SSC CGL 2025 (Held On: 13 Sept, 2025 Shift 2) (Previous Year Paper)

The logic followed here is:

Logic: The first word is the Profession and the second word is the Work done by that profession.

Architect : Design → An Architect creates designs.

Chef : Cook → A Chef creates recipe.

Note: The relationship here is 'Professional : Master Blueprint' (an Architect creates a Design), so the correct match for a Chef must be a 'Recipe' rather than 'Cook', which is just a general daily action.

Hence, the correct answer is Option 3.

The logic followed here is:

Logic: The first word represents an Ethical Theory and the second word represents its Main Principle.

Utilitarianism : Consequence → Utilitarianism judges actions based on their Consequences.

Deontology : Duty → Deontology judges actions based on Duty.

Hence, the correct answer is Option 2.

The logic followed here is:

Hence, the correct answer is "Option 2".

The logic followed here is:

.

Hence, the correct answer is "Option 3".

The logic followed here is:

Hence, the correct answer is "Option 4".

The logic followed here is:

Hence, the correct answer is "Option 1".

The logic followed here is:

Hence, the correct answer is "Option 4".

The logic followed here is:

Each number is multiplied by 0.8.

120 × 0.8 = 96

96 × 0.8 = 76.8

76.8 × 0.8 = 61.44

61.44 × 0.8 = 49.152

Hence, the correct answer is Option 4.

The given word: CONTRAST

The letter that is first from the left is A.

The letter that is fifth from the right is O.

Now, the number of letters between A and O in the English alphabetical series are:

B, C, D, E, F, G, H, I, J, K, L, M, N

Total letters = 13

Hence, the correct answer is Option 2.

Given address:

Thus, addresses 1 and 2 are exactly identical to each other.

Hence, the correct answer is "Option (1)".

The address given in the problem is,

Sunil Kumar

21/4, Ashok Vihar

Delhi, 110052

Let's compare each option with the given address.

Thus, only option (2) is exactly identical to the given address.

Hence, the correct answer is "Option (3)".

The logic followed here is:

CAB : EDF

Similarly,

GIH : ?

Hence, the correct answer is Option 3.

The logic followed here is:

DP : XZ

D (4) + 20 = X (24)

P (16) + 10 = Z (26)

Similarly,

EN : ?

E (5) + 20 = Y (25)

N (14) + 10 = X (24)

Hence, the correct answer is Option 4.

Given: Ten persons are sitting in two parallel rows containing five persons each, such that there is an equal distance between adjacent persons.

Line 1: Five men (A–E) are seated and all of them are facing South.

Line 2: Five women (M–Q) are seated and all of them are facing North.

1) D is opposite to M.

2) M is at one of the extreme ends of Line 2.

3) O is at the center position of Line 2.

there are 2 possibilty

Case 1) M is extreme left end

Case 2) M is extreme right end

M is sitting at the extreme end, so the second to the right of M will be the third position from M. As in case 2, there is no possible position for one who is second to the right of M, therefore is eliminated.

Case 1) M is extreme left end

That, O second to the right of M.

Hence, the correct answer is Option 1.

The logic followed here is:

AROUND : RAUODN

Similarly,

GROUND : ?

Hence, the correct answer is Option 1.

The logic followed here is:

QDXM : SFYN

UIOZ : ?

Hence, the correct answer is Option 4.

The logic followed here is:

1) Mon Feb 04 2008 00:00:00 GMT+0530 – Modern date, standard GMT offset.

2) Tue Mar 09 2027 00:00:00 GMT+0530 – Future date, standard GMT offset.

3) Fri May 25 0125 00:00:00 GMT+0553 – Ancient year (0125), unusual GMT offset (+0553), not standard.

4) Thu Apr 16 1936 00:00:00 GMT+0530 – Historical, but normal GMT offset (+0530).

Year is very old (0125)

GMT offset is non-standard (+0553)

Hence, the odd one is Option 3.

The logic followed here is:

Option 1) M13 : J10 : G7

Option 2) Z26 : W23 : T20

Option 3) D4 : A1 : X24

Option 4) H8 : E6 : B4

Thus, out of all 'H8 : E6 : B4' is the odd one out.

Hence, the correct answer is "Option 4".

The logic followed here is:

Z1$ : Y2# : X3@ → Letters decrease by 1, numbers increase by 1, symbols change 

W4% : V5^ : U6& → Letters decrease by 1, numbers increase by 1, symbols change 

T7* : S8( : R9) → Letters decrease by 1, numbers increase by 1, symbols change 

Q10$: P11$ : O12$ → Letters decrease by 1, numbers increase by 1, symbols do not change 

Hence, the correct answer is Option 4.

Given: 

1. B is the daughter of A.

2. C is the husband of B.

3. D is the son of C.

As the A's gender is not known yet, therefore A's relation with D cannot be determined.

Hence, the correct option is Option 4.

Decoding signs:

Given: 15 W 3 X 2 Y 4=?

15 W 3 X 2 Y 4=? ⇒ 15 ÷ 3 + 2 - 4=?

⇒ 15 ÷ 3 + 2 - 4=?

⇒ 5 + 2 - 4=?

⇒ 7 - 4=?

⇒ 3 = ?

Hence, "Option 2" is the correct answer.

Given: A's father's wife's only daughter is B.

A is the sibling of B

Hence, the correct option is Option 3.

From the given information

Given: Introducing a boy, a girl says, "He is the son of my mother's only son."

Thus, the boy is girl's nephew.

Hence, the correct option is Option 3.

Given:

1. A is the wife of B.

2. B is the brother of C.

3. C is the father of D.

Thus, A is D's aunt.

Hence, the correct answer is "Option 1".

Decoding signs:

Given: 50 + 10 - 5 × 5 = ?

50 + 10 - 5 × 5 = ? ⇒ 50 - 10 + 5 ÷ 5 = ?

⇒ 50 - 10 + 5 ÷ 5 = ?

⇒ 50 + 1 - 10 = ?

⇒ 51 - 10 = ?

⇒ 41 = ?

Hence, "Option 1" is the correct answer.

The correct answer is b-c-a.

Key Points

• Bharhut (2nd century BCE): It represents the earliest phase of Buddhist art in India, with its relief sculptures being relatively simple and narrative in nature. The themes revolve around the Jataka tales and early Buddhist motifs.
• Sanchi Stupa-1 (1st century BCE): Located in Madhya Pradesh, the Sanchi Stupa showcases more advanced artistic techniques, with intricate carvings, toranas (gateways), and a more evolved style of Buddhist representation.
• Mathura (1st-2nd century CE): The Mathura school of art is known for creating Buddha and Bodhisattva sculptures in human form, transitioning from symbolic representation to anthropomorphic depictions.
• The chronological sequence of the development of Buddhist sculptures starts with Bharhut, followed by Sanchi Stupa-1, and ends with Mathura, reflecting a clear evolution in artistic styles and techniques.
• This progression demonstrates the shift from symbolic to anthropomorphic representation of Buddha in Indian art, marking the stylistic evolution of Buddhist sculpture over centuries.

Additional Information

• Bharhut:
• Located in Madhya Pradesh, Bharhut is one of the earliest examples of Buddhist art and architecture.
• The Bharhut stupa features narrative reliefs depicting Jataka tales and early Buddhist symbols like the Bodhi tree, footprints, and wheels.
• The art here is relatively simple, with less emphasis on intricate detailing compared to later styles.
• Sanchi Stupa-1:
• The Great Stupa at Sanchi is a UNESCO World Heritage Site and one of the most significant monuments of Indian Buddhist architecture.
• The intricately carved toranas (gateways) and railings depict scenes from the life of Buddha, Jataka tales, and Buddhist symbols.
• The Sanchi sculptures show a transition toward more detailed and refined artistry compared to Bharhut.
• Mathura:
• The Mathura school of art developed during the Kushana period and is known for its anthropomorphic depictions of Buddha and Bodhisattvas.
• This style represents a clear shift from symbolic to human representation, with figures characterized by intricate detailing and a spiritual expression.
• Mathura art also influenced other Buddhist art forms, including the Gandhara school of art.

The correct answer is Lord Kallazhagar arrives on a golden horse.

Key Points

• The Chithirai Festival is one of the most famous festivals of Tamil Nadu, celebrated annually in Madurai during the Tamil month of Chithirai (April-May).
• The festival reenacts the celestial wedding of Goddess Meenakshi (an avatar of Parvati) and Lord Sundareswarar (an avatar of Shiva).
• A significant highlight of the festival is the arrival of Lord Kallazhagar (an avatar of Vishnu) to the Vaigai River on a golden horse to bless the wedding of his sister Meenakshi.
• Lord Kallazhagar's journey from his temple in Alagar Kovil to Madurai is celebrated with great fervor and devotion, drawing thousands of devotees.
• The event symbolizes the unity of the Shaivite (followers of Shiva) and Vaishnavite (followers of Vishnu) traditions in Tamil culture.

Additional Information

• Major Festivals
• Pongal: A four-day harvest festival giving thanks to the Sun God, featuring traditional cooking, cattle worship, and the famous Jallikattu bull sport in Madurai.
• Puthandu (Tamil New Year): Celebrated in mid-April (Chittirai month) with traditional customs and feasts.
• Thaipusam: Devotees carry milk pots to temples for Lord Murugan (Kartikeya).
• Karthigai Deepam: A ten-day festival marked by new clothes, gifts, and temple celebrations, with large lamps lit.
• Diwali (Deepavali): The festival of lights, celebrated with lamps, sweets, and family gatherings.

The correct answer is Bandish.

Key Points

• A Bandish is a fixed, melodic composition in Hindustani classical music, often set to a specific raga and taal.
• It serves as the foundation for improvisation and is typically performed in a specific rhythmic cycle, such as Teentaal or Ektaal.
• Bandish is composed of two main parts: the Sthayi (the initial part) and the Antara (the second part), which explore the raga's characteristics.
• It is usually written in a specific language, such as Hindi, Urdu, Braj Bhasha, or Sanskrit, and often has devotional or romantic themes.
• Bandishes are an integral part of the Gharana system, with each Gharana having its unique repertoire of compositions.

Additional Information

• Kriti
• A Kriti is a structured composition in Carnatic classical music, not Hindustani music.
• It consists of three main sections: Pallavi (opening), Anupallavi (middle section), and Charanam (ending).
• Kritis often emphasize devotional themes and are composed in South Indian languages like Tamil, Telugu, or Kannada.
• Famous Carnatic composers like Tyagaraja, Muthuswami Dikshitar, and Syama Sastri are known for their kritis.
• Gamaka
• Gamaka refers to the ornamentation or embellishment of notes in Indian classical music.
• It involves techniques such as oscillation, slides, and bends to bring out the essence of a raga.
• Gamaka is not a composition but a stylistic element used during performance.
• It varies in execution between Hindustani and Carnatic music styles.
• Rasa
• Rasa refers to the aesthetic or emotional experience evoked in the listener by a musical performance.
• Indian classical music often aims to evoke specific rasas, such as joy (Hasya), sorrow (Karuna), or peace (Shanta).
• It is a broader concept rooted in Indian aesthetics and applies to various art forms, including dance and drama.

The correct answer is Both 1 and 2.

