SSC CGL 2025 (Held On: 13 Sept, 2025 Shift 1) (Previous Year Paper)

The logic followed here is :-  

Given, Jurisprudence : Law 

Jurisprudence is the study or philosophy of Law.

Similarly, Epistemology : ?

Epistemology is the study or philosophy of Knowledge.

So, Epistemology : Knowledge

Hence, the correct answer is Option 3.

The logic followed here is :-  

Let’s check the letters between the first vowel and the last consonant in "SIGNATURE":

First vowel = I → 2nd letter 

Last consonant = R → 8th letter 

Now, the letters between them are:

G (3rd), N (4th), A (5th), T (6th), U (7th) → total 5 letters

Hence, the correct answer is "Option 3".

The logic followed here is:

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

Hence, the correct answer is "Option 3".

The logic followed here is:

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

Hence, the correct answer is "Option 4".

The logic followed here is :-  

Given sequence:

Chaos → Conflict → Negotiation → Resolution → ?

Let’s understand the flow:

Chaos – Disorder

Conflict – Open disagreement

Negotiation – Attempt to solve the conflict

Resolution – Conflict is solved

Thus, the natural outcome is peace and balance.

Now, checking the given options:

Option 1) Justice

Justice – may or may not follow

Option 2) Harmony

Harmony – state of peace after resolution

Option 3) Stability

Stability – possible but broader term

Option 4) Law

Law – not directly part of the sequence.

Here, The most logical continuation is Harmony.

Hence, the correct answer is Option 2.

The logic followed here is :-  

Given:

Hibiscus – a flower

Tulip – a flower

Apple – a fruit

Daisy – a flower

Hibiscus, Tulip, and Daisy are flowers, while Apple is a fruit.

So, the word that does not belong to the category is: Apple

Hence, the correct answer is Option 2.

The logic followed here is: n³ - n

⇒ 1³ - 1 = 1 - 1 = 0

⇒ 2³ - 2 = 8 - 2 = 6

⇒ 3³ - 3 = 27 - 3 = 24

⇒ 4³ - 4 = 64 - 4 = 60

⇒ 5³ - 5 = 125 - 5 = 120

⇒ 6³ - 6 = 216 - 6 = 210 = ?

Thus, "210" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

thus, "143" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given Statements:

No student of Class X failed the science exam this year.

Conclusions:

I. The science exam was very easy - False  (Even if no one failed, it doesn’t prove the exam was easy.

Students might have prepared well, or grading may have been lenient.).

II. All students of Class X passed science - True (If no student failed, it logically means all students passed.).

So, Only Conclusion II follows

Hence, the correct answer is "Option 2".

Given:

Statement: Although recent advancements in quantum computing have significantly improved processing speeds, the stability of quantum states remains a critical barrier to developing large-scale quantum computers for practical applications.

Assumptions:

I. Quantum computing holds the potential for faster data processing than classical computing - True (Because the statement explicitly mentions that advancements in quantum computing have significantly improved processing speeds. This assumes that faster processing is an important potential advantage of quantum computing.)

II. Quantum states are inherently unstable, making large-scale quantum computing unfeasible - False (Because the statement says that the stability of quantum states is a critical barrier, but it does not state that quantum states are inherently unstable or that large-scale quantum computing is completely unfeasible. The issue is presented as a current technological challenge.)

III. Recent advancements in quantum computing have resolved the issue of quantum state stability - False (Because the statement explicitly states that quantum-state stability remains a critical barrier. Therefore, the stability problem has not been resolved by recent advancements.)

So, only Assumption I is implicit.

Hence, the correct answer is "Option 3".

The logic followed here is :-  

Given Word: GARDEN

Letters: G, A, R, D, E, N

Alphabetical order: A, D, E, G, N, R

New letter cluster: A D E G N R

Thus, Fifth from the left → N

Third from the right → G

Here,  Alphabetical order: G, H, I, J, K, L, M, N

Letters between G and N: H, I, J, K, L, M

So, Total letters = 6

Hence, the correct answer is Option 3.

The logic followed here is :-  

Given Word: "ELEVATE"

1st letter → E

2nd letter → L

4th letter → V

5th letter → A

So, the letters are: E, L, V, A

Thus, Possible meaningful words:

VEAL

VALE

LAVE

All three are valid English words.

Therefore, total meaningful four-letter words = 3

Hence, the correct answer is Option 3.

The logic followed here is:

Logic: The address should match exaxtly.

Given Addrtess:

Rajesh Verma 15B, Sapphire Towers, Jaipur, 302018

Now, checking all the given options:

1. Rajesh Verma 15B, Sapphire Towers, Jaipur, 302018

⇒ Exactly the same.

2. Rajesh Verma 15B, Sapphire Towers, Jaipur, 302019

⇒ PIN code is different (302019 instead of 302018).

3. Rajesh Verma 15B, Sapphire Tower, Jaipur, 302018

⇒ "Tower" is singular instead of "Towers".

4. Rajesh Verma 15B, Sapphire Towers, Jaipur, 302017

⇒ PIN code is different (302017 instead of 302018).

Hence, Option 2 is the correct answer.

The logic followed here is :-

EK :: MS

Similarly, FP : ?

Hence, the correct answer is "Option 3".

