SSC CGL 2025 (Held On: 23 Sept, 2025 Shift 2) (Previous Year Paper)

Given: Flat No. 401, Lotus Enclave,JP Nagar, Bangalore, Karnataka - 560078

Option 1) Flat 401, Lotus Enclave, JP Nagar, Bengaluru, Karnataka - 560078 → Incorrect

Option 2) Flat No. 401, Lotus Enclave, JP Nagar, Bangalore, Karnataka - 560078 → Correct

Option 3) Flat No. 401, Lotus Enclave, J.P. Nagar, Bangalore, Karnataka - 560087 → Incorrect

Option 4) Flat No. 410, Lotus Enclave, JP Nagar, Bangalore, Karnataka - 560078 → Incorrect

Hence, "Option 2" is the correct answer.

According to the question:

Thus, 6 days after  the seminar in Raipur did the seminar in Tiruchirappalli occur.

Hence, "Option 1" is the correct answer.

Given:

2 × 3 = 8

3 × 4 = 15

Pattern Used:

a × b = a × (a + 2)    (when b = a + 1)

Calculation:

4 × 5 = 4 × (4 + 2)

⇒ 4 × 6

⇒ 24

∴The correct answer is option (3).

logic use here:

a × b = (a × b) + a

Calculation:

1 × 2

⇒ (1 × 2) + 1

⇒ 2 + 1 = 3

2 × 3

⇒ (2 × 3) + 2

⇒ 6 + 2 = 8

3 × 4

⇒ (3 × 4) + 3

⇒ 12 + 3 = 15

4 × 5

⇒ (4 × 5) + 4

⇒ 20 + 4 = 24

∴ The correct answer is option (1).

Given:

Product of two numbers = 2025

HCF = 15

Formula used:

Product of two numbers = HCF × LCM

Calculation:

⇒ 2025 = 15 × LCM

⇒ LCM = 2025 ÷ 15

⇒ LCM = 135

∴ The LCM of the two numbers = 135.

Given:

X + Y = 2Z

Z + W = 2X

Formula used:

Use substitution and addition of equations

Calculation:

X + Y = 2Z

⇒ Y = 2Z − X

Z + W = 2X

⇒ W = 2X − Z

Add Y and W

⇒ Y + W = (2Z − X) + (2X − Z)

⇒ Y + W = Z + X

⇒ X + Z = Y + W

∴ The correct answer is option (1).

Identify the category of each item:

Linux ⇒ Operating System

Windows ⇒ Operating System

macOS ⇒ Operating System

Intel ⇒ Processor / Hardware company

⇒ All the given options are Operating system except 'Intel', which is a hardware.

∴ The different one is Intel (Option 2).

Given:

Distance = 60 km

Time = 45 minutes

Formula used:

Speed = Distance ÷ Time

Calculation:

Time = 45 minutes = (45/60) = 3/4 hours

So, Speed = 60 ÷ (3/4) 

⇒ Speed = 60 × 4/3

⇒ Speed = 20 × 4 = 80 km/h

∴ The speed of the train = 80 km/h.

logic used here:

Number ÷ 7

Calculation:

21

⇒ 21 ÷ 7 = 3

574

⇒ 574 ÷ 7 = 82

∴ Missing number is 82.

Given:

Digital root = Sum of digits repeatedly until one digit remains

Number = 999999

Formula used:

Digital Root = Repeated sum of digits

Calculation:

999999

⇒ 9 + 9 + 9 + 9 + 9 + 9 = 54

⇒ 5 + 4 = 9

Digital root = 9

∴ Roll number with digital root 9 = 999999.

Logic used here:

Centre code = First three digits of roll number

Calculation:

817129 ⇒ 817

817131 ⇒ 817

⇒ Same centre code

184843 ⇒ 184

184320 ⇒ 184

⇒ Same centre code

908325 ⇒ 908

908187 ⇒ 908

⇒ Same centre code

Thus pairs with same examination centre:

817129 & 817131

184843 & 184320

908325 & 908187

∴ The correct answer is option (1).

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

Conclusions:

I. Some humans are creators → Follow (As some humans are artists and all artists are creators. As some part of human is common with artist which whole comes in creators, therefore it is true.)

II. All creators are artists → Does not follow (As all artists are creators. So it is possible but not definite.)

∴ Here, Only I follows.​

Hence, the correct answer is "Option 1".

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

Conclusions:

I. No A is D → Does not follow (As some A are B, all B are C and no C is D. As no direct relation is given between A and D, therefore it is false.)

II. Some B are not D → Follow (As all B are C and no C is D. As whole B comes in C which cannot be D, therefore it is true.)

∴ Here, Only II follows.​

Hence, the correct answer is "Option 2".

Given:

Series: D, H, L, P, ?

Logic used here:

Alphabet position increases by +4 each step

⇒ P + 4 = T

∴ The correct answer is option (2).

Given:

Word = INTELLIGENCE

A word can be formed only if its letters exist in the given word with equal or fewer frequency.

Calculation:

INTELLIGENCE

Check Gene

⇒ All letters available

⇒ O not present ❌

Check Can

⇒ A not present ❌

Check Come

⇒ O not present ❌

∴ The word formed Gene (Option 1).

Given:

GANDHI → HBOEIJ

Logic used here:

Each letter shifts +1 forward in the English alphabet.

Similarly for NEHRU:

N → O , E → F , H → I , R → S , U → V

⇒ NEHRU → OFISV

∴ The correct answer is option (2).

The logic followed here is:

Hence, the correct answer is "Option 4".

Given:

Series = 5, 11, 23, 47, ?

Logic use here:

Pattern: Multiply by 2 and add 1

Calculation:

5 × 2 + 1 = 11

11 × 2 + 1 = 23

23 × 2 + 1 = 47

⇒ 47 × 2 + 1

⇒ 94 + 1

⇒ 95

∴ The corrct answer is option (4).

Given:

256 : 4096 :: 225 : ?

Logic used here:

⇒ Pattern: n2 → n3

256 = 16²

4096 = 16³

225 = 15²

⇒ 15³ = 15 × 15 × 15

⇒ 3375

∴ The corrct answer is option (1).

Given:

Series = 31, 63, 127, 255, 511, ?

Logic used here:

Pattern: Multiply by 2 and add 1

Calculation:

31 × 2 + 1 = 63

63 × 2 + 1 = 127

127 × 2 + 1 = 255

255 × 2 + 1 = 511

⇒ 511 × 2 + 1

⇒ 1022 + 1

⇒ 1023

∴ The correct answer is option (1).

According to the question:

Thus, 'apple grapes banana' is coded as 'abc, jkl, def'.

Hence, "Option 3" is the correct answer.

Given:

Cost Price (CP) = ₹80

Profit = 20%

Formula used:

Selling Price = CP × (100 + Profit%) / 100

Calculation:

SP = 80 × (100 + 20) / 100

⇒ 80 × 120 / 100

⇒ 96

∴ The correct answer is option (1).

Given:

Initial people = 9

Contribution per person = ₹45

People left = 3

Remaining people = 9 − 3 = 6

Formula used:

Total amount = Number of people × Contribution

Calculation:

Total amount collected initially

⇒ 9 × 45

⇒ 405

Remaining people = 6

Required contribution per person

⇒ 405 ÷ 6

⇒ 67.5

∴ Each person must now pay ₹67.5

Given:

Series = 1, 3, 6, 10, 15, ?

Logic used here:

Difference of between numbers 2,3,4,5,6.

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

⇒ 15 + 6 = 21

∴ The corrct answer is option (2).

Logic used here:

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

Calculation:

COLD

⇒ COLD → DPME

HEAT

⇒ HEAT → IFBU

∴ HEAT will be coded as IFBU.

The correct answer is Monitors space debris and protects Indian space assets.

Key Points

• Space Situational Awareness (SSA) is essential for tracking and monitoring space debris, which poses a threat to operational satellites and other space assets.
• SSA helps in collision avoidance by predicting potential impacts between operational satellites and debris, ensuring safe operations.
• It plays a critical role in protecting India's space assets, including satellites used for communication, navigation, weather forecasting, and defense purposes.
• ISRO's SSA program includes efforts to map space debris and provide real-time updates to mitigate risks associated with space operations.
• The program contributes to global efforts in maintaining space sustainability, ensuring long-term usability of the space environment.

Additional Information

• Space Debris:
• Space debris includes defunct satellites, spent rocket stages, and fragments from collisions or explosions in orbit.
• As of 2023, there are over 34,000 pieces of debris larger than 10 cm and millions of smaller fragments orbiting Earth.
• Debris travels at high speeds, making even small fragments capable of causing significant damage.
• ISRO's SSA Program:
• Launched as part of ISRO's Directorate of Space Situational Awareness and Management (DSSAM).
• Includes initiatives like Netra (Network for Space Object Tracking and Analysis), which monitors objects in low-Earth orbit and geostationary orbit.
• Netra uses ground-based radar and optical telescopes to track space objects and debris.
• Global Context:
• SSA programs are part of international efforts to ensure space sustainability.
• Organizations like NASA, ESA (European Space Agency), and JAXA (Japan Aerospace Exploration Agency) also have active SSA programs.
• The United Nations Office for Outer Space Affairs (UNOOSA) promotes guidelines to reduce space debris.
• Challenges in Space Debris Management:
• Identifying and tracking millions of debris fragments is technically challenging.
• Developing cost-effective methods for debris removal remains a major hurdle.
• International collaboration is required to ensure debris mitigation standards are followed universally.

Important Points

• India's Space Assets:
• India has a growing number of satellites in orbit, including GSAT series for communication, NavIC satellites for navigation, and IRS series for Earth observation.
• Protecting these assets is crucial for national security, economic development, and scientific progress.
• Collision Avoidance Strategies:
• Satellites can be maneuvered to avoid collisions using propulsion systems.
• Real-time tracking and prediction are critical for timely decision-making.
• International Agreements:
• India is a signatory to treaties like the Outer Space Treaty (1967), which promotes peaceful use of space.
• Global cooperation is key to addressing space debris and ensuring sustainable exploration.

The correct answer is India.

Key Points

• Chabahar Port is located in Iran, and its development has been significantly supported by India.
• The port holds strategic importance for India as it provides a direct trade route to Afghanistan and Central Asia, bypassing Pakistan.
• India signed an agreement with Iran in May 2016 for the development of the port. The collaboration includes investment of $500 million from India.
• The operational control of the port was handed over to India in December 2018, making it India's first overseas port project.
• Chabahar Port is considered a key element in India's International North-South Transport Corridor (INSTC), which aims to enhance connectivity between India, Russia, and Europe.

Additional Information

• Location and Strategic Importance:
• Chabahar Port is located in the Sistan-Baluchestan Province of Iran, along the Gulf of Oman.
• The port's location offers India access to trade routes connecting Afghanistan, Central Asia, and beyond.
• It serves as an alternative to Pakistan's Gwadar Port, located just 80 kilometers away.
• India's Investment:
• India has committed $500 million for the development of Chabahar Port, which includes infrastructure upgrades and equipment procurement.
• Under the agreement, India has also pledged to establish two terminals and cargo berths at the port.
• International North-South Transport Corridor (INSTC):
• The INSTC is a multimodal transportation route connecting India, Russia, and Europe through Iran.
• Chabahar Port plays a pivotal role in this corridor, enabling seamless trade and reducing transportation costs and time.
• Impact on Regional Trade:
• Chabahar Port facilitates India's trade with Afghanistan without relying on Pakistan.
• It promotes economic development in the region and strengthens India's geopolitical influence.
• The port is also instrumental in transporting humanitarian aid and goods to war-torn Afghanistan.
• Comparison with Gwadar Port:
• While Gwadar Port is developed and managed by China, Chabahar Port is India's answer to counter China's influence in the region.
• India's involvement in Chabahar balances the strategic dynamics in the Arabian Sea and the Indian Ocean.

Important Points

• Timeline:
• India signed the agreement with Iran in May 2016.
• The port's operational control was handed over to India in December 2018.
• Geopolitical Significance:
• Chabahar Port strengthens India's connectivity with Central Asia and counters China's Belt and Road Initiative (BRI).
• It enhances India's geopolitical leverage in the region by bypassing Pakistan.
• Economic Benefits:
• The port reduces transportation costs for Indian exports to Central Asia and Europe.
• It fosters regional economic integration and increases India's trade prospects.

The correct answer is Providing financial assistance to beneficiaries for the construction, purchase, or enhancement of affordable housing.

Key Points

• The Pradhan Mantri Awas Yojana - Urban (PMAY-U) was launched in 2015 by the Government of India to address the urban housing shortage, especially among the economically weaker sections (EWS), low-income groups (LIG), and middle-income groups (MIG).
• The scheme includes a component of providing financial assistance to eligible beneficiaries for constructing, purchasing, or enhancing affordable housing units. This is a key approach to achieving the scheme's objectives.
• The financial assistance is provided through a Credit Linked Subsidy Scheme (CLSS), under which beneficiaries can avail of interest subsidies on housing loans based on their income categories.
• Under PMAY-U, the focus is on creating “Housing for All” by 2022, with the target of constructing over 1 crore houses in urban areas.
• The scheme also encourages public-private partnerships (PPP) and state-level implementation to ensure effective execution and broader reach.

Additional Information

• Key Components of PMAY-U:
• In-situ Slum Redevelopment (ISSR): Focuses on using land as a resource to provide housing to slum dwellers with the participation of private developers.
• Affordable Housing in Partnership (AHP): Encourages partnerships with private and public sectors to provide affordable housing for the EWS category.
• Beneficiary-led Construction (BLC): Provides financial assistance to eligible individuals to construct or enhance their houses on their own land.
• Credit Linked Subsidy Scheme (CLSS): Offers interest subsidies on housing loans to EWS, LIG, and MIG beneficiaries.
• Eligibility Criteria:
• The applicant must belong to the EWS, LIG, or MIG categories, with annual income ranging from up to ₹3 lakh (EWS) to ₹18 lakh (MIG).
• The beneficiary should not own a pucca house in their name or any family member’s name.
• Women, Scheduled Castes (SCs), Scheduled Tribes (STs), and other vulnerable groups are given priority.
• Achievements:
• As of 2023, over 1.2 crore houses have been sanctioned under PMAY-U, with significant progress in construction and delivery.
• The scheme has successfully mobilized private investment and created employment opportunities in the construction sector.
• Challenges:
• Land acquisition and high real estate costs in urban areas pose challenges to affordable housing projects.
• Ensuring the timely delivery of houses and maintaining quality standards remain critical issues.

