is [R ÷ (100 + R) × 100]% less than A. This helps to easily reverse percentage comparisons.

Shortcut Trick

Percentage of profit remaining for distribution after X's fee = 100% − 15% = 85%

Remaining Divisible Profit = 85% of 4,00,000 = 3,40,000

Investment ratio of X : Y : Z = 2 : 3 : 5 (Total = 10 parts)

Y's share = (3 ÷ 10) × 3,40,000

Y's share = 1,02,000

∴ The correct answer is Rs. 1,02,000.

Alternate Method

Given:

Total profit at the end of the year = Rs. 4,00,000

Ratio of investments X : Y : Z = 2 : 3 : 5

X's management fee = 15% of the total annual profit

Total Profit: 4,00,000

X's 15% Fee: 60,000

Divisible: 3,40,000

X (2 parts) 68,000

Y (3 parts) 1,02,000

Z (5 parts) 1,70,000

Calculations:

⇒ Management fee of X = (15 ÷ 100) × 4,00,000 = Rs. 60,000

⇒ Divisible profit to be distributed = Total Profit − Management Fee

⇒ Divisible profit = 4,00,000 − 60,000 = Rs. 3,40,000

⇒ Total ratio parts = 2 + 3 + 5 = 10 parts

⇒ Y's share in the divisible profit = (3 ÷ 10) × 3,40,000

⇒ Y's share = 3 × 34,000 = Rs. 1,02,000

∴ The correct answer is Rs. 1,02,000.

Additional Information

Active vs Sleeping Partner

A partner who manages the business operations is an active partner and typically receives a fixed salary or a certain percentage of profit as a management fee before the rest of the profit is shared among all partners.

Partnership Profit Distribution Formula

When partners invest different amounts for specific time periods, the divisible profit is distributed in the ratio of the product of their respective capital and time: Ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃).

Divisible Profit Concept

Total gross profit is first reduced by common expenses, taxes, and any individual partner's remuneration. The remaining balance (Divisible Profit) is allocated to all partners based on their effective investment ratio.

Shortcut Trick

Area of a path constructed inside a rectangle = 2w(L + B − 2w)

⇒ Path area = 2 × 2 × (40 + 30 − 4) = 4 × 66 = 264 m²

Total area of the lawn = 40 × 30 = 1200 m²

⇒ Percentage occupied by path = (264 ÷ 1200) × 100 = 22%

∴ The correct answer is 22%.

Alternate Method

Given:

Length of the rectangular lawn (L) = 40 m

Breadth of the rectangular lawn (B) = 30 m

Width of the inside path (w) = 2 m

Outer Lawn (40 m × 30 m) 30 m

Inner Area (36 m × 26 m)

2m

2m

Calculations:

⇒ Total area of the lawn = 40 × 30 = 1200 m²

⇒ Length of the inner rectangle = 40 − (2 × 2) = 40 − 4 = 36 m

⇒ Breadth of the inner rectangle = 30 − (2 × 2) = 30 − 4 = 26 m

⇒ Area of the inner rectangle = 36 × 26 = 936 m²

⇒ Area of the path = Total area of lawn − Area of the inner rectangle

⇒ Area of the path = 1200 − 936 = 264 m²

⇒ Required percentage = (Area of path ÷ Total area) × 100

⇒ Required percentage = (264 ÷ 1200) × 100 = 22%

∴ The correct answer is 22%.

Additional Information

Path Inside a Rectangle

When a path of uniform width w is constructed all around the inside of a rectangle of length L and breadth B, the area of the path = 2w(L + B − 2w).

Path Outside a Rectangle

When a path of uniform width w is constructed all around the outside of a rectangle of length L and breadth B, the area of the path = 2w(L + B + 2w).

Cross Path in the Middle

When two mutually perpendicular paths of uniform width w are drawn through the center of a rectangle, parallel to its length and breadth, the area of the paths = w(L + B − w).

