SSC CGL 2025 (Held On: 20 Sept, 2025 Shift 2) (Previous Year Paper)

Vowel series: 

(Left) A, E, I, O , U (Right)

Clearly, 'A' is the 5th vowel from the right in the alphabet.

Hence, the correct answer is "Option 1".

Thus, option 3 is exactly the same as the one given.

Hence, the correct answer is "Option 3".

Statements:

I. People are preferring UPI payments over cash.

II. UPI is more convenient and widely accepted.

If UPI is more convenient and widely accepted, it naturally leads to people preferring it over cash.

Statement II provides a reason explaining Statement I.

Thus, II is the cause and I is the effect.

Hence, the correct answer is "Option 2".

Given sequence: f_hk_gh_fg_k

1. g,f,k,h → fghk / fghk / fghk

'fghk' is repeated.

2. g,k,h,f → fghk / kghh / fgfk
3. k,g,f,h → fkhk / gghf / fghk
4. k,g,h,f → fkhk / gghh / fgfk

Thus, 'fghk' is repeated in option 1.

Hence, the correct answer is "Option 1".

Given series: 1, 4, 16, 64, 324

The logic followed here is:

So, 324 is the wrong number in the series.

Hence, the correct answer is "Option 4".

BODMAS Rule:

Decoding the given information:

Given equation: 12 × 6 ÷ 3 - 2 + 5 = ?

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 12 + 6 - 3 ÷ 2 × 5 = ?

⇒ 12 + 6 - 1.5 × 5 = ?

⇒ 12 + 6 - 7.5 = ?

⇒ 18 - 7.5 = ?

⇒ 10.5 = ?

Hence, the correct answer is "Option 2".

BODMAS Rule:

Let's check each option one by one.

1. × and +

Given: 22 - 4 × 2 ÷ 2 + 6 = 20

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 22 - 4 + 2 ÷ 2 × 6 = 20

⇒ 22 - 4 + 1 × 6 = 20

⇒ 22 - 4 + 6 = 20

⇒ 28 - 4 = 20

⇒ 24 ≠ 20

⇒ LHS ≠ RHS

2. - and ÷

Given: 22 - 4 × 2 ÷ 2 + 6 = 20

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 22 ÷ 4 × 2 - 2 + 6 = 20

⇒ 5.5 × 2 - 2 + 6 = 20

⇒ 11 - 2 + 6 = 20

⇒ 17 - 2 = 20

⇒ 15 ≠ 20

⇒ LHS ≠ RHS

3. ÷ and -

Given: 22 - 4 × 2 ÷ 2 + 6 = 20

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 22 ÷ 4 × 2 - 2 + 6 = 20

⇒ 5.5 × 2 - 2 + 6 = 20

⇒ 11 - 2 + 6 = 20

⇒ 17 - 2 = 20

⇒ 15 ≠ 20

⇒ LHS ≠ RHS

4. + and -

Given: 22 - 4 × 2 ÷ 2 + 6 = 20

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 22 + 4 × 2 ÷ 2 - 6 = 20

⇒ 22 + 4 × 1 - 6 = 20

⇒ 22 + 4 - 6 = 20

⇒ 26 - 6 = 20

⇒ 20 = 20

⇒ LHS = RHS

Hence, the correct answer is "Option 4".

The pattern given is:

• Initial of the State/UT + digits in a particular pattern
• Example: MH10967 → MH stands for Maharashtra

Now we check the options:

1. MH20987 → MH = Maharashtra
2. UP19021 → UP = Uttar Pradesh
3. GJ10243 → GJ = Gujarat
4. RJ30391 → RJ = Rajasthan

Since UP is the correct abbreviation for Uttar Pradesh, the roll number that is likely from Uttar Pradesh is UP19021.

Hence, the correct answer is "Option 2".

The logic followed here is:

1. Apple → Fruit

2. Orange → Fruit

3. Litchi → Fruit

4. Marigold → Flower

Thus, all options except Marigold are fruits, while Marigold is a flower.

Hence, the correct answer is "Option 4".

Family Chart:

Given: Pointing to a woman, John said, "She is my mother's only daughter."

Clearly, the woman is siter of John.

Hence, the correct answer is "Option 2".

Family Chart:

Given: In a family of six members A, B, C, D, E and F:

• A and B are a married couple.
  
• C is the only son of A.
  
• D is the daughter of B.
  
• F is the mother of B.

Clearly, F is the grandmother of C.

Hence, the correct answer is "Option 4".

Given series: 0, 1, 1, 2, 3, 5, 8, 13, ?, ?

The logic followed here is:

Logic: This is the Fibonacci series.

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

13 + 21 = 34

So, 21, 34 comes next.

Hence, the correct answer is "Option 4".

The logic followed here is:

1. 289 → 17²
2. 347 → 347 is not a perfect square.
3. 361 → 192
4. 441 → 212

All options except 347 are perfect squares. 

So, 347 is the odd one out.

Hence, the correct answer is "Option 2".

Given: Eight persons Dhruv, Dipak, Khush, Sapna, Sudhir, Vaibhav, Vimal and Vivek on eight different floors of 8-storey building from bottom to top such as ground floor is numbered as 1st and top floor is numbered as 8th.

1. Khush lives on 3rd floor.  2. Vivek lives below Khush's floor.

3. Vaibhav lives on even numbered floor but not on 6th floor. 4. Vaibhav lives just above Vimal's floor.

5. Sudhir lives two floors above Dipak's floor. Sudhir does not live on topmost floor.

6. Dhruv lives on an even numbered floor but not adjacent to Sudhir's floor. 

This statement is violated in Case I, so it is eliminated.

7. Sapna's floor is not adjacent to Dipak's floor.

This statement is violated in Case II (B), so it is eliminated.

Thus, Khush lives two floors below Sapna's floor.

Hence, the correct answer is "Option 3".

Family Chart:

Decoding the Symbols:

Given: Z # M $ C & D & E

It means: M is sister of Z, C is husband of M, D is son of C, E is son of D.

Clearly, E is grandson of M.

Hence, the correct answer is "Option 1".

Statement: All oranges are fruits. No fruit is a vegetable. All vegetables are green.

The least possible Venn diagram for the given statement is:

Conclusions:

I. No orange is green. - False (No orange being green is a possibility, but it is not definite. So, the conclusion is false)

II. No orange is a vegetable. - True (All orange being fruit is not a vegetable as No fruit is a vegetable)

Hence, the correct answer is "Option 2".

Given word: Hollowness

Let's check each option:

1. Hole → Hollowness → Can be formed 2. whale → Hollowness → Cannot be formed as "a" is missing. 3. Hang → Hollowness → Cannot be formed as "a" and "g" are missing. 4. HUng → Hollowness → Cannot be formed as "u" and "g" are missing.

so, Hole can be formed from the letters of the given word.

Hence, the correct answer is "Option 1".

The position and positional values or English Alphabets are:

Given series: AB, BC, CD, DE, ?

The logic followed here is:

So, EF comes next.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

The logic followed here is:

Similarly,

So, SUGAR is written as RTFZQ.

Hence, the correct answer is "Option 1".

The logic followed here is:

1. Balance of Trade → Economic concept (difference between exports and imports of goods)

2. Balance of Payments → Economic concept (record of all international transactions)

3. Current Account Deficit → Economic concept (when imports + outflows exceed exports + inflows)

4. Food Security Act → Government legislation/law

All options except "Food Security Act" are economic/financial terms, while Food Security Act is a law/policy.

Hence, the correct answer is "Option 4".

Given: As a judge, you notice a lawyer influencing a witness. 

1. Ignore unless it repeats

Ignoring such behavior is inappropriate because influencing a witness directly affects the fairness of the trial and the integrity of the judicial process.

2. Hold the lawyer in contempt

This is the correct action. Influencing a witness is professional misconduct and obstruction of justice. A judge has the authority to hold the lawyer in contempt of court to maintain fairness and discipline in proceedings.

3. Adjourn the case permanently

Permanently adjourning the case is not a proper response. The issue should be addressed by disciplinary action against the lawyer, not by stopping the case entirely.

4. Request the witness to stay silent

Silencing the witness does not solve the problem. The correct step is to address the lawyer’s misconduct, not restrict the witness.

Hence, the correct answer is "Option 2".

Given equation: 8 × 3 + 4 = 28

BODMAS Table:

Let's check each option one by one.

1. + and ×, 3 and 4

Given: 8 × 3 + 4 = 28

After interchanging the signs and numbers we get;

⇒ 8 + 4 × 3 = 28

⇒ 8 + 12 = 28

⇒ 20 ≠ 28

⇒ LHS ≠ RHS

2. • and ×, 8 and 3

Given: 8 × 3 + 4 = 28

After interchanging the signs and numbers we get;

⇒ 3 + 8 × 4 = 28

⇒ 3 + 32 = 28

⇒ 35 ≠ 28

⇒ LHS ≠ RHS

3. • and ×, 8 and 4

Given: 8 × 3 + 4 = 28

After interchanging the signs and numbers we get;

⇒ 4 + 3 × 8 = 28

⇒ 4 + 24 = 28

⇒ 28 = 28

⇒ LHS = RHS

4. • and ×, 4 and 3

Given: 8 × 3 + 4 = 28

After interchanging the signs and numbers we get;

⇒ 8 + 4 × 3 = 28

⇒ 8 + 12 = 28

⇒ 20 ≠ 28

⇒ LHS ≠ RHS

Hence, the correct answer is "Option 3".

Given: 100, 81, 64, 49, ?, 25

The logic followed here is:

10² = 100

92 = 81

82 = 64

72 = 49

62 = 36

52 = 25

So, 36 will come at the place of question mark.

Hence, the correct answer is "Option 2".

Given:

Total members = 9

Members to be selected = 4

Formula used:

Combination = ^_nC_r

nCr = n! ÷ [r! × (n − r)!]

Calculation:

Number of sub-committees = 9C4

⇒ 9C4 = 9! ÷ (4! × 5!)

⇒ (9 × 8 × 7 × 6 × 5!) ÷ (4 × 3 × 2 × 1 × 5!)

⇒ (9 × 8 × 7 × 6) ÷ (4 × 3 × 2 × 1)

⇒ 3024 ÷ 24

⇒ 126

∴ Number of different sub-committees = 126.

The position and positional values or English Alphabets are:

The logic followed here is:

Similarly,

So, GBNJMZ is coded as HCOKNA.

Hence, the correct answer is "Option 1".

The correct answer is PM eVIDYA.

Key Points

• PM eVIDYA was launched by the Government of India in May 2020 under the Atmanirbhar Bharat Abhiyan to unify all efforts aimed at digital education.
• The initiative primarily focuses on improving learning outcomes and promoting digital literacy in schools across India, especially during the COVID-19 pandemic.
• It includes the launch of "One Nation, One Digital Platform" and "One Class, One Channel" for providing access to quality education.
• The scheme aims to cater to diverse learners by providing multilingual content through platforms like Diksha, a national digital infrastructure for school education.
• PM eVIDYA integrates various initiatives such as TV channels, e-content, QR-coded textbooks, podcasts, and radio programs, ensuring inclusive access to education for students across socioeconomic backgrounds.
• The program targets both students and teachers, offering resources for teacher training and professional development.

Additional Information

• Digital India Mission:
• Launched in 2015, the Digital India initiative aims to transform India into a digitally empowered society and knowledge economy.
• It focuses on providing digital infrastructure, delivering governance services electronically, and promoting digital literacy among citizens.
• Although it overlaps with PM eVIDYA in promoting digital literacy, it is broader in scope and includes initiatives like e-Governance, e-Kranti, and digital payments.
• Swachh Bharat Abhiyan:
• Launched on 2 October 2014, this campaign focuses on improving sanitation and eliminating open defecation in India.
• It does not directly address learning outcomes or digital literacy but contributes to better health and hygiene in schools through initiatives like building toilets.
• PM Gati Shakti National Master Plan:
• Launched in October 2021, this initiative aims to revolutionize infrastructure development and logistics efficiency in India.
• Its focus is on economic growth and connectivity, and it does not overlap with educational or digital literacy goals.
• Importance of PM eVIDYA:
• During the COVID-19 pandemic, PM eVIDYA played a critical role in ensuring continuity in education despite lockdowns and school closures.
• The initiative promotes equity by providing educational resources to students in remote areas through radio, TV, and online platforms.
• Teacher training programs under PM eVIDYA enhance the quality of education by equipping educators with the tools to deliver effective digital lessons.

Important Points

• The Diksha platform, under PM eVIDYA, offers over 80,000 digital content pieces in multiple languages and is accessible to students, teachers, and parents.
• SWAYAM Prabha channels provide quality educational content through 32 DTH TV channels, ensuring access to students without internet connectivity.
• The QR-coded textbooks introduced under PM eVIDYA allow students to access supplementary digital content using smartphones.
• The initiative also supports special education needs by offering content for students with disabilities through platforms like NISHTHA and Inclusive Education Portal.

The correct answer is Expert witnesses such as forensic analysts.

Key Points

• Expert witnesses are individuals with specialized knowledge, skills, or expertise in a particular field relevant to a legal case. They are formally recognized under the BSA (Bharatiya Sakshya Adhiniyam), 2023.
• The role of expert witnesses includes providing technical, scientific, or professional opinions to help the court understand complex evidence or facts.
• Fields of expertise commonly considered include forensics, medicine, ballistics, engineering, accounting, and digital forensics.
• The BSA, 2023 highlights the importance of expert opinions in cases involving scientific analysis, forensic evidence, and technical data.
• Unlike lay witnesses, expert witnesses are allowed to offer opinions based on their expertise rather than limiting their testimony to factual observations. This makes their input critical in modern legal proceedings.