Important Points

• Option 1: While rental housing for migrant workers is essential, it is not the primary focus of PMAY-U. Instead, it is covered under schemes like the Affordable Rental Housing Complexes (ARHCs).
• Option 3: The scheme does not mandate the exclusive use of indigenous building materials. It allows flexibility to ensure cost-effectiveness and sustainability.
• Option 4: PMAY-U focuses on all urban areas, including Tier-1, Tier-2, and Tier-3 cities, rather than concentrating solely on Tier-1 and Tier-2 cities.

The correct answer is Nagara (Kalinga).

Key Points

• The Jagannath Temple in Puri is an example of the Nagara style of architecture, specifically the Kalinga sub-style, which is prevalent in Odisha.
• The Nagara style is characterized by its curvilinear tower (Shikhara), often adorned with intricate carvings and sculptures, commonly found in northern and eastern India.
• The Kalinga sub-style is a distinct variation of Nagara architecture, featuring a rekha deula (curvilinear tower), pidha deula (pyramidal roof), and khakhara deula (wagon-shaped roof).
• The Jagannath Temple was built in the 12th century by King Anantavarman Chodaganga Deva of the Eastern Ganga Dynasty.
• The temple complex includes structures such as the Main Shrine (Vimana), Jagamohana (Audience Hall), Natamandapa (Dance Hall), and Bhoga Mandapa (Offering Hall), showcasing the distinct elements of Kalinga architecture.

Additional Information

• Nagara Style of Architecture:
• The Nagara style is primarily found in North India. It is distinct from the Dravidian style, which is prevalent in South India.
• It typically features a Shikhara (tower) that rises vertically and is often crowned by an amalaka (ribbed circular stone) and a kalasha (finial).
• The temples are often constructed on a raised platform, and the sanctum (Garbhagriha) is centrally located.
• Examples include the Kandariya Mahadeva Temple in Khajuraho and the Sun Temple at Konark.
• Kalinga Sub-style:
• The Kalinga sub-style is specific to Odisha and is further divided into three types: rekha deula, pidha deula, and khakhara deula.
• The rekha deula represents the tall, curvilinear tower (Shikhara), as seen in the main shrine of the Jagannath Temple.
• The pidha deula features a pyramidal roof, often used for ancillary structures like the audience hall.
• The khakhara deula, characterized by a wagon-shaped roof, is typically used in temples dedicated to goddesses.
• Jagannath Temple:
• The Jagannath Temple is one of the Char Dham pilgrimage sites in Hinduism and is dedicated to Lord Jagannath, a form of Lord Vishnu.
• It was built in the 12th century by King Anantavarman Chodaganga Deva.
• The temple is known for its Rath Yatra, an annual chariot festival that draws millions of devotees.
• The temple complex includes various subsidiary shrines and is surrounded by a 20-foot-high wall called the Meghanada Pacheri.
• Other Architectural Styles:
• Dravidian Style: Found in South India, characterized by pyramidal towers (Gopuram), intricate carvings, and large temple complexes (e.g., Brihadeeswarar Temple).
• Maratha Style: Known for its simplicity and practicality, often featuring flat-roofed structures and courtyards (e.g., Shaniwar Wada).
• Vesara Style: A blend of Dravidian and Nagara styles found in the Deccan region (e.g., Chalukyan temples).

Important Points

• Historical Significance: The Jagannath Temple plays a crucial role in preserving the Kalinga architectural heritage and is a major cultural and spiritual landmark in India.
• Religious Importance: As one of the Char Dham pilgrimage sites, it holds immense significance in Hinduism, attracting devotees from across the globe.
• Architectural Innovation: The temple's design reflects the advanced architectural techniques and aesthetic sensibilities of the Eastern Ganga Dynasty.
• UNESCO Recognition: The nearby Sun Temple at Konark, another example of Kalinga architecture, is a UNESCO World Heritage Site, emphasizing the global recognition of this architectural style.

The correct answer is The issue of allowing Indian judges to try European offenders in criminal cases.

Key Points

• The Ilbert Bill, introduced in 1883 by the then Viceroy of India, Lord Ripon, sought to allow Indian judges to preside over cases involving European offenders in criminal matters.
• During the colonial period, European residents in India were subject to a separate legal framework, which barred Indian judges from trying them. The Ilbert Bill aimed to remove this racial distinction in the judiciary system.
• The bill faced severe opposition, especially from the European community in India, who viewed it as a threat to their perceived racial superiority. This led to widespread protests and debates.
• The controversy highlighted the deep-seated racial tensions in colonial India and revealed the discriminatory practices prevalent in the British administration.
• The opposition forced the British government to amend the bill by introducing a compromise, where European offenders could opt to be tried by a jury consisting of at least 50% Europeans. This diluted the original intent of the bill.

Additional Information

• Historical Context:
• The Ilbert Bill was part of a series of reforms introduced by Lord Ripon, who is often referred to as the "Father of Local Self-Government in India."
• It was introduced at a time when Indians were increasingly demanding equality and justice under British rule.
• Impact on Indian Society:
• The Ilbert Bill controversy brought Indians together, as they saw this as an opportunity to demand equality in the judiciary system.
• It helped to strengthen the nascent Indian National Movement, as leaders like Surendranath Banerjee and others used this issue to mobilize public opinion against colonial injustice.
• Racial Discrimination in Colonial India:
• The opposition to the Ilbert Bill revealed the inherent racial biases of the British administration, where Europeans were placed above Indian citizens in matters of law and governance.
• It emphasized the need for a fair and impartial judiciary that treated all citizens equally, irrespective of their race.
• Outcome of the Controversy:
• Although the bill was amended and diluted, the controversy surrounding it became a significant milestone in India's struggle for independence.
• The Ilbert Bill controversy also demonstrated the power of collective action, as both Indians and Europeans organized protests to voice their opinions.

Important Points

• Lord Ripon's Reforms:
• Lord Ripon is known for introducing the Local Self-Government Act in 1882, which empowered local bodies such as municipalities and district boards.
• His tenure was marked by progressive policies aimed at involving Indians in administrative roles and promoting education.
• European Opposition:
• The European community in India argued that Indian judges were not competent to try Europeans due to cultural and racial differences.
• This opposition was rooted in the belief in the racial superiority of Europeans over Indians.
• Legacy of the Ilbert Bill:
• The controversy set the stage for future debates on equality and justice in colonial India.
• It also exposed the limitations of British liberalism, which claimed to promote equality but often failed to apply it in practice.

The correct answer is Direct and related to what the witness saw or heard.

Key Points

• Under the BSA, 2023, oral evidence must be direct, meaning it should pertain to what the witness personally observed or heard, rather than relying on assumptions or hearsay.
• Oral evidence plays a crucial role in the judicial process, as it provides firsthand accounts that can help establish the facts of the case.
• Evidence based on assumptions or speculation is inadmissible under the law, as it lacks reliability and factual accuracy. Hence, statement 2 is incorrect.
• Language is not a criterion for admissibility of oral evidence under the BSA, 2023. Evidence can be recorded in any language, provided it is accurately translated if necessary. Hence, statement 3 is incorrect.
• Hearsay evidence, or evidence not based on direct observation, is typically inadmissible except under certain exceptions outlined in the law. Hence, statement 4 is incorrect.
• Direct evidence is considered more reliable and credible in legal proceedings compared to indirect or hearsay evidence, as it is firsthand and less prone to distortion or misinterpretation.

Additional Information

• Direct Evidence:
• Direct evidence is testimony or information provided by a witness who has personally experienced the event in question.
• It is considered highly credible and is admissible in court without requiring further corroboration.
• Example: A witness testifying that they saw a person commit a crime.
• Hearsay Evidence:
• Hearsay evidence refers to information that a witness heard from someone else, rather than directly observing the event.
• It is generally inadmissible under the legal framework, except in specific exceptions such as dying declarations or admissions.
• Example: A witness stating that they were told by another person about the crime.
• Exceptions to Hearsay Rule:
• Certain exceptions allow hearsay evidence to be admitted, such as statements made under imminent fear of death (dying declaration).
• Other exceptions include business records, statements against interest, or declarations made during the course of business.
• Importance of Oral Evidence:
• Oral evidence is a key element in trials, as it provides the court with a direct account of events.
• It must be scrutinized for reliability, consistency, and credibility during cross-examination.

Important Points

• Admissibility Criteria:
• Oral evidence must be relevant, material, and directly related to the facts of the case.
• It is subject to cross-examination to test the witness's credibility and the accuracy of their testimony.
• Legal Basis:
• The Indian Evidence Act, 1872 governs admissibility and relevance of oral evidence in India.
• Similar provisions exist globally, ensuring that direct evidence is prioritized for reliability.

The correct answer is B.V.R. Subrahmanyam.

Key Points

• B.V.R. Subrahmanyam was appointed as the CEO of NITI Aayog in February 2023, succeeding Parameswaran Iyer.
• He is a 1987 batch Indian Administrative Service (IAS) officer of the Chhattisgarh cadre.
• Prior to his appointment, B.V.R. Subrahmanyam served as Chairman and Managing Director of the India Trade Promotion Organisation (ITPO), which is responsible for managing India’s largest exhibition complex, Pragati Maidan.
• He has held several key positions in the Indian government, including serving as the Chief Secretary of Jammu and Kashmir, where he played a crucial role in the region's administrative and developmental changes post its reorganization in 2019.
• His appointment aligns with the vision of NITI Aayog to foster development and policy innovation, leveraging his vast administrative experience.

Additional Information

• NITI Aayog:
• NITI Aayog stands for the National Institution for Transforming India and was established on January 1, 2015, replacing the Planning Commission.
• It serves as the policy think tank of the Government of India and provides strategic and technical advice to various ministries.
• The organization focuses on cooperative federalism by engaging states in policy formulation and implementation.
• It is chaired by the Prime Minister of India, and the CEO role facilitates operational execution and strategy implementation.
• Key roles previously held by B.V.R. Subrahmanyam:
• He has worked in the Prime Minister’s Office (PMO) during the tenure of multiple Prime Ministers, offering administrative and policy expertise.
• As Chief Secretary of Jammu and Kashmir, he oversaw critical developmental initiatives and administrative restructuring post the region’s reorganization.
• India Trade Promotion Organisation (ITPO):
• ITPO is a government agency responsible for promoting trade and commerce by organizing exhibitions and events.
• B.V.R. Subrahmanyam contributed to the modernization of Pragati Maidan, including the construction of the new Integrated Exhibition-cum-Convention Centre in Delhi.
• Significance of NITI Aayog’s CEO Role:
• The CEO drives the implementation of flagship programs such as Atal Innovation Mission, Digital India, and Make in India.
• He/she works to enhance India's global competitiveness by facilitating policy reforms and innovation.

Important Points

• Other notable individuals in NITI Aayog:
• Parameswaran Iyer: Former CEO of NITI Aayog, known for his contributions to the Swachh Bharat Mission.
• Amitabh Kant: Preceded Parameswaran Iyer and played a significant role in driving initiatives like Ease of Doing Business and Startup India.
• V.K. Paul: Member of NITI Aayog, focused on health policy and initiatives such as Ayushman Bharat.
• Vision of NITI Aayog: NITI Aayog aims to promote sustainable development, innovation, and inclusive growth through transformative policies and programs.

The correct answer is Vagbhata.

Key Points

• Vagbhata was an ancient Indian scholar and physician, known as one of the principal contributors to Ayurveda, the traditional system of medicine in India.
• He authored two renowned texts on Ayurveda: the Ashtanga Hridaya and the Ashtanga Sangraha.
• The Ashtanga Sangraha is a comprehensive guide to the eight limbs (ashtanga) of Ayurveda and provides detailed insights into clinical diagnosis and treatment protocols.
• This text is written in a prose style and includes extensive information on therapeutic procedures, herbal formulations, and surgical interventions.
• Vagbhata's works are considered foundational in the field of Ayurveda, alongside Charaka's and Sushruta's contributions.

Additional Information

• Charaka:
• Charaka was an ancient Indian physician who authored the Charaka Samhita, one of the foundational texts of Ayurveda.
• The Charaka Samhita focuses primarily on internal medicine and emphasizes preventive healthcare, lifestyle, and diet.
• He is often regarded as the father of Indian medicine for his contributions to understanding human anatomy and physiology.
• Sushruta:
• Sushruta is known as the father of surgery and wrote the Sushruta Samhita, which is a key treatise in Ayurveda.
• This text contains detailed descriptions of surgical techniques, instruments, and procedures, including rhinoplasty and cataract surgery.
• Sushruta's work has influenced both Indian and global surgical practices.
• Bhaskaracharya:
• Bhaskaracharya, also known as Bhaskara II, was a 12th-century Indian mathematician and astronomer.
• He is renowned for his work on calculus and algebra and authored the book “Siddhanta Shiromani.”
• His contributions were primarily in the field of mathematics and not related to medicine or Ayurveda.
• Vagbhata:
• Vagbhata was a later contributor to Ayurveda, following in the footsteps of Charaka and Sushruta.
• He emphasized the integration of theory and practice in Ayurvedic medicine, with a focus on holistic healing and lifestyle modifications.
• The Ashtanga Sangraha and Ashtanga Hridaya continue to be widely studied in Ayurvedic medical education today.