Shortcut Trick

Total Area = Area of Rectangle 1 + Area of Rectangle 2

⇒ Total Area = (10 × 6) + (6 × 4) = 60 + 24 = 84 m²

Total Cost = Total Area × Rate per m²

⇒ Cost = 84 × 350 = 29,400

∴ The correct answer is ₹29,400.

Alternate Method

10m 6m 6m 4m

Area = 60 sq.m Area = 24 sq.m

Given:

⇒ Dimensions of first rectangle = 10m × 6m

⇒ Dimensions of second rectangle = 6m × 4m

⇒ Cost of tiling = ₹350 per m²

Formula Used:

⇒ Area of a rectangle = length × breadth

⇒ Total Cost = Total Area × Rate per unit area

Calculations:

⇒ Area of the L-shaped floor = Area of first rectangle + Area of second rectangle

⇒ Total Area = (10 × 6) + (6 × 4)

⇒ Total Area = 60 + 24 = 84 m²

⇒ Total Cost = Total Area × Cost per m²

⇒ Total Cost = 84 × 350

⇒ Total Cost = 29,400

∴ The correct answer is ₹29,400.

Additional Information

Area of Composite Shapes

To find the area of an irregular or composite shape, divide it into simple, non-overlapping basic geometric figures (like rectangles or squares), calculate their separate areas, and add them together.

Area of a Rectangle

The 2D space occupied by a rectangle is given by the product of its dimensions: Area = Length × Breadth.

Total Cost Calculation

For surface-level work such as flooring, painting, or laying tiles, the total expenditure is directly proportional to the total area. Total Cost = Total Area × Rate per unit square.

Shortcut Trick

A regular hexagon is composed of 6 smaller equilateral triangles of the same side length.

Let the side of the large triangle be a and the hexagon be b.

Since areas are equal: Area of large triangle = 6 × Area of one small triangle of side b.

(√3 ÷ 4) × a² = 6 × (√3 ÷ 4) × b²

⇒ a² = 6 × b² ⇒ a : b = √6 : 1

∴ The correct answer is √6 : 1.

Alternate Method

Side = a Area = A

Side = b Area = A

Equal Areas

Given:

Area of a regular triangle = Area of a regular hexagon

Formula Used:

Area of an equilateral triangle = (√3 ÷ 4) × (side)²

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

Calculations:

Let the side of the regular triangle be a and the side of the regular hexagon be b.

⇒ Area of the regular triangle = (√3 ÷ 4) × a²

⇒ Area of the regular hexagon = 6 × (√3 ÷ 4) × b²

According to the question, both areas are equal:

⇒ (√3 ÷ 4) × a² = 6 × (√3 ÷ 4) × b²

Dividing both sides by (√3 ÷ 4):

⇒ a² = 6 × b²

⇒ a² ÷ b² = 6 ÷ 1

Taking square root on both sides:

⇒ a ÷ b = √6 ÷ 1

∴ The correct answer is √6 : 1.

Additional Information

Properties of Equilateral Triangle

All three sides are equal, and each internal angle is 60°. Its height is given by h = (√3 ÷ 2) × a.

Properties of Regular Hexagon

It consists of 6 smaller equilateral triangles. The interior angle is 120°, and the circumradius is equal to its side length.

Inradius & Circumradius Ratio

For an equilateral triangle, the ratio of circumradius (R) to inradius (r) is strictly 2 : 1. For a regular hexagon, R : r = 2 : √3.

Shortcut Trick

When two items are sold at the same price, and there is a profit of x% on one and a loss of x% on the other, there is always an overall loss.

Overall Loss % = (x²) ÷ 100 = (25 × 25) ÷ 100 = 6.25% = 1/16

If Total CP is 16 units, Loss is 1 unit, making Total SP = 16 − 1 = 15 units.