Additional Information

• Types of Witnesses:
• Fact Witnesses: These witnesses testify about events or facts they have directly observed or experienced.
• Character Witnesses: These individuals provide testimony about the character, reputation, or behavior of a person involved in the case. However, they are not involved in providing technical or scientific opinions.
• Expert Witnesses: Recognized under the BSA, 2023, they provide specialized knowledge or analysis in areas like medical forensics, ballistics, handwriting analysis, and digital technology.
• Role of Expert Witnesses in the Legal System:
• They assist judges or juries in understanding scientific, technical, or complex evidence.
• For example, a forensic analyst may testify about DNA evidence in a criminal case or a medical expert may analyze injury patterns to ascertain the cause of harm.
• Expert witnesses also provide written reports summarizing their analysis and conclusions, which may be submitted as evidence in court.
• Case Examples:
• Nirbhaya Case (2012): Forensic experts played a crucial role in analyzing DNA samples and other evidence, which were pivotal in securing convictions.
• Arushi Talwar Case: Expert witnesses analyzed forensic evidence, including bloodstains and post-mortem reports, to provide detailed findings.
• Legal Provisions under BSA, 2023:
• The Act explicitly recognizes the contribution of expert witnesses in modern legal proceedings, ensuring that their opinions are admissible in court.
• Provisions ensure that expert testimony is subjected to scrutiny and cross-examination to validate its reliability and relevance.
• Experts must adhere to ethical standards and ensure that their opinions are based on verified data and established methodologies.

Important Points

• Key Terminology:
• Forensic Analysis: The application of scientific techniques to investigate crimes, including DNA testing, ballistics, and toxicology.
• Ballistics: The study of projectiles and firearms, often used to analyze bullet trajectories and identify weapons used in crimes.
• Digital Forensics: The process of recovering, analyzing, and interpreting electronic data, such as emails or deleted files, to support legal investigations.
• Cross-Examination: The process of questioning a witness by the opposing party to test the credibility or reliability of their testimony.
• Recent Updates:
• The inclusion of expert witnesses in the BSA, 2023 aligns with global practices, ensuring that Indian courts can effectively handle cases involving modern technology and science.
• Training programs for forensic experts and digital analysts are being emphasized to maintain high standards of evidence quality in legal proceedings.

The correct answer is Limestones.

Key Points

• The Vindhyan system of rocks is one of the largest rock systems in India, spanning across the central and northern regions of the country.
• The Vindhyan rocks are primarily known for their extensive deposits of limestone, which is a key raw material in industries like cement production and chemical manufacturing.
• These rocks are of sedimentary origin and were formed during the Proterozoic era, making them geologically significant.
• The limestone deposits found in the Vindhyan system are of high quality and are extensively used in construction, agriculture, and industrial applications.
• In addition to limestone, the Vindhyan system also contains minor deposits of minerals like sandstone and shale, but it is primarily recognized for its limestone production.

Additional Information

• Geographical Extent of Vindhyan System:
• The Vindhyan system spans across Madhya Pradesh, Uttar Pradesh, Rajasthan, and Bihar.
• The system forms the Vindhyan mountain range, which is a significant geological feature in central India.
• Economic Importance:
• The limestone extracted from the Vindhyan rocks is extensively used in the cement industry.
• It is also used as a flux in iron and steel production and as a soil conditioner in agriculture.
• Other Rocks in the Vindhyan System:
• In addition to limestone, the Vindhyan system contains sandstone, which is used in construction and building materials.
• Shale, another component of the Vindhyan rocks, is used in industries like ceramics and brick manufacturing.
• Historical Significance:
• The Vindhyan rocks are geologically ancient and represent one of the earliest formations of sedimentary rocks in India.
• The region is also known for its contribution to archaeological studies, as it contains fossils and evidence of early life forms.

Important Points

• Other Rock Systems in India:
• Aravalli System: Known for minerals like lead, zinc, and copper.
• Dharwar System: Known for iron ore, manganese, and gold deposits.
• Gondwana System: Famous for coal deposits in regions like Jharkhand and Odisha.
• Applications of Limestone:
• Widely used in the cement industry for producing cement and concrete.
• Used as a raw material in the chemical industry for producing lime and calcium-based compounds.
• Essential in agriculture as a soil conditioner to neutralize acidic soils.

The correct answer is Puthari.

Key Points

• Puthari is a traditional harvest festival celebrated by the Kodava community in the Coorg (Kodagu) region of Karnataka, India.
• The festival marks the harvest of paddy crops and is celebrated with great enthusiasm and reverence for nature and agriculture.
• The ceremonial paddy harvest is accompanied by chants of "Poli, Poli, Deva," which invoke blessings from Lord Iguthappa, the deity of harvest and rains.
• The festival is characterized by rituals, traditional dances, and the preparation of special dishes made from freshly harvested rice.
• Puthari fosters a sense of community bonding and gratitude towards the land and the gods for a bountiful harvest.

Additional Information

• Significance of Lord Iguthappa:
• Lord Iguthappa is regarded as the chief deity of the Kodava people and is associated with harvest and rains.
• The Igguthappa Temple in Kakkabe village, Coorg, is a prominent pilgrimage site for Kodavas and other devotees.
• Devotees seek blessings from Lord Iguthappa for agricultural prosperity and well-being during the festival.
• Rituals of Puthari:
• The celebration begins with the lighting of lamps in homes and community spaces.
• The head of the family performs a ceremonial cutting of paddy sheaves in the fields, marking the start of the harvest.
• Paddy is brought home and worshipped, symbolizing gratitude for the harvest.
• Traditional Activities:
• During Puthari, Kodavas engage in folk dances, songs, and cultural performances that reflect their rich heritage.
• Special dishes like "Thambuttu" and "Pandi Curry" are prepared and enjoyed during the festivities.
• Other Harvest Festivals in India:
• Onam: Celebrated in Kerala, Onam marks the harvest season and is associated with the legend of King Mahabali.
• Baishakhi: A harvest festival celebrated in Punjab, marking the wheat harvest.
• Makar Sankranti: Celebrated across India, it signifies the harvest and the transition of the sun into Capricorn (Makara Rashi).

Important Points

• Kodava Culture:
• Kodavas are known for their unique traditions, including their distinct attire, language (Kodava Takk), and martial heritage.
• Puthari is one of their most important festivals, reflecting their agrarian roots and spiritual connection to nature.
• Celebration in Modern Times:
• While the essence of Puthari remains rooted in tradition, modern celebrations also include community gatherings and cultural programs.
• The festival serves as an opportunity for younger generations to connect with their heritage and learn about traditional agricultural practices.

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

Key Points

• Assertion (A): The sculptures of the Nalanda School are three-dimensional and delicately ornamented.
• The Nalanda School of art is renowned for its exquisite sculptural art that displays delicate ornamentation and three-dimensional depth.
• The sculptures often depict Buddhist deities and figures, featuring intricate details like jewelry, drapery, and facial expressions, emphasizing their artistic finesse. Hence, the statement is correct.
• Reason (R): The Nalanda sculptural tradition was influenced by Sarnath, local Bihar styles, and central Indian art.
• The Nalanda School drew significant inspiration from the Sarnath School of Art, known for its simplicity, elegance, and spiritual themes.
• Local art traditions of Bihar and influences from central Indian sculptural techniques also contributed to its unique style, blending regional and external elements. Hence, the statement is correct.
• Relationship between A and R:
• The three-dimensional and delicately ornamented nature of Nalanda sculptures is directly linked to the influences from Sarnath, local Bihar, and central Indian traditions, which shaped its distinct artistic characteristics. Therefore, R is the correct explanation of A.

Additional Information

• Nalanda School of Art:
• The Nalanda School of Art flourished from the 5th to the 12th century CE and is closely associated with the Nalanda Mahavihara, an ancient Buddhist monastic university in Bihar, India.
• It is characterized by sculptures of Buddhist deities like Buddha, Avalokiteshvara, and Tara, often made from black stone, bronze, and stucco.
• The sculptures showcase a combination of realism and idealism, with fine detailing and lifelike expressions.
• The art style also reflects a strong spiritual essence, emphasizing the importance of meditation and enlightenment in Buddhist philosophy.
• Sarnath School of Art:
• Originating during the Gupta period, the Sarnath School influenced the Nalanda tradition with its focus on simplicity, grace, and refined workmanship.
• It emphasized spiritual expression over materialistic ornamentation, which is evident in the calm and serene faces of Buddha statues.
• The famous Sarnath Buddha statue, depicting Buddha in the Dharmachakra Mudra, is a key example of its artistic legacy.
• Central Indian Art Influences:
• Central Indian art, especially from regions like Mathura, contributed to the iconography and stylistic elements of Nalanda sculptures.
• Features like well-defined musculature, detailed drapery, and intricate floral motifs were integrated into the Nalanda tradition.
• Notable Examples of Nalanda Art:
• The bronze sculptures of Nalanda, discovered during excavations, are considered masterpieces of Indian art, showcasing advanced metallurgical skills.
• The black stone sculptures of Buddhist deities, such as Avalokiteshvara and Tara, highlight the intricate carvings and symbolic details of the Nalanda School.

Important Points

• The art of Nalanda represents a significant phase in Buddhist artistic heritage, blending regional and external influences to create a unique style.
• Excavations at Nalanda have unearthed numerous sculptures, manuscripts, and inscriptions that provide insights into its cultural and artistic achievements.
• The Nalanda School played a vital role in spreading Buddhist art and culture across South and Southeast Asia.

The correct answer is Fellow feeling.

Key Points

• Metta, often translated as "loving-kindness" or "fellow feeling," is one of the four Brahmaviharas (sublime states) in Buddhist teachings.
• It is the practice of cultivating an attitude of unconditional goodwill, care, and compassion toward all living beings without discrimination.
• The term "Metta" originates from the Pali language and represents one of the primary qualities of an enlightened being in Theravāda Buddhism.
• In practice, Metta is typically developed through a structured meditation technique called Metta Bhavana, where one begins by directing loving-kindness toward oneself and gradually extends it to others, including friends, strangers, and even adversaries.
• The concept emphasizes the importance of fellow feeling and empathy, encouraging individuals to overcome hatred, jealousy, and ill will.

Additional Information

• Brahmaviharas (Sublime States):
• The Brahmaviharas are four mental states that Buddhists are encouraged to cultivate: Metta (loving-kindness), Karuna (compassion), Mudita (sympathetic joy), and Upekkha (equanimity).
• These qualities are considered essential for achieving spiritual liberation and fostering harmonious relationships with others.
• Metta Bhavana Meditation:
• This is a structured meditation technique for developing loving-kindness. Practitioners systematically extend feelings of goodwill to themselves, loved ones, neutral individuals, and adversaries.
• The ultimate goal of this practice is to develop an attitude of universal benevolence.
• Metta in Buddhist Texts:
• The concept of Metta is extensively discussed in Buddhist scriptures, such as the Metta Sutta (Discourse on Loving-kindness), which provides guidance on cultivating this quality in daily life.
• In the Metta Sutta, the Buddha emphasizes the importance of radiating loving-kindness to all beings, just as a mother loves her only child.
• Benefits of Practicing Metta:
• Practicing Metta is believed to reduce stress, anger, and ill will, and promote inner peace and emotional resilience.
• It fosters healthy relationships by encouraging empathy and reducing conflicts.
• Metta Across Buddhist Traditions:
• While the concept of Metta is central to Theravāda Buddhism, it is also emphasized in Mahayana and Vajrayana traditions, often in slightly different forms.
• In Mahayana, it is closely linked to the practice of Bodhicitta (the aspiration to attain enlightenment for the benefit of all beings).

Important Points

• Difference from Mindfulness: While Metta emphasizes cultivating positive emotions toward others, mindfulness focuses on maintaining present-moment awareness without judgment.
• Universal Application: Metta is not limited to Buddhists; it is a universally applicable practice that can benefit individuals of all faiths and belief systems.
• Scientific Perspective: Research has shown that practicing loving-kindness meditation can increase positive emotions, improve mental health, and enhance social connections.
• Historical Context: The Buddha taught Metta as an antidote to anger and hatred, making it a central element of his teachings on ethical living and personal transformation.

The correct answer is 2.44 m.

Key Points

• The standard height of a football goal post is defined by the Laws of the Game, as set by the International Football Association Board (IFAB).
• The height of the goal post is exactly 2.44 meters (8 feet) from the ground to the underside of the crossbar.
• The width of the goal is 7.32 meters (24 feet), measured between the inside edges of the vertical goalposts.
• The dimensions of the goal post are standardized for all levels of professional football, as per FIFA regulations.
• The goalposts and crossbar must be made of suitable material, typically metal or other approved materials, and must have a consistent shape, often circular or elliptical.

Additional Information

• Laws of the Game (Football):
• The goal dimensions are outlined in Law 1 of the Laws of the Game, which governs the field of play.
• The goalpost’s height and width are consistent across all international matches, tournaments, and competitions to ensure uniformity.
• The area around the goal is marked by the goal area and the penalty area, which are also regulated by the Laws of the Game.
• Material and Safety Standards:
• Goalposts are required to be firmly anchored and must not pose any hazard to players.
• They are typically painted white, ensuring visibility during matches and in different lighting conditions.
• Historical Context:
• The first standardized dimensions for football goals were introduced in 1863, when the Football Association (FA) was formed in England.
• Prior to this, dimensions varied widely, leading to inconsistent play.
• Variations for Youth and Amateur Football:
• Youth and amateur leagues often use smaller goal dimensions to suit the age and skill level of players.
• For example, under-12 football matches may feature goalposts that are 1.83 meters (6 feet) high and 5.49 meters (18 feet) wide.

The correct answer is 1937.

Key Points

• The Federal Court of India was established under the provisions of the Government of India Act, 1935, which was a landmark legislation enacted by the British Parliament to provide a new constitutional framework for India.
• The Federal Court was formally inaugurated on 1 October 1937, making it the first apex court of India before the Supreme Court was established post-independence.
• The court had jurisdiction to resolve disputes between provinces and princely states, as well as to hear appeals against decisions of the High Courts in certain cases.
• It comprised a Chief Justice and not more than six judges. The first Chief Justice of the Federal Court was Sir Maurice Gwyer.
• The Federal Court functioned until 28 January 1950, when it was replaced by the Supreme Court of India under the Constitution of India.