Key Points

• Statement 1 is Correct: In Judo, the red-and-white belt (Kōhaku obi) is worn by judokas holding the ranks of Sixth Dan (6th), Seventh Dan (7th), and Eighth Dan (8th).
• Statement 2 is Correct: According to the International Judo Federation (IJF) competition rules, when two competition areas are adjacent, a shared safety zone of at least 3 metres must be maintained between them to ensure the safety of the competitors.
• The safety zone provides adequate space for athletes who may move beyond the contest area during a bout, reducing the risk of injuries.
• Hence, both statements are correct.

Additional Information

• The competition area (contest area) in Judo is generally square and is surrounded by a designated safety zone.
• Judokas holding the Ninth Dan (9th) and Tenth Dan (10th) traditionally wear a solid red belt.
• The belt system in Judo reflects a practitioner's technical skill, experience, and contribution to the sport.

Important Points

• Red-and-White Belt: 6th Dan to 8th Dan.
• Red Belt: 9th Dan and 10th Dan.
• Shared Safety Zone: Minimum 3 metres between adjoining competition areas.
• Governing Body: International Judo Federation (IJF).
• Exam Tip: Remember that the red-and-white belt is awarded to 6th–8th Dan judokas, while a 3-metre shared safety zone is mandatory between adjoining competition areas.
• Correct Answer: Both 1 and 2.

The correct answer is Gulveer Singh.

Key Points

• Gulveer Singh broke the 30-year-old Asian Games men's 5,000m meet record during the National Open Athletics Championships in 2024.
• He clocked an impressive time, surpassing all previous records set in the Asian Games men's 5,000m event.
• Gulveer Singh is recognized as one of India's emerging long-distance runners, gaining accolades for his performance in national and international events.
• The record-breaking achievement highlights India's growing prominence in athletics, especially in long-distance running.
• His performance has placed him among the ranks of elite Indian athletes contributing to the nation's success in global athletics.

Additional Information

• Bahadur Singh:
• Bahadur Singh is a renowned former Indian shot putter who represented India in multiple international events, including the Olympics.
• He held national records in shot put for several years and was a pivotal figure in Indian athletics during his active years.
• While he excelled in field events, he is not associated with long-distance running or the 5,000m event.
• Neeraj Chopra:
• Neeraj Chopra is India's iconic javelin thrower, known for winning gold medals at the Olympics and Asian Games.
• He became the first Indian to win an Olympic gold medal in athletics during the Tokyo 2020 Olympics.
• Although he is highly celebrated in throwing events, he does not compete in the 5,000m race.
• Hima Das:
• Hima Das is an Indian sprinter popularly known as the "Dhing Express" for her achievements in short-distance running.
• She specializes in 400m and 200m events and has won numerous medals in international competitions.
• Her expertise lies in sprinting, not long-distance events like the 5,000m race.
• Asian Games Men's 5,000m Event:
• The Asian Games men's 5,000m is a long-distance track event that tests endurance, strategy, and speed.
• It has historically seen records set by athletes from various nations, showcasing their athletic prowess.
• Breaking a 30-year-old record signifies immense skill, dedication, and training, marking the athlete's place in history.

The correct answer is Option 1.

Key Points

• BRICS Science, Technology and Innovation (STI) cooperation is established on the foundation of mutual benefit, public-good-driven innovation, and sustainable development.
• The cooperation framework is designed to address common challenges faced by BRICS nations, such as energy security, healthcare, and climate change, while promoting inclusive growth.
• It is not solely governed through annual Heads of State Summits but includes several ministerial meetings, working groups, and collaborative initiatives to ensure effective coordination and implementation.
• BRICS STI cooperation emphasizes joint research projects, knowledge sharing, and capacity building among member nations: Brazil, Russia, India, China, and South Africa.
• This initiative fosters collaboration in areas like technology transfer, innovation policy exchange, and the development of advanced technologies.

Additional Information

• Statement 2: It is governed only through annual Heads of State Summits
• This statement is incorrect as BRICS STI cooperation is not limited to annual Heads of State Summits.
• The framework involves various mechanisms, including ministerial-level meetings, expert-level working groups, and joint initiatives in STI fields.
• Moreover, the BRICS STI Steering Committee plays a significant role in formulating strategies and monitoring progress.
• About BRICS STI Cooperation
• The initiative was launched to enhance collaboration between BRICS nations in science, technology, and innovation to solve global and regional challenges.
• Key focus areas include energy, climate change, water resources, biotechnology, and information and communication technologies (ICTs).
• The BRICS Young Scientist Forum (YSF) is an integral part of this cooperation, aiming to foster the exchange of knowledge among young researchers.
• BRICS STI Framework Program promotes joint multilateral research and development (R&D) projects in priority areas of mutual interest.

The correct answer is Dec 2023.

Key Points

• The inaugural Khelo India Para Games were held from December 2 to December 6, 2023, in New Delhi, India.
• The event was organized to promote inclusivity and provide a platform for para-athletes to showcase their talent.
• Seven sporting disciplines were featured, highlighting the government’s commitment to promoting sports among persons with disabilities (PwDs).
• The event took place at three venues in New Delhi, including Jawaharlal Nehru Stadium and others.
• The Khelo India initiative aims to promote grassroots sports and encourage young athletes to participate in national and international events.

Additional Information

• Khelo India Initiative:
• The Khelo India program is a flagship initiative by the Ministry of Youth Affairs and Sports, Government of India, aimed at promoting sports and fitness among youth.
• It includes various events like the Khelo India Youth Games, Khelo India University Games, and Khelo India Para Games.
• The initiative also provides financial assistance to athletes, infrastructure development, and training programs to nurture sports talent in the country.

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

Key Points

• Assertion (A): Last Man in Tower by Aravind Adiga primarily explores the moral and ethical dilemmas faced by residents during rapid urban development and real estate pressures in Mumbai.
• This novel highlights the conflicting aspirations of residents in a cooperative housing society when confronted with a lucrative buyout offer from a real estate developer.
• The characters in the story grapple with their conscience, values, and the implications of urbanization on their lives and relationships.
• Reason (R): The novel does not focus on the historical evolution of Mumbai's architecture or its colonial past. Instead, it centers on contemporary issues related to urbanization and societal changes.
• Hence, while the assertion (A) is true, the reason (R) is false as it does not accurately explain the assertion.

Additional Information

• Main Theme of the Novel:
• Last Man in Tower is a social and moral commentary on the effects of rapid urbanization and real estate development in modern India.
• The novel sheds light on the human cost of progress, such as strained relationships, greed, and the erosion of communal values in the face of economic temptations.
• It portrays the tension between traditional values and modern aspirations in a rapidly changing urban landscape.

The correct answer is Stalagmite - Formed by dripping water from cave ceiling.

Key Points

• Stalagmites are mineral formations that rise from the floor of caves, formed due to the deposition of minerals from dripping water.
• They are created when water containing dissolved calcium carbonate or other minerals drips from the cave ceiling, leaving deposits on the cave floor.
• The process involves evaporation of water and deposition of minerals over time, leading to upward growth of the stalagmite.
• Stalagmites typically grow in a conical shape, but their appearance can vary depending on environmental conditions and mineral concentration.
• The formation is a slow process, often taking thousands of years to develop significant size.

Additional Information

• Stalactites
• Stalactites are mineral formations that hang from the ceiling of caves, formed by dripping water rich in calcium carbonate or other minerals.
• Water drips down from the ceiling, leaving behind small deposits of minerals that accumulate over time.
• The growth direction of stalactites is downward and they often appear like icicles.
• Stalactites and stalagmites can sometimes join together to form a column or pillar.
• Pillars
• Pillars, also known as columns, are formed when a stalactite and a stalagmite grow until they meet and connect.
• This process can take hundreds of thousands of years depending on the rate of mineral deposition.
• Pillars symbolize the unification of upward (stalagmite) and downward (stalactite) mineral formation.
• General Cave Formations
• Cave formations, also known as speleothems, are created by the deposition of minerals from water.
• The most common minerals involved are calcium carbonate, forming structures like stalactites, stalagmites, and pillars.
• The process of mineral deposition is influenced by factors such as water flow rate, mineral concentration, temperature, and humidity within the cave.
• Cave formations are important for understanding geological processes and the history of Earth's climate.

The correct answer is It boosts India's navigation system by offering precise position and timing data.

Key Points

• NVS-02 is part of the NavIC (Navigation with Indian Constellation) satellite system, designed to enhance India's regional navigation capabilities.
• The satellite was launched aboard the GSLV-F15 rocket by the Indian Space Research Organisation (ISRO).
• It primarily strengthens India's indigenous satellite-based navigation system by providing accurate positional and timing data.
• The satellite ensures improved accuracy for navigation services across India and a region extending up to 1,500 km beyond Indian borders.
• NVS-02 is equipped with advanced rubidium atomic clocks for precise timekeeping, a critical component for navigation systems.

Additional Information

• India's NavIC System
• NavIC (Navigation with Indian Constellation) is India's independent regional navigation satellite system developed by ISRO.
• It provides accurate position information services to users in India and the surrounding region.
• NavIC is comparable to global navigation systems like GPS (USA), GLONASS (Russia), Galileo (EU), and BeiDou (China).
• It has applications in terrestrial, aerial, and marine navigation, disaster management, vehicle tracking, and more.

The correct answer is Option 3: A is true, but R is false.

Key Points

• Assertion (A): Population growth can be measured in absolute numbers and percentage change per year.
• This statement is correct as population growth is commonly assessed using both absolute numbers (total increase in population) and percentage change (growth rate relative to the total population).
• Absolute numbers give the actual increase in population size, while percentage change provides a standardized measure to compare growth rates across regions or time periods.
• Reason (R): Measuring growth only in absolute numbers provides detailed insights into population trends over time.
• This statement is incorrect because measuring growth only in absolute numbers does not provide a complete or detailed picture of population trends. For example, it does not account for variations in population size or growth rates across regions or timeframes.
• Therefore, while the assertion is true, the reason does not correctly explain it, making Option 3 the correct answer.

Additional Information

• Population Growth Measurement
• Population growth can be influenced by factors such as birth rate, death rate, immigration, and emigration.
• Absolute numbers are useful for understanding the total increase in population size but do not account for population distribution or density.
• Percentage change provides a relative measure, enabling comparison across regions or time periods.

The correct answer is Option 1) 1 and 2 only.

Key Points

• K. Sanjay Murthy took office as the Comptroller and Auditor General (CAG) of India in November 2024, succeeding Girish Chandra Murmu.
• The CAG is a constitutional authority and is appointed by the President of India as per Article 148 of the Indian Constitution.
• The CAG is responsible for auditing the accounts of the Union and State governments and public sector organizations to ensure accountability.
• The term of the CAG is six years, or until the age of 65, whichever is earlier. The CAG is ineligible for further office under the Government of India or any State Government after ceasing to hold office.
• Statements 1 and 2 are correct, while Statement 3 is incorrect because the term is six years (not five).

Additional Information

• Role of the CAG:
• The CAG audits all receipts and expenditure of the Government of India and the state governments, including those of bodies and authorities substantially financed by the government.
• The CAG reports are laid before the Parliament and state legislatures, ensuring transparency and accountability in public finances.
• The CAG plays a key role in ensuring financial discipline and adherence to financial rules and regulations.
• Constitutional Provisions:
• Article 148 of the Indian Constitution provides for the establishment of the CAG as an independent authority.
• The CAG is considered the guardian of the public purse and serves as a critical institution for democracy.

The correct answer is Protein modification.