Given, Total SP = 500 + 500 = Rs. 1000

Loss (1 unit) = 1000 ÷ 15 = 200/3 ≈ Rs. 66.67

∴ The correct answer is Rs. 66.67 loss.

Alternate Method

Given:

Selling Price (SP) of each shirt = Rs. 500

Profit on 1st shirt = 25%

Loss on 2nd shirt = 25%

Formula Used:

Cost Price (CP) = SP × [100 ÷ (100 + Profit%)]

Cost Price (CP) = SP × [100 ÷ (100 − Loss%)]

Loss = Total CP − Total SP

Shirt 1 (Gain 25%)

CP = Rs. 400

SP = Rs. 500

• Rs. 100 Gain

Shirt 2 (Loss 25%)

CP = Rs. 666.67 (or 2000/3)

SP = Rs. 500 − Rs. 166.67 Loss

Total SP = Rs. 1000 | Total CP ≈ Rs. 1066.67 | Net Loss ≈ Rs. 66.67

Calculations:

⇒ Total SP = 500 + 500 = Rs. 1000

⇒ CP of 1st shirt = 500 × [100 ÷ (100 + 25)] = 500 × (100/125) = Rs. 400

⇒ CP of 2nd shirt = 500 × [100 ÷ (100 − 25)] = 500 × (100/75) = Rs. 2000/3

⇒ Total CP = 400 + (2000/3) = (1200 + 2000)/3 = Rs. 3200/3

⇒ Loss = Total CP − Total SP

⇒ Loss = (3200/3) − 1000 = (3200 − 3000)/3

⇒ Loss = 200/3 = Rs. 66.67

∴ The correct answer is Rs. 66.67 loss.

Additional Information

Case of Same Selling Price

If two items are sold at the same price (SP), one at a profit of x% and the other at a loss of x%, there is always an overall loss given by x²/100 %.

Case of Same Cost Price

If two items have the same cost price (CP), and one is sold at x% profit while the other is sold at x% loss, there is no overall net profit or net loss.

Successive Percentage Change

When a value is subjected to two successive percentage changes a% and b%, the net effective change is given by (a + b + ab/100)%.

Shortcut Trick

Let the Marked Price (MP) be 100 units.

Selling Price (SP) = 90 units, Cost Price (CP) = 80 units.

Profit = SP − CP = 90 − 80 = 10 units.

Profit % = (10 ÷ 80) × 100 = 12.5%.

∴ The correct answer is 12.5%.

Alternate Method

Given:

Selling Price (SP) = 90% of Marked Price (MP)

Cost Price (CP) = 80% of Marked Price (MP)

Formula Used:

Profit = SP − CP

Profit % = (Profit ÷ CP) × 100

CP = 80

SP = 90

MP = 100

Profit +10

Discount −10

Profit % = (10 ÷ 80) × 100 = 12.5%

Calculations:

⇒ Let Marked Price (MP) = 100 units

⇒ Selling Price (SP) = (90 ÷ 100) × 100 = 90 units

⇒ Cost Price (CP) = (80 ÷ 100) × 100 = 80 units

⇒ Profit = SP − CP = 90 − 80 = 10 units

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = (10 ÷ 80) × 100 = 100 ÷ 8 = 12.5%

∴ The correct answer is 12.5%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Successive Discounts

For two successive discounts of x% and y%, the net discount percentage is given by (x + y − xy ÷ 100)%.

Dishonest Dealer

If a dealer sells at CP but uses false weight, Profit % = [Error ÷ (True Value − Error)] × 100.

Shortcut Trick

The overall average of 10 numbers is 50.

The first 6 numbers have an average of 45, causing a net deviation of 6 × (45 − 50) = −30.

To balance the overall average to 50, the last 4 numbers must have a compensating deviation of +30.

The average deviation for the last 4 numbers = +30 ÷ 4 = +7.5.

Average of the last 4 numbers = 50 + 7.5 = 57.5.

∴ The correct answer is 57.5.