Additional Information

• Government of India Act, 1935:
• The act is regarded as the most comprehensive legislation enacted by the British Parliament for governing India. It introduced significant constitutional reforms, including the establishment of the Federal Court.
• It provided for a federal structure, dyarchy at the provincial level, and direct elections for legislative bodies.
• Jurisdiction of the Federal Court:
• The Federal Court had original jurisdiction to settle disputes between the provinces and princely states.
• It also had appellate jurisdiction over decisions of the High Courts in cases involving the interpretation of the Government of India Act, 1935.
• However, decisions of the Federal Court could be appealed to the Judicial Committee of the Privy Council in London.
• Composition:
• The Federal Court consisted of the Chief Justice and a maximum of six other judges appointed by the British Crown.
• The judges were selected based on their legal expertise and experience in judicial service.
• Replacement by Supreme Court:
• With the enforcement of the Constitution of India on 26 January 1950, the Federal Court ceased to exist and was replaced by the Supreme Court of India.
• The Supreme Court inherited the jurisdiction and functions of the Federal Court, along with additional powers outlined in the Constitution.

Important Points

• Sir Maurice Gwyer, the first Chief Justice of the Federal Court, played a pivotal role in shaping the judicial legacy of the Federal Court and establishing its independence.
• The Federal Court served as an important precedent for the creation of the independent judiciary in India, highlighting the concept of separation of powers and the rule of law.
• The Federal Court's original jurisdiction was limited, as it primarily dealt with inter-governmental disputes and constitutional matters under the 1935 Act.
• Though the Federal Court was an important institution, its decisions were often subject to review by the Privy Council, reflecting the colonial nature of governance during that period.

The correct answer is The Lotus Mahal shows clear Persian(indo-islamic) architectural influence.

Key Points

• The Lotus Mahal in the Vijayanagara Empire is a fine example of Indo-Islamic architecture. It displays a fusion of Indian and Persian architectural styles, with its symmetrical arches and domes.
• This structure was likely used as a royal pavilion or council chamber, showcasing the empire's openness to incorporating diverse cultural elements into its architectural designs.
• The arched design and pyramidal roof of the Lotus Mahal are indicative of Persian influence, blending with traditional Indian architectural patterns.
• The building is part of the Royal Centre of the Vijayanagara Empire in Hampi and remains an iconic representation of the empire's grandeur.
• Its unique style demonstrates the cultural exchanges and syncretism that took place during the Vijayanagara period. Hence, statement 2 is correct.

Additional Information

• The Hazara Rama Temple:
• The Hazara Rama Temple, located in the Royal Centre of Hampi, was not a public temple. It was exclusively meant for the royal family.
• It is renowned for its intricate bas-relief carvings depicting scenes from the Ramayana. These carvings also illustrate the lifestyle of the Vijayanagara people.
• Hence, statement 1 is incorrect.
• The Mahanavami Dibba:
• The Mahanavami Dibba is a massive platform in the Royal Centre. It was used for royal ceremonies, processions, and celebrations during the Mahanavami festival.
• It does not have a stone superstructure on top. Instead, it is adorned with intricate carvings on its sides, depicting processions and festivities. Hence, statement 3 is incorrect.
• Sacred Centre of Vijayanagara:
• The sacred centre of the city lies south of the Tungabhadra River, where most of the temples and religious structures are located.
• This area includes important sites such as the Virupaksha Temple and the Vittala Temple. Hence, statement 4 is incorrect.
• Indo-Islamic Architecture in Vijayanagara:
• The Vijayanagara Empire is notable for its blend of Hindu and Islamic architectural styles. Structures like the Lotus Mahal and the Queen's Bath exhibit such influences.
• This syncretic style reflects the empire's interactions with Islamic kingdoms and traders.

Important Points

• Vijayanagara Empire:
• The Vijayanagara Empire (14th–17th century) was one of the most powerful empires in South India, known for its administrative efficiency, military prowess, and cultural achievements.
• Hampi, the capital of Vijayanagara, is a UNESCO World Heritage Site, showcasing a rich blend of architectural, cultural, and historical significance.
• The empire played a crucial role in preserving Hindu culture during a period of significant political and religious changes in India.
• Architectural Legacy:
• Vijayanagara architecture is characterized by massive stone structures, intricate carvings, and a blend of Dravidian and Indo-Islamic styles.
• Key structures include the Vittala Temple, Virupaksha Temple, Hazara Rama Temple, and the Lotus Mahal.
• The empire's emphasis on temple-building reinforced its role as a center of religion and culture.

The correct answer is 1.

Key Points

• A relay race team consists of 6 members in total, out of which 4 are running members. The additional members typically serve as substitutes or reserves for the team. Hence, statement 1 is correct.
• The baton exchange zone in relay races is 20 meters long, not 50 meters. This is a standard rule in athletics to ensure fair exchanges between runners. Hence, statement 2 is incorrect.
• The baton used in relay races must weigh between 50 to 60 grams, not exactly 100 grams. This weight is standardized to ensure ease of handling during the race. Hence, statement 3 is incorrect.
• The length of the baton is 28-30 cm, much longer than 10-12 cm. This ensures that the baton is visible and easy to pass during the exchange. Hence, statement 4 is incorrect.

Additional Information

• Baton Exchange Rules:
• The baton must be passed within the designated 20-meter exchange zone. Failure to do so results in disqualification.
• The runner receiving the baton cannot begin running outside of the acceleration zone, which is an additional 10 meters before the exchange zone.
• The baton must be held firmly and cannot be dropped during the race; otherwise, it may lead to disqualification unless the baton is picked up by the same runner.
• Relay Race Types:
• Common relay races include the 4x100 meters and 4x400 meters, where each runner completes one leg of the race before passing the baton.
• Mixed-gender relay races, such as the 4x400m mixed relay, have gained prominence in international competitions like the Olympics.
• Important Relay Race Competitions:
• Relay races are a key feature of major events like the Olympics, World Athletics Championships, and the Asian Games.
• Countries like Jamaica, USA, and Kenya are known for their dominance in relay events.
• Historical Context:
• The first recorded relay race took place in the early 20th century as part of athletic meets in the USA.
• Relay races were included in the Olympic Games for the first time during the 1908 London Olympics.

Important Points

• Technical Aspects of the Baton:
• The baton is typically made of lightweight materials like aluminum or plastic.
• It must not have any sharp edges or modifications that could harm the runners or give an unfair advantage.
• The diameter of the baton is 4 cm, ensuring a comfortable grip for athletes of different hand sizes.
• Relay Team Roles:
• The first runner, or the starter, needs a strong start and acceleration.
• The second and third runners handle the curve and straight portions of the track, requiring coordination and speed.
• The final runner, known as the anchor, is typically the fastest and most experienced team member.
• Famous Relay Records:
• The Jamaican men’s team holds the world record in the 4x100 meters relay with a time of 36.84 seconds, set during the 2012 London Olympics.
• The USA women’s team set the 4x100 meters relay record at 40.82 seconds during the 2012 London Olympics.

The correct answer is HADR + logistics via LEMOA.

Key Points

• Exercise Tiger Triumph: It is a biennial India-US military exercise launched in 2019 with a focus on Humanitarian Assistance and Disaster Relief (HADR).
• The 2025 edition of Exercise Tiger Triumph will expand its scope to include logistics cooperation under the Logistics Exchange Memorandum of Agreement (LEMOA).
• HADR: The exercise continues to emphasize disaster relief operations to enhance interoperability between the Indian and US armed forces for quick response during natural disasters.
• LEMOA: The logistics element in the 2025 exercise builds on the strategic agreement signed between India and the US in 2016, allowing mutual access to each other's bases for resupply, repair, and logistics support.
• The evolved focus reflects a shift toward integrating logistics and operational readiness alongside traditional HADR drills to strengthen military cooperation between the two nations.

Additional Information

• Logistics Exchange Memorandum of Agreement (LEMOA):
• LEMOA is one of the foundational agreements signed between India and the US in 2016 to facilitate logistics support, supplies, and services.
• It provides access to each other’s military facilities for refueling and replenishment, enhancing strategic and operational flexibility.
• Humanitarian Assistance and Disaster Relief (HADR):
• HADR operations involve providing aid and relief during natural disasters such as earthquakes, tsunamis, and cyclones.
• The India-US Tiger Triumph exercise began in 2019 to improve coordination and readiness in such operations.
• 2019 Edition of Tiger Triumph:
• The inaugural exercise was a tri-service exercise involving the Indian Army, Navy, and Air Force along with their US counterparts.
• It focused on amphibious drills and HADR scenarios to improve inter-operability.
• Strategic Importance:
• The exercise strengthens the India-US defense partnership, which is crucial in the Indo-Pacific region for addressing emerging security and humanitarian challenges.
• It aligns with India's focus on building regional capacity for disaster response and the US's strategy of fostering strong military alliances.

Important Points

• India-US Strategic Agreements:
• LEMOA (2016): Focuses on logistics support and interoperability.
• COMCASA (2018): Facilitates secure communication between militaries.
• BECA (2020): Enables sharing of geospatial intelligence for improved targeting and navigation.
• Indo-Pacific Relevance:
• The Indo-Pacific region faces challenges like natural disasters, piracy, and geopolitical tensions.
• Exercises like Tiger Triumph enhance India-US cooperation in ensuring regional stability and disaster management.

The correct answer is Known for experimental style and social themes.

Key Points

• 'Bonkers!' by Natasha Sharma is celebrated for its experimental literary style and its focus on social themes, making it a unique piece in contemporary children's literature.
• The book utilizes a playful and humorous narrative to explore complex social situations, presenting them in a manner that is accessible to children.
• The author’s innovative storytelling approach combines quirky characters, unusual settings, and relatable dilemmas to engage young readers and provoke thought.
• Through its narrative, 'Bonkers!' addresses themes of self-discovery, family dynamics, and individuality, promoting discussions on emotional intelligence and social awareness.
• Natasha Sharma has been recognized for her ability to balance humor with deeper messages, making her work resonate with both children and adults.

Additional Information

• About Natasha Sharma:
• Natasha Sharma is a renowned Indian author specializing in children's literature, known for her engaging and humorous writing style.
• She has written several award-winning books, including 'The Good Indian Child’s Guide to Eating Mangoes' and 'Squiggle Takes a Walk.'
• Her works often combine imaginative elements with real-life issues, encouraging children to think critically and creatively.
• Literary Style:
• Sharma’s books are characterized by their whimsical humor, creative wordplay, and memorable characters.
• She often uses unconventional storytelling techniques, such as integrating illustrations and interactive elements, to engage her readers.
• Children’s Literature in India:
• India has a rich tradition of children's literature, with authors like Ruskin Bond, Sudha Murty, and Anushka Ravishankar contributing significantly to the genre.
• Books like 'Bonkers!' highlight the growing trend of modern, relatable stories that address contemporary issues in a child-friendly manner.
• Social Themes in Children's Literature:
• Modern children’s books often tackle topics like diversity, bullying, environmental awareness, and emotional well-being.
• Authors like Natasha Sharma use humor and relatable scenarios to subtly introduce these themes, making them digestible for young minds.

Important Points

• Significance of 'Bonkers!':
• The book has been praised for its ability to resonate with young readers while also entertaining adults who explore its deeper messages.
• It stands as a testament to the evolution of children's literature in India, moving beyond traditional tales to address modern issues and emotions.
• Other Works by Natasha Sharma:
• 'Squiggle Takes a Walk': A humorous take on punctuation and grammar.
• 'Rooster Raga': A delightful story about individuality and finding one’s voice.
• 'The Good Indian Child’s Guide to Eating Mangoes': A witty exploration of cultural practices.

The correct answer is Mansukh Mandaviya.

Key Points

• Mansukh Mandaviya, the Union Minister of Health and Family Welfare, officially announced that the Khelo India University Games 2025 will be held in Rajasthan this November.
• The announcement showcases the government's commitment to promoting sports culture and providing a platform for young talent to showcase their skills.
• The Khelo India University Games is a part of the Khelo India initiative, which aims to nurture grassroots sports talent and provide opportunities for young athletes.
• The decision to host the event in Rajasthan is significant as it highlights the state's growing role in sports development and infrastructure.
• The event will witness the participation of university athletes from across the country, promoting unity and sportsmanship.

Additional Information

• Khelo India Initiative:
• The Khelo India Programme was launched by the Government of India in 2018 to revive the sports culture in the country.
• It aims to promote mass participation in sports and identify and nurture sports talent at the grassroots level.
• Under the initiative, athletes are provided with financial assistance of Rs. 5 lakhs per annum for training and development.
• Previous Khelo India University Games:
• The inaugural edition of the Khelo India University Games was held in Bhubaneswar, Odisha, in 2020.
• The second edition was hosted by Bengaluru, Karnataka, in 2022.
• The games aim to bring together university athletes from all over the country and showcase their sporting prowess.
• Rajasthan's Role in Sports:
• Rajasthan has been actively developing its sports infrastructure, including stadiums and training facilities, to promote sports culture in the state.
• The state has hosted several national and international sporting events, contributing to its reputation as an emerging sports hub in India.
• Significance of Khelo India University Games:
• The games provide an important platform for young athletes to gain exposure and experience at a competitive level.
• They contribute to the identification of talent for national and international competitions, including the Olympics and Commonwealth Games.
• The event also fosters unity and cultural exchange among participants from different parts of the country.

The correct answer is Only B is correct.