Important Points

• Ayurveda:
• Ayurveda is a traditional system of medicine originating in India over 5,000 years ago. It focuses on balancing the body, mind, and spirit to maintain health.
• The term "Ayurveda" is derived from the Sanskrit words "Ayur" (life) and "Veda" (science or knowledge), meaning the "science of life."
• It comprises eight branches (Ashtanga), including surgery (Shalya Tantra), pediatrics (Kaumarabhritya), and toxicology (Agada Tantra).
• Legacy of Vagbhata:
• Vagbhata's works are still relevant in contemporary Ayurveda for clinical practice and research.
• His texts serve as a bridge between ancient and modern medical practices, highlighting the importance of diet, exercise, and mental well-being.
• Modern Relevance:
• Ayurveda is recognized by the World Health Organization (WHO) as a traditional medicine system and is practiced globally today.
• Institutions in India, such as the All India Institute of Ayurveda (AIIA), promote the integration of Ayurvedic principles into modern healthcare.

The correct answer is Statement I is correct, but Statement II is incorrect..

Key Points

• Statement I: The city of Chennai is popularly known as the Detroit of India.
• Chennai is referred to as the "Detroit of India" due to its thriving automobile industry.
• It is home to major automobile manufacturers such as Hyundai, Ford, BMW, and Renault Nissan, alongside numerous auto component industries.
• The city has emerged as a prominent hub for automobile production, with its factories contributing significantly to both domestic and export markets. This statement is correct.
• Statement II: The city is a hub of the petrochemical industry and a leader in petrochemical product manufacturing in India.
• While Chennai has industries dealing with chemicals, it is not predominantly known for being a hub for the petrochemical industry.
• Petrochemical hubs in India are more prominently located in regions such as Gujarat (Jamnagar) and Maharashtra (Mumbai), owing to the proximity to oil refineries and ports.
• Hence, attributing Chennai as a leader in petrochemical product manufacturing is incorrect. This statement is incorrect.

Additional Information

• Chennai as the "Detroit of India":
• Chennai contributes over 40% of India’s automobile exports due to its well-established automobile ecosystem.
• The city hosts production facilities for international automobile brands and auto-component industries, making it a central hub for vehicle manufacturing.
• It is also home to institutions focused on automobile engineering and innovation, further strengthening its position in the industry.
• Examples of automobile companies include Hyundai Motor India, Ford India, Mitsubishi Motors, and Ashok Leyland.
• Petrochemical hubs in India:
• Gujarat: The Jamnagar refinery, owned by Reliance Industries, is the world’s largest refining hub, contributing significantly to petrochemical production.
• Maharashtra: Mumbai and surrounding areas host several petrochemical industries due to proximity to ports.
• Odisha: Paradip hosts a major refinery by Indian Oil Corporation, contributing to petrochemical output.
• Automobile Industry in India:
• India is one of the largest producers of automobiles globally, with major hubs located in Chennai, Pune, Gurugram, and Gujarat.
• Chennai's focus on automobiles stems from its strategic location and infrastructure designed to support the industry.
• The automobile industry in India contributes over 7.1% of the GDP and is a major source of employment.

Important Points

• Significance of Chennai’s automobile industry:
• Chennai’s industrial ecosystem supports vehicle production with facilities for design, assembly, and export.
• The city benefits from its port facilities, which ease the export of vehicles and components.
• Chennai’s automobile sector is supported by numerous small and medium enterprises (SMEs) specializing in auto components.
• Petrochemical Industry Overview:
• India’s petrochemical sector is driven by refinery products, with major hubs located in coastal regions for ease of import/export.
• Petrochemical products include polymers, synthetic fibers, and elastomers, which serve industries such as textiles and packaging.
• Government initiatives like Petroleum, Chemicals, and Petrochemicals Investment Regions (PCPIR) promote growth in the sector.

The correct answer is 1 and 3 only.

Key Points

• Statement 1: Muslims outnumber all other minorities in India, and they form a majority in the state of Jammu and Kashmir.
• The Census of India 2011 highlights that the Muslim population is the largest minority group in India, accounting for approximately 14.2% of the total population.
• They constitute a majority in the state of Jammu & Kashmir. Hence, this statement is correct.
• Statement 2: Christian population is distributed mostly in urban areas of India.
• The Christian population in India is spread across both urban and rural areas, with a significant concentration in northeastern states like Nagaland, Mizoram, and Meghalaya and southern states like Kerala and Tamil Nadu.
• This statement is incorrect as Christians are predominantly found in rural regions of the northeastern and southern states.
• Statement 3: Jains have a major concentration in the urban areas of Rajasthan, Gujarat, and Maharashtra, while the Buddhists are concentrated mostly in Maharashtra.
• Jains are primarily concentrated in urban areas, particularly in states like Rajasthan, Gujarat, and Maharashtra. This community is known for its business-oriented population, largely residing in metropolitan areas.
• Buddhists, on the other hand, have a significant presence in Maharashtra, especially among Scheduled Castes, due to the influence of Dr. B.R. Ambedkar's Dalit Buddhist Movement. Hence, this statement is correct.
• Hence, the correct answer is 1 and 3 only.

Additional Information

• Religious Demographics in India (Census 2011):
• Hindus: Constitute approximately 79.8% of the population, making them the majority religious group in India.
• Muslims: Represent 14.2%, the largest minority group in the country.
• Christians: Account for 2.3%, with significant representation in the northeastern states and Kerala.
• Sikhs: Constitute 1.7%, predominantly found in Punjab.
• Buddhists: Represent 0.7%, primarily concentrated in Maharashtra due to the Dalit Buddhist movement.
• Jains: Comprise 0.4%, mostly found in urban centers of western India.
• Significance of Census Data:
• The Census of India provides critical data on demographics, including religion, language, literacy, and other socio-economic factors.
• This data is essential for policy formulation and allocation of resources by the government.
• Urban and Rural Distribution:
• Religious communities like Jains are concentrated in urban areas due to their business-oriented lifestyle.
• Others, like Christians and Buddhists, have significant populations in rural areas, particularly in northeastern states and Maharashtra.
• Historical Influence:
• The religious distribution in India is influenced by historical factors such as migrations, colonial rule, and social reform movements like the Dalit Buddhist Movement.
• For instance, the conversion of Dalits to Buddhism under Dr. B.R. Ambedkar significantly impacted the demographics in Maharashtra.

Important Points

• Jainism: Known for its emphasis on non-violence (Ahimsa), Jainism originated in India and has a rich tradition of philosophy and ethics.
• Buddhism: Founded by Gautama Buddha, Buddhism focuses on the teachings of the Four Noble Truths and the Eightfold Path.
• Dalit Buddhist Movement: Initiated by Dr. B.R. Ambedkar, this movement encouraged Dalits to convert to Buddhism to escape caste-based discrimination.
• Religious Diversity: India is known for its unique blend of religious pluralism, housing multiple faiths coexisting harmoniously.
• Census Importance: The 2011 Census remains a critical source for understanding India's demographic trends and planning welfare schemes accordingly.

The correct answer is Kriti.

Key Points

• Kriti is the most commonly performed composition in Carnatic music concerts. It is a structured musical form that focuses on both melody and rhythm, making it a centerpiece of the performance.
• Kriti compositions often showcase the composer's creativity and express devotion, philosophy, or emotions. They are known for their lyrical richness and intricate musical details.
• The structure of a Kriti typically includes three sections: Pallavi (opening), Anupallavi (middle section), and Charanam (concluding section), allowing for thematic development and elaboration.
• Renowned composers like Thyagaraja, Muthuswami Dikshitar, and Shyama Sastri have created timeless Kriti compositions, which are central to Carnatic music repertoire.
• In concerts, the rendition of a Kriti is often accompanied by improvisational techniques like Niraval and Kalpana Swaras, enhancing the performance and engaging the audience.

Additional Information

• Varnam:
• Varnam is a type of Carnatic music composition used as a warm-up piece at the beginning of concerts. It emphasizes and provides a foundation for practicing vocal techniques.
• Varnam compositions are used to highlight a particular raga and its melodic nuances but are less elaborate than Kritis.
• Raga Alapana:
• Raga Alapana is a purely improvisational segment in Carnatic music where the artist explores the melodic framework of a raga without rhythm or lyrics.
• It is usually performed before a Kriti and showcases the artist's creativity and understanding of the raga.
• Tillana:
• Tillana is a lively and rhythmic composition often performed towards the end of a concert. It focuses on percussion-rich patterns and dance-like elements.
• It usually serves as a concluding piece to leave the audience energized.
• Characteristics of Carnatic Music:
• Carnatic music is one of the two major traditions of Indian classical music, originating in South India.
• The music is deeply rooted in spiritual themes, with compositions often dedicated to Hindu deities.
• It emphasizes raga (melody) and tala (rhythm), with improvisation being a key feature.

Important Points

• Key Composers: Saint Thyagaraja, Muthuswami Dikshitar, and Shyama Sastri are regarded as the Trinity of Carnatic Music.
• Kriti's Role: Kriti compositions act as the primary medium for ragas and tala exposition during concerts.
• Improvisation: Techniques such as Kalpana Swaras and Niraval enhance Kriti renditions, showcasing the performer's skill and creativity.
• Popular Ragas: Ragas like Kalyani, Todi, and Bhairavi are often chosen for Kriti compositions.
• Concert Sequence: A typical Carnatic concert begins with a Varnam, followed by Kritis interspersed with improvisational segments and ends with a Tillana.

The correct answer is It follows a panchayatana layout and is an early example of Nagara style..

Key Points

• The Deogarh temple, also known as the Dashavatara Temple, was constructed in the early 6th century CE. It is one of the earliest surviving examples of Hindu temple architecture.
• The temple follows a panchayatana layout, a style where the main shrine is surrounded by four subsidiary shrines. This layout is a hallmark of early Hindu temple design.
• The architectural style of the temple is classified as Nagara style, which is characterized by its curvilinear shikhara (tower) and intricate carvings.
• The temple is dedicated to Lord Vishnu, as evidenced by the presence of sculptures depicting Vishnu's incarnations (Dashavatara).
• The temple's ornamentation and sculptural reliefs are considered masterpieces of ancient Indian art and provide insights into the religious themes and artistic conventions of the time.

Additional Information

• Deogarh Temple's Panchayatana Layout:
• The term panchayatana refers to a layout where a central shrine is surrounded by four subsidiary shrines, symbolizing the presence of other deities alongside the main deity.
• This architectural design reflects the syncretic nature of Hinduism, accommodating multiple deities within a single sacred complex.
• Nagara Style Architecture:
• The Nagara style is predominantly found in northern India and is marked by its beehive-shaped shikhara (tower) rising over the sanctum sanctorum.
• The style emphasizes verticality and intricate carvings, often depicting mythological and religious motifs.
• The Deogarh temple is an early example of this style and showcases its formative features.
• Historical Context:
• The temple was likely constructed during the reign of the Gupta Empire, a period renowned for cultural and artistic achievements.
• The Gupta period is often referred to as the "Golden Age of India", and the Deogarh temple reflects the sophisticated architectural and artistic norms of the era.
• Other Options Explained:
• Option 2: The temple was not built by the Chandela rulers, as they came much later (9th-13th century CE). Incorrect.
• Option 3: Its main deity is Vishnu, not Shiva as Mahadeo. Incorrect.
• Option 4: The temple is not entirely rock-cut; it is constructed using stone blocks. Incorrect.

Important Points

• Gupta Period Art and Architecture:
• Known for its refinement and sophistication, the Gupta era marked significant advancements in temple construction, sculpture, and painting.
• Religious themes, particularly those associated with Hinduism, Buddhism, and Jainism, dominated artistic expressions.
• Significance of the Deogarh Temple:
• The temple provides valuable insights into the development of Hindu temple architecture and early iconography.
• It is a key resource for understanding the transition from wooden structures to stone temples in ancient India.

The correct answer is Option 3.

Key Points

• The biomechanical coordination during the throwing phase in discus throw involves the bending of the right knee after forward extension for balance, which stabilizes the body and enhances the accuracy of the throw. Hence, statement 3 is correct.
• The shoulders and trunk do not continue rotating for 1 minute after the discus is released. The rotation stops immediately after the release to prevent unnecessary loss of energy. Hence, statement 1 is incorrect.
• The left foot does not remain fixed during the throwing phase. Both feet contribute to rotational force, and the movement of the feet is crucial for generating momentum. Hence, statement 2 is incorrect.
• The body’s weight does not shift entirely to the left foot while the right leg stabilizes the upper body lean. The weight distribution varies throughout the phases, ensuring proper momentum and balance during the throw. Hence, statement 4 is incorrect.
• The correct biomechanical technique involves bending the right knee after forward extension to maintain balance and maximize the throw’s efficiency. This technique is crucial for proper stabilization. Hence, statement 3 is correct.

Additional Information

• Biomechanics in Discus Throw:
• The discus throw involves coordinated movements of the lower limbs, trunk, and upper limbs to generate rotational force and release the discus at an optimal angle.
• The right knee bending after forward extension is crucial for maintaining balance and stabilizing the body during the critical release phase.
• The shoulders and trunk rotation play a significant role in generating angular momentum, but the rotation stops immediately after the throw.
• Phases of Discus Throw:
• It includes the wind-up phase, where the thrower builds initial momentum by rotating the trunk and arms.
• The transition phase involves shifting the weight between the feet to prepare for rotational acceleration.
• The delivery phase is where the discus is released, requiring precise coordination and balance.
• The follow-through phase ensures that the thrower maintains balance and avoids injury.
• Key Techniques in Discus Throw:
• The thrower should focus on maintaining a low center of gravity for stability during rotational movements.
• The discus is released at an optimal angle, typically between 35-45 degrees, for maximum distance.
• Proper foot placement and rotation are essential for generating torque and transferring energy to the discus.
• Applications in Sports Science:
• Understanding biomechanics in discus throw helps athletes improve their performance and minimize the risk of injuries.
• It allows coaches to analyze movements and suggest corrective measures for better technique.
• Sports scientists use motion capture technology to study the kinematics and dynamics of the throw.