Key Points

• The Golgi apparatus is an essential organelle in eukaryotic cells responsible for processing and modifying proteins received from the endoplasmic reticulum (ER).
• It functions by adding carbohydrates, lipids, or other molecules to proteins, forming glycoproteins and lipoproteins.
• Proteins are sorted, packaged, and directed to their specific destinations, such as lysosomes, the plasma membrane, or secretion outside the cell.
• The Golgi consists of cisternae (flattened membrane-bound sacs) arranged in a stack, with the cis-face receiving proteins and the trans-face dispatching them.
• It also plays a role in the synthesis of polysaccharides used in the cell wall of plants and extracellular matrix components in animal cells.

Additional Information

• ATP Production:
• ATP production primarily occurs in the mitochondria through cellular respiration, not the Golgi apparatus.
• Mitochondria are known as the "powerhouse of the cell" and produce ATP via oxidative phosphorylation during the electron transport chain.
• ATP serves as the main energy currency for cellular processes, including muscle contraction, active transport, and biosynthesis.
• Lipid Synthesis:
• Lipid synthesis occurs predominantly in the smooth endoplasmic reticulum (SER), not the Golgi apparatus.
• The SER is responsible for synthesizing phospholipids, cholesterol, and steroid hormones essential for cellular membranes and signaling molecules.
• DNA Replication:
• DNA replication occurs in the nucleus during the S-phase of the cell cycle, not the Golgi apparatus.
• It involves unwinding the DNA double helix and synthesizing a complementary strand using enzymes like DNA polymerase.

The correct answer is Three-fourths majority.

Key Points

• Article 279A of the Indian Constitution establishes the Goods and Services Tax (GST) Council, responsible for making recommendations to the Union and State governments on key GST issues.
• Decisions of the GST Council require a three-fourths majority of the weighted votes of the members present and voting.
• The voting weightage is split between the Union Government and State Governments—one-third of the total votes are allocated to the Union Government, and two-thirds to the State Governments.
• A quorum of at least 50% of the total members of the GST Council is required to conduct a meeting, ensuring sufficient representation for decision-making.
• The GST Council’s decisions are pivotal in determining tax rates, exemptions, thresholds, and other GST-related matters, fostering cooperative federalism in India.

Additional Information

• Simple majority
• A simple majority refers to a decision being passed if more than 50% of the members present and voting approve it.
• Many parliamentary decisions in India, such as passing ordinary bills, require a simple majority.
• However, the GST Council requires a higher threshold to ensure wider consensus among members.
• Two-thirds majority
• A two-thirds majority is required for certain critical decisions, such as constitutional amendments (under Article 368 of the Indian Constitution).
• This majority ensures a larger agreement among members for significant changes to the legal framework.
• However, GST Council decisions require an even higher threshold of three-fourths majority.
• GST Council 
• The GST Council is a constitutional body formed under Article 279A to create a unified tax structure for India.
• It comprises the Union Finance Minister as the Chairperson, the Union Minister of State for Revenue, and representatives from all states and union territories.
• It plays a crucial role in promoting cooperative federalism by ensuring participation from both the central and state governments in tax-related decisions.

The correct answer is Omission of prior recommendation doesn't invalidate law if assent is given.

Key Points

• Article 255 of the Indian Constitution states that the absence of prior recommendation or sanction required by the Constitution does not invalidate a law if it has received the assent of the President or the Governor.
• The provision ensures that minor procedural lapses do not obstruct the legislative process, provided the bill has been duly assented to.
• For example, even if a recommendation from the President or Governor is not obtained at the time of introducing a bill, the law remains valid if subsequent assent is granted.
• This article upholds the principle of legislative supremacy by ensuring that procedural errors do not nullify valid legislative acts.
• It applies to both state and central legislation, ensuring uniformity in the treatment of procedural lapses related to assent.

Additional Information

• Constitutional Articles (Part VI, Chapter II)
• Article 153: Establishes the office of the Governor for each state (one person can be Governor for two or more states).
• Article 154: Vests the executive power of the state in the Governor.
• Article 155: Appointment by the President (warrant under hand and seal).
• Article 156: Term of office (holds office during President's pleasure).
• Article 157: Qualifications for appointment (Citizen of India, 35 years old).
• Article 163: Council of Ministers to aid and advise the Governor (except in discretionary matters).
• Article 164: Other provisions as to Ministers (appointment, responsibility to the Assembly).

The correct answer is Frequency.

Key Points

• When light transitions from one medium to another, its frequency remains constant because frequency is determined by the source of the light and does not change during refraction.
• The speed of light changes depending on the refractive index of the medium, resulting in a change in wavelength to maintain the constant frequency.
• Frequency is the number of oscillations or cycles per second of a wave and is measured in Hertz (Hz).
• According to the equation v = f × λ (where v is the speed, f is the frequency, and λ is the wavelength), frequency must remain unchanged during refraction to preserve energy conservation.
• Since the frequency is directly linked to the energy of photons, maintaining constant frequency ensures that the energy of light remains unchanged during its transition between different media.

Additional Information

• Wavelength:
• The wavelength of light changes when it enters a different medium because its speed changes.
• As per the formula λ = v / f, a decrease in speed due to a higher refractive index leads to a shorter wavelength.
• This is why light bends when transitioning from air to a denser medium (e.g., glass or water).
• Speed:
• Light travels at different speeds depending on the refractive index of the medium.
• In air or vacuum, light travels at its maximum speed (approximately 3 × 10⁸ m/s), but in denser media, the speed decreases.
• The relationship between speed and refractive index is given by v = c / n, where c is the speed of light in a vacuum and n is the refractive index.
• Amplitude:
• The amplitude of light represents the intensity or brightness of the wave.
• While amplitude can change due to absorption or scattering in the medium, it is unrelated to the refractive properties of the medium.
• Amplitude affects how bright or dim the light appears, but does not influence its frequency or wavelength.
• Concept of Refraction:
• Refraction occurs when light changes direction upon entering a medium with a different refractive index.
• The refractive index is a measure of how much the medium slows down light compared to a vacuum.
• The change in speed and wavelength during refraction leads to bending of light, as described by Snell's Law: n₁ sin θ₁ = n₂ sin θ₂.

The correct answer is Forensic science and digital technology.

Key Points

• The Bharatiya Sakshya Adhiniyam (Indian Evidence Act) aims to modernize evidence laws by incorporating advancements in forensic science and digital technology.
• The amendments address the increasing reliance on digital evidence, such as electronic records, emails, social media posts, and CCTV footage, in legal proceedings.
• Forensic science advancements, including DNA profiling, fingerprint analysis, and other scientific methods, are being integrated to ensure the reliability and accuracy of evidence.
• The Act seeks to align Indian legal practices with global standards by focusing on scientific evidence and digital tools to improve the justice delivery system.
• The government aims to combat cybercrime, fraud, and other crimes effectively through the use of advanced forensic and digital technologies.

Additional Information 

• The Bharatiya Sakshya Adhiniyam focuses on modernizing legal evidence practices by adopting scientific and technological advancements.
• It aims to strengthen the credibility of legal evidence by integrating forensic science and digital technology into the Indian judicial system.
• This modernization is crucial for addressing contemporary challenges like cybercrime, fraud, and the growing reliance on digital data.

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

Key Points

• The Rashtriya Vigyan Puraskar is indeed a civilian award designed to honor individuals for their contributions in the field of science.
• It is a prestigious recognition aimed at promoting scientific research, innovation, and excellence in India.
• The award is conferred by the Government of India and is not directly associated with the Ministry of Culture.
• The Ministry of Science and Technology is typically the administering authority for science-related awards in India.
• Hence, while the assertion (A) about the nature of the award is correct, the reason (R) attributing it to the Ministry of Culture is incorrect.

Additional Information

• Ministry of Science and Technology
• The Ministry of Science and Technology is responsible for promoting scientific research and innovation in India.
• It administers various awards and recognitions to encourage excellence in science and technology.
• These awards aim to foster a culture of innovation and research among scientists and researchers.

The correct answer is Politics, Economy, Health, Education.

Key Points

• The Global Gender Gap Report, published annually by the World Economic Forum (WEF), measures gender-based disparities in four key areas: Politics, Economy, Health, and Education.
• It uses a comprehensive framework to evaluate countries on how effectively they are closing gender gaps across these four dimensions.
• The report assigns a Gender Gap Index score between 0 (complete inequality) and 1 (complete equality) for each country.
• In the political domain, factors such as the percentage of women in parliamentary and ministerial positions are considered.
• In the economic domain, the report assesses metrics like workforce participation, wage equality, and leadership roles.
• The health dimension examines life expectancy and the gender gap in survival rates, while education focuses on literacy rates and access to education at all levels.

Additional Information

• World Economic Forum (WEF) releases several major annual reports, including: 
• Global Risks Report: Identifies and analyzes the most critical global risks, drawing from expert surveys.
• Future of Jobs Report: Explores how technology and socio-economic shifts are reshaping jobs and required skills.
• Global Gender Gap Report: Benchmarks progress on gender equality in four key areas (Economic Participation, Education, Health, Politics).
• Top 10 Emerging Technologies: Spotlights technologies with the potential to positively impact society.
• Global Cybersecurity Outlook: Examines cybersecurity trends affecting economies and societies.
• Global Competitiveness Report: Assesses national competitiveness.

The correct answer is Manipur.

Key Points

• Thang-Ta is a traditional martial art dance form originating from Manipur, a northeastern state of India.
• It is an integral part of the Manipuri martial arts system known as Huyen Langlon, which combines armed and unarmed combat techniques.
• The term Thang-Ta is derived from the Manipuri words "Thang" (sword) and "Ta" (spear), symbolizing the weapons used in this martial art.
• This martial art was historically practiced by the Meitei warriors for self-defense and warfare.
• Thang-Ta also has a spiritual dimension and is performed as a dance-drama to depict the valor and heroism of Manipuri warriors.
• The art form has evolved into a cultural performance and is now showcased at various festivals and events to preserve its traditional heritage.

Additional Information

• Assam
• Assam is known for its classical dance form called Sattriya, which was introduced by the Vaishnavite saint Srimanta Sankardev.
• Sattriya is a blend of dance, drama, and music and is performed in monasteries called "Satras."
• The state is also famous for Bihu dance, a joyous folk dance performed during the Bihu festival.
• Nagaland
• Nagaland is renowned for its rich tribal culture, and the Naga warrior dances are an important tradition.
• The dances reflect the bravery and unity of different Naga tribes, often performed during festivals like Hornbill Festival.
• The tribes of Nagaland use vibrant costumes and rhythmic steps to celebrate their heritage.
• Tripura
• Tripura is home to the famous tribal dance called Hojagiri, performed by the Reang community.
• Hojagiri involves intricate balancing acts and is performed by women, showcasing their skill and grace.
• The dance is an integral part of Tripura's culture and is performed during the Hojagiri Festival.

The correct answer is Shipbuilding and Automobile.

Key Points

• Heavy industries are characterized by the production of large-scale goods or machinery requiring significant capital investment, energy, and raw materials.
• Shipbuilding and Automobile industries are correctly classified as heavy industries due to their large-scale operations, heavy machinery usage, and production of durable goods.
• Shipbuilding involves the construction of ships and other sea vessels, requiring extensive infrastructure, labor, and advanced engineering techniques.
• The automobile industry focuses on manufacturing vehicles, which involves complex processes such as designing, assembling, and testing heavy machinery and components.
• Both industries significantly contribute to the economic development and infrastructure of a nation, providing employment opportunities and driving technological advancements.