Key Points

• The first Asian Games were held in 1951 in New Delhi, India. The venue for the games was the National Stadium, Delhi, not Jawaharlal Nehru Stadium. Hence, statement A is incorrect.
• A total of 11 nations participated in the first Asian Games, not 12. These nations were Afghanistan, Burma (now Myanmar), Ceylon (now Sri Lanka), India, Indonesia, Iran, Japan, Nepal, the Philippines, Singapore, and Thailand. Hence, statement A is incorrect.
• Japan topped the medal tally in the 1951 Asian Games with the highest number of medals, while India secured the second position. This is in line with the information provided in statement B. Hence, statement B is correct.
• The Asian Games were held from 4th to 11th March 1951, marking the beginning of a significant sporting event for Asia.
• The first Asian Games were organized under the guidance of the Asian Games Federation, with the Indian Olympic Association playing a crucial role in their execution.

Additional Information

• History of the Asian Games:
• The idea of the Asian Games was first proposed during the 1948 London Olympics by Indian sports administrator Guru Dutt Sondhi.
• The Asian Games Federation was formed in February 1949 to oversee the organization of the games.
• Participation in the 1951 Asian Games:
• A total of 489 athletes from 11 nations competed in the games.
• There were 6 sports disciplines in the first edition: athletics, aquatics (swimming, diving, and water polo), basketball, cycling, football, and weightlifting.
• Medal Tally (1951 Asian Games):
• Japan won the most medals with 24 gold, 21 silver, and 15 bronze.
• India secured the second position with 15 gold, 16 silver, and 20 bronze.
• Legacy of the 1951 Asian Games:
• The games set the foundation for multi-sport events in Asia and are now held every four years.
• The Asian Games are considered the second-largest multi-sport event after the Olympics.

Important Points

• National Stadium vs. Jawaharlal Nehru Stadium:
• The National Stadium, Delhi, was the venue for the 1951 Asian Games.
• The Jawaharlal Nehru Stadium, often confused as the venue, was built later in 1982 for the 1982 Asian Games.
• Evolution of the Asian Games:
• The games have grown significantly since 1951, now featuring over 40 sports disciplines and participation from over 40 nations.
• The Olympic Council of Asia (OCA) oversees the organization of the games today.

The correct answer is Union.

Key Points

• The Union List in Schedule VII of the Indian Constitution consists of subjects of national importance that require uniform policy across the country.
• It includes 97 items (originally 100), such as defence, foreign affairs, atomic energy, railways, banking, and currency, which are exclusively under the legislative jurisdiction of the Parliament of India.
• The subjects in the Union List are of such nature that they require centralized decision-making to ensure uniformity in administration and functioning across the nation. Hence, matters like defence and foreign affairs are included in this list.
• Article 246 of the Constitution specifies that Parliament has the exclusive power to make laws on any subject enumerated in the Union List.
• Examples of topics in the Union List include naval, military, and air forces, international treaties, and interstate trade and commerce.

Additional Information

• State List:
• The State List in Schedule VII contains subjects of local or state importance, such as police, public health, and agriculture. Only the state legislatures can make laws on these matters.
• It originally consisted of 66 items, but some subjects have been moved to other lists.
• Examples of State List topics include markets and fairs, burial grounds, and libraries.
• Concurrent List:
• The Concurrent List contains subjects of common interest to both the Union and States, such as education, marriage, and adoption.
• Both Parliament and state legislatures can legislate on these subjects, but in case of conflict, the Union law prevails.
• It originally consisted of 47 items.
• Residuary Powers:
• Any subject not mentioned in the Union, State, or Concurrent Lists comes under the residuary powers, which are vested exclusively in the Parliament.
• This provision ensures that unforeseen or new subjects like cybersecurity fall under the Union Government’s jurisdiction.
• Separation of Powers:
• The division of powers between the Union and State governments is a hallmark of India’s federal structure.
• This separation ensures a balance of governance, with the Union List focusing on national interests and the State List catering to regional needs.

Important Points

• Historical Context:
• The division of powers was heavily influenced by the Government of India Act, 1935, which served as the blueprint for Schedule VII.
• The framers of the Constitution intended to create a system ensuring both autonomy to states and strong central governance.
• Amendments to Schedule VII:
• The Union, State, and Concurrent Lists have undergone changes through Constitutional Amendments, reflecting evolving governance needs.
• For instance, subjects like education have been moved from the State List to the Concurrent List.
• Judicial Interpretation:
• The judiciary often interprets disputes related to the scope of legislative powers under different lists, ensuring the constitutional balance of power.
• Examples of Union List Usage:
• The Union List has been invoked for national-level initiatives like the Goods and Services Tax (GST) and international agreements.

The correct answer is To reduce structural weight while maintaining performance.

Key Points

• Generative-design software is extensively used in launch vehicle (LV) structural engineering to reduce the structural weight of components while ensuring that they meet the required performance, strength, and safety standards.
• It utilizes advanced algorithms and artificial intelligence to explore a wide range of design possibilities, aiming to identify the most efficient and lightweight structures.
• This software can simulate and analyze stress, load distribution, and other mechanical properties of a structure to ensure that it can withstand extreme conditions during launch and space travel.
• By minimizing weight, generative design contributes to improving the payload capacity of the launch vehicle, allowing for more equipment, satellites, or scientific instruments to be carried.
• Generative design also plays a critical role in optimizing material usage, leading to cost savings and environmentally sustainable practices in aerospace manufacturing.

Additional Information

• What is Generative Design?
• Generative design is an advanced design methodology that uses AI algorithms and computational models to create optimized structures based on specific parameters like weight, material strength, and performance.
• It is widely employed in industries such as aerospace, automotive, and architecture, where weight and efficiency are critical.
• Importance of Weight Reduction in Launch Vehicles:
• Reducing weight is crucial in aerospace engineering, as every extra kilogram adds to the cost of launching and reduces payload capacity.
• Lightweight structures improve fuel efficiency and enable the inclusion of additional scientific instruments or payloads.
• Applications of Generative Design in Aerospace:
• It is used for designing rocket fins, fuel tanks, brackets, and other critical components of launch vehicles.
• Generative design ensures that these components can handle high stress, extreme temperatures, and dynamic forces during launch and space travel.
• Example of Generative Design in Aerospace:
• NASA has used generative design techniques to develop lightweight and durable brackets for the Mars Rover, optimizing for strength and cost efficiency.
• SpaceX uses such technologies to create high-performance components for its Falcon and Starship programs.

Important Points

• Key Features of Generative Design in Structural Engineering:
• Ability to generate multiple design iterations based on input constraints like weight, material, and load capacity.
• Integration with simulation tools to analyze real-world performance under various conditions.
• Reduction in design time by automating the initial concept generation.
• Limitations and Challenges:
• Requires high computational power and advanced software tools.
• Designs may require specialized manufacturing techniques, such as 3D printing, which can increase initial costs.

The correct answer is Both 1 and 2.

Key Points

• Consolidated Fund of the State:
• The Consolidated Fund of the State is defined under Article 266(1) of the Indian Constitution.
• All revenues received by the State Government, loans raised, and money received in repayment of loans form part of this fund.
• No money can be withdrawn from this fund without the prior approval of the State Legislature. Hence, statement 1 is correct.
• Contingency Fund of the State:
• The Contingency Fund of the State is defined under Article 267(2) of the Constitution.
• It is a fund used to meet unforeseen expenditures, and it is placed at the disposal of the Governor.
• Expenditure from this fund is made in accordance with the rules prescribed by the State Legislature. Hence, statement 2 is correct.

Additional Information

• Public Account of the State:
• Defined under Article 266(2), the Public Account includes transactions where the government acts as a banker, such as provident funds, small savings collections, and deposits.
• Money from the Public Account does not require legislative approval for withdrawal.
• Structure of State Funds:
• The funds of the State are divided into three parts: the Consolidated Fund, the Contingency Fund, and the Public Account.
• Each fund serves a specific purpose and operates under distinct constitutional provisions.
• Role of the Governor:
• The Governor plays a pivotal role in managing the Contingency Fund, ensuring that unforeseen expenditures are met promptly.
• However, the legislative assembly must approve replenishment of this fund.
• Legislative Oversight:
• Funds such as the Consolidated Fund require legislative approval, ensuring transparency and accountability in financial management.
• The State Legislature has the authority to regulate the operation of these funds through appropriate laws and rules.

Important Points

• Key Articles:
• Article 266: Consolidated Fund and Public Account of the State.
• Article 267: Contingency Fund of the State.
• Examples of Usage:
• The Consolidated Fund is used for routine expenditures such as salaries, pensions, and development activities.
• The Contingency Fund is used during emergencies like natural disasters or unforeseen expenses.
• Key Features:
• Both funds ensure financial discipline and accountability in the State's governance.
• They are managed as per the constitutional framework, maintaining a balance between executive action and legislative oversight.

The correct answer is Sand displaces, reducing pushback.

Key Points

• Sand displaces under pressure: When you walk on sand, the particles beneath your feet shift and move apart. This displacement reduces the ability of the sand to provide a firm surface for pushback, making walking more difficult. Hence, this explains the correct answer.
• Concrete provides a rigid surface: Unlike sand, concrete is a solid, immovable surface. It does not shift or deform under pressure, allowing for greater pushback and making walking easier.
• Lack of resistance on sand: Sand does not offer as much resistance as concrete because it is composed of loose particles that can easily rearrange. This lack of resistance makes walking on sand more effort-intensive.
• Energy dissipation on sand: When you walk on sand, much of the energy you exert is absorbed by the shifting particles. This energy dissipation requires you to exert more force to propel yourself forward.
• Weight distribution: On sand, your weight causes localized displacement of particles, leading to instability and a need for constant adjustments in your gait. This does not occur on a solid surface like concrete.

Additional Information

• Coefficient of static friction:
• The coefficient of static friction is the measure of how much frictional force exists between two surfaces before motion begins.
• Sand typically has a lower coefficient of static friction compared to concrete, meaning it provides less grip. This makes walking on sand less stable compared to concrete.
• Density of sand vs. concrete:
• Sand is less dense than concrete, but this has no direct impact on gravitational pull. Gravitational pull depends on the mass of the Earth and the individual walking, not the density of the surface.
• Magnetic properties of sand:
• Sand particles are not inherently magnetic under normal conditions. Hence, there is no opposing magnetic force affecting your ability to walk on sand.
• Energy efficiency on different surfaces:
• Walking on sand requires 2–2.5 times more energy compared to walking on a hard surface like concrete due to the loose nature of sand and its energy-absorbing properties.
• Practical examples:
• Walking along a beach is more tiring than walking on a sidewalk due to the reasons outlined above.
• Sports like beach volleyball are more challenging than their court-based counterparts due to the added effort required to move on sand.

Important Points

• Concrete pavements: Concrete pavements are designed to offer high stability, firmness, and resistance to pressure. This makes them easier to walk on compared to natural, loose surfaces like sand.
• Energy transfer and mechanics: The mechanics of walking involve transferring energy to the surface for pushback. On sand, this energy is absorbed and dissipated due to particle displacement, while on concrete, it is efficiently transferred back for forward motion.
• Biomechanics of walking: Walking on sand requires adjustments in stride length and gait to maintain balance, leading to increased muscular effort and fatigue.

The correct answer is Khasi.

Key Points

• Ka Shad Suk Mynsiem is a traditional dance of the Khasi tribe, a major indigenous community of Meghalaya in Northeast India.
• This dance is performed as a thanksgiving ceremony to the divine power for blessings received during the past year.
• The term "Ka Shad Suk Mynsiem" translates to "Dance of the Happy Heart" in the Khasi language, symbolizing joy, gratitude, and harmony.
• The dance is usually performed during the spring season, marking the beginning of the sowing season and celebrating the bounties of nature.
• The event is characterized by traditional attire: men wear dhotis, sleeveless coats, and turbans, while women wear elegant silk dresses with gold and silver ornaments, reflecting their cultural heritage.
• The dance is accompanied by the rhythmic beats of the drums and pipes, creating a spiritual ambiance.

Additional Information

• Khasi Tribe:
• The Khasi tribe is one of the three major tribes of Meghalaya, along with the Jaintia and Garo tribes.
• The Khasi people are known for their matrilineal society, where lineage and inheritance are traced through the mother.
• Their traditional religion, called Niam Khasi, revolves around nature worship and ancestral reverence.
• Traditional Dances of Meghalaya:
• Other notable dance forms of Meghalaya include Wangala Dance of the Garo tribe and Behdienkhlam Dance of the Jaintia tribe.
• These dances are deeply rooted in agricultural practices, spiritual beliefs, and the socio-cultural life of the communities.
• Spring Festivals in India:
• Spring is celebrated across various parts of India with different festivals and dances, such as Bihu in Assam, Baisakhi in Punjab, and Holi in North India.
• These celebrations often include traditional dances, music, and rituals, symbolizing gratitude for nature’s bounties.
• Meghalaya's Cultural Significance:
• Known as the "Abode of Clouds," Meghalaya is celebrated for its rich cultural heritage, biodiversity, and scenic landscapes.
• The state hosts several festivals and events that showcase the unique traditions of its indigenous communities.

Important Points

• Cultural Preservation:
• Efforts have been made to preserve the Khasi traditions, including dances like Ka Shad Suk Mynsiem, through government programs, tourism, and local community initiatives.
• These dances are not just cultural expressions but also a way to pass on values, history, and identity to future generations.
• Tourism and Recognition:
• Ka Shad Suk Mynsiem and other traditional dances attract both domestic and international tourists, boosting Meghalaya’s cultural tourism sector.
• The dance has gained recognition for its aesthetic appeal and spiritual significance, often performed at cultural events and festivals.

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

Key Points

• Human Development Index (HDI): The HDI is a composite index used to measure a country's overall achievement in its social and economic dimensions. It combines three key dimensions: life expectancy at birth (health), mean years of schooling and expected years of schooling (education), and Gross National Income (GNI) per capita (income).
• Assertion (A) - The Human Development Index can increase even if income drops: This statement is true as the HDI is a composite measure. Even if the income component declines, improvements in life expectancy or education can compensate for the income drop, resulting in a net increase in HDI.
• Reason (R) - Improvements in life expectancy and education can offset income decline: This statement is also true. In the HDI calculation, all three components (health, education, and income) are weighted equally. A significant improvement in life expectancy or education can balance or even outweigh the adverse effects of a decline in income.
• R is the correct explanation of A: Since the HDI is calculated using a weighted approach, any improvement in one or more dimensions (like health or education) can counterbalance a drop in income, validating that Reason (R) explains Assertion (A).