Alternate Method

Given:

Total numbers = 10

Average of 10 numbers = 50

Number of first items = 6

Average of first 6 numbers = 45

Formula Used:

Sum of items = Average × Total number of items

First 6 Numbers Avg = 45 | Sum = 270

Last 4 Numbers Avg = ? | Sum = 230

Total 10 Numbers (Overall Average = 50) Total Sum = 10 × 50 = 500

Calculations:

⇒ Total sum of all 10 numbers = 10 × 50 = 500

⇒ Sum of the first 6 numbers = 6 × 45 = 270

⇒ Sum of the last 4 numbers = Total sum − Sum of first 6 numbers

⇒ Sum of the last 4 numbers = 500 − 270 = 230

⇒ Average of the last 4 numbers = 230 ÷ 4

⇒ Average of the last 4 numbers = 57.5

∴ The correct answer is 57.5.

Additional Information

Weighted Average

When combining two different groups of sizes n₁ and n₂ with averages A₁ and A₂, the combined average is calculated as (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Deviation Method

The net sum of deviations of all observations from their arithmetic mean is always exactly zero. Balancing the positive and negative deviations provides the fastest way to compute missing averages without heavy multiplication.

Replacement Method

When a new observation replaces an old one in a set, the value of the new observation can be directly calculated as: New Value = Old Value + (Increase/Decrease in Average × Total Number of Items).

Shortcut Trick

Area of both bases = Total Surface Area − Lateral Surface Area

Lateral Surface Area = Base Perimeter × Height = 30 × 7 = 210 cm²

Area of both bases = 282 − 210 = 72 cm²

∴ The correct answer is 72 cm².

Alternate Method

Given:

Base perimeter = 30 cm

Height (h) = 7 cm

Total Surface Area (TSA) = 282 cm²

Formula Used:

Lateral Surface Area (LSA) = Base perimeter × Height

Total Surface Area (TSA) = LSA + Area of both bases

h = 7 cm

Base Perimeter = 30 cm TSA = 282 cm² LSA = 30 × 7 = 210 cm² Bases Area = TSA − LSA

Calculations:

⇒ LSA = 30 × 7 = 210 cm²

⇒ 282 = 210 + Area of both bases

⇒ Area of both bases = 282 − 210 = 72 cm²

∴ The correct answer is 72 cm².

Additional Information

Total Surface Area of a Prism

The total surface area includes the lateral surface area and the areas of the two parallel bases: TSA = LSA + Area of both bases.

Volume of a Prism

The volume of any uniform prism is calculated as the product of its base area and its height: Volume = Base Area × Height.

Euler's Formula for Polyhedrons

For any convex polyhedron (including all prisms), the number of faces (F), vertices (V), and edges (E) are related by the formula: F + V = E + 2.

Shortcut Trick

When identical items are purchased, the effective discount is independent of their actual price.

Assume the Marked Price (MP) of each book is 1 unit.

Total MP for 5 books = 5 units.

Discount offered = 50% on the 5th book = 0.5 units.

Effective Discount % = (Discount ÷ Total MP) × 100 = (0.5 ÷ 5) × 100 = 10%.

∴ The correct answer is 10%.

Alternate Method

Given:

Marked Price (MP) of each book = Rs. 480

Number of books bought at full price = 4

Discount on the 5th book = 50%

Formula Used:

Discount = Total MP − Total SP

Discount % = (Discount ÷ Total MP) × 100

Total Marked Price = 5 × 480 = Rs. 2400

Full Full Full Full 50% Off

SP = 4 × 480 = Rs. 1920

SP = 240

Total Selling Price = Rs. 2160 | Discount = Rs. 240

Calculations:

⇒ Total Marked Price of 5 books = 5 × 480 = Rs. 2400

⇒ Selling Price of the first 4 books = 4 × 480 = Rs. 1920

⇒ Discount on the 5th book = 50% of 480 = Rs. 240

⇒ Selling Price of the 5th book = 480 − 240 = Rs. 240

⇒ Total Selling Price for 5 books = 1920 + 240 = Rs. 2160

⇒ Total Discount = Total MP − Total SP = 2400 − 2160 = Rs. 240

⇒ Effective Discount % = (Total Discount ÷ Total MP) × 100

⇒ Effective Discount % = (240 ÷ 2400) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Buy X Get Y Free Equivalent

In promotional offers where you get partial or full discounts on specific items, the effective discount percentage must always be calculated on the total marked price of all items combined.