Important Points

• Historical Context:
• The discus throw has its origins in ancient Greece and was part of the Olympic Games as early as 708 BC.
• Modern techniques were developed in the 19th century, with emphasis on rotational movements.
• Physics in Discus Throw:
• The throw involves concepts such as angular momentum, torque, and projectile motion.
• The distance covered by the discus depends on the release velocity, angle, and height.
• Training Essentials:
• Athletes undergo strength training to develop core stability and rotational power.
• They practice drills to enhance footwork and improve timing during the throw.

The correct answer is Rayas.

Key Points

• In the context of the Vijayanagara Empire, the term "narapati", which translates to "lord of men," was used to refer to the Rayas, the rulers of the empire.
• The Vijayanagara Empire, established in 1336, was primarily ruled by dynasties like the Sangama, Saluva, Tuluva, and Aravidu. The kings of these dynasties were referred to as Rayas.
• The Rayas were considered the supreme authority in the empire, holding political, military, and religious leadership.
• Prominent rulers such as Krishnadevaraya, a Tuluva dynasty king, were known for their administrative acumen, military conquests, and promotion of arts and literature.
• The title "narapati" symbolized their responsibility to govern and protect their people, reflecting their role as the central figure in the hierarchical structure of the empire.

Additional Information

• Vijayanagara Empire:
• The Vijayanagara Empire was located in the Deccan region of South India and flourished between the 14th and 17th centuries.
• It was founded by Harihara I and Bukka Raya I of the Sangama dynasty in 1336.
• The empire served as a bastion of Hindu culture and traditions during a period of Islamic invasions in the Indian subcontinent.
• Prominent Rayas:
• Krishnadevaraya (1509–1529): One of the most famous rulers of the Vijayanagara Empire, known for his military conquests, economic reforms, and patronage of literature and the arts.
• Krishnadevaraya authored the Telugu work "Amuktamalyada", showcasing his literary prowess.
• Administrative Structure:
• The Vijayanagara Empire was divided into provinces, each governed by a Nayaka (feudal lord) who reported to the Raya.
• The Rayas maintained a centralized administration while delegating local governance to Nayakas, ensuring efficient management of resources and military strength.
• Cultural Contributions:
• The empire was known for its architectural marvels, such as the Virupaksha Temple and Vittala Temple in Hampi.
• The Rayas supported diverse art forms, including dance, music, and literature in languages like Kannada, Telugu, Tamil, and Sanskrit.

Important Points

• The Vijayanagara Empire played a crucial role in resisting the advancements of the Deccan Sultanates and preserving Hindu traditions.
• The term "Raya" is derived from the Sanskrit word "Raja", emphasizing the sovereign authority of the Vijayanagara rulers.
• The title "narapati" highlights the king's role as a protector and benefactor of his subjects, aligning with the Indian concept of a just and noble ruler.
• The empire's capital, Hampi, is a UNESCO World Heritage Site and serves as a testament to the architectural and cultural achievements of the Rayas.

The correct answer is Blocks 16-19 inches from line, rear toe and front heel aligned.

Key Points

• The "bunch start" position in sprinting is a common starting stance used by sprinters to optimize their launch speed. It is characterized by having the starting blocks placed 16-19 inches from the starting line. This measurement ensures that the sprinter can generate maximum force during the initial push-off.
• In this position, the rear toe aligns with the front heel. This alignment is crucial as it allows the sprinter to maintain balance while simultaneously optimizing forward thrust power.
• The design of the "bunch start" position focuses on creating the shortest reaction time and explosive movement off the blocks, which is essential for short-distance sprints such as the 100m and 200m races.
• Unlike other starting techniques, the "bunch start" ensures that the sprinter's body is positioned closer to the starting line, enabling a quicker transition into the acceleration phase.
• This technique is especially suitable for athletes who possess strong legs and explosive power, as it requires a rapid and forceful push-off to propel the body forward efficiently.

Additional Information

• Other Starting Positions in Sprinting:
• Medium Start: In this position, the starting blocks are set further back from the starting line compared to the "bunch start." The rear knee is aligned with the front heel. It provides a balanced compromise between power and control, suitable for athletes with moderate explosive strength.
• Elongated Start: The starting blocks are positioned farthest from the starting line. The rear knee is positioned behind the front foot. This start focuses more on gradual acceleration rather than immediate explosive speed, making it suitable for longer sprint races.
• Importance of Starting Position in Sprinting:
• The starting position significantly impacts a sprinter's performance as it dictates the efficiency of the initial push-off and transition into the acceleration phase.
• A proper start minimizes reaction time and ensures optimal utilization of muscle groups, especially the quadriceps and hamstrings, for a powerful launch.
• Starting techniques are often tailored to an individual sprinter's physical attributes, such as leg strength, flexibility, and body proportion.
• Key Characteristics of Sprinting:
• Sprinting involves high-intensity, short-distance running events, typically ranging from 60m to 400m.
• The primary focus is on generating maximum speed and maintaining it for the duration of the race.
• Key muscle groups used in sprinting include the quadriceps, hamstrings, glutes, and calves.
• Training for sprinting includes a combination of explosive drills, strength training, and technique improvement for optimal performance.
• Equipment Used in Sprinting:
• Starting Blocks: These are crucial for providing a stable base for the sprinter to push off at the start. They are adjustable to accommodate different starting styles.
• Spikes: Special sprinting shoes with spikes enhance grip and traction on the track, allowing for better acceleration and speed.
• Track Surface: Modern sprinting tracks are made from synthetic materials like polyurethane, which provide a balance of grip and shock absorption.
• Famous Sprinters:
• Usain Bolt: Holds the world record for the 100m (9.58 seconds) and 200m (19.19 seconds) sprints.
• Carl Lewis: A legendary sprinter and long jumper who dominated the Olympics during the 1980s and 1990s.
• Florence Griffith-Joyner: Known for her record-breaking performances in the 100m and 200m events, which still stand today.

The correct answer is Option 1.

Key Points

• Statement 1: The novel 'The Golden Gate' by Vikram Seth is written entirely in verse and was shortlisted for the Whitbread Book Award. This showcases the innovative poetic form of the book and its literary significance. Hence, this statement is correct.
• Statement 2: The claim that the book won the Booker Prize is incorrect. 'The Golden Gate' did not win the Booker Prize, as the book primarily reflects themes of contemporary life in California rather than Indian society and culture. Hence, this statement is incorrect.
• Statement 3: The book was not honored with the Sahitya Akademi Award. While Vikram Seth's work is widely recognized in Indian English literature, this specific award does not apply to 'The Golden Gate.' Hence, this statement is incorrect.
• Therefore, only Statement 1 is correct, and the correct answer is Option 1.

Additional Information

• 'The Golden Gate' by Vikram Seth:
• Published: The novel was published in 1986.
• Form: The entire book is written in verse, specifically in the format of sonnets, inspired by Alexander Pushkin's "Eugene Onegin."
• Theme: It explores themes such as love, friendship, politics, and loneliness, set in the backdrop of modern-day California.
• Recognition: It received critical acclaim for its inventive style but was not awarded the Booker Prize or Sahitya Akademi Award.
• Vikram Seth:
• Vikram Seth is a renowned Indian author and poet, widely recognized for his works like 'A Suitable Boy' and 'An Equal Music.'
• His contributions to English literature have earned him numerous accolades, including the Padma Shri in 2001.
• Whitbread Book Awards:
• These awards, now known as the Costa Book Awards, recognize outstanding literary works from the UK and Ireland.
• 'The Golden Gate' was shortlisted, highlighting its global recognition despite being authored by an Indian writer.
• Booker Prize:
• One of the most prestigious literary awards, given annually for the best original novel written in English.
• Vikram Seth's 'A Suitable Boy' was longlisted for the award but 'The Golden Gate' did not win this prize.
• Sahitya Akademi Award:
• This is a literary honor presented by India's National Academy of Letters, recognizing works in Indian languages.
• 'The Golden Gate' is not an awardee of the Sahitya Akademi, as it is primarily a work about life in the US, written in English verse.

Important Points

• Vikram Seth's Writing Style:
• Seth is celebrated for his versatility, writing across genres including poetry, novels, and essays.
• His ability to craft narratives in verse, as demonstrated in 'The Golden Gate', sets him apart in contemporary literature.
• Significance of 'The Golden Gate':
• It is considered a groundbreaking work for its use of sonnet form in a modern novel.
• The novel remains relevant for its exploration of universal human emotions and societal dynamics.

The correct answer is Vadodara, Gujarat.

Key Points

• The 2nd Para Table Tennis National Ranking Championships for 2024-25 was held in Vadodara, Gujarat, a city known for its vibrant sports culture and facilities.
• This championship is an essential event in the Indian para table tennis circuit, aimed at promoting inclusivity and providing a competitive platform for athletes with disabilities.
• Vadodara, Gujarat, has hosted several national and international sports events due to its well-developed sports infrastructure and accessibility for athletes.
• The event witnessed participation from para table tennis players across India, showcasing their talent and determination in the sport.
• Gujarat has been a progressive state in supporting sports for differently-abled individuals, making it a suitable venue for such championships.

Additional Information

• Para Table Tennis Overview:
• Para table tennis is a variant of table tennis designed for athletes with physical disabilities.
• The sport is governed by the International Table Tennis Federation (ITTF) and follows classification rules to ensure fair competition.
• India has been actively promoting para sports, including table tennis, under various government schemes and initiatives.
• Vadodara, Gujarat:
• Vadodara is the third-largest city in Gujarat and is known for its rich cultural heritage and modern infrastructure.
• It is equipped with facilities like indoor stadiums and accessibility features that cater to athletes with disabilities.
• Vadodara has previously hosted events like the National Games and other sports championships.
• Promoting Para Sports in India:
• The Government of India has introduced several initiatives to promote para sports, such as financial assistance, infrastructure development, and coaching facilities.
• Events like the Para Table Tennis National Ranking Championships play a crucial role in identifying and nurturing talent for international competitions.
• Other Host Cities Considered:
• Jaipur, Rajasthan: Known for its tourist attractions and occasional sports events, but not selected for this championship.
• Pune, Maharashtra: A hub for sports academies and events but not the venue for this particular championship.
• Indore, Madhya Pradesh: Famous for hosting sports tournaments but not chosen for the 2nd Para Table Tennis National Ranking Championships.

Important Points

• Para Sports Initiatives:
• The Indian government supports para sports through schemes like the Khelo India Program and the Sports Authority of India (SAI).
• Para table tennis players are encouraged to participate in international events like the Paralympic Games and Asian Para Games.
• Significance of Championships:
• National ranking championships help in creating a pool of skilled athletes for global competitions.
• Such events provide exposure and boost the confidence of players, contributing to their overall growth.

The correct answer is Beta Decay.

Key Points

• Beta Decay is a type of nuclear decay in which a beta particle (electron or positron) is emitted from the nucleus.
• During beta decay, a neutron in the nucleus is converted into a proton (or vice versa), leading to a change in the atomic number but not the mass number.
• The mass number remains unchanged because the total number of nucleons (protons + neutrons) stays the same.
• Beta decay increases or decreases the atomic number by 1, thus transforming the atom into a different element.
• This process helps unstable nuclei achieve a more stable configuration by adjusting the proton-to-neutron ratio.

Additional Information

• Alpha Decay:
• Alpha decay involves the emission of an alpha particle (2 protons and 2 neutrons) from the nucleus.
• This reduces the atomic number by 2 and the mass number by 4.
• It occurs in heavy nuclei, such as uranium or radium, to achieve stability.
• Gamma Decay:
• Gamma decay involves the emission of gamma rays (high-energy photons) from an excited nucleus.
• It does not change the atomic number or mass number but releases excess energy from the nucleus.
• This often accompanies alpha or beta decay.
• Fission:
• Nuclear fission is the splitting of a heavy nucleus into two lighter nuclei, accompanied by the release of energy and neutrons.
• It significantly changes both the atomic and mass numbers of the original nucleus.
• Fission is commonly used in nuclear reactors and weapons.
• Key Characteristics of Beta Decay:
• There are two types: Beta-minus (β⁻) and Beta-plus (β⁺) decay.
• In β⁻ decay, a neutron is converted into a proton, emitting an electron and an antineutrino.
• In β⁺ decay, a proton is converted into a neutron, emitting a positron and a neutrino.
• Beta decay plays a critical role in processes like radioactive decay chains and nuclear reactions in stars.

Important Points

• Applications of Beta Decay:
• Used in medical imaging and treatment, such as in PET scans (Positron Emission Tomography).
• Helps in carbon dating to determine the age of archaeological artifacts (e.g., Carbon-14 decay).
• Used in studying nuclear reactions and understanding stellar processes.
• Examples of Beta Decay:
• Carbon-14 (C-14): Undergoes β⁻ decay to form Nitrogen-14 (N-14).
• Potassium-40 (K-40): Undergoes β⁺ decay to form Argon-40 (Ar-40).

The correct answer is colorful robes.

Key Points

• Dumhal dance is a traditional folk dance performed by the Rauf tribe in Jammu and Kashmir, India.
• During the performance, Dumhal dancers wear colorful robes which are intricately designed to reflect the cultural heritage of the region.
• The colorful robes serve as a visual representation of the tribe’s tradition and add vibrancy to the performance, making it more engaging.
• The dance is performed to the accompaniment of drums and singing, and the attire complements the overall aesthetic of the event.
• These robes are designed specifically to accommodate the energetic movements and ceremonial nature of the Dumhal dance.