Additional Information

• Components of HDI:
• Health: Measured by life expectancy at birth, it reflects the ability to lead a long and healthy life.
• Education: Calculated using two indicators – mean years of schooling (average number of years of education received by people aged 25 and older) and expected years of schooling (years of education a child entering school can expect).
• Income: Measured by Gross National Income (GNI) per capita (adjusted for purchasing power parity), reflecting the standard of living.
• Calculation of HDI: HDI is calculated as the geometric mean of normalized indices for each of the three dimensions (health, education, and income).
• Weightage of Dimensions: Each dimension contributes equally (one-third weightage) to the overall HDI value. This ensures that no single dimension dominates the calculation.
• Country Comparison: HDI enables the comparison of development levels between countries, focusing on human well-being rather than just economic growth.
• Global Trends: Countries like Norway, Switzerland, and Ireland often top the HDI rankings due to their strong performance in health, education, and income indicators.

Important Points

• Criticism of HDI: HDI does not capture inequality, poverty, or environmental sustainability. To address these gaps, the Inequality-adjusted HDI (IHDI) was introduced.
• HDI vs GDP: While GDP focuses solely on economic output, HDI provides a broader perspective, incorporating health and education dimensions.
• India’s HDI Rank: As per the latest Human Development Report, India ranks 132 among 191 countries, with significant improvements in life expectancy and education over the years.
• Example of Offset: In countries with declining income but improving healthcare and education (e.g., post-communist nations), HDI values may still rise.

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

Key Points

• Assertion (A): In macroeconomics, treating a single commodity as representative of all goods allows simplification of aggregate analysis.
• This statement is true because macroeconomics often uses a representative commodity to analyze aggregate trends, such as inflation, output, or consumption.
• Using a single representative good or aggregate measure simplifies the complexities of dealing with numerous individual goods.
• For example, the concept of an "aggregate price level" is used to study inflation, even though it combines the prices of multiple goods and services.
• Reason (R): This is because all individual goods always behave identically under inflationary trends.
• This statement is false because different goods behave differently under inflationary trends.
• For instance, luxury goods, essential goods, and non-essential goods may show varied price elasticity and responses during inflation.
• The demand and supply dynamics of individual goods are influenced by factors such as consumer preferences, production costs, and market conditions, which do not always align uniformly under inflation.
• Since the assertion is correct but the reason provided does not justify the assertion, the correct option is A is true, but R is false.

Additional Information

• Macroeconomics and Aggregation:
• Macroeconomics studies the economy as a whole, focusing on aggregate indicators like GDP, inflation, and unemployment.
• To simplify analysis, macroeconomics often uses aggregate variables like "aggregate demand," "aggregate supply," and "price levels."
• The concept of treating a single commodity as representative is a methodological tool to make the analysis of complex economies more manageable.
• Inflationary Trends:
• Inflation refers to the general increase in prices of goods and services over a period of time.
• Not all goods are affected equally by inflation; for example, prices of necessities like food and fuel may rise faster compared to luxury items.
• Inflation impacts consumer purchasing power, but the degree of impact varies across different sectors and goods.
• Representative Commodity:
• The use of a representative commodity or index (e.g., Consumer Price Index) helps study the overall behavior of the economy.
• Indices like CPI and GDP Deflator are examples of aggregate measures created by combining data from multiple goods and services.
• Micro vs. Macro Behavior:
• In microeconomics, the behavior of individual goods is studied, which often varies based on market structure, demand elasticity, and other factors.
• In macroeconomics, the focus shifts to aggregate behavior, which is a summary of the entire economy, rather than individual goods.

Important Points

• Consumer Price Index (CPI):
• CPI measures changes in the price level of a basket of consumer goods and services purchased by households.
• It is widely used as an indicator of inflation and cost of living.
• Price Elasticity of Demand:
• Price elasticity measures how sensitive the quantity demanded of a good is to a change in its price.
• Essential goods tend to have low elasticity, while luxury goods often have higher elasticity.
• Inflation Types:
• Demand-pull inflation occurs when demand outpaces supply.
• Cost-push inflation arises due to increased production costs, such as raw materials and wages.

The correct answer is Replace foreign imports with domestic production.

Key Points

• Import substitution policy was a strategy adopted by India during its early planning era, particularly under the Second Five-Year Plan (1956-1961), to promote self-reliance by reducing dependence on foreign imports.
• The primary objective was to develop domestic industries to produce goods that were previously imported, thereby fostering economic self-sufficiency.
• This policy involved the use of high tariffs and quotas on imports to protect nascent domestic industries from foreign competition.
• It was part of a larger strategy aimed at achieving industrialization and reducing India's reliance on foreign products, particularly in the wake of its colonial history.
• Import substitution was also aligned with the idea of creating a planned economy where the state played a central role in directing industrial and economic growth.
• Examples include the development of industries such as steel, chemicals, textiles, and heavy machinery, which were prioritized to replace imports.

Additional Information

• Key Features of Import Substitution Policy:
• The policy aimed at creating a self-reliant economy by reducing dependence on foreign goods and focusing on domestic production.
• It involved the use of protectionist measures such as import tariffs, quotas, and bans to restrict imports.
• Encouraged the growth of public sector enterprises in key industries like steel, heavy machinery, and energy production.
• Promoted the development of indigenous industries, particularly in the areas of manufacturing and capital goods.
• Focused on reducing the outflow of foreign exchange by minimizing imports of consumer goods.
• Outcomes of Import Substitution:
• India successfully developed a strong base in heavy industries and infrastructure during the early planning years.
• However, the policy led to reduced competition, resulting in low-quality goods and inefficiency in production.
• It created a license raj system, which involved complex bureaucratic controls and regulations, often stifling private sector growth.
• The policy contributed to technological stagnation and limited innovation in the industrial sector due to a lack of competition from foreign firms.
• Transition from Import Substitution:
• By the early 1990s, India shifted its focus from import substitution to export-oriented industrialization as part of economic liberalization and globalization policies.
• The 1991 economic reforms marked a significant shift in trade policy, reducing tariffs, removing import quotas, and promoting foreign direct investment (FDI).
• This transition was aimed at integrating India into the global economy, increasing competitiveness, and fostering economic growth.

Important Points

• Historical Context:
• India’s focus on import substitution was influenced by its colonial experience, during which the country was highly dependent on foreign goods and lacked a strong industrial base.
• The strategy was inspired by the Soviet model of planned economic development, emphasizing state control and self-reliance.
• Examples of Import Substitution in India:
• The establishment of Bhilai Steel Plant in collaboration with the Soviet Union in 1955 was a key milestone in building India’s industrial base.
• The indigenous production of goods like fertilizers, cement, and pharmaceuticals reduced dependency on imports significantly during the planned economy era.
• The Hindustan Aeronautics Limited (HAL) was set up to manufacture aircraft domestically, reducing reliance on foreign technology.
• Criticism of Import Substitution:
• Critics argue that the policy led to inefficiency, lack of innovation, and limited competitiveness in the global market.
• The restrictive trade policies also resulted in a lack of foreign exchange reserves and foreign investment.

The correct answer is Uses bulky inputs to produce heavy goods.

Key Points

• Heavy industries are characterized by their use of bulky raw materials and heavy machinery to produce large-scale and heavy goods such as steel, machinery, and vehicles.
• These industries often require large capital investments and involve significant industrial processes, including mining, metallurgy, and production of heavy engineering equipment.
• The raw materials used, such as iron ore, coal, and limestone, are bulky and require extensive transportation and handling.
• The products of heavy industries are typically used as inputs for other industries, such as construction, automobile manufacturing, and shipbuilding, making them foundational to industrial development.
• Examples of heavy industries include the steel industry, oil refining, shipbuilding, and machinery manufacturing.

Additional Information

• Characteristics of Heavy Industries:
• They produce capital goods rather than consumer goods, such as machinery, tools, and construction materials.
• They require a large workforce and create significant employment opportunities in related sectors.
• Often located near raw material sources or transportation hubs to reduce costs.
• They play a critical role in a country’s infrastructure development and economic growth.
• Examples of Heavy Industries:
• The steel industry, which produces steel used in construction and manufacturing.
• Automobile and shipbuilding industries, which create large vehicles and ships for transportation and trade.
• The mining industry, which extracts and processes raw materials like coal and iron ore.
• The chemical industry, which produces industrial chemicals, fertilizers, and other heavy goods.
• Difference between Heavy and Light Industries:
• Heavy industries use bulky raw materials and produce goods that are often used in further industrial processes.
• Light industries, on the other hand, use light raw materials and produce consumer goods such as textiles, cosmetics, and electronics.
• Global Importance:
• Countries like China, the USA, Germany, and India have invested heavily in heavy industries to boost their economic infrastructure and industrial growth.
• Heavy industries form the backbone of export economies, contributing significantly to a nation’s GDP.

Important Points

• The Industrial Revolution of the 18th and 19th centuries marked the rapid growth of heavy industries, particularly in Europe and North America.
• Many heavy industries are considered energy-intensive and contribute significantly to greenhouse gas emissions, making sustainability a key concern in the modern era.
• Policies such as Make in India and Atmanirbhar Bharat aim to promote the growth of heavy industries in India to reduce dependence on imports and boost self-reliance.
• Technological advancements, such as the use of automation and AI in heavy industries, are improving efficiency and reducing production costs.

The correct answer is Only 1 and 2 are correct.

Key Points

• PRITHvi Vigyan Mission is an initiative aimed at unifying ocean, weather, and geoscience observations under a single framework to improve India's scientific capabilities. Hence, statement 1 is correct.
• The mission plays a pivotal role in supporting India's early warning systems for natural disasters like tsunamis, cyclones, and extreme weather events, enhancing disaster preparedness and response. Hence, statement 2 is correct.
• The mission is implemented by the Ministry of Earth Sciences (MoES), not the Ministry of Health and Family Welfare. This makes statement 3 incorrect.
• The PRITHvi Vigyan Mission is designed to integrate data from various disciplines, including oceanography, meteorology, and geophysics, to boost India's environmental and disaster management systems.
• The focus is on using advanced technologies such as artificial intelligence, data analytics, and high-resolution modeling for better predictions and understanding of natural phenomena.

Additional Information

• Ministry of Earth Sciences (MoES):
• The PRITHvi Vigyan Mission is spearheaded by the Ministry of Earth Sciences, which oversees India's initiatives in oceanography, meteorology, and geosciences.
• The ministry is responsible for programs like the Indian Tsunami Early Warning System, which has successfully monitored and predicted tsunamis in the Indian Ocean region.
• Early Warning Systems:
• India has developed advanced early warning systems for disasters like cyclones and tsunamis, supported by missions like PRITHvi Vigyan.
• The system uses data from satellites, ocean buoys, and seismic sensors to provide real-time alerts and minimize the loss of life and property.
• Applications of Geosciences:
• Geoscience plays a vital role in disaster management, exploration of natural resources, and environmental preservation.
• PRITHvi Vigyan integrates geoscience with meteorological data to improve climate predictions and resource management.
• Technological Integration:
• The mission uses cutting-edge technology like AI and machine learning to process large datasets for accurate weather and ocean predictions.
• High-performance computing systems are also employed to run complex models for climate and disaster forecasting.

Important Points

• Key Objectives: The mission aims to enhance the understanding of the Indian Ocean region, improve disaster forecasting, and strengthen India's preparedness for environmental challenges.
• Significant Achievements: The Indian Tsunami Early Warning System and cyclone prediction models have significantly reduced the impact of natural disasters in India.
• Global Collaborations: The mission aligns with international efforts like the United Nations Decade of Ocean Science for Sustainable Development (2021–2030).

The correct answer is World Bank and IMF.

Key Points

• The 1991 Economic Reforms in India were a result of a severe balance of payments crisis. To address this, India approached the World Bank and International Monetary Fund (IMF) for financial assistance.
• The IMF provided a loan of $2.2 billion in exchange for implementing structural reforms, including liberalization, privatization, and globalization (LPG model).
• World Bank supported India’s structural adjustment programs by recommending policies to deregulate industries, reduce trade barriers, and encourage foreign investment.
• The reforms introduced in 1991 under the leadership of Prime Minister P.V. Narasimha Rao and Finance Minister Dr. Manmohan Singh included industrial deregulation, financial sector reforms, tax reforms, and trade liberalization.
• The role of the World Bank and IMF was instrumental in shaping India’s economic liberalization, enabling the country to stabilize its economy and initiate growth-oriented policies.

Additional Information

• World Bank:
• The World Bank is an international financial institution established in 1944 and headquartered in Washington, D.C.
• Its primary goals include reducing poverty and fostering sustainable development through financial and technical assistance to developing countries.
• The World Bank provides low-interest loans, interest-free credits, and grants to support projects in education, healthcare, infrastructure, agriculture, and environmental protection.
• India is one of the largest beneficiaries of the World Bank’s assistance, particularly for infrastructure and poverty alleviation programs.
• IMF:
• The International Monetary Fund (IMF) was established in 1944 and is headquartered in Washington, D.C.
• Its primary objective is to ensure global monetary cooperation, secure financial stability, facilitate international trade, promote high employment, and reduce poverty.
• The IMF provides short-term financial assistance to countries facing balance of payments issues to stabilize their economies.
• India has been a member of the IMF since its inception, and the institution played a pivotal role in assisting India during the 1991 economic crisis.
• 1991 Economic Reforms:
• The reforms introduced in 1991 are often referred to as the LPG reforms, representing Liberalization, Privatization, and Globalization.
• Liberalization: Reduced government control over industries, removed licensing requirements, and encouraged private sector participation.
• Privatization: Disinvestment in public sector enterprises and promotion of private ownership in key industries.
• Globalization: Reduced trade barriers, encouraged foreign direct investment (FDI), and integrated the Indian economy with global markets.
• Impact of Reforms:
• The reforms led to an increase in GDP growth rates, higher foreign investments, and a significant expansion of the services sector.
• India became one of the world’s fastest-growing economies, with a focus on sectors like IT, telecom, and pharmaceuticals.
• However, challenges like income inequality, unemployment, and agrarian distress remain critical issues to address.