Marked Price & Discount

A discount is universally applied on the Marked Price (MP) of an article. The Selling Price (SP) is obtained after subtracting the discount: SP = MP × [(100 − Discount %) ÷ 100].

Successive Discounts

If two successive discounts of a% and b% are given, the single equivalent overall discount percentage is given by the formula (a + b − ab ÷ 100)%.

Shortcut Trick

Overall average of 5 stocks = 9%

Average of 4 stocks = 8%

Deviation of the 4 stocks from the overall average = 4 × (8% − 9%) = −4%

To balance the overall average to 9%, the 5th stock must compensate with a +4% deviation.

Return on 5th stock = 9% + 4% = 13%

∴ The correct answer is 13%.

Alternate Method

Given:

Total number of stocks = 5

Average return of 4 stocks = 8%

Overall average return of 5 stocks = 9%

Formula Used:

Sum of observations = Average × Number of observations

4 Stocks (Total = 32%)

5th Stock = ?

Overall 5 Stocks (Total = 5 × 9% = 45%)

45% − 32% = 13%

Calculations:

⇒ Total return of all 5 stocks = 5 × 9% = 45%

⇒ Total return of the 4 tracked stocks = 4 × 8% = 32%

⇒ Return on 5th stock = (Total return of 5 stocks) − (Total return of 4 stocks)

⇒ Return on 5th stock = 45% − 32%

⇒ Return on 5th stock = 13%

∴ The correct answer is 13%.

Additional Information

Average Formula

Average = Sum of all observations ÷ Total number of observations.

Net Deviation Method

The sum of deviations of all observations from their true average is always zero. This concept helps in calculating an unknown observation directly.

Weighted Average

Used to find the combined average when different groups have different individual averages. Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Shortcut Trick

The expression to find is the exact algebraic expansion of (sinA + cosA)².

We are given that sinA + cosA = √3 / 2.

Simply square the given value directly:

(√3 / 2)² = 3/4

∴ The correct answer is 3/4.

Alternate Method

Given:

sinA + cosA = √3 / 2

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ We need to find the value of sin²A + cos²A + 2sinAcosA

⇒ By applying the algebraic identity, we know that:

⇒ sin²A + cos²A + 2sinAcosA = (sinA + cosA)²

⇒ Substitute the given value of (sinA + cosA) into the equation:

⇒ (sinA + cosA)² = (√3 / 2)²

⇒ (√3)² / (2)² = 3 / 4

∴ The correct answer is 3/4.

Additional Information

Pythagorean Trigonometric Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Double Angle Formulas

sin(2θ) = 2sinθcosθ
cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1

Algebraic Identities

(a + b)² = a² + b² + 2ab
(a − b)² = a² + b² − 2ab

Shortcut Trick

If secθ + tanθ = x, then secθ − tanθ = 1/x.

Subtracting both equations yields: 2tanθ = x − 1/x.

Given x = 5, we can directly find tanA:

⇒ 2tanA = 5 − 1/5

⇒ 2tanA = 24/5

⇒ tanA = 12/5

∴ The correct answer is 12/5.