Additional Information

• Major Industry Sectors in India
• Primary Sector (Extraction & Agriculture)
• Agriculture & Allied: Food processing, dairy, fishing, forestry, horticulture.
• Mining & Minerals: Coal, iron ore, steel, cement, oil & gas.
• Secondary Sector (Manufacturing & Construction)
• Manufacturing: Automotive, textiles, electronics, pharmaceuticals, chemicals, engineering goods.
• Construction: Real estate, infrastructure (roads, ports).
• Tertiary Sector (Services)
• IT & BPM: Software services, IT outsourcing, tech consulting.
• Financial Services: Banking, insurance, FinTech, NBFCs.
• Healthcare & Education: Hospitals, medical devices, training.
• FMCG: Fast-moving consumer goods (food, personal care).
• Telecom & Media: Mobile networks, digital content.
• Tourism & Hospitality: Hotels, travel, aviation.

The correct answer is Appointed by the President.

Key Points

• The Governor of a state in India is appointed by the President of India under Article 155 of the Indian Constitution.
• The Governor acts as the nominal head of the state, while the real executive powers lie with the Chief Minister and the Council of Ministers.
• The Governor serves a term of 5 years, but they hold office at the pleasure of the President, meaning they can be removed earlier.
• The appointment process ensures that the Governor is a representative of the central government in the state and maintains coordination between the center and state.
• The role of the Governor includes duties such as granting assent to bills, appointing the Chief Minister, and overseeing the formation of the state government after elections.

Additional Information

• Key Appointments by the President: 
• Prime Minister (PM) & Council of Ministers: Appoints the PM (usually leader of the majority in Lok Sabha) and other ministers on the PM's advice.
• Judiciary: Chief Justice of India, Supreme Court Judges, and High Court Judges (on consultation).
• Election Commission: Chief Election Commissioner and other Commissioners.
• Union Public Service Commission (UPSC): Chairman and Members.
• Attorney General of India: The government's chief legal advisor.
• Comptroller and Auditor General (CAG): Guardian of public finances.
• Governors: Appointed for each state.
• Administrators: For Union Territories.
• National Commissions: Chairpersons & members for Scheduled Castes, Scheduled Tribes, Minorities, etc.

The correct answer is Introduction of invasive alien species.

Key Points

• Invasive alien species disrupt ecosystems by outcompeting native species, altering habitat structure, and sometimes causing extinctions.
• The Western Ghats are home to numerous endemic species that are highly vulnerable to invasive species like Lantana camara, Chromolaena odorata, and Parthenium hysterophorus.
• Invasive plants and animals often modify the soil chemistry and nutrient cycles, making it harder for native flora and fauna to thrive.
• Species such as the African Giant Snail and Tilapia fish have been introduced, impacting indigenous species and ecosystems in the Western Ghats.
• Research indicates that invasive alien species are considered one of the most significant threats to biodiversity globally, and the Western Ghats are no exception.

Additional Information

• Western Ghats Biodiversity Hotspot:
• The Western Ghats is one of the eight "hottest hotspots" of biological diversity in the world.
• It is home to numerous endemic species, including plants, amphibians, reptiles, and mammals.
• Efforts to conserve this biodiversity include biosphere reserves, sanctuaries, and national parks.
• Key threats include habitat loss, climate change, invasive alien species, and human activities.

The correct answer is Polar sun-synchronous orbit.

Key Points

• NISAR is a collaborative Earth observation satellite jointly developed by NASA and ISRO.
• The satellite will operate in a polar sun-synchronous orbit, which allows it to cover the entire Earth while maintaining consistent lighting conditions.
• A sun-synchronous orbit passes over the same region at the same local solar time, making it ideal for imaging applications like environmental monitoring, disaster management, and agricultural assessments.
• The orbit's altitude is approximately 747 kilometers above Earth, ensuring optimal radar imaging capabilities.
• NISAR aims to provide high-resolution data for measuring changes in Earth’s surface, such as ice sheet collapses, land subsidence, and vegetation patterns.

Additional Information

• Geostationary Transfer Orbit (GTO):
• A GTO is an elliptical orbit used to transfer satellites from low Earth orbit to geostationary orbit.
• This orbit is commonly used for communication satellites that need to maintain a fixed position relative to Earth.
• However, NISAR does not use this orbit because it requires global coverage, which a geostationary satellite cannot provide.
• Geostationary Earth Orbit (GEO):
• Satellites in GEO orbit at an altitude of approximately 35,786 km and match Earth's rotation to stay over a fixed point.
• GEO is ideal for communication, weather monitoring, and broadcasting satellites.
• NISAR, designed for Earth observation, requires frequent revisits to different regions, making GEO unsuitable.
• Low Earth Equatorial Orbit (LEEO):
• LEEO is a type of low Earth orbit where satellites travel close to the equator.
• It is commonly used for regional observation and communication satellites.
• NISAR needs to monitor global phenomena, making polar sun-synchronous orbit a better choice.
• Polar Sun-Synchronous Orbit:
• This orbit combines polar coverage with consistent lighting conditions, making it suitable for imaging and remote sensing satellites like NISAR.
• It enables monitoring of the entire Earth’s surface, including polar regions, over time.
• Such orbits are used extensively in environmental studies, disaster monitoring, and climate research.

The correct answer is Option 1: Only 1.

Key Points

• The Governor of a state has the constitutional authority to dissolve the State Legislative Assembly under Article 174(2)(b) of the Indian Constitution, usually on the advice of the Chief Minister or in certain circumstances like a no-confidence motion.
• The President of India, on the other hand, has the power to dissolve the Lok Sabha under Article 85(2)(b) of the Indian Constitution. Therefore, Statement 2 is incorrect.
• The dissolution of the State Legislative Assembly is an executive action of the Governor and is generally done to pave the way for fresh elections.
• The President's power to dissolve the Lok Sabha is exercised on the advice of the Council of Ministers headed by the Prime Minister.
• Thus, only Statement 1 is correct, as the Governor has the power to dissolve the State Legislative Assembly.

Additional Information

• Governor's Role and Powers:
• The Governor is the constitutional head of the state and acts on the advice of the state's Chief Minister and Council of Ministers.
• The Governor can summon, prorogue, or dissolve the State Legislative Assembly under Article 174 of the Constitution.
• In extraordinary circumstances (e.g., no-confidence motion passed), the Governor can dissolve the assembly without the advice of the Chief Minister, but this is subject to judicial review.
• President's Power to Dissolve Lok Sabha:
• The President of India can dissolve the Lok Sabha under Article 85(2)(b) of the Constitution.
• This power is exercised on the advice of the Prime Minister and the Council of Ministers.
• The dissolution of the Lok Sabha paves the way for general elections in the country.
• Difference Between Dissolution and Prorogation:
• Dissolution: Refers to the complete termination of the current House of Parliament or State Legislative Assembly. It necessitates fresh elections.
• Prorogation: Refers to the termination of a session of the House without dissolving it. It simply ends the proceedings for the session.

Shortcut Trick

For a series of a repeated digit 'd' in the form d + dd + ddd + ... up to n terms (where n ≤ 9):

Sum = d × 123...n

Here, the digit d = 4 and there are 5 terms.

Sum = 4 × 12345 = 49380

∴ The correct answer is 49380.

Alternate Method

Given:

Expression = 4 + 44 + 444 + 4444 + 44444

Calculation:

Factoring out 4 from the expression:

⇒ 4 × (1 + 11 + 111 + 1111 + 11111)

Adding the numbers inside the bracket step-by-step:

⇒ 1 + 11 = 12

⇒ 12 + 111 = 123

⇒ 123 + 1111 = 1234

⇒ 1234 + 11111 = 12345

Now, multiply the sum by 4:

⇒ 4 × 12345

Breaking it down:

⇒ 4 × (10000 + 2000 + 300 + 40 + 5)

⇒ 40000 + 8000 + 1200 + 160 + 20 = 49380

∴ The correct answer is 49380.

Additional Information

Sum of Repeated 1s

The sum of a series 1 + 11 + 111 + ... up to n terms (for n ≤ 9) is always a sequential number 123...n.

Sum of Repeated 9s

For a series like 9 + 99 + 999 + ... up to n terms, rewrite each term as (10 − 1) + (100 − 1) + (1000 − 1) ... to simplify the calculation to (10 + 100 + 1000 + ...) − n.

Column Addition Shortcut for Repeated Digits

To add repeated digit terms vertically, count the digits in each column. The units column has n terms, tens has n−1 terms, and so on. Multiply the digit by the count and sequentially carry over the extra value to the next place.

Shortcut Trick

Convert all mixed fractions directly into decimals for faster calculation.

1 ²/₅ = 1 + 0.4 = 1.4

1 ³/₄ = 1 + 0.75 = 1.75

⇒ (1.4 + 2.6) − 1.75

⇒ 4.0 − 1.75 = 2.25

∴ The correct answer is 2.25.

Alternate Method

Given:

(1 ²/₅ + 2.6) − 1 ³/₄

Method Used:

Convert mixed fractions and decimals into improper fractions and simplify using BODMAS.

Calculations:

⇒ 1 ²/₅ = ^{(1 × 5 + 2)}/₅ = ⁷/₅

⇒ 2.6 = ²⁶/₁₀ = ¹³/₅

⇒ 1 ³/₄ = ^{(1 × 4 + 3)}/₄ = ⁷/₄

⇒ Expression = (⁷/₅ + ¹³/₅) − ⁷/₄

⇒ Expression = (²⁰/₅) − ⁷/₄

⇒ Expression = 4 − ⁷/₄

⇒ Expression = ^{(16 − 7)}/₄

⇒ Expression = ⁹/₄

⇒ Expression = 2.25

∴ The correct answer is 2.25.

Additional Information

BODMAS Rule

Simplification problems always follow the BODMAS order: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

Converting mixed fractions directly to decimals (e.g., ²/₅ = 0.4 and ³/₄ = 0.75) avoids LCM calculations and saves significant time in exams.

Mixed to Improper Fraction

A mixed fraction A ^B/_C can be converted to an improper fraction using the standard formula (A × C + B) ÷ C.

Shortcut Trick

Rewrite 3 as √9 and 2 as √4 to see the continuous pattern.

Rationalizing 1/(√a − √b) gives √a + √b when a − b = 1.

The given expression evaluates to: (√9 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + √4)

All intermediate terms cancel out, leaving only the first and last parts: √9 + √4 = 3 + 2 = 5

∴ The correct answer is 5.

Alternate Method

Given:

1/(3 − √8) − 1/(√8 − √7) + 1/(√7 − √6) − 1/(√6 − √5) + 1/(√5 − 2)

Formula Used:

(a − b)(a + b) = a² − b²

Calculations:

⇒ Rationalize the first term:

⇒ 1/(3 − √8) = (3 + √8) / (3² − (√8)²) = (3 + √8) / (9 − 8) = 3 + √8

⇒ Rationalize the second term:

⇒ 1/(√8 − √7) = (√8 + √7) / ((√8)² − (√7)²) = (√8 + √7) / (8 − 7) = √8 + √7

⇒ Rationalize the third term:

⇒ 1/(√7 − √6) = (√7 + √6) / (7 − 6) = √7 + √6

⇒ Rationalize the fourth term:

⇒ 1/(√6 − √5) = (√6 + √5) / (6 − 5) = √6 + √5

⇒ Rationalize the fifth term:

⇒ 1/(√5 − 2) = (√5 + 2) / ((√5)² − 2²) = (√5 + 2) / (5 − 4) = √5 + 2

⇒ Substitute all these simplified values back into the original expression:

⇒ (3 + √8) − (√8 + √7) + (√7 + √6) − (√6 + √5) + (√5 + 2)

⇒ Expand by applying the minus signs:

⇒ 3 + √8 − √8 − √7 + √7 + √6 − √6 − √5 + √5 + 2

⇒ Cancel out the opposite matching pairs:

⇒ 3 + 0 + 0 + 0 + 0 + 2 = 5

∴ The correct answer is 5.