Shortcut Trick

For √9801: The nearest base square is 100² = 10000. Since 9801 is slightly less and ends in 1, its square root is 99.

For √1681: The nearest base square is 40² = 1600. Since 1681 is slightly more and ends in 1, its square root is 41.

Expression value = 99 − 41 = 58.

∴ The correct answer is 58.

Alternate Method

Given:

The expression is √9801 − √1681

Calculation:

⇒ Let us find the square root of 9801 by prime factorization.

⇒ 9801 = 9 × 1089

⇒ 9801 = 9 × 9 × 121

⇒ 9801 = 9² × 11²

⇒ √9801 = 9 × 11 = 99

⇒ Now let us find the square root of 1681.

⇒ Since 40² = 1600 and 1681 is slightly greater than 1600 with a unit digit of 1, the number must be 41.

⇒ 41 × 41 = 1681 ⇒ √1681 = 41

⇒ Now, substitute the extracted values back into the expression:

⇒ √9801 − √1681 = 99 − 41

⇒ 99 − 41 = 58

∴ The correct answer is 58.

Additional Information

Unit Digits of Perfect Squares

A perfect square can only end in 0, 1, 4, 5, 6, or 9. If a perfect square ends in 1, its square root will always have a unit digit of either 1 or 9.

Estimating Square Roots

Find the nearest multiples of 10 whose squares bracket the given number. For example, 1681 lies just above 1600 (which is 40²), immediately indicating the root is close to 40.

Base 100 Square Trick

For a number written as (100 − a), its square is formed by writing (100 − 2a) followed by a² (in two digits). For 99 (where a=1), (100 − 2) = 98 and 1² = 01, giving exactly 9801.

Shortcut Trick

Let the initial investments be 6x, 7x, and 9x.

Equating the scaling factors of the new ratio for P and Q: (6x + 30000) ÷ 8 = (7x + 25000) ÷ 9

⇒ 54x + 270000 = 56x + 200000 ⇒ 2x = 70000 ⇒ x = 35000

Value of 1 part in new ratio = (6 × 35000 + 30000) ÷ 8 = 30000

R's added amount = New Investment − Old Investment = (11 × 30000) − (9 × 35000) = 15000

∴ The correct answer is ₹15,000.

Alternate Method

Given:

Initial ratio of investments of P, Q, and R = 6 : 7 : 9

Amount added by P = ₹30,000

Amount added by Q = ₹25,000

New ratio of investments of P, Q, and R = 8 : 9 : 11

Formula Used:

Ratio = Value of Part 1 ÷ Value of Part 2

Partner P Partner Q Partner R

Initial: 6x = 210,000

Initial: 7x = 245,000

Initial: 9x = 315,000

• 30,000

• 25,000

• 15,000

Final: 8 parts = 240,000

Final: 9 parts = 270,000

Final: 11 parts = 330,000

Amount added by R = 330,000 − 315,000 = 15,000

Calculations:

Let the initial investments of P, Q, and R be 6x, 7x, and 9x respectively.

After 1 year, new investments of P and Q are (6x + 30000) and (7x + 25000).

According to the given new ratio for P and Q:

⇒ (6x + 30000) ÷ (7x + 25000) = 8 ÷ 9

⇒ 9 × (6x + 30000) = 8 × (7x + 25000)

⇒ 54x + 270000 = 56x + 200000

⇒ 56x − 54x = 270000 − 200000

⇒ 2x = 70000

⇒ x = 35000

Initial investment of R:

⇒ 9x = 9 × 35000 = 315000

New investment of P:

⇒ 6x + 30000 = 6 × 35000 + 30000 = 240000

Since the new ratio for P is 8 parts, we can find the exact value of 1 ratio part:

⇒ 1 part = 240000 ÷ 8 = 30000

New investment of R:

⇒ 11 parts = 11 × 30000 = 330000

Amount added by R:

⇒ New investment − Initial investment

⇒ 330000 − 315000 = 15000

∴ The correct answer is ₹15,000.

Additional Information

Partnership Ratios

Profit in a partnership is distributed in proportion to the product of the capital invested and the time period of investment. (Profit Ratio = C₁ × T₁ : C₂ × T₂).

Changes in Capital

When a partner adds or withdraws capital midway through the term, the total investment weight is calculated by multiplying each active capital amount by the respective time duration it remained invested.

Ratio Equating Method

When fixed amounts are added to or subtracted from components of an original ratio to form a new ratio, setting up a cross-multiplication equation between any two quantities helps efficiently find the common scaling factor.

Shortcut Trick

Let the total salary of the employee be 100%.

Remaining after 20% charity = 100% − 20% = 80%.

Remaining after 40% investment = 80% × (100% − 40%) = 48%.

Travel portion = 48% × [5 ÷ (7 + 5)] = 20%.

Since 20% ≡ ₹6000, 100% = ₹6000 × 5 = ₹30,000.

∴ The correct answer is ₹30,000.

Alternate Method

Given:

Contribution to charity = 20%

Investment allocation = 40% of the remaining amount

Education to travel ratio = 7 : 5

Expense on travel = ₹6000

Formula Used:

Remaining Value = Initial Value × (1 − Percentage ÷ 100)

Salary 100%

−20%

Rem. 80%

−40%

Rest 48%

Edu (28%)

Travel (20%)

Travel = 20% = ₹6000 → Total (100%) = ₹30,000

Calculations:

⇒ Let total salary be S.

⇒ Remaining amount after charity = S × (100% − 20%) = 0.8S

⇒ Amount left after investment = 0.8S × (100% − 40%) = 0.48S

⇒ Share of travel in the rest amount = 5 ÷ (7 + 5) = 5/12

⇒ Expense on travel = 0.48S × (5 ÷ 12) = 0.2S

⇒ 0.2S = 6000

⇒ S = 6000 ÷ 0.2 = 30000

∴ The correct answer is ₹30,000.

Additional Information

Successive Percentage Change

When multiple percentage reductions occur successively on a remaining value, apply the multiplying factor method: Final Value = Initial × (1 − R₁/100) × (1 − R₂/100).

Base Change in Percentages

Always note whether a percentage is applied to the original total or the remaining amount. A 40% reduction on an 80% remaining balance means a net reduction of 32% (40% of 80%) from the original base.

Ratio Distribution

If an amount is distributed in the ratio a : b, the individual parts are calculated as (a ÷ (a + b)) and (b ÷ (a + b)) respectively of the total distributed amount.

Shortcut Trick

In Compound Interest, if a sum becomes n times in T years, it becomes n^m times in m × T years.

Here, the amount becomes 5 times in 10 years (n = 5, T = 10).

We need to find the time it takes to become 25 times, which is 5² times (m = 2).

Required time = m × T = 2 × 10 = 20 years.

∴ The correct answer is 20 years.

Alternate Method

P 5P 25P

0 Years 10 Years 20 Years

× 5 10 Years

× 5 10 Years

Compound Interest Timeline

Given:

Amount after 10 years = 5 × Initial Amount

Formula Used:

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

Where A is amount, P is principal, R is rate of interest, and T is time in years.

Calculations:

⇒ Let the initial amount be P.

⇒ After 10 years, Amount A = 5P

⇒ 5P = P(1 + R/100)¹⁰

⇒ 5 = (1 + R/100)¹⁰     --- (Equation 1)

⇒ We need to find the time T when Amount A = 25P

⇒ 25P = P(1 + R/100)^T

⇒ 25 = (1 + R/100)^T

⇒ 5² = (1 + R/100)^T

⇒ Substituting the value of 5 from Equation 1:

⇒ [(1 + R/100)¹⁰]² = (1 + R/100)^T

⇒ (1 + R/100)²⁰ = (1 + R/100)^T

⇒ By comparing the powers on both sides, T = 20 years.

∴ The correct answer is 20 years.

Additional Information

Compound Interest Multiplier Rule

If a given sum becomes n times in T years at a certain rate of compound interest, it will become n^m times in m × T years.

Simple Interest Equivalent

In simple interest, becoming 5 times means earning an interest of 4P in 10 years. Becoming 25 times means earning an interest of 24P, which would take (24 ÷ 4) × 10 = 60 years.

Rule of 72

It is a quick approximation formula to estimate the doubling time of an investment. Doubling time ≈ 72 ÷ R years, where R is the annual compound interest rate.

Shortcut Trick

Remaining licenses for sale = 800 − 50 = 750.

The offer 'Buy 9 get 1 free' implies 9 paid licenses out of every 10 distributed.

Total paid licenses = 750 × (9 ÷ 10) = 675.

Selling Price per license = ₹6000 × (1 − 0.15) = ₹5100.

Total SP = 675 × 5100 = ₹34,42,500. Total CP = ₹40,00,000.

Loss % = ((40,00,000 − 34,42,500) ÷ 40,00,000) × 100 = 13.9375% ≈ 13.94%.

∴ The correct answer is 13.94% loss.

Alternate Method

Given:

Total licenses = 800, Total CP = ₹4,000,000, Free to top beta testers = 50.

Remaining licenses = 750, Marked Price (MP) per license = ₹6000.

Discount = 15%, Promotional Offer = 1 free license for every 9 licenses bought.

Formula Used:

SP = MP × ((100 − Discount%) ÷ 100)

Loss = CP − SP

Loss % = (Loss ÷ CP) × 100

800 Licenses

-50 free

750 Remaining

9:1 ratio

675 Paid

CP: ₹4,000,000 SP: 675 × ₹5100 = ₹3,442,500 Loss = ₹557,500 (13.94%)

Calculations:

⇒ SP of each license after a 15% discount = 6000 × (85 ÷ 100) = ₹5100.

⇒ The offer '1 free for every 9 bought' means for every bundle of 10 licenses, only 9 are paid for.

⇒ Total bundles formed from 750 remaining licenses = 750 ÷ 10 = 75.

⇒ Number of paid licenses = 75 × 9 = 675.

⇒ Total Selling Price (SP) = 675 × 5100 = ₹3,442,500.

⇒ Total Loss = Total CP − Total SP = 4,000,000 − 3,442,500 = ₹557,500.

⇒ Loss % = (557,500 ÷ 4,000,000) × 100 = 13.9375%.

⇒ Rounding off to two decimal places, the loss is 13.94%.

∴ The correct answer is 13.94% loss.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price. Formula: SP = MP × (100 − D%) ÷ 100.

Buy X Get Y Free Concept

When a dealer offers Y items free on the purchase of X items, the total items given are (X + Y). The equivalent discount percentage is (Y ÷ (X + Y)) × 100.

Profit and Loss Percentage

Profit or Loss is always evaluated on the Cost Price (CP) unless specified otherwise. Loss % = (Loss ÷ CP) × 100.

Shortcut Trick

When the time taken to meet is constant, the ratio of distances covered is equal to the ratio of their speeds.

Distance ratio = Speed ratio = 5 : 3.

The difference in the distance ratio corresponds to the head start ⇒ 5 − 3 = 2 units.

Given head start = 24 km ⇒ 2 units = 24 km ⇒ 1 unit = 12 km.

Distance covered by the faster motorcyclist = 5 units = 5 × 12 = 60 km.

∴ The correct answer is 60 km.

Alternate Method

Start (Faster) Start (Slower) Overtake

Speed: 5x Speed: 3x

24 km Gap

Faster Motorcyclist Total Distance = ?

Given:

Ratio of speeds = 5 : 3

Speed of slower motorcyclist = 12 km/h

Head start distance = 24 km

Formula Used:

Relative Speed = Speed of faster − Speed of slower (moving in the same direction)

Time to Overtake = Head Start Distance ÷ Relative Speed

Distance = Speed × Time

Calculations:

⇒ Let the speeds be 5x and 3x respectively.

⇒ 3x = 12 km/h ⇒ x = 4 km/h.

⇒ Speed of faster motorcyclist = 5x = 5 × 4 = 20 km/h.

⇒ Relative speed = 20 − 12 = 8 km/h.

⇒ Time taken to overtake = 24 ÷ 8 = 3 hours.

⇒ Distance travelled by the faster motorcyclist = 20 × 3 = 60 km.

∴ The correct answer is 60 km.

Additional Information

Relative Speed

When two bodies travel in the same direction, their relative speed is the difference of their individual speeds (v₁ − v₂). When travelling in opposite directions, it is the sum of their speeds (v₁ + v₂).

Proportionality in Distance and Speed

When the time factor is constant for both objects, the distance travelled is directly proportional to their respective speeds (d₁ ÷ d₂ = s₁ ÷ s₂).

Overtaking Time

The exact time required to catch up to a moving target is calculated by dividing the initial gap distance by the relative speed of the chasing body.

Shortcut Trick

Time duration from 1st April to 1st September = 5 months = 5 ÷ 12 years.

Simple Interest = Principal × Rate × Time

SI = 7500 × 9% × (5 ÷ 12) = ₹281.25

Total Amount = Principal + SI = 7500 + 281.25 = ₹7781.25

∴ The correct answer is ₹7781.25.

Alternate Method

Given:

Principal (P) = ₹7500

Rate of Interest (R) = 9% per annum

Loan start date = 1 April 2025

Loan clear date = 1 September 2025

Formula Used:

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

Amount (A) = P + SI

1 Apr 2025 P = ₹7500

1 Sep 2025 Amount = ?

Time = 5 Months (5/12 Year)

Rate = 9% p.a.