Additional Information

• Origin of Dumhal Dance:
• The Dumhal dance is believed to have originated centuries ago and is deeply rooted in the traditions of the Rauf tribe.
• It is performed on special occasions, such as celebrations or festivals, and is considered a sacred ritual by the tribe.
• Features of Dumhal Performance:
• The dancers form a group and perform in a circular motion, symbolizing unity and harmony within the tribe.
• A ceremonial banner is planted into the ground at the beginning of the performance, marking the sacredness of the event.
• The dance movements are synchronized, and the performers' attire plays a significant role in enhancing the visual appeal.
• Other Cultural Aspects of Jammu and Kashmir:
• Jammu and Kashmir are known for their rich cultural heritage, including various folk dances, music, and handicrafts.
• Other prominent dances from the region include the Rouf dance and Bhand Pather, which are also performed during festivals and celebrations.
• The region’s art and culture reflect the influence of various dynasties and traditions over centuries.
• Significance of Dumhal Dance:
• The Dumhal dance is not just a form of entertainment but also a way to preserve the tribal traditions and history of the Rauf people.
• It serves as a medium to pass down stories, rituals, and customs from one generation to the next.
• Efforts are being made to promote and preserve such folk dances through cultural festivals and initiatives by the government.

Important Points

• Option Analysis:
• Masks: Masks are not typically worn during Dumhal performances. They are more commonly associated with other cultural dances or theatrical performances.
• Veils: Veils are often used in some Indian classical or folk dances but are not a part of the Dumhal dance attire.
• Bells: Bells are used in dances like Kathak to enhance rhythmic sound, but they are not part of Dumhal dancers’ attire.
• Preservation Efforts:
• Various cultural preservation programs are being conducted to keep traditional folk dances like Dumhal alive.
• Government initiatives in Jammu and Kashmir include organizing state-sponsored festivals and promoting local art forms.

The correct answer is It indicates borrowing for current spending.

Key Points

• Revenue deficit refers to the shortfall in revenue receipts compared to revenue expenditure, indicating that the government is unable to meet its routine expenses from its revenue income.
• This deficit signifies that the government is borrowing funds not for productive purposes like capital creation but for current consumption, such as paying salaries, subsidies, and interest payments.
• Borrowing for current spending leads to a burden of interest payments, which increases fiscal pressure and reduces funds available for developmental activities.
• A high revenue deficit is a concern as it reflects poor financial management and indicates the need for structural reforms to reduce non-productive expenses.
• It may also result in inflationary pressures due to increased borrowing and reduced public sector savings, further weakening economic stability.

Additional Information

• Revenue Deficit vs. Fiscal Deficit:
• Revenue Deficit: The excess of revenue expenditure over revenue receipts. It highlights the government's inability to finance its day-to-day operations.
• Fiscal Deficit: The total borrowing requirement of the government, including revenue deficit and capital expenditure.
• Impact of Revenue Deficit:
• Reduces the government's ability to fund long-term projects and infrastructure development.
• Increases dependency on borrowing, raising public debt levels.
• Can lead to higher inflation if the government resorts to monetizing the deficit.
• Measures to Reduce Revenue Deficit:
• Enhancing tax compliance and broadening the tax base to increase revenue receipts.
• Reducing non-essential subsidies and controlling unproductive expenditures.
• Encouraging public-private partnerships for infrastructure projects to reduce fiscal burden.
• Global Perspective:
• Countries with high revenue deficits often experience economic instability and lower credit ratings, impacting foreign investments.
• Governments worldwide aim to maintain a balanced budget to promote fiscal discipline and economic growth.

Important Points

• Examples of Revenue Deficit:
• India's revenue deficit has been a significant concern in recent years, driven by subsidies and interest payments.
• During the COVID-19 pandemic, revenue deficits increased globally as governments spent heavily on healthcare and economic stimulus while revenue collection fell.
• Government Guidelines:
• The Fiscal Responsibility and Budget Management (FRBM) Act in India mandates reducing revenue deficits to ensure fiscal health.
• Periodic audits and reviews of government expenditures are recommended to identify and reduce non-productive spending.
• Significance of Balanced Budgets:
• A balanced budget ensures economic stability, promotes growth, and boosts investor confidence.
• It helps reduce dependency on borrowing, lowering interest obligations and public debt.

The correct answer is Create and maintain an environment for enterprises to compete.

Key Points

• The World Competitiveness Index, published annually by the International Institute for Management Development (IMD), evaluates how well countries manage their resources and competencies to enable enterprises to achieve sustainable growth, innovation, and overall competitiveness.
• It primarily assesses a country's ability to create and maintain an environment for enterprises to compete effectively on a global scale, fostering economic growth and stability.
• The index considers various aspects such as economic performance, government efficiency, business efficiency, and infrastructure to determine competitiveness.
• Countries are ranked based on how they support businesses in terms of policy frameworks, regulatory environments, innovation capabilities, and human capital development.
• The goal of the index is to provide insights that help governments and organizations enhance their competitiveness by addressing identified weaknesses and leveraging strengths.

Additional Information

• Key Features of the World Competitiveness Index:
• The index is published by the IMD World Competitiveness Center, based in Switzerland, and has been released annually since 1989.
• It evaluates 64 economies worldwide, providing a comprehensive view of global competitiveness.
• The rankings are based on 333 criteria categorized into four major factors: economic performance, government efficiency, business efficiency, and infrastructure.
• Data is collected from national statistics, international organizations, and surveys conducted among business executives.
• Example: In the 2023 rankings, Denmark, Switzerland, and Singapore were the top-performing countries, showcasing their strong business environments and innovative capabilities.
• Factors Influencing Competitiveness:
• Economic Performance: Includes macroeconomic indicators like GDP growth, trade performance, and employment rates.
• Government Efficiency: Evaluates policies, regulations, and institutional frameworks that support businesses.
• Business Efficiency: Measures productivity, labor market conditions, and financial systems.
• Infrastructure: Focuses on physical infrastructure, scientific innovation, education, and healthcare systems.
• Other Global Indices Related to Competitiveness:
• Global Innovation Index: Ranks countries based on their innovation capabilities.
• Ease of Doing Business Index: Published by the World Bank, evaluates how easy it is to start and operate a business in a country.
• Global Competitiveness Report: Released by the World Economic Forum, examines the set of institutions, policies, and factors that determine productivity levels.
• Importance of Competitiveness:
• Promotes economic growth by enabling businesses to thrive.
• Encourages innovation and development in industries.
• Improves living standards by creating more job opportunities and increasing income levels.
• Helps countries adapt to global challenges and compete effectively in international markets.

Important Points

• Incorrect Options:
• Option 1: Attracting tourists is not the primary focus of the World Competitiveness Index, though tourism may indirectly contribute to competitiveness.
• Option 3: Innovation in cultural heritage is significant but does not form the core criteria of this index.
• Option 4: Conservation of natural resources is an important aspect of sustainability but is not the primary focus of the index.
• Option 5: This is left blank and does not contribute to the question or answer.
• The World Competitiveness Index provides actionable insights for policymakers to improve their country's economic and business environments, ensuring long-term sustainability and growth.

The correct answer is Expansion of public sector.

Key Points

• The Industrial Policy Resolution of 1956 was a landmark policy document that aimed to shape India's industrial development post-independence. It emphasized the need for state intervention and planning in industrial growth.
• The resolution advocated for the expansion of the public sector, giving the government a pivotal role in owning and managing key industries such as iron and steel, heavy engineering, coal, and power.
• It divided industries into three categories: Schedule A (exclusively under state control), Schedule B (joint control by the state and private entities), and Schedule C (industries left open for private sector participation).
• The focus on the public sector was intended to achieve self-reliance, reduce foreign dependence, and ensure equitable distribution of wealth.
• The policy highlighted the importance of creating an industrial base for economic development while promoting socialistic patterns of society, which were key goals during that era.

Additional Information

• Industrial Policy Resolution of 1956:
• This policy was an extension of the earlier Industrial Policy of 1948, but with more emphasis on state ownership and control of key industries.
• It aimed to establish a mixed economy, combining the strengths of both the public and private sectors.
• The policy was guided by the vision of creating a socialist pattern of society, as envisioned by leaders like Jawaharlal Nehru.
• Categories of Industries:
• Schedule A: Industries exclusively reserved for the state, including defense, atomic energy, and railways.
• Schedule B: Industries where the state would set up new enterprises but private entities could also participate.
• Schedule C: Industries left open for private sector participation, subject to government regulation.
• Impact of the Resolution:
• It laid the foundation for India's industrial development and fostered the growth of industries like steel, energy, and infrastructure.
• It helped establish institutions like the Steel Authority of India (SAIL) and BHEL, which became critical for India's economic progress.
• The emphasis on the public sector was criticized later for inefficiencies and lack of competitiveness but played a major role in early industrialization.
• Other Key Features:
• The policy promoted equitable regional development by encouraging industries to set up in less-developed regions.
• It aimed to reduce income disparities through state control over key resources and industries.
• The policy also encouraged foreign collaboration and technical assistance to strengthen India's industrial base.

Important Points

• Historical Context:
• The resolution was adopted during the Second Five-Year Plan (1956–1961), which focused on rapid industrialization.
• This period marked a shift from agrarian to industrial economy, with emphasis on heavy industries.
• Challenges and Criticism:
• The over-reliance on the public sector led to inefficiencies, corruption, and lack of innovation in some industries.
• Critics argued that the policy restricted private sector participation, which could have accelerated industrial growth.
• Long-Term Impact:
• The resolution influenced subsequent industrial policies and laid the groundwork for India's mixed economy model.
• Many public sector enterprises established during this era continue to play a vital role in India's economy.

The correct answer is Birsa Munda.

Key Points

• Birsa Munda was a prominent tribal leader from the Munda tribe in Jharkhand, renowned for his leadership during the 'Ulgulan' (Great Rebellion), which occurred during the late 19th century.
• He led the rebellion against British colonial exploitation and the oppressive feudal system imposed on tribal lands. His movement aimed to protect the indigenous identity and rights of the tribal people.
• The rebellion focused on resisting the British policies of land revenue and the exploitation of tribal communities by landlords, moneylenders, and British officials.
• Birsa Munda is considered a hero and a symbol of tribal resistance, and his efforts brought significant attention to the plight of tribal communities under British rule.
• He passed away at the young age of 25 years in 1900, but his contributions remain a source of inspiration for tribal rights movements in India.

Additional Information

• Birsa Munda's Early Life:
• He was born on 15 November 1875 in the village of Ulihatu, Jharkhand, and grew up in a rural tribal community.
• Birsa Munda received education from missionary schools, where he learned about Christianity and Western ideologies but later focused on tribal traditions and practices.
• 'Ulgulan' Movement:
• The term 'Ulgulan' translates to 'The Great Tumult', signifying the uprising of tribal communities against British rule and exploitation.
• It was primarily aimed at the restoration of tribal rights to land and the abolition of forced labor and oppressive systems.
• Birsa Munda advocated for the establishment of a self-governed tribal state free from British interference.
• Legacy:
• Birsa Munda is regarded as a freedom fighter and a symbol of tribal unity and resilience.
• The Jharkhand state was officially created in 2000, fulfilling the demand for tribal autonomy, which was inspired by leaders like Birsa Munda.
• His contributions are commemorated through Birsa Munda Airport in Ranchi and various institutions named after him.
• Other Tribal Leaders:
• Sidhu Murmu: Along with Kanhu Murmu, led the Santhal Rebellion (1855-1856) against British exploitation of the Santhal tribe's lands and resources.
• Kanhu Murmu: Co-leader of the Santhal Rebellion alongside Sidhu Murmu, focusing on tribal rights and resistance against British policies.
• Tilka Manjhi: The first tribal leader to revolt against British colonial rule in 1784, representing the tribal uprising in Bihar.

Important Points

• Impact on Tribal Movements:
• Birsa Munda's movement laid the foundation for several tribal rights movements in India, advocating for the preservation of indigenous culture, traditions, and land rights.
• The struggle highlighted the exploitation faced by tribal communities and remains an important part of India's freedom struggle history.
• Commemoration:
• Birsa Munda's birth anniversary, 15 November, is celebrated as Jharkhand Foundation Day, marking the creation of Jharkhand state.
• His life and achievements have been the subject of books, documentaries, and public memorials across India.

The correct answer is Led Agni-V project; began career at LRDE.

Key Points

• Dr. Tessy Thomas is popularly referred to as the 'Missile Woman of India' due to her significant contributions to India's missile development programs, particularly the Agni series of missiles.
• She played a pivotal role in leading the Agni-V project, one of India's most advanced intercontinental ballistic missiles (ICBMs) with a range of over 5,000 km. This missile added strategic deterrence capabilities to India's defense arsenal.
• Dr. Tessy began her career at the Electronics and Radar Development Establishment (LRDE), a lab of the Defense Research and Development Organization (DRDO), where she worked on the design and development of various radar systems.
• She has been associated with the DRDO for over three decades, specializing in missile guidance, control, and system integration. Her expertise and leadership have earned her national and international recognition.
• Dr. Tessy Thomas is the first woman to head a missile project in India, breaking stereotypes and inspiring women across the country to pursue careers in science and technology.

Additional Information

• Agni-V Missile:
• The Agni-V missile is an advanced nuclear-capable intercontinental ballistic missile developed by India under the Integrated Guided Missile Development Program (IGMDP).
• It has a range of 5,000 to 8,000 km, making it capable of striking targets far beyond India's immediate neighborhood, including regions in Asia, Europe, and Africa.
• The missile features a three-stage solid propulsion system and advanced technologies for precision targeting and survivability.
• The successful development of Agni-V enhances India's strategic deterrence capabilities and positions the country as a global power in missile technology.
• The project represents India's commitment to maintaining a credible minimum deterrent and strengthening its national security.
• Dr. Tessy Thomas's Early Life and Education:
• Dr. Tessy Thomas was born in April 1963 in Alappuzha, Kerala, India. She exhibited a keen interest in science and mathematics during her school years.
• She earned a B.Tech in Electrical Engineering from the Government Engineering College, Thrissur, and later pursued an M.Tech in Guided Missile Technology at the Institute of Armament Technology, Pune.
• Her academic excellence and passion for research led her to join the DRDO, where she contributed significantly to India's defense sector.
• Role in DRDO:
• Dr. Tessy Thomas joined the DRDO in 1988 and has been instrumental in the design, development, and deployment of strategic missile systems.
• She has worked on several projects, including the Agni-I, Agni-II, Agni-III, and Agni-IV missiles, before leading the Agni-V program.
• Her contributions have earned her numerous awards, including the DRDO Scientist of the Year Award and the prestigious Lal Bahadur Shastri National Award for Excellence in Public Administration.
• Notable Achievements:
• Dr. Tessy Thomas has been honored as a role model for women in STEM (Science, Technology, Engineering, and Mathematics) fields.
• She is widely recognized for her leadership and innovation in missile technology, contributing to India's self-reliance in defense systems.
• Her work has inspired many aspiring scientists and engineers, particularly women, to pursue careers in defense and technology.