Additional Information

Rationalization of Surds

To remove a square root surd from the denominator, multiply both the numerator and denominator by its conjugate. The conjugate of (√a − √b) is (√a + √b).

Difference of Squares Formula

The standard algebraic identity (x − y)(x + y) = x² − y² is widely applied during rationalization to eliminate square root symbols entirely from denominators.

Continuous Series Simplification

When solving a continuous sequence of surd fractions, locating patterns like differences of 1 will allow quick cancellation of intermediate terms, drastically reducing calculation time.

Shortcut Trick

Total Football players = 25 (Both) + 45 (Only Football) = 70

Total Basketball players = 25 (Both) + 35 (Only Basketball) = 60

Ratio of Football to Basketball players = 70 ÷ 60 = 7 : 6

∴ The correct answer is 7:6.

Alternate Method

Given:

Students playing both Football and Basketball = 25

Students playing Football but not Basketball = 45

Students playing Basketball but not Football = 35

Formula Used:

Total players of a sport = Players of only that sport + Players of both sports

Calculation:

⇒ Number of students who play Football = 45 + 25 = 70

⇒ Number of students who play Basketball = 35 + 25 = 60

⇒ Required Ratio = Total Football players : Total Basketball players

⇒ Required Ratio = 70 : 60

⇒ Required Ratio = 7 : 6

∴ The correct answer is 7:6.

Additional Information

Venn Diagram Fundamentals

In a two-set Venn diagram, the total number of elements in set A is calculated by n(A) = n(Only A) + n(A and B).

Union of Two Sets

The total number of elements in at least one set is found using the formula: n(A or B) = n(A) + n(B) − n(A and B).

Ratio & Proportion

A ratio compares two quantities of the same kind. The ratio a:b is in its simplest form when a and b have no common factor other than 1.

Shortcut Trick

Work with ratios in 10-thousands to simplify the calculation (1 unit = ₹10,000).

Tina's equivalent capital = 10 × 6 + 8 × 6 = 60 + 48 = 108

Meera's equivalent capital = 8 × 6 + 12 × 6 = 48 + 72 = 120

Profit Ratio = 108 : 120 = 9 : 10

∴ The correct answer is 9:10.

Alternate Method

Given:

Tina's initial investment = ₹1,00,000

Meera's initial investment = ₹80,000

After 6 months, Tina withdrew ₹20,000 and Meera added ₹40,000.

₹100k × 6m

₹80k × 6m Tina

₹80k × 6m

₹120k × 6m Meera

Total = 1080k Total = 1200k Final Ratio = 1080 : 1200 = 9 : 10

Formula Used:

Profit Ratio = (C₁ × T₁ + C₂ × T₂) : (C₃ × T₃ + C₄ × T₄)

Calculations:

⇒ For the first 6 months, Tina's investment = ₹1,00,000

⇒ For the next 6 months, Tina's investment = ₹1,00,000 − ₹20,000 = ₹80,000

⇒ Total equivalent capital for Tina = (1,00,000 × 6) + (80,000 × 6) = 6,00,000 + 4,80,000 = ₹10,80,000

⇒ For the first 6 months, Meera's investment = ₹80,000

⇒ For the next 6 months, Meera's investment = ₹80,000 + ₹40,000 = ₹1,20,000

⇒ Total equivalent capital for Meera = (80,000 × 6) + (1,20,000 × 6) = 4,80,000 + 7,20,000 = ₹12,00,000

⇒ Profit sharing ratio = Tina's equivalent capital : Meera's equivalent capital

⇒ Profit sharing ratio = 10,80,000 : 12,00,000

⇒ Profit sharing ratio = 108 : 120 = 9 : 10

∴ The correct answer is 9:10.

Additional Information

Partnership Basic Rule

When investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time. P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Compound Partnership

If a partner's capital changes during the year, calculate the equivalent capital by multiplying each unique capital amount by the respective number of months it remained invested, then sum them up.

Active vs Sleeping Partner

An active partner takes part in running the business and typically gets a salary or a fixed percentage of the total profits before the remaining profit is divided based on their capital ratios.

Shortcut Trick

A's management fee = 20% of ₹60,000 = ₹12,000

Remaining profit to be shared = 60,000 − 12,000 = ₹48,000

Profit sharing ratio of A and B = 20,000 : 30,000 = 2 : 3

A's profit share = (2/5) × 48,000 = ₹19,200

A's total share = 12,000 + 19,200 = ₹31,200

∴ The correct answer is ₹31,200.

Alternate Method

Given:

A's investment = ₹20,000

B's investment = ₹30,000

Total profit = ₹60,000

A's management commission = 20% of total profit

Total Profit: ₹60,000

A's Mgmt: ₹12,000

Remaining: ₹48,000

A's Share: ₹19,200

B's: ₹28,800

A's Total: ₹31,200

Calculations:

⇒ Profit distribution ratio = Ratio of investments (since time period is equal)

⇒ Ratio of A : B = 20,000 : 30,000 = 2 : 3

⇒ A's management fee = (20 / 100) × 60,000 = ₹12,000

⇒ Remaining profit = 60,000 − 12,000 = ₹48,000

⇒ A's share from remaining profit = 48,000 × [2 ÷ (2 + 3)]

⇒ A's share = 48,000 × (2 / 5) = ₹19,200

⇒ Total share of A = Management fee + Profit share

⇒ Total share of A = 12,000 + 19,200 = ₹31,200

∴ The correct answer is ₹31,200.

Additional Information

Active vs Sleeping Partner

An active partner manages the business and receives a salary or a fixed percentage of the profit before the remainder is split. A sleeping partner only invests capital and takes a share of the remaining profit based on investment and time.

Partnership Profit Formula

Profit is directly proportional to the product of capital invested and the time period for which it is invested. Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Shortcut Trick

Net increase in average = 48 − 44 = 4 runs

Total extra runs scored above the old average = 132 − 44 = 88 runs

Total number of innings played (new) = Total extra runs ÷ Net increase = 88 ÷ 4 = 22

Earlier innings played = 22 − 1 = 21

∴ The correct answer is 21 innings.

Alternate Method

Given:

Initial average = 44

Score in the next inning = 132

New average = 48

Formula Used:

Average = Total Runs ÷ Number of Innings

Avg = 44x inningsDeficit = 4 × xAvg = 48Next inningScore = 132Surplus = 84New Avg = 48132 runs

Calculation:

Let the number of innings played earlier be x.

⇒ Total runs scored in earlier innings = 44 × x = 44x

⇒ Total runs after the next inning = 44x + 132

⇒ New average = (44x + 132) ÷ (x + 1)

⇒ 48 = (44x + 132) ÷ (x + 1)

⇒ 48 × (x + 1) = 44x + 132

⇒ 48x + 48 = 44x + 132

⇒ 48x − 44x = 132 − 48

⇒ 4x = 84

⇒ x = 21

∴ The correct answer is 21 innings.

Additional Information

Batting Average

Batting average is calculated as the total number of runs scored divided by the total number of times the batsman was dismissed. Formula: Total Runs = Average × Innings.

Bowling Average

Bowling average is the number of runs conceded per wicket taken. Formula: Bowling Average = Total Runs Conceded ÷ Total Wickets Taken.

Deviation Method for Averages

When a new observation is added, the new average changes based on the deviation of the new value from the old average. Change in Average = (New Value − Old Average) ÷ New Total Number of Observations.

Shortcut Trick

Net deviation of the first 10 numbers from the overall average 70 = 5 × (60 − 70) + 5 × (75 − 70) = −50 + 25 = −25.

To balance the overall average, the sum of the last 3 numbers must be = 3 × 70 − (−25) = 235.

Let the 13th number be x. Then 11th = x − 6, and 12th = x + 4.

Sum = (x − 6) + (x + 4) + x = 3x − 2 = 235 ⇒ x = 79.

The 11th number = 73, and the 12th number = 83. Their average = (73 + 83) ÷ 2 = 78.

∴ The correct answer is 78.

Alternate Method

Given:

Average of 13 numbers = 70

Average of first 5 numbers = 60

Average of next 5 numbers = 60 + (25% of 60) = 75

11th number = 13th number − 6

12th number = 13th number + 4

1st 5 Nums Avg: 60 Sum: 300

Next 5 Nums Avg: 75 Sum: 375

11th, 12th, 13th Avg: ? Sum: 235

Total Sum of 13 Nums = 910 (Avg = 70)

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Total sum of 13 numbers = 13 × 70 = 910

⇒ Sum of first 5 numbers = 5 × 60 = 300

⇒ Sum of next 5 numbers = 5 × 75 = 375

⇒ Sum of the remaining 3 numbers (11th, 12th, 13th) = 910 − (300 + 375) = 235

⇒ Let the 13th number be x.

⇒ 11th number = x − 6

⇒ 12th number = x + 4

⇒ Sum of these three numbers = (x − 6) + (x + 4) + x = 3x − 2

⇒ 3x − 2 = 235

⇒ 3x = 237

⇒ x = 79

⇒ 11th number = 79 − 6 = 73

⇒ 12th number = 79 + 4 = 83

⇒ Average of 11th and 12th numbers = (73 + 83) ÷ 2 = 156 ÷ 2 = 78

∴ The correct answer is 78.

Additional Information

Weighted Average

When different groups have different averages, the combined average is calculated as (n₁A₁ + n₂A₂ + ...) ÷ (n₁ + n₂ + ...).

Net Deviation Method

The sum of deviations of all observations from their arithmetic mean is always zero. This property is highly useful for finding missing observations in a data set without calculating heavy sum values.

Successive Percentage Increase

If a base value is increased by x%, the new value is obtained by multiplying the base by (1 + x/100). For example, a 25% increase on a base of 60 makes it 60 × 1.25 = 75.

Shortcut Trick

Let the total eligible voters be V.

V × 80% × 95% × 60% = 11400

V × (4/5) × (19/20) × (3/5) = 11400

V × (228/500) = 11400 ⇒ V = (11400 × 500) ÷ 228 = 25000

∴ The correct answer is 25000.

Alternate Method

Given:

Votes cast = 80% of total eligible voters

Invalid votes = 5% of cast votes

Winner's votes = 60% of valid votes = 11400

Total Eligible Voters = 100x

Votes Cast = 80x

Valid Votes = 76x

Winner's Votes = 45.6x = 11400

80% cast

95% valid

60% of valid

Calculations:

⇒ Let the total eligible voters be 100x.

⇒ Votes cast = 80% of 100x = 80x

⇒ Valid votes = 80x − (5% of 80x) = 80x − 4x = 76x

⇒ Winner's votes = 60% of 76x = 45.6x

⇒ We are given that Winner's votes = 11400

⇒ 45.6x = 11400

⇒ x = 11400 ÷ 45.6 = 250

⇒ Total eligible voters = 100x = 100 × 250 = 25000

∴ The correct answer is 25000.

Additional Information

Election Percentage Concept

Always maintain the chain: Total Voters → Votes Cast → Valid Votes → Winner's Votes. Subtract invalid votes from cast votes, not from total eligible voters.