SI = ₹281.25 → Amount = ₹7781.25

Calculations:

⇒ Total duration from 1 April 2025 to 1 September 2025 is exactly 5 full months.

⇒ Time (T) in years = 5 ÷ 12 years.

⇒ SI = (7500 × 9 × 5) ÷ (100 × 12)

⇒ SI = (75 × 45) ÷ 12

⇒ SI = 3375 ÷ 12

⇒ SI = ₹281.25

⇒ Total Amount = Principal + SI

⇒ Total Amount = 7500 + 281.25

⇒ Total Amount = ₹7781.25

∴ The correct answer is ₹7781.25.

Additional Information

Conversion of Time

When the time given is in months or days, it must be converted to years for simple interest formulas. Divide months by 12 and days by 365.

Counting Days

When calculating exact days for simple interest, it is standard practice to exclude either the day the money is borrowed or the day it is returned, but not both.

Simple Interest vs Compound Interest

For the first year (or unit time period), Simple Interest and Compound Interest are identical. From the second period onwards, Compound Interest is calculated on the accumulated interest as well as the principal.

Shortcut Trick

Whole numbers comprise 0 and all positive counting numbers (0, 1, 2, 3...).

Integers encompass all whole numbers along with their negative counterparts (...−2, −1, 0, 1, 2...).

Since the set of whole numbers is a subset of integers, any whole number is automatically an integer.

Scanning the options, 0 is the only number that fits into the whole number category.

∴ The correct answer is 0.

Alternate Method

Given:

Four numbers: −4, 0, 3.5, and 1/2

Concept Used:

Integers (Z) = {..., −3, −2, −1, 0, 1, 2, 3, ...}

Whole Numbers (W) = {0, 1, 2, 3, ...}

Calculation:

⇒ Evaluate Option 1 (−4): It is a negative number. It belongs to integers but not whole numbers.

⇒ Evaluate Option 2 (0): It is a non-negative number without a fractional or decimal part. It belongs to both integers and whole numbers.

⇒ Evaluate Option 3 (3.5): It is a decimal value. It is neither an integer nor a whole number.

⇒ Evaluate Option 4 (1/2): It is a fractional value. It is neither an integer nor a whole number.

∴ The correct answer is 0.

Additional Information

Natural Numbers

These are all positive counting numbers starting from 1. Set N = {1, 2, 3, 4, ...}. Remember that 0 is not a natural number.

Rational Numbers

Any number that can be expressed in the form of p/q where p and q are integers and q ≠ 0. For example, 1/2, −4/1, 0/1.

Irrational Numbers

Numbers that cannot be written as a simple fraction, such as √2, √3, or π. Their decimal expansions are non-terminating and non-repeating.

Shortcut Trick

Since the investment duration is identical for all partners, the profit is distributed exactly in the ratio of their investments.

Profit Ratio of A : B : C = 2 : 3 : 5

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

B's share = Total Profit × (B's parts ÷ Total parts) = 1,00,000 × (3 ÷ 10) = 30,000.

∴ The correct answer is Rs. 30,000.

Alternate Method

Given:

Ratio of investments of A, B, and C = 2 : 3 : 5

Total profit earned after a year = Rs. 1,00,000

Formula Used:

Profit Ratio = Capital × Time

When time periods are equal, Profit Ratio = Capital Ratio.

2 Parts A

3 Parts B

5 Parts C

Total Profit = Rs. 1,00,000 (10 Parts)

Calculations:

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

⇒ Sum of all parts (Total Profit) = 2x + 3x + 5x = 10x

⇒ We are given that the total profit is Rs. 1,00,000.

⇒ 10x = 1,00,000

⇒ x = 1,00,000 ÷ 10 = 10,000

⇒ B's share = 3x = 3 × 10,000 = Rs. 30,000

∴ The correct answer is Rs. 30,000.

Additional Information

Partnership Profit Ratio Formula

When investments and time periods differ, the profit is divided in the ratio of the product of their capitals and respective time periods: Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Active vs Sleeping Partner

An active partner manages the business and usually receives a fixed percentage of the total profit as a salary before the remaining profit is distributed among all partners according to their investment ratios. A sleeping partner only invests capital.

Joining Mid-Way

If a partner joins the business after a few months, their time period for investment is calculated only for the remaining months of the financial cycle (e.g., joining after 5 months means T = 7 months in a 1-year cycle).

Shortcut Trick

Total paint required = (Total Area ÷ Coverage per litre) × Increasing factor

Total paint = (48 ÷ 2) × 1.05

Total paint = 24 × 1.05 = 25.2 litres

∴ The correct answer is 25.2 litres.

Alternate Method

Given:

Total surface area of the solid pyramid = 48 m²

Paint coverage = 2 m²/litre

Extra paint required for retouch = 5%

Total Area 48 m²

÷ 2

Base Paint 24 L

• 5% Extra

Total Paint 25.2 L

Formula Used:

Base paint = Total surface area ÷ Coverage

Total paint = Base paint + (5% of Base paint)

Calculations:

⇒ Base paint required = 48 ÷ 2 = 24 litres

⇒ Extra paint = 5% of 24 = (5 ÷ 100) × 24 = 1.2 litres

⇒ Total paint needed = 24 + 1.2 = 25.2 litres

∴ The correct answer is 25.2 litres.

Additional Information

Total Surface Area of a Pyramid

The total surface area is the sum of the base area and the lateral surface area (sum of the areas of all the triangular side faces).

Volume of a Pyramid

The space enclosed by a solid pyramid is given by the formula: Volume = (1 ÷ 3) × Base Area × Vertical Height.

Percentage Increase

When a base value needs an extra R% addition, the new value can be quickly computed by multiplying the base value by the multiplying factor (1 + R ÷ 100).

Shortcut TrickTotal Area = (6 × 4) + (8 × 6) = 24 + 48 = 72 m2.Since 72 m2 > 70 m2, effective cost per m2 = 450 × 0.9 = ₹405.Total Cost = 72 × 405 = ₹29,160.∴ The correct answer is ₹29,160.Alternate MethodGiven:⇒ Dimensions of first rectangle = 6 m × 4 m⇒ Dimensions of second rectangle = 8 m × 6 m⇒ Rate of tiles = ₹450/m2⇒ Discount = 10% (applicable if total area > 70 m2)6 m4 m24 m²+8 m6 m48 m²=72 m²Total Area = 72 m² (Discount Applies)Calculations:⇒ Total Area = (6 × 4) + (8 × 6)⇒ Total Area = 24 + 48 = 72 m2⇒ Total cost without discount = 72 × 450 = ₹32,400⇒ Since 72 m2 > 70 m2, a 10% discount is applied to the total cost.⇒ Discount amount = 10% of 32,400 = (10 ÷ 100) × 32400 = ₹3,240⇒ Final Cost = 32400 − 3240 = ₹29,160∴ The correct answer is ₹29,160.Additional InformationArea of a RectangleThe total space enclosed by a rectangular shape is calculated as Length × Breadth.Calculating Total CostWhen a specific rate per unit area is given, Total Cost = Total Area × Rate per unit area.Discount PercentageWhen a discount of D% is applied on a value, Final Value = Initial Value × (100 − D) ÷ 100.

Shortcut Trick

Let the area of a single regular hexagon be 1 unit.

Since the combined shape is formed by exactly 6 such regular hexagons without overlapping, its total area is simply the sum of all their individual areas.

Total area of the combined shape = 6 × 1 = 6 units.

Ratio of the combined area to the area of one hexagon = 6 : 1.

∴ The correct answer is 6:1.

Alternate Method

Given:

Number of regular hexagons combined together = 6

Concept Used:

When multiple non-overlapping two-dimensional figures are joined together, the total area of the resulting figure is equal to the sum of the individual areas (Conservation of Area principle).

Center

Area = A

Combined Shape (Area = 6A) One Hexagon

Calculation:

⇒ Let the area of one regular hexagon = A

⇒ The large polygon is constructed by placing 6 regular hexagons around a common center in a tiled format.

⇒ Total Area of the new shape = Area of Hexagon 1 + Area of Hexagon 2 + ... + Area of Hexagon 6

⇒ Total Area = 6 × A = 6A

⇒ Required Ratio = (Total Area of combined shape) ÷ (Area of one hexagon)

⇒ Required Ratio = 6A / A = 6:1

∴ The correct answer is 6:1.

Additional Information

Area of a Regular Hexagon

The area of a regular hexagon with side length a is given by the formula: Area = (3√3 ÷ 2) × a².

Conservation of Area

In two-dimensional geometry, if a planar figure is divided into several non-overlapping pieces, the total area of the whole figure exactly equals the sum of the areas of those distinct pieces.

Tessellation of Hexagons

Regular hexagons are one of the only three regular polygons (along with equilateral triangles and squares) that can perfectly tile a flat plane without leaving gaps or creating overlaps.

Shortcut Trick

Semi-perimeter (s) = (13 + 14 + 15) ÷ 2 = 21 m.

Area using Heron's formula = √(21 × 8 × 7 × 6) = √7056 = 84 m².

Total Cost = Area × Rate = 84 × 5 = ₹420.

∴ The correct answer is ₹420.

Alternate Method

Given:

Sides of the triangular sign: a = 13 m, b = 14 m, c = 15 m.

Painting cost rate = ₹5 per m².

Triangular Traffic Sign

14 m 13 m 15 m

Formula Used:

Heron's Formula for Area of a Triangle = √[s(s − a)(s − b)(s − c)]

Where semi-perimeter, s = (a + b + c) ÷ 2

Calculation:

⇒ s = (13 + 14 + 15) ÷ 2

⇒ s = 42 ÷ 2 = 21 m

⇒ Area = √[21 × (21 − 13) × (21 − 14) × (21 − 15)]

⇒ Area = √(21 × 8 × 7 × 6)

⇒ Area = √(7 × 3 × 4 × 2 × 7 × 3 × 2)

⇒ Area = √(7² × 3² × 4²)

⇒ Area = 7 × 3 × 4 = 84 m²

⇒ Total Cost = Area × Rate per m²

⇒ Total Cost = 84 × 5 = 420

∴ The correct answer is ₹420.

Additional Information

Area of an Equilateral Triangle

For a triangle with all equal sides (a), Area = (√3 ÷ 4) × a².

Area of a Right-Angled Triangle

When base and perpendicular height are given, Area = (1 ÷ 2) × Base × Height.

Inradius of a Triangle

The radius of the largest circle that can fit inside a triangle is calculated as r = Area ÷ s.

Shortcut Trick

Given, sin θ = cos θ

Divide both sides of the equation by cos θ:

⇒ sin θ ÷ cos θ = 1

⇒ tan θ = 1

We know from standard trigonometric values that tan 45° = 1.

∴ The correct answer is 45°.

Alternate Method

θ Perpendicular Base Hypotenuse Isosceles Right Triangle (Base = Perpendicular)

Given:

sin θ = cos θ

Formulas Used:

sin θ = Perpendicular ÷ Hypotenuse

cos θ = Base ÷ Hypotenuse

Calculation:

⇒ Perpendicular ÷ Hypotenuse = Base ÷ Hypotenuse

⇒ Perpendicular = Base

In a right-angled triangle, if the perpendicular equals the base, the triangle is an isosceles right-angled triangle.

⇒ The sum of angles in a triangle is 180°, and the two non-right angles are equal.

⇒ θ + θ + 90° = 180°

⇒ 2θ = 90°

⇒ θ = 45°

∴ The correct answer is 45°.

Additional Information

Complementary Angles Property

We know that sin θ = cos (90° − θ). If sin θ = cos θ, then equating the angles gives θ = 90° − θ ⇒ 2θ = 90° ⇒ θ = 45°.

Standard Trigonometric Values

For standard angles, sin 30° = 1/2 and cos 30° = √3/2. However, for 45°, sin 45° = 1/√2 and cos 45° = 1/√2, hence their values are exactly equal.

Pythagorean Identity Method

Using the identity sin² θ + cos² θ = 1, substituting sin θ for cos θ gives 2 × sin² θ = 1 ⇒ sin θ = 1/√2, which yields θ = 45° in the first quadrant.

Shortcut Trick

The Volume of a prism is the product of its Base Area and Height (V = A × h).

Since the height of the prism remains constant, the Volume is directly proportional to the Base Area.

Therefore, a 25% increase in the Base Area directly results in a 25% increase in the Volume.

∴ The correct answer is 25%.

Alternate Method

Given:

Increase in Base Area = 25%

Height remains unchanged.

Formula Used:

Volume of Prism (V) = Base Area (A) × Height (h)

Percentage Increase = [(New Value − Original Value) ÷ Original Value] × 100

Original Prism Area = A Height = h V = A × h

New Prism Area = 1.25A Height = h V = 1.25Ah

Area +25% Volume +25%

Calculations:

⇒ Let Original Base Area = A and Original Height = h.

⇒ Original Volume (V₁) = A × h

⇒ New Base Area = A + (25 ÷ 100) × A = 1.25A

⇒ New Height = h

⇒ New Volume (V₂) = 1.25A × h = 1.25(A × h)

⇒ Increase in Volume = V₂ − V₁ = 1.25(A × h) − 1(A × h) = 0.25(A × h)

⇒ Percentage Increase = [0.25(A × h) ÷ (A × h)] × 100 = 0.25 × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Proportionality Rule

When a formula consists of a product of variables (like V = A × h), and one variable is kept constant, the percentage change in the result strictly equals the percentage change in the varying quantity.

Volume of a Prism

A prism is a 3D solid with identical parallel ends (bases) and flat side faces. The volume of any uniform solid is consistently calculated as: Volume = Base Area × Height.

Successive Percentage Change

If both the base area and the height were changed by x% and y% respectively, the overall percentage change in volume would be calculated using the successive change formula: x + y + (x × y) ÷ 100.

Shortcut Trick

LCM of given time intervals (3 mins and 4 mins) = 12 mins.

Water filled by the man in 12 mins = 4 × (12 ÷ 3) = 16 liters.