Important Points

• LRDE: The Electronics and Radar Development Establishment specializes in the design and development of radar and electronic systems. It plays a crucial role in India's defense infrastructure.
• DRDO: The Defense Research and Development Organization is India's premier agency for research and development in military technology. It operates under the Ministry of Defense.
• Gender Representation: Dr. Tessy Thomas's leadership in a male-dominated field underscores the importance of increasing gender diversity in science and technology.
• Strategic Importance: The development of missile systems like Agni-V reinforces India's defense capabilities and diplomatic leverage in international relations.
• Recognition: Her contributions have elevated India's global standing in missile technology and defense innovation.

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

Key Points

• Assertion (A): One objective of liberalisation was to remove barriers to entry and growth of firms.
• The policy of liberalisation was introduced in 1991 in India to promote economic reforms and reduce bureaucratic constraints that restricted the growth of firms.
• Liberalisation aimed at reducing restrictions such as licensing requirements, quotas, and other regulations that acted as barriers to entry and expansion for businesses. Hence, the assertion is correct.
• Reason (R): The license raj restricted entrepreneurial freedom and market competition.
• License raj refers to the system of excessive regulations and requirements for businesses to obtain licenses and approvals for various operations, which prevailed in India until the 1991 reforms.
• This system hindered entrepreneurial activity, stifled competition, and discouraged innovation due to bureaucratic delays and corruption. Hence, the reason is correct.
• Explanation:
• The assertion and reason are interlinked, as liberalisation sought to address the negative effects of license raj by removing barriers and fostering a competitive environment.
• The reason (R) correctly explains the objective mentioned in the assertion (A) since the removal of the license raj directly aligns with promoting entry and growth of firms. Hence, option 1 is correct.

Additional Information

• Liberalisation:
• Liberalisation refers to the relaxation of government restrictions and regulations in economic policies to allow private sector participation and foreign investments.
• It played a crucial role in India's economic reforms initiated in 1991 under the leadership of Prime Minister P.V. Narasimha Rao and Finance Minister Dr. Manmohan Singh.
• The reforms included measures such as the abolition of industrial licensing, reduction in tariffs, and promotion of foreign direct investment (FDI).
• License Raj:
• The term "license raj" refers to the system of excessive regulatory controls and licensing requirements that governed businesses in India before the 1991 economic reforms.
• Businesses had to obtain multiple licenses and approvals, leading to bureaucratic inefficiencies and corruption.
• This system was replaced with liberalisation reforms to encourage free-market principles and entrepreneurial activity.
• Impact of Liberalisation:
• Liberalisation led to increased competition, innovation, and efficiency in the market.
• It attracted foreign investments, boosted economic growth, and improved consumer choices.
• However, it also posed challenges such as unequal distribution of wealth and increased dependence on foreign markets.
• Key Government Initiatives:
• Industrial Policy of 1991: Focused on reducing government intervention and promoting private sector participation.
• FDI Policy: Allowed foreign investments in various sectors to boost economic growth and technology transfer.
• Disinvestment Policy: Encouraged privatisation of public sector enterprises to improve their efficiency and profitability.

Important Points

• Historical Context:
• India's economic policies post-independence were focused on self-reliance and protectionism, leading to the establishment of the license raj.
• The Balance of Payments (BoP) crisis in 1991 necessitated the adoption of liberalisation measures to stabilise the economy.
• Key Features of Liberalisation:
• Removal of industrial licensing and quotas.
• Reduction in import tariffs and promotion of exports.
• Encouragement of private and foreign investments in various sectors.
• Privatisation of public sector enterprises to improve efficiency.

Shortcut Trick

From the table, students in Section A with a height of 1.62 m = 14

Total students in Section A = 6 + 10 + 14 + 16 + 9 + 7 + 5 = 67

Required Ratio = 14 : 67

∴ The correct answer is 14:67.

Alternate Method

Given:

Number of students with 1.62 m height in Section A = 14

Formula Used:

Required Ratio = (Frequency of a specific category) ÷ (Total sum of all frequencies)

Calculations:

⇒ Total number of students in Section A = Sum of students across all given heights

⇒ Total students in Section A = 6 + 10 + 14 + 16 + 9 + 7 + 5

⇒ Total students in Section A = 67

⇒ Required ratio = (Students in Section A with height 1.62 m) : (Total students in Section A)

⇒ Required ratio = 14 : 67

∴ The correct answer is 14:67.

Additional Information

Ratio & Proportion

A ratio represents a quantitative relation between two or more values, indicating how many times one value is contained within the other. It is represented as a:b or a/b and is always simplified to its lowest terms.

Data Tabulation

Tabulation involves arranging numerical data systematically in a structured format with rows and columns, making it simpler to compare and analyze complex grouped frequencies.

Percentage vs Ratio

A direct way to convert any ratio a:b into a percentage format is by multiplying the fraction by 100, calculated as: (a ÷ b) × 100.

Shortcut Trick

Apply BODMAS directly to evaluate the expression inside the root.

⇒ √(5 × 196 ÷ 14 + 25 − 4)

⇒ √(5 × 14 + 21)

⇒ √(70 + 21)

⇒ √91

∴ The correct answer is √91.

Alternate Method

Given:

√(5 × 196 ÷ 14 + 25 − 4)

Concept Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculation:

⇒ Step 1: Perform division inside the bracket.

⇒ 196 ÷ 14 = 14

⇒ The expression becomes: √(5 × 14 + 25 − 4)

⇒ Step 2: Perform multiplication.

⇒ 5 × 14 = 70

⇒ The expression becomes: √(70 + 25 − 4)

⇒ Step 3: Perform addition and subtraction from left to right.

⇒ 70 + 25 = 95

⇒ 95 − 4 = 91

⇒ The final value is √91

∴ The correct answer is √91.

Additional Information

BODMAS Rule

It sets the order of mathematical operations: Brackets, Orders (roots/powers), Division, Multiplication, Addition, and Subtraction. Division is always calculated before multiplication.

Perfect Squares

Memorizing squares up to 30 helps in faster calculation. Recognizing that 196 is 14² allows immediate division without using rough work.

Simplification Strategy

When solving complex terms under a square root, always resolve the entire expression down to a single number before determining the square root.

Shortcut Trick

Simple interest for 2 years = Amount in 4 years − Amount in 2 years = ₹7,200 − ₹6,600 = ₹600

Simple interest for 1 year = ₹600 ÷ 2 = ₹300

Principal = Amount in 2 years − Interest for 2 years = ₹6,600 − ₹600 = ₹6,000

Rate = (Interest for 1 year ÷ Principal) × 100 = (300 ÷ 6000) × 100 = 5%

∴ The correct answer is 5%.

Alternate Method

Given:

Amount after 2 years (A₁) = ₹6,600

Amount after 4 years (A₂) = ₹7,200

Time gap = 4 − 2 = 2 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Amount = P + SI

Year 0 P = ?

Year 2 ₹6600

Year 4 ₹7200

• ₹600 (SI)

• ₹600 (SI)

Calculations:

⇒ Simple Interest for 2 years = ₹7,200 − ₹6,600 = ₹600

⇒ Simple Interest for 1 year = ₹600 ÷ 2 = ₹300

⇒ Amount after 2 years = Principal + SI for 2 years

⇒ ₹6,600 = Principal + ₹600

⇒ Principal (P) = ₹6,600 − ₹600 = ₹6,000

⇒ Rate of Interest (R) = (SI for 1 year × 100) ÷ P

⇒ R = (300 × 100) ÷ 6000

⇒ R = 30000 ÷ 6000 = 5

⇒ Rate of Interest = 5%

∴ The correct answer is 5%.

Additional Information

Simple Interest Constant Property

In simple interest, the interest earned every year remains identical because it is calculated entirely on the original principal sum over the whole duration.

Simple Interest Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate of Interest, and T is Time period.

Amount Calculation

The final amount is the total of the principal and accumulated interest. A = P + SI = P × (1 + (R × T) ÷ 100).

Shortcut Trick

Speed ratio = 20 : 18 = 10 : 9

Time ratio = 9 : 10 (since Distance is constant)

Difference in time ratio = 1 unit = 8 minutes.

Normal time (9 units) = 9 × 8 = 72 minutes = 72 / 60 hours.

Distance = Normal Speed × Normal Time = 20 × (72 / 60) = 24 km.

∴ The correct answer is 24 km.

Alternate Method

Given:

Normal Speed (S₁) = 20 kmph

Reduced Speed (S₂) = 18 kmph

Difference in time (ΔT) = 8 minutes = 8 / 60 hours = 2 / 15 hours

Formula Used:

Distance (D) = (S₁ × S₂) / (S₁ − S₂) × ΔT

Start End

Distance (D) = ?

Speed: 20 kmph | Time: T

Speed: 18 kmph | Time: T + 8 min

Calculations:

⇒ Let the required distance be D km.

⇒ Time taken at 18 kmph = D / 18 hours

⇒ Time taken at 20 kmph = D / 20 hours

⇒ According to the question, the difference in time is 8 minutes.

⇒ D / 18 − D / 20 = 2 / 15

⇒ (10D − 9D) / 180 = 2 / 15

⇒ D / 180 = 2 / 15

⇒ D = (2 × 180) / 15

⇒ D = 360 / 15

⇒ D = 24 km

∴ The correct answer is 24 km.

Additional Information

Relative Speed

When two objects move in the same direction, Relative Speed = S₁ − S₂. When they move in opposite directions, Relative Speed = S₁ + S₂.

Average Speed

If equal distances are covered at speeds S₁ and S₂, Average Speed = (2 × S₁ × S₂) / (S₁ + S₂).

Distance Formula with Time Difference

If a person travels a distance at speed S₁ and S₂ with a time difference ΔT, the distance can be calculated directly using: Distance = (S₁ × S₂) / (S₁ − S₂) × ΔT.

Shortcut Trick

In a 30°-60°-90° right-angled triangle, the sides opposite to the angles are always in the ratio 1 : √3 : 2.

The altitude is opposite to the 30° angle, which corresponds to 1 unit = 75 m.

The string acts as the hypotenuse (opposite to 90°), which corresponds to 2 units.

Length of string = 2 × 75 m = 150 m.

∴ The correct answer is 150 m.

Alternate Method

30° 75 m String (L) P G K (Kite)

Given:

Altitude of the kite from the ground (Perpendicular, KG) = 75 m

Angle of elevation of the string (θ) = 30°

Let the length of the string (Hypotenuse, KP) be L.

Formula Used:

sin(θ) = Perpendicular ÷ Hypotenuse

Calculation:

⇒ sin(30°) = KG ÷ KP

⇒ 1/2 = 75 ÷ L

⇒ L = 75 × 2

⇒ L = 150 m

∴ The correct answer is 150 m.

Additional Information

Standard Trigonometric Ratios Trick

In any right-angled triangle with angles measuring 30°, 60°, and 90°, the lengths of the sides opposite to these respective angles are strictly in the ratio 1 : √3 : 2.

Angle of Elevation

The angle formed by the line of sight with the horizontal when an observer looks at an object situated above the horizontal line of sight.

Pythagoras Theorem

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

Shortcut Trick

Rise in water level = Total volume of submerged objects ÷ Base area of tank

⇒ Rise = [5 × (4/3) × π × 5³] ÷ [100 × 50]

⇒ Rise = (2500π / 3) ÷ 5000

⇒ Rise = π/6 cm

∴ The correct answer is π/6 cm.

Alternate Method

Given:

Length of the tank (L) = 100 cm

Breadth of the tank (B) = 50 cm

Initial height of water = 20 cm

Radius of each spherical stone (r) = 5 cm

Number of stones = 5

Formula Used:

Volume of a sphere = 4/3 × π × r³

Volume of a cuboid = L × B × h

Base Area = 100 × 50 = 5000 sq. cmRise = h20 cm5 spheres (r=5)

Calculations:

⇒ Volume of 1 spherical stone = 4/3 × π × (5)³

⇒ Volume of 1 stone = 500π / 3 cm³

⇒ Total volume of 5 stones = 5 × 500π / 3 = 2500π / 3 cm³

⇒ Let the rise in water level be h cm.

⇒ Volume of displaced water = Base Area of tank × Rise in height

⇒ Volume of displaced water = (100 × 50) × h = 5000h cm³

⇒ As the stones are completely submerged, Volume of displaced water = Total volume of stones

⇒ 5000h = 2500π / 3

⇒ h = 2500π ÷ (3 × 5000)

⇒ h = π/6 cm

∴ The correct answer is π/6 cm.

Additional Information

Archimedes' Principle

When an object is fully submerged in a fluid, the volume of the displaced fluid is exactly equal to the volume of the submerged object.

Volume and Surface Area of a Sphere

For a sphere of radius r: Volume = 4/3 × π × r³ | Surface Area = 4 × π × r²

Volume of a Cuboid

Volume = Length × Breadth × Height. In liquid displacement problems involving rectangular tanks, only the liquid's height changes, so Rise in Volume = Base Area × Rise in Height.

Shortcut Trick

Profit ratio = Ratio of investments × Ratio of time periods

Investment Ratio = 60,000 : 90,000 = 2 : 3

Time Ratio = 6 : 4 = 3 : 2

Profit Ratio = (2 × 3) : (3 × 2) = 6 : 6 = 1 : 1

∴ The correct answer is 1:1.