Alternate Method

Given:

secA + tanA = 5

Formula Used:

sec²A − tan²A = 1

tanA = Opposite / Adjacent

Calculations:

⇒ (secA − tanA)(secA + tanA) = 1

⇒ secA − tanA = 1/5

Adding the two equations:

⇒ (secA + tanA) + (secA − tanA) = 5 + 1/5

⇒ 2secA = 26/5

⇒ secA = 13/5

Since secA = Hypotenuse / Adjacent = 13 / 5, we can represent this using a right-angled triangle where the Hypotenuse is 13 and the Adjacent side is 5:

A C B 5 12 13 A

By Pythagoras theorem:

⇒ Opposite² + Adjacent² = Hypotenuse²

⇒ BC² + 5² = 13²

⇒ BC² + 25 = 169

⇒ BC = √144 = 12

Now, substitute the values to find tanA:

⇒ tanA = Opposite / Adjacent

⇒ tanA = 12 / 5

∴ The correct answer is 12/5.

Additional Information

Trigonometric Identities

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

Reciprocal Pairs

If secθ + tanθ = x, then its related conjugate pair is secθ − tanθ = 1/x. This same direct logic also applies to cosec and cot where cosecθ + cotθ = x gives cosecθ − cotθ = 1/x.

Pythagorean Triplets

A set of three positive integers (a, b, c) satisfying a² + b² = c² is known as a Pythagorean triplet. Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Shortcut Trick

The total angle of a full circle is 2π radians.

Fraction of the circle = (π/2) ÷ 2π = 1/4

Required Percentage = (1/4) × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Angle of the sector = π/2 radians

Formula Used:

Percentage of full circle = (Given Angle ÷ Total Angle) × 100

π/2 rad 25% Total Circle = 2π rad 100%

Calculations:

⇒ Total angle of a full circle = 2π radians

⇒ Given angle = π/2 radians

⇒ Fraction of the circle = (π/2) ÷ 2π

⇒ Fraction = 1/4

⇒ Percentage = (1/4) × 100%

⇒ Percentage = 25%

∴ The correct answer is 25%.

Additional Information

Radians to Degrees Conversion

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

Degrees to Radians Conversion

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

Common Circle Fractions

π radians = 180° (50%), π/2 radians = 90° (25%), and π/4 radians = 45° (12.5%).

Shortcut Trick

If tan A = cot B, then A + B = 90°.

Substituting A = x and B = 3x − 30°:

⇒ x + (3x − 30°) = 90°

⇒ 4x = 120° ⇒ x = 30°

∴ The correct answer is 30°.

Alternate Method

Given:

tan x = cot(3x − 30°)

Concept Used:

cot θ = tan(90° − θ)

Calculation:

⇒ tan x = tan(90° − (3x − 30°))

Comparing the angles on both sides:

⇒ x = 90° − (3x − 30°)

⇒ x = 90° − 3x + 30°

⇒ x + 3x = 120°

⇒ 4x = 120°

⇒ x = 30°

∴ The correct answer is 30°.

Additional Information

Complementary Angles Rule

When the sum of two angles is 90°, they are known as complementary angles. The values of their respective trigonometric co-functions are always equal.

Trigonometric Co-function Identities

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

Important Values at 30°

tan 30° = 1/√3 and cot 30° = √3

Shortcut Trick Direct Common Tangent (DCT) = √[d2 − (R − r)2] Given distance (d) = 10, larger radius (R) = 6, and smaller radius (r) = 2. DCT = √[102 − (6 − 2)2] = √(100 − 16) DCT = √84 = 2√21 cm ∴ The correct answer is 2√21 cm.

Alternate Method

C1 C2

10 cm

DCT

A B

6 cm 2 cm

P

4 cm

Given: ⇒ Radius of larger circle (R) = 6 cm ⇒ Radius of smaller circle (r) = 2 cm ⇒ Distance between centers (d) = 10 cm Formula Used: ⇒ Length of Direct Common Tangent (DCT) = √[d2 − (R − r)2] Calculation: ⇒ Difference in radii = R − r = 6 − 2 = 4 cm ⇒ DCT = √[102 − 42] ⇒ DCT = √[100 − 16] ⇒ DCT = √84 = 2√21 cm ∴ The correct answer is 2√21 cm.