Water filled by the woman in 12 mins = 3 × (12 ÷ 4) = 9 liters.

Total water filled by both together in 12 mins = 16 + 9 = 25 liters.

Time required to fill 100 liters = (100 ÷ 25) × 12 = 48 min.

∴ The correct answer is 48 min.

Alternate Method

Given:

Man pours 4 liters every 3 minutes.

Woman pours 3 liters every 4 minutes.

Total capacity of the tank = 100 liters.

Formula Used:

Total Time = Total Work ÷ Combined Efficiency

Calculations:

⇒ Time taken by the man to independently fill 100 liters = (100 ÷ 4) × 3 = 75 min

⇒ Time taken by the woman to independently fill 100 liters = (100 ÷ 3) × 4 = 400/3 min

⇒ Combined efficiency = 4/3 + 3/4

⇒ Combined efficiency = (16 + 9) ÷ 12 = 25/12 liters/min

⇒ Total time required to fill the tank = Total Work ÷ Combined efficiency

⇒ Total time = 100 ÷ (25/12)

⇒ Total time = (100 × 12) ÷ 25

⇒ Total time = 4 × 12 = 48 min

∴ The correct answer is 48 min.

Additional Information

Pipes & Cisterns Formula

If an inlet pipe fills a tank in x hours, its 1 hour work is 1/x. For two pipes taking x and y hours independently, their combined 1 hour work is given by (1/x + 1/y).

Rate of Work Principle

Total Work = Efficiency (Rate of Work) × Time. When multiple entities work together continuously without interruption, their individual volumetric efficiencies per unit time can simply be added to find the overall combined rate.

Alternate Working (Pipes)

If two individuals or pipes work on an alternate cycle basis, calculate the total volume filled in one complete cycle (e.g., 2 minutes) to deduce how many full cycles are required before calculating the remainder of the work.

Shortcut Trick

Total sum = Average × 3 = 100 × 3 = 300.

Total ratio units = 2 + 3 + 5 = 10 units.

Value of 1 unit = 300 ÷ 10 = 30.

Difference in ratio between largest (5) and smallest (2) = 3 units.

Required difference = 3 × 30 = 90.

∴ The correct answer is 90.

Alternate Method

Given:

Ratio of the three numbers = 2 : 3 : 5

Average of the numbers = 100

Formula Used:

Sum of elements = Average × Number of elements

Total Sum = 100 × 3 = 300 (10 units)

2 units 3 units 5 units

Value = 60 Value = 90 Value = 150

Difference = 150 − 60 = 90

Calculation:

⇒ Let the three numbers be 2x, 3x, and 5x.

⇒ Sum of the numbers = 2x + 3x + 5x = 10x

⇒ Average = 10x ÷ 3 = 100

⇒ 10x = 300

⇒ x = 30

⇒ The smallest number = 2x = 2 × 30 = 60

⇒ The largest number = 5x = 5 × 30 = 150

⇒ Difference between the largest and smallest = 150 − 60 = 90

∴ The correct answer is 90.

Additional Information

Average (Mean) Concept

The total sum of all given entities in a group is always equal to the average multiplied by the number of entities: Sum = Average × N.

Ratio Distribution

To find individual values from a given ratio, you must first divide the total sum of the entities by the sum of their corresponding ratio units to determine the exact value of a single unit.

Direct Difference Shortcut

Whenever you need to find the difference or sum between quantities in a ratio, operate on their ratio unit values directly first, then multiply by the calculated base value of a single unit. This avoids the extra step of calculating both exact numerical values.

Shortcut Trick

Average CP per litre = (40 × 50 + 60 × 60) ÷ 100 = Rs. 56

Selling Price (SP) per litre = Rs. 70

Profit per litre = SP − CP = 70 − 56 = Rs. 14

Total profit on 20 litres sold = 20 × 14 = Rs. 280

∴ The correct answer is Rs. 280 profit.

Alternate Method

Given:

Quantity of Can 1 = 40 litres at Rs. 50/litre

Quantity of Can 2 = 60 litres at Rs. 60/litre

Quantity sold = 20 litres at Rs. 70/litre

Formula Used:

Average CP = Total Cost ÷ Total Quantity

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

CP of Can 1 CP of Can 2

₹50

₹60

₹56 Mean CP

2 Ratio (40L)

3 Ratio (60L)

Calculations:

⇒ Total quantity of mixture = 40 + 60 = 100 litres

⇒ Total CP of 100 litres = (40 × 50) + (60 × 60) = 2000 + 3600 = Rs. 5600

⇒ Average CP per litre = 5600 ÷ 100 = Rs. 56

⇒ CP of 20 litres of mixture = 20 × 56 = Rs. 1120

⇒ SP of 20 litres = 20 × 70 = Rs. 1400

⇒ Profit = 1400 − 1120 = Rs. 280

∴ The correct answer is Rs. 280 profit.

Additional Information

Alligation Method

It is a rule that enables finding the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired average price.

Rule of Alligation

Quantity of Cheaper / Quantity of Dearer = (Price of Dearer − Mean Price) / (Mean Price − Price of Cheaper)

Profit and Loss Basic Rules

Profit occurs when Selling Price > Cost Price. Profit = SP − CP. Loss occurs when Cost Price > Selling Price. Loss = CP − SP.

Shortcut Trick

The problem directly asks for the standard double-angle trigonometric identity.

According to the standard identity: cos2A = cos²A − sin²A.

Since cos²A = 1 − sin²A, substituting this directly gives:

cos2A = (1 − sin²A) − sin²A = 1 − 2sin²A

∴ The correct answer is 1−2sin²A.

Alternate Method

Given:

We need to find the equivalent formula for cos2A based on the options.

Formula Used:

Standard trigonometric identities:

cos(X + Y) = cosX × cosY − sinX × sinY

sin²X + cos²X = 1

Calculations:

⇒ Let X = A and Y = A in the cosine addition formula.

⇒ cos(A + A) = cosA × cosA − sinA × sinA

⇒ cos2A = cos²A − sin²A

⇒ From the Pythagorean identity, we know that cos²A = 1 − sin²A.

⇒ Substitute this value of cos²A into the cos2A equation:

⇒ cos2A = (1 − sin²A) − sin²A

⇒ cos2A = 1 − 2sin²A

∴ The correct answer is 1−2sin²A.

Additional Information

Double Angle Formulas for Sine and Tangent

sin2A = 2sinA × cosA = (2tanA) ÷ (1 + tan²A)

Other Forms of cos2A

cos2A = 2cos²A − 1 = (1 − tan²A) ÷ (1 + tan²A)

Pythagorean Trigonometric Identities

sin²A + cos²A = 1, sec²A − tan²A = 1, csc²A − cot²A = 1

Shortcut Trick

A straight line must pass through both given points. We can find the correct equation by substituting the given coordinates into the options.

Test point (1, 2) in the options:

⇒ Option 1 gives y = 3(1) − 1 = 2 (Valid equation)

⇒ Option 3 gives y = 4(1) − 2 = 2 (Valid equation)

Now test point (3, 8) in the valid options: Option 1 gives y = 3(3) − 1 = 8 (Valid) whereas Option 3 gives y = 4(3) − 2 = 10 (Invalid).

∴ The correct answer is y = 3x − 1.

Alternate Method

Given:

Two points on the line are (x₁, y₁) = (1, 2) and (x₂, y₂) = (3, 8).

Formula Used:

The slope (m) of a line passing through two points is: m = (y₂ − y₁) ÷ (x₂ − x₁)

The equation of a straight line passing through a point (x₁, y₁) is: y − y₁ = m(x − x₁)

X Y

1 3 2 8

A(1, 2)

B(3, 8) y = 3x − 1

Calculations:

⇒ Slope m = (8 − 2) ÷ (3 − 1)

⇒ m = 6 ÷ 2 = 3

⇒ Now, substitute m = 3 and the point (1, 2) into the point-slope formula:

⇒ y − 2 = 3 × (x − 1)

⇒ y − 2 = 3x − 3

⇒ y = 3x − 3 + 2

⇒ y = 3x − 1

∴ The correct answer is y = 3x − 1.

Additional Information

Slope-Intercept Form

The equation of a straight line can be written directly as y = mx + c, where 'm' represents the slope and 'c' represents the y-intercept.

Point-Slope Form

The linear equation for a line with slope 'm' passing through a specific coordinate (x₁, y₁) is given by y − y₁ = m(x − x₁).

Collinear Points

Three or more points are considered collinear if they lie on the same straight line, which implies that the slope between any two pairs of these points is exactly equal.

Intercept Form

The equation of a line making intercepts 'a' and 'b' on the x-axis and y-axis respectively is written as (x ÷ a) + (y ÷ b) = 1.

Shortcut Trick

If sin A = cos B, then A + B = 90°.

⇒ (3x + 10°) + (2x − 5°) = 90°

⇒ 5x + 5° = 90°

⇒ 5x = 85°

⇒ x = 17°

∴ The correct answer is 17°.

Alternate Method

A B C

3x+10°

2x−5° sin A = BC/AC = cos C

Given:

⇒ sin(3x + 10°) = cos(2x − 5°)

Formula Used:

⇒ cos θ = sin(90° − θ)

Calculations:

⇒ sin(3x + 10°) = sin(90° − (2x − 5°))

Comparing the angles on both sides:

⇒ 3x + 10° = 90° − 2x + 5°

⇒ 3x + 10° = 95° − 2x

⇒ 3x + 2x = 95° − 10°

⇒ 5x = 85°

⇒ x = 17°

∴ The correct answer is 17°.

Additional Information

Complementary Angle Relations

Two angles are complementary if their sum is 90°. The relations are: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Pythagorean Identities

The fundamental identity is sin²θ + cos²θ = 1. Others include sec²θ − tan²θ = 1 and cosec²θ − cot²θ = 1.

Equality of Sine and Cosine

For acute angles A and B, if sin A = cos B, then A + B = 90°. This geometric property stems from the two non-right angles in a right-angled triangle.

Shortcut Trick

For any convex polygon with 'n' sides, the number of triangles formed by drawing all possible diagonals from a single vertex is given directly by the formula (n − 2).

An octagon has 8 sides, so n = 8.

Number of triangles = 8 − 2 = 6.

∴ The correct answer is 6.

Alternate Method

Given:

A polygon with 8 sides (Octagon).

Concept Used:

When drawing diagonals from one vertex of a polygon to all other possible non-adjacent vertices, the polygon gets divided completely into non-overlapping triangles.

Vertex

1 2 3 4 5 6

n = 8 sides 5 diagonals 6 Triangles

Calculations:

⇒ Total vertices in an octagon = 8

⇒ If we choose one specific vertex, we cannot draw a diagonal to itself or to the 2 vertices adjacent to it.

⇒ Number of valid diagonals from one vertex = 8 − 3 = 5 diagonals.

⇒ Each newly drawn diagonal divides the remaining area, essentially adding one more triangle to the total count.

⇒ Total number of triangles formed = Number of diagonals + 1

⇒ Total number of triangles = 5 + 1 = 6

∴ The correct answer is 6.

Additional Information

Total Number of Diagonals

The total number of diagonals that can be drawn in a convex polygon with n sides from all vertices is n(n − 3) ÷ 2.

Sum of Interior Angles

The sum of the interior angles of any n-sided polygon is calculated using the formula (n − 2) × 180°.

Sum of Exterior Angles

The sum of the exterior angles (taken one at each vertex) of any convex polygon is always exactly 360°.

Interior Angle of a Regular Polygon

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

Shortcut Trick

In a right-angled triangle, the circumcenter always lies exactly at the midpoint of the hypotenuse.

The given points P(0, 0), Q(8, 0), and R(0, 6) form a right triangle at the origin P(0, 0).

The hypotenuse is the line segment joining the other two vertices Q(8, 0) and R(0, 6).

Midpoint of hypotenuse = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2) = ((8 + 0) ÷ 2, (0 + 6) ÷ 2) = (4, 3).

∴ The correct answer is (4, 3).

Alternate Method

P(0, 0) Q(8, 0) R(0, 6) C(4, 3) 8 units 6 units 10 units

Given:

Vertices of the triangle are P(0, 0), Q(8, 0), and R(0, 6).

Formula Used:

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint formula: M(x, y) = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Calculations:

⇒ Length of PQ = √((8 − 0)² + (0 − 0)²) = √(64) = 8

⇒ Length of PR = √((0 − 0)² + (6 − 0)²) = √(36) = 6

⇒ Length of QR = √((0 − 8)² + (6 − 0)²) = √(64 + 36) = √100 = 10

⇒ Here, PQ² + PR² = 8² + 6² = 64 + 36 = 100.

⇒ Since PQ² + PR² = QR², triangle PQR satisfies Pythagoras' theorem and is right-angled at P.

⇒ For a right-angled triangle, the circumcenter is located exactly at the midpoint of its hypotenuse (QR).

⇒ Circumcenter coordinates = Midpoint of QR = ((8 + 0) ÷ 2, (0 + 6) ÷ 2)

⇒ Circumcenter = (4, 3)

∴ The correct answer is (4, 3).

Additional Information

Circumcenter of Different Triangles

For an acute triangle, the circumcenter lies strictly inside the triangle. For an obtuse triangle, it lies outside. For a right-angled triangle, it lies exactly at the midpoint of the hypotenuse.

Orthocenter in Right Triangles

The orthocenter (intersection of altitudes) of a right-angled triangle is exactly at the vertex containing the right angle. In this question, the orthocenter is directly at P(0, 0).

Circumradius

The radius of the circumcircle (circumradius) for a right-angled triangle is exactly half the length of the hypotenuse. Here, Circumradius = 10 ÷ 2 = 5 units.

Shortcut Trick

Factorize the numbers inside the roots into perfect squares:

√12 = √(4 × 3) = 2√3

√75 = √(25 × 3) = 5√3

Add the similar surds: 2√3 + 5√3 = 7√3

∴ The correct answer is 7√3.