Successive Percentage

When a value is subjected to multiple percentage changes sequentially, Final Value = Initial Value × (1 ± p₁/100) × (1 ± p₂/100).

Base Change in Percentage

The base for the winner's 60% share is 'Valid Votes' and not 'Total Voters'. Always correctly identify the base "of what" the percentage is calculated.

Shortcut Trick

Effective rate for the 1st year = 18%

Effective rate for the next 1/2 year = 18% ÷ 2 = 9%

Total effective Compound Interest % = 18 + 9 + (18 × 9) / 100 = 28.62%

According to the question, 28.62% of Principal = ₹2862

Principal = (2862 ÷ 28.62) × 100 = 10000

∴ The correct answer is ₹10,000.

Alternate Method

Given:

Compound Interest (CI) = ₹2862

Rate of Interest (R) = 18% p.a.

Time (T) = 1.5 years

Formula Used:

For time = n + f years (where n is the complete year and f is the fraction), compounded yearly:

Amount (A) = P × (1 + R/100)ⁿ × (1 + fR/100)

Compound Interest (CI) = A − P

100% (P)

+18% (1 Year)

118%

+9% (1/2 Year)

128.62%

Total CI = 28.62% = ₹2862

Calculations:

⇒ Amount (A) = P × (1 + 18/100) × (1 + (1/2 × 18)/100)

⇒ A = P × (118/100) × (109/100)

⇒ A = P × 1.18 × 1.09

⇒ A = 1.2862 P

⇒ CI = A − P = 1.2862 P − P = 0.2862 P

⇒ 0.2862 P = 2862

⇒ P = 2862 ÷ 0.2862

⇒ P = 10000

∴ The correct answer is ₹10,000.

Additional Information

Fractional Year Compounding

When time is given in fractions (e.g., n full years and f fraction of a year), interest for the fractional year is applied as simple interest on the compounded amount: A = P × (1 + R/100)ⁿ × (1 + fR/100).

Effective Interest Rate

For two successive periods with different rates R₁ and R₂, the net effective rate is given by the formula: R₁ + R₂ + (R₁ × R₂) / 100.

Half-Yearly & Quarterly Compounding

If compounded half-yearly, the rate becomes R/2 and the time periods become 2T. If quarterly, the rate is R/4 and the time is 4T.

Shortcut Trick

Rate = 16 2/3% = 1/6.

Let Principal = 6³ = 216 units.

Amount after 3 years = 7³ = 343 units ⇒ CI = 343 − 216 = 127 units.

Given CI = ₹1270 ⇒ 127 units = 1270 ⇒ 1 unit = ₹10.

SI for 3 years = 216 × (1/6) × 3 = 108 units = 108 × 10 = ₹1080.

∴ The correct answer is ₹1080.

Alternate Method

Given:

Rate (R) = 16 2/3% = 50/3% per annum

Time (T) = 3 years

Compound Interest (CI) = ₹1270

Formula Used:

Amount (A) = P × (1 + R/100)^T

CI = A − P

SI = (P × R × T) ÷ 100

Principal = ₹2160

Year 1 SI = ₹360

Year 2 SI = ₹360

Year 3 SI = ₹360

Total SI = ₹1080

Calculations:

⇒ Let the Principal be P.

⇒ Amount (A) = P × (1 + 1/6)³

⇒ A = P × (7/6)³ = P × (343/216)

⇒ CI = A − P = P × (343/216) − P = P × (127/216)

⇒ P × (127/216) = 1270

⇒ P = 1270 × (216/127) = 10 × 216 = 2160

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

⇒ SI = (2160 × 50/3 × 3) ÷ 100

⇒ SI = (2160 × 50) ÷ 100 = 1080

∴ The correct answer is ₹1080.

Additional Information

Difference between CI and SI for 2 years

The difference between compound interest and simple interest for 2 years on a certain sum is directly calculated by: D = P(R/100)².

Difference between CI and SI for 3 years

For a 3-year period, the difference formula extends to: D = P(R/100)² × (3 + R/100).

Multiplying Factor Method

When the rate of interest fraction is 1/x, the principal is usually assumed as x³ for 3 years to simplify compound interest calculations and avoid decimals.

Shortcut Trick Calculate the net profit or loss per unit directly using individual deviations from the final Selling Price: Loss on Darjeeling tea = (300 − 260) = ₹40/kg ⇒ Total = 15 × 40 = ₹600 (Loss) Profit on Assam tea = (260 − 240) = ₹20/kg ⇒ Total = 25 × 20 = ₹500 (Profit) Net overall effect = 500 (Profit) − 600 (Loss) = −100 (Net Loss) ∴ The correct answer is Loss of ₹100.

Alternate Method Given: Quantity of Darjeeling tea = 15 kg at ₹300/kg Quantity of Assam tea = 25 kg at ₹240/kg Selling price of the entire mixture = ₹260/kg

Formula Used: Total Cost Price (CP) = Sum of (Quantity × Rate per kg) Total Selling Price (SP) = Total Quantity × SP per kg Profit or Loss = Total SP − Total CP

Darjeeling Tea 15 kg × ₹300 = ₹4500

Assam Tea 25 kg × ₹240 = ₹6000

Total CP = ₹10500

Loss = ₹100

40 kg × ₹260 Total SP = ₹10400

Calculations: ⇒ Total CP of Darjeeling tea = 15 × 300 = ₹4500 ⇒ Total CP of Assam tea = 25 × 240 = ₹6000 ⇒ Total Cost Price (CP) of the mixture = 4500 + 6000 = ₹10500 ⇒ Total quantity of the mixture = 15 + 25 = 40 kg ⇒ Total Selling Price (SP) of the mixture = 40 × 260 = ₹10400 ⇒ Since Total CP > Total SP, the seller incurs a loss. ⇒ Loss = Total CP − Total SP = 10500 − 10400 = ₹100 ∴ The correct answer is Loss of ₹100.

Additional Information Alligation Method Used to find the ratio in which two ingredients are mixed to produce a mixture of a desired mean price. The ratio of quantities is inversely proportional to the differences in their prices from the mean price. Weighted Average To find the mean price directly without calculating overall totals, use the formula: Mean Price = (Q1 × P1 + Q2 × P2) ÷ (Q1 + Q2). Profit and Loss Percentage Profit % = (Profit ÷ Total CP) × 100. Loss % = (Loss ÷ Total CP) × 100. Both are always calculated on the total Cost Price unless specifically mentioned otherwise.

Shortcut Trick

To avoid decimal calculations, multiply the SP ratio 4 : 5 : 6 by 9 to get assumed SP values: 36, 45, and 54.

Total SP = 36 + 45 + 54 = 135 units.

CP of A = 36 ÷ 0.90 = 40, CP of B = 45 ÷ 1.50 = 30, and CP of C = 54 (no profit/loss).

Total CP = 40 + 30 + 54 = 124 units.

Profit = 135 − 124 = 11 units. ⇒ Profit % = (11 ÷ 124) × 100 ≈ 8.87% (approx. 8.88%).

∴ The correct answer is profit of 8.88%.

Alternate Method

Given:

Selling price (SP) ratio of A, B, and C = 4 : 5 : 6

Loss on item A = 10%

Profit on item B = 50%

No profit or loss on item C

Formula Used:

Cost Price (CP) = (SP × 100) ÷ (100 ± Profit/Loss %)

Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Total Cost Price (CP) = 124 unitsTotal Selling Price (SP) = 135 unitsProfit = 11Overall Profit % = (11 ÷ 124) × 100 ≈ 8.87%

Calculations:

⇒ Let the SP of items A, B, and C be 36x, 45x, and 54x respectively. (Multiplying the ratio by 9x to avoid fractional CP)

⇒ Total SP = 36x + 45x + 54x = 135x

⇒ CP of A = (36x × 100) ÷ 90 = 40x

⇒ CP of B = (45x × 100) ÷ 150 = 30x

⇒ CP of C = 54x (Since he broke even on item C, CP = SP)

⇒ Total CP = 40x + 30x + 54x = 124x

⇒ Overall Profit = Total SP − Total CP

⇒ Overall Profit = 135x − 124x = 11x

⇒ Profit % = (11x ÷ 124x) × 100

⇒ Profit % = 1100 ÷ 124 ≈ 8.87%

⇒ The closest calculated value matches option 2.

∴ The correct answer is profit of 8.88%.

Additional Information

Marked Price & Discount

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

Dishonest Dealer

When a dealer sells goods at CP but uses false weights, the profit percentage is calculated as: Profit % = [(True Weight − False Weight) ÷ False Weight] × 100.

Successive Discounts

The equivalent single discount for two successive discounts of a% and b% is given by: Equivalent Discount % = (a + b − ab ÷ 100)%.

Shortcut Trick

Remaining mixture volume = 75 − 15 = 60 liters.

Since the ratio is 3:2, Milk = 60 × (3 / 5) = 36 liters and Water = 60 × (2 / 5) = 24 liters.

After adding 10 liters of pure milk, the new volume of Milk = 36 + 10 = 46 liters.

The volume of Water remains the same at 24 liters.

New ratio of Milk to Water = 46 : 24 = 23 : 12.

∴ The correct answer is 23:12.

Alternate Method

Given:

Total initial volume of mixture = 75 liters

Initial ratio of Milk to Water = 3 : 2

Volume of mixture removed = 15 liters

Volume of pure milk added = 10 liters

60 L Mix Milk: 36 L Water: 24 L

10 L Pure Milk

=

70 L Final Milk: 46 L Water: 24 L

Mixture Replacement Visualization Final Ratio = 46 : 24 = 23 : 12

Calculations:

⇒ Quantity of mixture remaining after removal = 75 − 15 = 60 liters

⇒ In the remaining 60 liters, the ratio of Milk to Water remains 3 : 2

⇒ Quantity of milk in the remaining mixture = 60 × (3 / 5) = 36 liters

⇒ Quantity of water in the remaining mixture = 60 × (2 / 5) = 24 liters

⇒ Now, 10 liters of pure milk is added to the mixture

⇒ Total milk in the final mixture = 36 + 10 = 46 liters

⇒ Total water in the final mixture remains strictly unchanged = 24 liters

⇒ Final ratio of Milk to Water = 46 : 24

⇒ Simplifying the ratio by dividing by 2 = 23 : 12

∴ The correct answer is 23:12.

Additional Information

Ratio Stability Principle

When any portion of a uniform mixture is drawn out, the ratio of its constituents remains exactly the same in both the drawn-off part and the remaining mixture.

Single Component Addition

If pure milk is explicitly added to a milk-water mixture, the absolute quantity of water naturally remains fully unchanged from its previous state, altering only the total total volume and the milk fraction.

Successive Replacement Formula

When a volume 'v' of a mixture of total volume 'V' is repeatedly drawn out and completely replaced by a pure component 'n' times, the remaining quantity of the other untouched component is given by Initial Quantity × (1 − v/V)ⁿ.

Shortcut Trick

Applying successive percentage multipliers on the base cost C yields the final selling price:

Final SP = C × 1.5 (markup) × 0.8 (discount) × 1.4 (markup) × 0.9 (discount)

⇒ 18144 = C × 1.2 × 1.26

⇒ 18144 = 1.512C

⇒ C = 18144 ÷ 1.512 = 12000

∴ The correct answer is ₹12,000.