Additional Information Transverse Common Tangent (TCT) The formula for the length of a transverse common tangent is TCT = √[d2 − (R + r)2]. Number of Common Tangents If d > R + r, there are 4 common tangents (2 direct, 2 transverse). If d = R + r (circles touch externally), there are exactly 3 common tangents. Intersecting Circles If R − r < d < R + r, the circles intersect at two points and have exactly 2 direct common tangents, but no transverse tangents.

Shortcut Trick

The angle subtended by an arc at the center is exactly double the angle subtended by it at any point on the remaining part of the circle.

Angle at the major segment = (Angle at the center) ÷ 2

⇒ Angle = 120° ÷ 2 = 60°

∴ The correct answer is 60°.

Alternate Method

Given:

⇒ Angle formed by the chord at the center = 120°

O P A B 120° 60°

Formula Used:

⇒ Angle subtended at the major segment = Central angle ÷ 2

Calculations:

⇒ Let the given chord be AB and the center of the circle be O.

⇒ The central angle subtended by the chord AB is ∠AOB = 120°.

⇒ Let P be any specific point situated in the major segment of the circle.

⇒ According to the fundamental circle theorem, the angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle.

⇒ ∠APB = ∠AOB ÷ 2

⇒ ∠APB = 120° ÷ 2

⇒ ∠APB = 60°

∴ The correct answer is 60°.

Additional Information

Cyclic Quadrilateral

If all four vertices of a quadrilateral lie perfectly on the circumference of a circle, it is known as a cyclic quadrilateral. The sum of either pair of opposite angles is exactly 180°.

Angle in a Semicircle

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

Angle in the Minor Segment

If the angle subtended at the center is θ, the angle subtended at the major segment is θ ÷ 2, and the corresponding angle subtended at the minor segment is exactly 180° − (θ ÷ 2).

Shortcut Trick

According to the properties of a circle, equal chords subtend equal angles at the circumference.

Given that chord AB is exactly equal to chord CD.

The angle subtended by AB at the circumference is given as 65°.

Therefore, the angle subtended by CD at the circumference will be identical.

∴ The correct answer is 65°.

Alternate Method

Given:

⇒ Length of Chord AB = Length of Chord CD

⇒ Angle subtended by AB at circumference = 65°

O

65° ?

A B C D P Q

Concept Used:

⇒ Equal chords of a circle subtend equal angles at the centre.

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

Calculation:

⇒ Let the angle subtended by AB at the centre be θ.

⇒ Angle subtended by AB at the circumference = θ ÷ 2

⇒ θ ÷ 2 = 65°

⇒ θ = 130°

⇒ Since AB = CD, the angle subtended by CD at the centre is also equal to θ = 130°.

⇒ Angle subtended by CD at the circumference = 130° ÷ 2 = 65°

∴ The correct answer is 65°.

Additional Information

Angle at the Centre Theorem

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

Angles in the Same Segment

Angles subtended by the same arc at any two points on the major (or minor) segment of a circle are always equal.

Equal Chords Property

Equal chords of a circle are equidistant from the centre and subtend identical angles at the centre.

Perpendicular from Centre

The perpendicular drawn from the centre of a circle to a chord bisects the chord perfectly into two equal halves.

Shortcut Trick

The angle subtended by a chord at the major arc of the circumference relates directly to the angle between the tangents at its endpoints.

Angle at circumference = 90° − (∠APB ÷ 2)

⇒ Angle = 90° − (40° ÷ 2)

⇒ Angle = 90° − 20° = 70°

∴ The correct answer is 70°.