Alternate Method

Given:

Investment of partner X = Rs. 60,000

Investment of partner Y = Rs. 90,000

Time period of partner X = 6 months

Time period of partner Y = 4 months

Formula Used:

Profit Ratio = (Capital of X × Time of X) : (Capital of Y × Time of Y)

Partner X Cap: Rs. 60,000 Time: 6 months

Partner Y Cap: Rs. 90,000 Time: 4 months Profit Ratio = (60k × 6) : (90k × 4) = 1 : 1

Calculations:

⇒ Profit share of X = 60000 × 6 = 360000

⇒ Profit share of Y = 90000 × 4 = 360000

⇒ Profit Ratio = 360000 : 360000

⇒ Profit Ratio = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Working vs Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or a specific percentage of the total profit. A sleeping partner only invests capital. The remaining profit is then divided according to the ratio of their effective investments (Capital × Time).

Multiple Partners

If three partners invest C₁, C₂, and C₃ for respective time periods T₁, T₂, and T₃, then their profit sharing ratio is determined directly by C₁×T₁ : C₂×T₂ : C₃×T₃.

Capital Addition or Withdrawal

When a partner adds or withdraws capital during the year, their total equivalent investment is calculated by multiplying each distinct capital amount by the respective number of months it remained in the business, and then summing all segments.

Shortcut Trick

The sum of interior and exterior angles of any regular polygon is always 180°.

Exterior angle = (180° − 100°) ÷ 2 = 40°.

Number of sides (n) = 360° ÷ Exterior angle = 360° ÷ 40° = 9.

∴ The correct answer is 9.

Alternate Method

Given:

Difference between interior and exterior angles = 100°

Formula Used:

Interior Angle (I) + Exterior Angle (E) = 180°

Number of sides (n) = 360° ÷ E

Interior (I)

Exterior (E)

Vertex Regular Polygon Angle Property I + E = 180°

Calculations:

⇒ Let the interior angle be I and the exterior angle be E.

⇒ I − E = 100° (Given)

⇒ I + E = 180° (Linear pair)

⇒ Subtracting the first equation from the second equation:

⇒ (I + E) − (I − E) = 180° − 100°

⇒ 2E = 80°

⇒ E = 40°

⇒ The sum of all exterior angles of a polygon is 360°.

⇒ Number of sides (n) = 360° ÷ 40°

⇒ n = 9

∴ The correct answer is 9.

Additional Information

Sum of Interior Angles

For a convex polygon with n sides, the sum of all interior angles is given by (n − 2) × 180°.

Each Interior Angle

In a regular polygon with n sides, each interior angle measures [(n − 2) × 180°] ÷ n.

Number of Diagonals

The total number of diagonals in an n-sided polygon can be calculated using the formula [n × (n − 3)] ÷ 2.

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon is always 360°, regardless of its number of sides.

Shortcut Trick

To find the required percentage decrease after an increase of R%:

Decrease % = [R ÷ (100 + R)] × 100%

⇒ Decrease % = [30 ÷ (100 + 30)] × 100%

⇒ Decrease % = (30 ÷ 130) × 100% = 300 ÷ 13% ≈ 23.08%

∴ The correct answer is 23.08%.

Alternate Method

Given:

A value is initially increased by 30%.

Calculations:

Let the original value be 100.

⇒ Value after 30% increase = 100 + (30% of 100) = 130

100 130 +30% Increase ? % Decrease

To return to the original value, the new value (130) must be decreased back to 100.

⇒ Required decrease amount = 130 − 100 = 30

⇒ Percentage decrease required = (Decrease amount ÷ New base value) × 100

⇒ Percentage decrease required = (30 ÷ 130) × 100

⇒ Percentage decrease required = 300 ÷ 13

⇒ Percentage decrease required ≈ 23.08%

∴ The correct answer is 23.08%.

Additional Information

Base Value Concept

When calculating percentage change, the base value is always the initial value before the change. While reversing a percentage change, the new base is the updated value.

Reverse Percentage Formula

If value A is x% more than value B, then B is strictly less than A by a percentage of [x ÷ (100 + x)] × 100.

Successive Percentage Change

If an amount undergoes successive changes of a% and b%, the net percentage change is given by the formula: a + b + (a × b ÷ 100).

Shortcut Trick

The area of a square can be directly calculated using its diagonal (d) with the formula: Area = d² ÷ 2

Given the diagonal, d = √98 m

Area = (√98)² ÷ 2

Area = 98 ÷ 2 = 49

∴ The correct answer is 49 m².

Alternate Method

d = √98 m Side = a Side = a

Given:

Diagonal of the square field (d) = √98 m

Formula Used:

Diagonal of a square = √2 × a (where a is the side of the square)

Area of a square = a²

Calculations:

⇒ √2 × a = √98

⇒ a = √(98 ÷ 2)

⇒ a = √49 = 7 m

⇒ Area = a²

⇒ Area = 7 × 7 = 49 m²

∴ The correct answer is 49 m².

Additional Information

Perimeter of a Square

The total length of the boundary of a square is its perimeter. It is calculated using the formula P = 4 × a.

Area of a Rhombus Property

Since every square is a special type of rhombus with equal diagonals, its area can also be derived from the rhombus formula: Area = (d₁ × d₂) ÷ 2.

Incircle and Circumcircle

For a square of side 'a', the radius of its inscribed circle is a ÷ 2, and the radius of its circumscribed circle is a ÷ √2.

Shortcut Trick

Initial mixture ratio of Acid : Water = 3 : 2 (Total = 5 parts).

Since 5 parts = 50 L, we get 1 part = 10 L.

To make the new ratio 1 : 1, the amount of water must equal the amount of acid.

Acid remains constant at 3 parts, so water must increase from 2 parts to 3 parts.

Increase in water = 3 − 2 = 1 part = 10 L.

∴ The correct answer is 10 L.

Alternate Method

Initial Mixture (50 L)

Acid: 30 L

Water: 20 L

Final Mixture (Ratio 1:1)

Acid: 30 L

Water: 20 L

• 10 L Added

Given:

Total volume of initial mixture = 50 L

Initial ratio of Acid to Water = 3 : 2

Calculation:

⇒ Total ratio parts = 3 + 2 = 5 parts

⇒ 5 parts = 50 L

⇒ 1 part = 10 L

⇒ Initial volume of Acid = 3 × 10 = 30 L

⇒ Initial volume of Water = 2 × 10 = 20 L

Let the volume of water added be x liters.

⇒ New volume of Water = (20 + x) L

Since the new ratio of Acid to Water is 1 : 1, their volumes must be equal:

⇒ Acid volume = New Water volume

⇒ 30 = 20 + x

⇒ x = 30 − 20 = 10 L

∴ The correct answer is 10 L.

Additional Information

Constant Quantity Principle

When only one component (such as water) is added to a mixture, the total absolute amount of the other component (such as acid) remains strictly constant. Equating the constant component is the fastest way to solve.

Rule of Alligation

Used to find the specific ratio in which two ingredients or mixtures are combined to get a desired concentration. The formula is: (Quantity of Cheaper) / (Quantity of Dearer) = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper).

Successive Replacement

If from a vessel containing V liters of pure liquid, x liters are replaced by water 'n' times, the remaining pure liquid is calculated as: V × (1 − x / V)ⁿ.

Shortcut Trick

Let the initial cost price be CP₁ and the new cost price be CP₂.

The new selling price is increased by 25%, so SP₂ = 125% of (120% of CP₁) = 1.5 × CP₁.

Since the new profit is 30%, the new selling price also equals 130% of CP₂ = 1.3 × CP₂.

∴ 1.3 × CP₂ = 1.5 × CP₁ ⇒ CP₂ / CP₁ = 15 / 13.

Percentage increase in CP = (2 / 13) × 100% = 15.38%.

∴ The correct answer is 15.38%.

Alternate Method

Given:

Initial profit percentage = 20%

Increase in selling price = 25%

New profit percentage = 30%

CP₁ = 100SP₁ = 120SP₂ = 150CP₂ = 115.38SP₂ = 150+20%+25%+30% Profit+15.38%

Calculations:

⇒ Let the initial cost price (CP₁) be 100 units.

⇒ Initial selling price (SP₁) = 100 + 20% of 100 = 120 units.

⇒ New selling price (SP₂) increases by 25%, so SP₂ = 120 + (25 / 100) × 120 = 150 units.

⇒ The new profit is given as 30%, which means SP₂ is 130% of the new cost price (CP₂).

⇒ 1.30 × CP₂ = 150 units

⇒ CP₂ = 150 ÷ 1.30 = 1500 / 13 ≈ 115.38 units.

⇒ Increase in CP = (1500 / 13) − 100 = 200 / 13 units.

⇒ Percentage increase in CP = [(200 / 13) ÷ 100] × 100 = 200 / 13 % = 15.38%.

∴ The correct answer is 15.38%.

Additional Information

Profit and Loss Essentials

Profit is always calculated on the base Cost Price. The fundamental relation is: SP = CP × (1 + Profit% ÷ 100).

Successive Changes

When a value is changed multiple times sequentially, the net percentage change for two successive changes a% and b% is found using: (a + b + ab ÷ 100)%.

Shortcut Trick

Total expenditure of the group = Rs. 1200

Amount spent by the remaining 5 students = 1200 − 300 = Rs. 900

Average spending of the 5 students = 900 ÷ 5 = Rs. 180

∴ The correct answer is Rs. 180.

Alternate Method

Given:

Total spending of 6 students = Rs. 1200

Amount spent by 1 student = Rs. 300

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Total Spending of 6 Students = Rs. 1200

1 Student: Rs. 300

5 Students: Rs. 900 Average = 900 ÷ 5 = Rs. 180

Calculation:

⇒ Amount spent by the other 5 students = Total spending − Amount spent by 1 student

⇒ Amount spent by the other 5 students = 1200 − 300 = Rs. 900

⇒ Since the remaining 5 students spent equally, their average spending is the total spent divided by 5.

⇒ Average spending = 900 ÷ 5 = Rs. 180

∴ The correct answer is Rs. 180.

Additional Information

Basic Average Formula

The average (or arithmetic mean) is calculated by adding all values and dividing by the count: Average = Sum of observations ÷ Total number of observations.

Combined Average

If two sets of sizes n₁ and n₂ have averages A₁ and A₂ respectively, Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Change in Average

If a value is added or removed and the new average changes by a specific amount, the impact can be seen directly on the total sum: New value = Old average ± (Total numbers × Change in average).

Shortcut Trick

Let the Market Price (MP) be 100%.

Selling Price after 15% discount = 100% − 15% = 85%.

According to the question, 85% = Rs. 7650

100% (Market Price) = (7650 ÷ 85) × 100 = 90 × 100 = 9000

∴ The correct answer is Rs. 9000.

Alternate Method

MP = ?

SP = 7650

-15% Discount

MP × 85% = SP

Given:

Selling Price (SP) = Rs. 7650

Discount = 15%

Formula Used:

SP = MP × (100 − Discount%) ÷ 100

Calculations:

⇒ 7650 = MP × (100 − 15) ÷ 100

⇒ 7650 = MP × 85 ÷ 100

⇒ MP = (7650 × 100) ÷ 85

⇒ MP = 90 × 100

⇒ MP = 9000

∴ The correct answer is Rs. 9000.

Additional Information

Marked Price and Discount

Discount is always offered on the Marked Price (or List Price) of an article. The relationship is simple: Selling Price = Marked Price − Discount.

Successive Discounts

The single equivalent discount for two successive discounts of x% and y% is given by the formula (x + y − xy ÷ 100)%.

CP to MP Ratio

The direct relationship between Cost Price and Marked Price is: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Shortcut Trick

For a Men OR b Women completing a work in D days, the time taken by c Men AND d Women is given by:

⇒ Required Days = D ÷ (c/a + d/b)

⇒ Required Days = 170 ÷ (20/10 + 20/15)

⇒ Required Days = 170 ÷ (2 + 4/3) = 170 ÷ (10/3)

⇒ Required Days = 170 × 3 ÷ 10 = 51 days

∴ The correct answer is 51 days.

Alternate Method

Given:

Time taken by 10 Men = 170 days

Time taken by 15 Women = 170 days

Let total work be the LCM of the given days (170 and 170) = 170 units.

⇒ Efficiency of 10 Men = 1 unit/day

⇒ Efficiency of 1 Man = 1/10 units/day

⇒ Efficiency of 15 Women = 1 unit/day

⇒ Efficiency of 1 Woman = 1/15 units/day

⇒ Combined efficiency of 20 Men and 20 Women = (20 × 1/10) + (20 × 1/15)

⇒ Combined efficiency = 2 + 4/3 = 10/3 units/day

⇒ Required Days = Total Work ÷ Combined Efficiency

⇒ Required Days = 170 ÷ (10/3)

⇒ Required Days = 170 × 3 ÷ 10 = 51 days

∴ The correct answer is 51 days.

Additional Information

MDH Rule (Chain Rule)

Formula: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where M is Men, D is Days, H is Hours, and W is Work done.

Total Work using Efficiency

Total Work = Efficiency × Number of Days. The ratio of efficiencies is inversely proportional to the ratio of days taken to complete the work.

Work and Wages

Wages are always distributed in direct proportion to the amount of work done by each person, or in the ratio of their efficiencies if they work for the same number of days.

Shortcut Trick

Side of the square = Perimeter ÷ 4

⇒ Side = 100 ÷ 4 = 25 cm

Area of square = (Side)²

⇒ Area = 25 × 25 = 625 cm²

∴ The correct answer is 625 cm².

Alternate Method

Given:

Perimeter of the square = 100 cm

Formula Used:

Perimeter of a square = 4 × a (where a is the side length)

Area of a square = a²

Side (a) = 25 cm Area = 625 sq cm Perimeter = 100 cm

a = 25 cm

Calculations:

⇒ 4 × a = 100

⇒ a = 100 ÷ 4

⇒ a = 25 cm

⇒ Area = 25 × 25

⇒ Area = 625 cm²

∴ The correct answer is 625 cm².