Alternate Method

Given:

Factory base cost = ₹ C

Wholesaler markup = 50%, Trade discount = 20%

Retailer markup = 40%, Customer discount = 10%

Final amount paid by customer = ₹18,144

Base Cost₹ CRetailer CP1.2 CFinal SP1.512 CMarkup 50%Discount 20%Markup 40%Discount 10%1.512 C = ₹18,144

Calculations:

⇒ Wholesaler Marked Price (MP) = C × (1 + 50 ÷ 100) = 1.5C

⇒ Retailer Cost Price (CP) = 1.5C × (1 − 20 ÷ 100) = 1.5C × 0.8 = 1.2C

⇒ Retailer Marked Price (MP) = 1.2C × (1 + 40 ÷ 100) = 1.2C × 1.4 = 1.68C

⇒ Final Selling Price (SP) = 1.68C × (1 − 10 ÷ 100) = 1.68C × 0.9 = 1.512C

Equating this to the final amount paid:

⇒ 1.512C = 18144

⇒ C = 18144 ÷ 1.512

⇒ C = 12000

∴ The correct answer is ₹12,000.

Additional Information

Successive Percentage Change

When a value undergoes multiple sequential percentage changes (like markup followed by discount), the final value is found by multiplying the initial base value by the combined multiplying factors (1 + R ÷ 100) or (1 − R ÷ 100).

Marked Price & Discount

Discount is always calculated on the Marked Price (also known as Printed Price or Label Price) and never on the Cost Price. Formula: Selling Price = Marked Price × (1 − Discount%).

Effective Discount Formula

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

Shortcut Trick

Total Work = LCM(16, 12, 4) = 48 units

Efficiencies: A = 48 ÷ 16 = 3, B = 48 ÷ 12 = 4, A + B + C = 48 ÷ 4 = 12

Efficiency of C = 12 − (3 + 4) = 5 units/day

Time taken by C alone = 48 ÷ 5 = 9 ³/₅ days

∴ The correct answer is 9 ³/₅ days.

Alternate Method

Given:

⇒ Time taken by A = 16 days

⇒ Time taken by B = 12 days

⇒ Time taken by A, B and C together = 4 days

Formula Used:

⇒ Total Work = Efficiency × Time

Calculations:

⇒ Total efficiency of A, B and C = 12 units/day

⇒ Efficiency of A = 3 units/day

⇒ Efficiency of B = 4 units/day

⇒ Efficiency of C = Total efficiency − (Efficiency of A + Efficiency of B)

⇒ Efficiency of C = 12 − (3 + 4) = 5 units/day

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time taken by C alone = 48 ÷ 5 = 9 ³/₅ days

∴ The correct answer is 9 ³/₅ days.

Additional Information

Work & Wages

Wages are distributed in direct proportion to the work done. If individuals work for the same number of days, their wages are divided exactly in the ratio of their daily efficiencies.

Pipes & Cisterns

This concept operates similarly to time and work, but inlet pipes have a positive efficiency (filling) while outlet pipes or leaks have a negative efficiency (emptying).

Alternate Days Work

When workers operate on alternate days, find the combined work done in a complete cycle (e.g., 2 days for 2 workers) and multiply to approach the total work limit before calculating the remaining days.

Shortcut Trick

Cost Price (CP) of mixture = 55 × (100 ÷ 110) = ₹50/kg.

Using the net deviation method from the average CP (₹50):

Net deviation = 2 × (40 − 50) + 3 × (50 − 50) + 5 × (x − 50) = 0

⇒ −20 + 0 + 5(x − 50) = 0

⇒ 5(x − 50) = 20 ⇒ x − 50 = 4 ⇒ x = 54

∴ The correct answer is ₹54.00.

Alternate Method

Given:

Prices of the first two varieties = ₹40/kg and ₹50/kg

Ratio of the three varieties = 2 : 3 : 5

Selling Price (SP) of the mixture = ₹55/kg at 10% profit

Mixture CP = ₹55 × (100 ÷ 110) = ₹50/kg2 parts₹40/kg3 parts₹50/kg5 parts₹x/kgTotal = 10 partsAvg CP = ₹50/kg

Formula Used:

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

Average Cost = (Total Cost) ÷ (Total parts)

Calculations:

⇒ CP of the mixture = (55 × 100) ÷ (100 + 10) = ₹50/kg

⇒ Let the price of the third variety be ₹x/kg.

⇒ Total parts of the mixture = 2 + 3 + 5 = 10 parts

⇒ Total CP for 10 parts = 10 × 50 = ₹500

⇒ Sum of individual costs = (2 × 40) + (3 × 50) + (5 × x)

⇒ 80 + 150 + 5x = 500

⇒ 230 + 5x = 500

⇒ 5x = 500 − 230 = 270

⇒ x = 270 ÷ 5 = 54

∴ The correct answer is ₹54.00.

Additional Information

Rule of Alligation

When two ingredients are mixed, the ratio of their quantities is inversely proportional to the difference in their prices from the average price. Quantity of cheaper / Quantity of dearer = (CP of dearer − Mean price) / (Mean price − CP of cheaper).

Weighted Average Method

For multiple mixtures, the weighted average cost is calculated as A_w = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n), where w represents the weight or ratio and x represents the value.

Cost Price from Selling Price

When profit percentage (P%) is given, Cost Price (CP) = SP × 100 ÷ (100 + P%). All mixture calculations must always be done on Cost Price, never on Selling Price directly.

Shortcut Trick

Let total work be LCM(8, 3) = 24 units.

Efficiency of A = 24 ÷ 8 = +3 units/day, Efficiency of B = 24 ÷ 3 = −8 units/day (negative as B breaks the wall).

Work done in first 4 days by A = 4 × 3 = +12 units.

Work done in next 2 days by A and B together = 2 × (3 − 8) = −10 units.

Total work completed = 12 − 10 = 2 units. Remaining work = 24 − 2 = 22 units.

Time for A to build the remaining work = 22 ÷ 3 = 7¹/₃ days.

∴ The correct answer is 7¹/₃ days.

Alternate Method

Given:

Time taken by A to build the wall = 8 days

Time taken by B to break the wall = 3 days

Calculations:

⇒ Work completed by A alone in the first 4 days = 4 × 3 = 12 units

⇒ Combined efficiency of A and B when working together = 3 − 8 = −5 units/day

⇒ Work done by A and B together in 2 days = 2 × (−5) = −10 units

⇒ Total work completed by the end of 6 days = 12 − 10 = 2 units

⇒ Remaining work to be completed = 24 − 2 = 22 units

⇒ Time required by A alone to build the remaining wall = Remaining work ÷ Efficiency of A

⇒ Time required = 22 ÷ 3 = 7¹/₃ days

∴ The correct answer is 7¹/₃ days.

Additional Information

Pipes & Cisterns Analogy (Negative Work)

When a person breaks a wall or a pipe empties a tank, their work is considered negative. Combined work = Positive Efficiency − Negative Efficiency.

Alternate Day Work

When people work on alternate days, calculate the work done in a complete cycle of days, then find the remaining work and assign it sequentially to the person whose turn it is next.

Work & Wages

Wages are strictly distributed in the ratio of the total work done by each individual. If the time worked by all is the same, the wages are simply distributed in the ratio of their efficiencies.

Shortcut Trick

When distance is constant, speed is inversely proportional to time.

Time ratio = 5 : 1

Speed ratio = 1 : 5

Given 1 unit of speed = 240 km/h

Required speed = 5 units = 5 × 240 = 1200 km/h.

∴ The correct answer is 1200 km/h.

Alternate Method

Case 1: 5 Hours

Distance = D Speed = 240 km/h | Time = 5 h

Case 2: 1 Hour

Distance = D Speed = ? | Time = 1 h

Distance is constant in both cases

Given:

Initial speed (S₁) = 240 km/h

Initial time (T₁) = 5 hours

Required time (T₂) = 1 hour

Formula Used:

Distance = Speed × Time

Speed = Distance ÷ Time

Calculations:

⇒ Total distance covered = S₁ × T₁

⇒ Distance = 240 × 5 = 1200 km

⇒ To cover the same distance in 1 hour, required speed (S₂) = Distance ÷ T₂

⇒ S₂ = 1200 ÷ 1 = 1200 km/h

∴ The correct answer is 1200 km/h.

Additional Information

Unit Conversion

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

Average Speed

Average Speed = Total Distance ÷ Total Time. If two equal distances are covered at speeds a and b respectively, then the average speed is given by (2 × a × b) ÷ (a + b).

Relative Speed

When two objects move in the same direction, their relative speed is S₁ − S₂. When they move in opposite directions towards each other, their relative speed is S₁ + S₂.

Shortcut Trick

Distance = [ (S₁ × S₂) / |S₁ − S₂| ] × Δt

Here, S₁ = 50 km/h, S₂ = 30 km/h.

Time difference Δt = 1 hour early + 1 hour late = 2 hours.

⇒ Distance = [ (50 × 30) / (50 − 30) ] × 2

⇒ Distance = [ 1500 / 20 ] × 2 = 150 km.

∴ The correct answer is 150 km.

Alternate Method

City A City B

Distance = d km

50 km/h → Arrives 1 hr early

30 km/h → Arrives 1 hr late

Given:

Speed 1 (S₁) = 50 km/h, Time 1 (t₁) = t − 1 hours

Speed 2 (S₂) = 30 km/h, Time 2 (t₂) = t + 1 hours

Formula Used:

Distance = Speed × Time

Calculations:

Let the usual time taken to cover distance d be t hours.

⇒ d = 50 × (t − 1) --- (Equation 1)

⇒ d = 30 × (t + 1) --- (Equation 2)

Equating both the distances since the actual distance is constant:

⇒ 50 × (t − 1) = 30 × (t + 1)

⇒ 5t − 5 = 3t + 3

⇒ 2t = 8

⇒ t = 4 hours

Substituting the value of t in Equation 1:

⇒ d = 50 × (4 − 1)

⇒ d = 50 × 3 = 150 km

∴ The correct answer is 150 km.

Additional Information

Average Speed

When a person travels equal distances at two different speeds S₁ and S₂, the average speed for the entire journey is given by (2 × S₁ × S₂) / (S₁ + S₂).

Relative Speed

If two bodies are moving in the same direction, relative speed = |S₁ − S₂|. If they are moving in opposite directions, relative speed = S₁ + S₂.

Time Difference Formula

When travelling the same distance with varying speeds resulting in early or late arrivals, the direct distance is always (S₁ × S₂) / |S₁ − S₂| × Total Time Difference.

Shortcut Trick

The percentage area of a ring relative to the full circle is independent of π.

Required Percentage = [(R² − r²) ÷ R²] × 100

⇒ Required Percentage = [(10² − 6²) ÷ 10²] × 100

⇒ Required Percentage = [(100 − 36) ÷ 100] × 100 = 64%

∴ The correct answer is 64%.

Alternate Method

r = 6 cm

R = 10 cm Metallic Ring Area

Given:

Outer radius of the metallic sheet (R) = 10 cm

Inner radius of the metallic sheet (r) = 6 cm

Formula Used:

Area of a circle = π × R²

Area of the ring = Area of outer circle − Area of inner circle

Calculation:

⇒ Area of the full outer circle = π × 10² = 100π cm²

⇒ Area of the inner empty circle = π × 6² = 36π cm²

⇒ Area of the ring = 100π − 36π = 64π cm²

⇒ Percentage of the full circle's area = (Area of ring ÷ Area of full circle) × 100

⇒ Percentage = (64π ÷ 100π) × 100

⇒ Percentage = 64%

∴ The correct answer is 64%.