Alternate Method

Given:

∠APB = 40°

Formula Used:

1. The sum of opposite angles of a quadrilateral formed by two tangents and two radii is 180°: ∠AOB + ∠APB = 180°
2. The angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle: ∠AOB = 2 × ∠ACB

O P A B C

40° 70° 140°

Calculations:

⇒ In quadrilateral OAPB, ∠OAP = 90° and ∠OBP = 90° (Radius is perpendicular to the tangent at the point of contact)

⇒ ∠AOB + ∠APB = 180°

⇒ ∠AOB + 40° = 180°

⇒ ∠AOB = 140°

⇒ Let C be a point on the major arc of the circle. The angle subtended by chord AB at the circumference is ∠ACB.

⇒ ∠ACB = ∠AOB ÷ 2

⇒ ∠ACB = 140° ÷ 2 = 70°

∴ The correct answer is 70°.

Additional Information

Tangent Properties

The tangent at any point of a circle is perpendicular to the radius through the point of contact (∠OAP = 90° and ∠OBP = 90°).

Angle Subtended by Arc/Chord

The angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle (∠AOB = 2 × ∠ACB).

Angle in the Minor Segment

If a point is taken on the minor arc, the angle subtended by the chord is 180° − (Angle in major segment) due to cyclic quadrilateral properties.

Shortcut Trick

The radius of a circle is always perpendicular to the tangent at the point of contact, making a 90° angle.

The angle between the radius and the given chord is 90° − 30° = 60°.

The triangle formed by the center of the circle and the chord consists of two equal radii, making it an isosceles triangle.

Since the base angles of this isosceles triangle are 60°, it is an equilateral triangle with all angles being 60°.

In an equilateral triangle, all sides are equal, so the radius equals the length of the chord, which is 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

O

A

B

T

30°

60°

8 cm R R

Given:

Length of chord AB = 8 cm

Angle between the tangent at A and the chord AB = 30°

Calculations:

⇒ Let O be the center of the circle, and OA and OB be the radii (R).

⇒ The radius drawn to the point of tangency is always perpendicular to the tangent line.

⇒ Total Angle between radius OA and the tangent = 90°

⇒ Angle OAB = Angle between OA and tangent − Angle between chord and tangent

⇒ Angle OAB = 90° − 30° = 60°

⇒ In triangle OAB, the sides OA and OB are equal because both are radii of the same circle (OA = OB = R).

⇒ Since angles opposite to equal sides are also equal, Angle OBA = Angle OAB = 60°.

⇒ The sum of angles in a triangle is 180°, so Angle AOB = 180° − (60° + 60°) = 60°.

⇒ Since all three interior angles are exactly 60°, triangle OAB is an equilateral triangle.

⇒ In an equilateral triangle, all three sides are equal in length.

⇒ Therefore, Radius (OA) = Radius (OB) = Chord (AB) = 8 cm.

∴ The correct answer is 8 cm.

Additional Information

Alternate Segment Theorem

The angle between a tangent and a chord through the point of contact is strictly equal to the angle subtended by the chord in the alternate segment of the circle.

Tangent-Radius Property

A tangent to a circle is always perpendicular to the radius drawn from the center of the circle to the exact point of tangency.

Equilateral Triangle Characteristics

An isosceles triangle containing any interior angle of 60° will always result in an equilateral triangle where all three sides and angles are completely equal.

The correct answer is 'represents'.

Key Points

• The subject of the sentence is "The pursuit of space exploration." The headword of this phrase is "pursuit," which is a singular noun.
• In English grammar, a singular subject requires a singular verb. In the simple present tense, singular verbs (for third-person subjects) typically end in '-s'. Therefore, "represents" is the grammatically correct form.
• "Represent" (Option 1) is a plural verb form and does not agree with the singular subject "pursuit."
• "Representing" (Option 3) is a present participle and cannot function as the main verb of the sentence without an auxiliary verb (like 'is').
• "Represented" (Option 4) is the past tense. However, the passage is written in the present tense to discuss a general truth and current human aspirations (as seen by other verbs in the text like "escalates," "foster," and "lie").