Additional Information

Diagonal of a Square

The length of the diagonal (d) of a square is given by d = a√2, where a is the side length of the square.

Area using Diagonal

The area can also be calculated directly from the diagonal length using the formula: Area = d² ÷ 2.

Incircle and Circumcircle

The radius of a circle inscribed within a square is a ÷ 2, and the radius of the circumscribed circle passing through its vertices is a ÷ √2.

Shortcut Trick

To find the value of an included observation, use the direct deviation shortcut:

New Score = Old Average + (Total New Observations × Change in Average)

⇒ Score on 8th test = 72 + (8 × 1)

⇒ Score = 72 + 8 = 80

∴ The correct answer is 80 marks.

Alternate Method

Given:

⇒ Average score of 7 tests = 72

⇒ Average score after 8 tests = 73

Formula Used:

⇒ Total Score = Average × Number of tests

Deviation Method (See-Saw Balance) Net Deviation = 0 ⇒ 7 × (−1) + 1 × (x − 73) = 0 ⇒ x = 80

Avg = 73

7 Tests Avg: 72

8th Test Score: 80

−1

Deviation: +7

Calculations:

⇒ Total score of 7 tests = 7 × 72 = 504

⇒ Total score of 8 tests = 8 × 73 = 584

⇒ Score on the 8th test = Total score of 8 tests − Total score of 7 tests

⇒ Score on the 8th test = 584 − 504 = 80

∴ The correct answer is 80 marks.

Additional Information

1. Inclusion of a New Value

When a new value is added to a group: New Value = Old Average + [New Total Count × Increase in Average]. If the average decreases, use a negative sign for the change.

2. Exclusion of a Value

When a value is removed from a group: Excluded Value = Old Average − [New Total Count × Increase in Average]. This logic reverses the inclusion formula.

3. Combined Average

To find the average of two groups combined: A = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n represents the item count and A represents the average of each respective group.

Shortcut Trick

The purchase amount of Rs. 25,000 exceeds the Rs. 10,000 threshold, so a flat 30% discount is applied.

Amount after 30% discount = 70% of 25,000 = Rs. 17,500.

Since Rs. 17,500 is greater than Rs. 15,000, an extra Rs. 1,000 discount is triggered.

Final amount to be paid = 17,500 − 1,000 = Rs. 16,500.

∴ The correct answer is Rs. 16,500.

Alternate Method

Given:

Total Purchase Amount = Rs. 25,000

Condition 1: 30% discount if the total purchase is above Rs. 10,000

Condition 2: Additional Rs. 1,000 off if the total after the first discount is above Rs. 15,000

Purchase Rs. 25,000

• 30% Off

After 30% Rs. 17,500

• Rs. 1000

Final Pay Rs. 16,500

Calculations:

⇒ The initial purchase value is Rs. 25,000.

⇒ Since 25,000 > 10,000, Condition 1 is satisfied and the 30% discount is applied.

⇒ First Discount = 30% of 25000

⇒ Discounted Amount = 25000 × (70 ÷ 100)

⇒ Discounted Amount = Rs. 17,500

⇒ Next, we check Condition 2. The discounted amount is Rs. 17,500, which is strictly greater than Rs. 15,000.

⇒ Since Condition 2 is satisfied, the customer gets the additional discount of Rs. 1,000.

⇒ Final Amount Paid = 17500 − 1000

⇒ Final Amount Paid = Rs. 16,500

∴ The correct answer is Rs. 16,500.

Additional Information

Successive Discounts

When two discounts d₁% and d₂% are given one after another, the effective or equivalent discount is calculated using the formula: Effective Discount% = (d₁ + d₂ − d₁×d₂ ÷ 100)%.

Selling Price vs Marked Price

Discount is always calculated on the marked price of the items. The mathematical relationship is: Selling Price = Marked Price × (100 − Discount%) ÷ 100.

Conditional Flat Discounts

These are dynamic discounts mapped to specific spending thresholds. They encourage larger purchases by offering higher percentage deductions or flat absolute cashbacks when the invoice total crosses specific levels.

Shortcut Trick

Initial salt = 60% of 100 L = 60 L. When water is added, the salt quantity remains constant.

To reduce concentration to 40%, the new total volume must be = 60 × (100 ÷ 40) = 150 L.

Water added = 150 − 100 = 50 L. The mixture now has 90 L water and 60 L salt.

To increase concentration back to 50%, the amount of salt must equal the amount of water.

Since water is 90 L, salt must also become 90 L. Pure salt to add = 90 − 60 = 30 L.

∴ The correct answer is 50 L of water, then 30 L of salt.

Alternate Method

Given:

Initial solution = 100 L

Initial salt concentration = 60%

Concept Used:

When water is added, the amount of salt remains constant. When pure salt is added, the amount of water remains constant.

100 L Salt: 60 L Water: 40 L 60% Salt + 50 L W 150 L Salt: 60 L Water: 90 L 40% Salt + 30 L S 180 L Salt: 90 L Water: 90 L 50% Salt Step-by-step Mixture Process

Calculations:

⇒ Initial amount of salt = 60% of 100 = 60 L

⇒ Initial amount of water = 100 − 60 = 40 L

⇒ Let water added be w. The new salt concentration is 40%.

⇒ [60 ÷ (100 + w)] × 100 = 40

⇒ 6000 = 4000 + 40w

⇒ 40w = 2000 ⇒ w = 50 L

⇒ Now, total volume = 150 L (Salt = 60 L, Water = 90 L).

⇒ Let salt added be s. The final salt concentration is 50%.

⇒ [(60 + s) ÷ (150 + s)] × 100 = 50

⇒ 6000 + 100s = 7500 + 50s

⇒ 50s = 1500 ⇒ s = 30 L

∴ The correct answer is 50 L of water, then 30 L of salt.

Additional Information

Constant Component Rule in Mixtures

When a pure liquid (e.g., water) is added to a mixture, the quantity of the other component (e.g., salt) remains constant. Equating the constant component's quantity before and after helps find the new overall volume directly.

50% Concentration Rule

For a mixture of two components to have a 50% concentration of one component, both components must be present in exactly equal quantities (a 1:1 ratio).

Successive Replacement Formula

If from V litres of pure liquid, x litres are drawn and replaced by water n times, the remaining pure liquid = V × (1 − x ÷ V)ⁿ.

Shortcut Trick

Given sinA = 4/5, we know Perpendicular = 4 and Hypotenuse = 5.

Base = √(5² − 4²) = 3.

Since A is in the 2nd quadrant, tanA is negative, so tanA = −4/3.

Expression: (1 + tanA)(1 − tanA) = 1 − tan²A

⇒ 1 − (−4/3)² = 1 − 16/9 = −7/9.

∴ The correct answer is −7/9.

Alternate Method

Given:

sinA = 4/5

Angle A lies in the 2nd quadrant.

Formula Used:

sinA = Perpendicular / Hypotenuse

tanA = Perpendicular / Base

Pythagoras Theorem: (Hypotenuse)² = (Perpendicular)² + (Base)²

(a + b)(a − b) = a² − b²

A Base = 3 Perpendicular = 4 Hypotenuse = 5

Calculations:

⇒ Let Perpendicular (P) = 4 and Hypotenuse (H) = 5.

⇒ (Base)² = H² − P²

⇒ Base = √(5² − 4²) = √(25 − 16) = √9 = 3

⇒ tanA = P / Base = 4 / 3 (magnitude)

⇒ Since A lies in the 2nd quadrant, only sine and cosecant are positive. Thus, tangent is negative.

⇒ tanA = −4/3

⇒ We need to find the value of (1 + tanA)(1 − tanA).

⇒ Using algebraic identity: (1 + tanA)(1 − tanA) = 1 − tan²A

⇒ 1 − (−4/3)²

⇒ 1 − (16/9)

⇒ (9 − 16) / 9 = −7/9

∴ The correct answer is −7/9.

Additional Information

ASTC Rule (Quadrant Signs)

1st Quad: All positive | 2nd Quad: sin, cosec positive | 3rd Quad: tan, cot positive | 4th Quad: cos, sec positive.

Pythagorean Triplets

Sets of integers satisfying a² + b² = c², such as (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Algebraic Identities in Trigonometry

Using standard formulas like (a + b)(a − b) = a² − b² simplifies complex trigonometric expressions significantly.

Shortcut Trick

The standard conversion value of 1 radian is approximately 57.3°.

To convert an angle from radians to degrees, multiply the given value by 57.3°.

Angle in degrees = 0.5 × 57.3°

⇒ Angle = 28.65°

∴ The correct answer is 28.65°.

Alternate Method

Given:

Angle in radians = 0.5 rad

Formula Used:

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

Calculations:

⇒ Angle in degrees = 0.5 × (180° / π)

⇒ Angle in degrees = 90° / π

Substituting the approximate value of π ≈ 3.1416:

⇒ Angle in degrees = 90° / 3.1416

⇒ Angle in degrees ≈ 28.6478°

⇒ Rounding off to two decimal places, we get 28.65°.

∴ The correct answer is 28.65°.

Additional Information

Degrees to Radians Conversion

To convert an angle from degrees to radians, multiply it by (π / 180°). For example, 90° = 90 × (π / 180) = π/2 radians.

Definition of Radian

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

Common Angles

Key angular relationships frequently used: 180° = π radians, 360° = 2π radians, 60° = π/3 radians, and 45° = π/4 radians.

Shortcut Trick

When sin A = cos B, the sum of angles A and B is always 90°.

Given: A = x and B = (x + 20°)

⇒ x + (x + 20°) = 90°

⇒ 2x = 70°

⇒ x = 35°

∴ The correct answer is 35°.

Alternate Method

Given:

sin x = cos(x + 20°)

x

x + 20° A B C In right triangle: sin(C) = cos(A) = AB / AC

Formula Used:

cos θ = sin(90° − θ)

Calculations:

⇒ sin x = sin(90° − (x + 20°))

Comparing the angles on both sides:

⇒ x = 90° − x − 20°

⇒ x + x = 70°

⇒ 2x = 70°

⇒ x = 35°

∴ The correct answer is 35°.

Additional Information

Complementary Angles

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

Basic Trigonometric Identities

1. sin²θ + cos²θ = 1
   
2. sec²θ − tan²θ = 1
   
3. cosec²θ − cot²θ = 1

Specific Angle Values

The values at common angles are frequently tested: sin 30° = cos 60° = 1/2, sin 45° = cos 45° = 1/√2, sin 60° = cos 30° = √3/2.

Shortcut Trick The orthocenter is the point where all three altitudes of a triangle intersect. For an acute-angled triangle, the orthocenter lies inside the triangle. For a right-angled triangle, it lies exactly on the right-angled vertex. For an obtuse-angled triangle (one interior angle > 90°), the orthocenter always lies outside the triangle. ∴ The correct answer is Outside the triangle.

Alternate Method Given: A triangle having one interior angle greater than 90°.

Formula / Concept Used: ⇒ Orthocenter: The exact point of intersection of the three altitudes of a triangle. ⇒ Altitude: A perpendicular line segment dropped from a vertex to the opposite side (or its extension).

A B C (>90°)

Orthocenter Outside Triangle

Calculations: ⇒ Let ΔABC be an obtuse-angled triangle where the interior angle at vertex C is greater than 90°. ⇒ Due to the obtuse angle, the altitude drawn from vertex A must fall on the extended side BC, meeting it at a point entirely outside the bounded region. ⇒ Similarly, the altitude drawn from vertex B falls on the extended side AC, also meeting it at an external point. ⇒ When these two external altitudes are extended further downward, they intersect with the extended altitude from vertex C at a common point. ⇒ Because two of the altitudes lie completely outside the area of the triangle, their intersection point (the Orthocenter) must also lie completely outside the triangle. ∴ The correct answer is Outside the triangle.

Additional Information Orthocenter of an Acute Triangle For any acute-angled triangle (where all interior angles are < 90°), all three altitudes fall inside the boundaries, placing the orthocenter permanently inside the triangle. Orthocenter of a Right-Angled Triangle For a right-angled triangle, the two perpendicular legs forming the 90° angle naturally act as altitudes. Hence, the orthocenter rests exactly on the right-angled vertex. Circumcenter Shift Rule The circumcenter (intersection of perpendicular bisectors) follows a similar rule: Inside for acute triangles, at the midpoint of the hypotenuse for right triangles, and entirely outside for obtuse triangles. Centroid and Incenter Regardless of the triangle classification (acute, right, or obtuse), both the centroid (intersection of medians) and the incenter (intersection of angle bisectors) consistently remain safely inside the triangle.

Shortcut Trick

The radius, distance from the center, and half of the chord form a right-angled triangle.

We can use the standard Pythagorean triplet (3, 4, 5) scaled by 3 → (9, 12, 15).

Since the hypotenuse (radius) is 15 cm and the perpendicular (distance) is 9 cm, the base (half-chord) must be 12 cm.

Total length of the chord = 2 × 12 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given:

Radius (r) = 15 cm

Distance of chord from center (d) = 9 cm

Formula Used:

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

Using Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

O

A B

M 9 cm

15 cm x

Calculations:

⇒ Let the half-length of the chord be x.

⇒ r² = d² + x²

⇒ 15² = 9² + x²

⇒ 225 = 81 + x²

⇒ x² = 225 − 81

⇒ x² = 144

⇒ x = √144 = 12 cm

⇒ Full length of the chord = 2 × x

⇒ Length = 2 × 12 = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Perpendicular Bisector Theorem

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

Properties of Chords

Chords of equal length in a circle are equidistant from the center. Conversely, chords equidistant from the center are equal in length.

Diameter

The longest chord of a circle passes through the center and is known as the diameter (Length = 2 × Radius).

Shortcut Trick

The radius, tangent length, and distance from the center form a right-angled triangle.

We can apply the direct formula: r = √(d² − t²)

⇒ r = √((√125)² − 10²)

⇒ r = √(125 − 100)

⇒ r = √25 = 5 cm

∴ The correct answer is 5 cm.