Additional Information

Concentric Circles

Two or more circles that share the exact same center point but have different radii are called concentric circles.

Area of an Annulus (Ring)

The flat shape bounded by two concentric circles is known as an annulus. Its area is calculated using the formula π(R² − r²).

Width of the Ring

The uniform width or thickness of any circular ring is strictly the difference between the outer radius and the inner radius: Width = R − r.

Shortcut Trick

The area of the pond is directly given as 452.16 m².

Area of a circular pond = π × r²

⇒ 3.14 × r² = 452.16

⇒ r² = 452.16 ÷ 3.14 = 144

⇒ r = √144 = 12 m

∴ The correct answer is 12 m.

Alternate Method

Given:

Area of the pond = 452.16 m²

Total area of pond and path = 616 m²

Width of the path = 2 m

Formula Used:

Area of a circle = π × radius²

     r   2 m   R  Pond Area = 452.16 m² Total Area = 616 m²

Calculation:

Method 1: Using Pond Area

⇒ Area of Pond = π × r²

⇒ 3.14 × r² = 452.16

⇒ r² = 452.16 ÷ 3.14

⇒ r² = 144

⇒ r = √144 = 12 m

Method 2: Using Total Area

Let the outer radius be R. (where R = r + 2)

⇒ Total Area = π × R² = 616

⇒ (22 ÷ 7) × R² = 616

⇒ R² = (616 × 7) ÷ 22

⇒ R² = 28 × 7 = 196

⇒ R = √196 = 14 m

Since R = r + 2,

⇒ r = R − 2 = 14 − 2 = 12 m

Both calculations confirm the radius of the pond.

∴ The correct answer is 12 m.

Additional Information

Area of a Circular Path

The area of a circular ring or path is calculated as π × (R² − r²), where R is the outer radius and r is the inner radius.

Circumference of a Circle

The total length of the outer boundary of a given circle is computed using the formula C = 2 × π × r.

Area of a Sector

The area of a circle's sector with a central angle θ (in degrees) is derived by the formula (θ ÷ 360) × π × r².

Shortcut Trick

The ratio of the circumferences of two circles is directly equal to the ratio of their radii.

Ratio of radii = 3 : 5

3 parts = 9 m (radius of the smaller flower bed)

1 part = 9 ÷ 3 = 3 m

5 parts = 5 × 3 = 15 m (radius of the larger flower bed)

∴ The correct answer is 15 m.

Alternate Method

Given:

Ratio of circumferences of two circular flower beds = 3 : 5

Radius of the smaller flower bed (r₁) = 9 m

Formula Used:

Circumference of a circle (C) = 2πr

Ratio of Circumferences = 3 : 5

r1 = 9 m

r2 = ?

Calculations:

⇒ Let the circumferences be C₁ and C₂, and their respective radii be r₁ and r₂.

⇒ The ratio of their circumferences is given as:

⇒ C₁ / C₂ = 2πr₁ / 2πr₂ = r₁ / r₂

⇒ Therefore, r₁ / r₂ = 3 / 5

⇒ Substituting r₁ = 9 m in the equation:

⇒ 9 / r₂ = 3 / 5

⇒ 3 × r₂ = 9 × 5

⇒ 3 × r₂ = 45

⇒ r₂ = 45 / 3

⇒ r₂ = 15 m

∴ The correct answer is 15 m.

Additional Information

Relation between Circumference and Radius

Since the circumference formula is C = 2πr, the value of C is directly proportional to the radius. Therefore, the ratio of circumferences of any two circles is always equal to the ratio of their radii (C₁ : C₂ = r₁ : r₂).

Relation between Areas

The area of a circle is calculated using A = πr². The ratio of the areas of two circles is the square of the ratio of their radii (A₁ : A₂ = r₁² : r₂²).

Relation with Diameter

Since the diameter is twice the radius (D = 2r), the ratio of the diameters of two circles is also identical to the ratio of their radii and circumferences (D₁ : D₂ = r₁ : r₂).

Shortcut Trick

The line passes through the y-axis at (0, 3), which means its y-intercept (c) is 3.

The given slope (m) is −1.

Using the slope-intercept form equation: y = mx + c

Substituting the given values: y = (−1)x + 3 ⇒ y = −x + 3

∴ The correct answer is y = −x + 3.

Alternate Method

Given:

Point on the line (x₁, y₁) = (0, 3)

Slope of the line (m) = −1

Formula Used:

The point-slope form of a linear equation is: y − y₁ = m(x − x₁)

X-axisY-axisO(0, 3)(3, 0)y = −x + 3

Calculations:

⇒ y − y₁ = m(x − x₁)

⇒ y − 3 = −1 × (x − 0)

⇒ y − 3 = −x

⇒ y = −x + 3

∴ The correct answer is y = −x + 3.

Additional Information

Slope-Intercept Form

The equation of a straight line is represented as y = mx + c, where m is the slope of the line and c is the y-intercept.

Intercept Form

When a line makes intercepts a and b on the x and y axes respectively, the equation is given by x/a + y/b = 1.

Two-Point Form

When two distinct points (x₁, y₁) and (x₂, y₂) are known, the equation is given by y − y₁ = [(y₂ − y₁) ÷ (x₂ − x₁)] × (x − x₁).

Shortcut Trick

If x + 1/x = k, then x² + 1/x² = k² − 2.

Here, k = 5.

x² + 1/x² = 5² − 2 = 25 − 2 = 23

∴ The correct answer is 23.

Alternate Method

Given:

x + 1/x = 5

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ x + 1/x = 5

Squaring both sides, we get:

⇒ (x + 1/x)² = 5²

⇒ x² + 1/x² + 2 × x × (1/x) = 25

⇒ x² + 1/x² + 2 = 25

⇒ x² + 1/x² = 25 − 2

⇒ x² + 1/x² = 23

∴ The correct answer is 23.

Additional Information

Minus Identity Shortcut

If x − 1/x = k, then x² + 1/x² = k² + 2.

Cubic Identity (Plus)

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k.

Cubic Identity (Minus)

If x − 1/x = k, then x³ − 1/x³ = k³ + 3k.

Power of 4 Identity

If x + 1/x = k, then x⁴ + 1/x⁴ = (k² − 2)² − 2.

The correct answer is '4' i.e. Book-lover .

Key Points

• The word "Bibliophile" refers to a person who has a great love or fondness for books.
• Among the given options, option 4, "Book-lover", is the most appropriate synonym for the word "Bibliophile".
• Other options like Publisher, Bookkeeper, or Librarian refer to professions or roles related to books but do not convey the sense of having a deep love for books.

Correct meaning: A bibliophile is someone who loves and collects books.

Additional Information

• Bibliophile Origin:
• Root Words: Derived from the Greek words "biblion" (book) and "philos" (loving).
• Meaning: A person with a passionate interest in books, often collecting rare or special editions.
• Examples of Usage:
• John is a true bibliophile; his library is filled with rare books from around the world.
• As a bibliophile, Maria spends most of her weekends exploring second-hand bookstores.

The correct answer is '3' i.e. He has an additional skill or option to rely on .

Key Points

• The idiom "has two strings to his bow" refers to having an additional skill or option to rely on.
• It originates from archery, where having two strings for a bow ensures a backup if one string fails.
• This idiom is often used figuratively to describe someone with a backup plan or an extra skill that can be utilized when needed.

Correct interpretation: "He has an additional skill or option to rely on."

Additional Information

• Option 1: "He is indecisive in crucial matters" - This is incorrect as it does not relate to the meaning of the idiom.
• Option 2: "He has two romantic interests simultaneously" - This is incorrect and unrelated to the idiom's meaning.
• Option 4: "He uses deception to achieve goals" - This is incorrect as the idiom does not imply deceit.

The correct answer is '4' i.e. Jovial .

Key Points

• Lachrymose means tearful or mournful, often associated with sadness.
• The antonym of lachrymose should represent the opposite of sadness or tearfulness.
• Among the options, Jovial means cheerful and happy, which is the most appropriate antonym.

Correct answer: Jovial.

Additional Information

• Synonyms of Lachrymose: Tearful, mournful, weepy.
• Antonyms of Lachrymose: Jovial, cheerful, happy.

The correct answer is '3' i.e. Consciencious.

Key Points

• Option 3, Consciencious, is the misspelt word.
• The correct spelling is Conscientious, which means being diligent and careful.
• The other options are correctly spelt:
• Acquiescence: Refers to passive agreement without protest.
• Bourgeoisie: Refers to the middle class, typically with materialistic values.
• Epitome: Refers to a perfect example or embodiment of something.

Correct sentence: The correct spelling for option 3 is Conscientious.

Additional Information

• Spelling errors are common in English, especially for words with complex structures or similar sounds.
• To avoid such errors, you can use tools like dictionaries or spell-check software.
• Regular practice and reading can help improve spelling skills.

The correct answer is '3' i.e. Cytochrome.

Key Points

• Cytochrome is the correct spelling of the electron-transport protein.
• It is a class of heme proteins involved in electron transfer in cellular respiration and photosynthesis.
• Other options provided in the question contain spelling errors or inconsistencies.

Correct answer: Cytochrome.

Additional Information

• Cytochromes play a crucial role in the electron transport chain, facilitating the transfer of electrons between complexes.
• They are essential for energy production in mitochondria and chloroplasts.

The correct answer is '3' i.e. Abeyance .

Key Points

• The term Abeyance refers to a state of temporary disuse or suspension.
• It is used when something is temporarily set aside or paused.
• The other options do not match the exact meaning:
• Intermission: A pause or break between acts or events, often in a performance or meeting.
• Recess: A short break during work or school hours.
• Dormancy: A state of inactivity, often used for living organisms like plants or animals.

Correct usage in a sentence: The project was held in abeyance while the team awaited further instructions.

Additional Information

• Synonyms of Abeyance: Suspension, pause, dormancy, intermission.
• Antonyms of Abeyance: Continuation, activity, operation.
• Example: The negotiations were in abeyance due to disagreements among the parties.

The correct answer is '1' i.e. too .

Key Points

• The sentence reads: "The analyst found the correlation to be ____ marginal to influence policy." The blank requires a word that signifies an extent that prevents an action.
• The word "too" is used to indicate that something is excessive and thus prevents the intended outcome.
• In this context, the correlation is excessively marginal, meaning it is insufficient to influence policy.

Correct sentence: The analyst found the correlation to be too marginal to influence policy.

Additional Information

• Explanation of Options:
• Option 1: too - Correct. "Too" is the correct word choice to indicate an excessive degree.
• Option 2: very - Incorrect. "Very" emphasizes intensity but does not convey excess that prevents an outcome.
• Option 3: so - Incorrect. "So" requires a concluding clause (e.g., "so...that").
• Option 4: as - Incorrect. "As" does not fit the context of excess or degree.
• Option 5: Blank - Incorrect. Leaving it blank would make the sentence incomplete.

The correct answer is '3' i.e. nevertheless .

Key Points

• The sentence conveys a contrast between the compelling nature of the argument and the fact that it is based on outdated data.
• The word "nevertheless" is used to introduce a contrasting idea while maintaining coherence in the statement.
• The other options do not fit the context as they fail to provide a logical contrast:
• Option 1: "otherwise" suggests an alternative condition, which is not relevant here.
• Option 2: "conversely" indicates a reversal of a situation, which is not applicable.
• Option 4: "subsequently" refers to time sequence, which is irrelevant to the contrast being presented.