The logic followed here is :-

Logic: Swap letters in pairs → (1↔2), (3↔4), (5↔6)

Given:

NUMBER : UNBMRE

NUMBER Positions: 

Rearranged as: 2 1 4 3 6 5 ⇒ U N B M R E

Similarly, GHOSTS : ?

GHOSTS Positions: 

Rearranged as: 2 1 4 3 6 5 ⇒ H G S O S T

Hence, the correct answer is "Option 3".

Family Chart:

Given: 

“He is the son of my mother's only son.”

Means: Your mother’s only son = your brother

So, Brother’s son = Nephew.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: All numbers are factorials (!)

These numbers look like factorials:

2 = 2!

6 = 3!

24 = 4!

120 = 5!

720 = 6! 

So, Next should be: 7! = 5040

thus, Instead of 5040, the series has 1000, which is not a factorial number.

Hence, the correct option is "Option 3".

The logic followed here is:

Logic: In the first three sets, each number is double the previous number.

Given,

(11, 22, 44), (13, 26, 52), (15, 30, 60), (14, 28, 43)

Now, checking the options:

Option 1) (11,22,44)

⇒ 11 × 2 = 22

22 × 2 = 44

Option 2) (13,26,52)

⇒ 13 × 2 = 26

26 × 2 = 52

Option 3) (15,30,60)

⇒ 15 × 2 = 30

30 × 2 = 60

Option 4) (14,28,43)

⇒14 × 2 = 28

28 × 2 = 56 (but given 43)

 Hence, the correct option is "Option 4".

The logic followed here is:

Logic: the pattern is a @ b = a² + b

Given:

WRITE is coded as YTKVG,

Similarly,  READ = ?

Hence, the correct answer is "Option 1."

The logic followed here is:

Logic: (1st number × 2nd number) + (2nd number +1) = 3rd number.

Given:

If 5 # 3 = 19,

⇒ 5 # 3 = (5 × 3) + (3 + 1) = 15 + 4 = 19 

and 7 # 2 = 17,

⇒ 7 # 2 = (7 × 2) + (2 + 1) = 14 + 3 = 17

Similarly, 9 # 1?

⇒ 9 # 1 = 9 × 1 + (1 + 1) = 9 + 2 = 11

Hence, the correct answer is "Option 4."

Shortcut Trick

Initial Ratio (Milk : Water) = 7 : 3 (Total = 10 units)

Final Ratio = 1 : 1 → Equate Milk (since only water is added) → 7 : 7

Increase in Water units = 7 − 3 = 4 units

Given 10 units = 60 L ⇒ 1 unit = 6 L

Added Water = 4 units × 6 L = 24 L

∴ The correct answer is 24 L.

Alternate Method

Given: Total mixture = 60 L, Initial ratio (M:W) = 7 : 3, Final ratio (M:W) = 1 : 1.

⇒ Total parts in initial mixture = 7 + 3 = 10 units

⇒ Quantity of Milk = (7 ÷ 10) × 60 = 42 L

⇒ Quantity of Water = (3 ÷ 10) × 60 = 18 L

⇒ In the final 1 : 1 mixture, let the water added be x liters.

⇒ New Water content = 18 + x

⇒ Milk : Water = 42 : (18 + x) = 1 : 1

⇒ 42 = 18 + x

⇒ x = 42 − 18 = 24 L

∴ The correct answer is 24 L.

Additional Information

Constant Quantity Concept

When one component of a mixture is added or removed, the quantity of the other component remains constant. Always equate the parts of the constant component in the ratio calculations.

Ratio Balancing

To simplify calculations, multiply the final ratio by a factor that makes the non-changing substance equal in both ratios (e.g., if milk is 7 parts initially, make it 7 parts in the final ratio).

Percentage Method

Initial Milk % = 70%. In 1:1 ratio, Milk % = 50%. Using inverse variation: Total Volume₁ × %₁ = Total Volume₂ × %₂ ⇒ 60 × 70 = V_new × 50 ⇒ V_new = 84 L.

Family Chart:

Given: 

“She is the daughter of the only child of my grandmother.”

Aryan’s grandmother has only one child.

That child must be Aryan’s parent (either father or mother).

Therefore, the girl is Aryan’s Sister.

Hence, the correct answer is "Option 3".

Family Chart:

Given: 

My father’s wife → This is Ramesh’s mother. 

The two sons of my father’s wife → This means Ramesh and his brother, because both are sons of the same mother

The boy is the younger of the two sons → So the boy must be Ramesh’s younger brother. 

So, the football-playing boy is Ramesh’s brother.

Hence, the correct answer is "Option 1".

Given:

A = 11 / 19

B = 2 / 9

We need to find: A − B

Formula Used: a/b − c/d = (ad − bc) / bd

Calculation:

A − B = (11/19) − (2/9)

Take LCM of 19 and 9:

19 × 9 = 171

Now convert into like fractions:

(11 × 9 − 2 × 19) / 171

= (99 − 38) / 171

= 61 / 171

Now convert to decimal:

61 ÷ 171 ≈ 0.356 ≈ 0.36

∴ The correct answer is "Option 1."

Shortcut Trick

Step 1: Calculate the sum of the given numbers: 35 + 46 = 81.

Step 2: Multiply the result by 2 as per the requirement: 81 × 2 = 162.

∴ The correct answer is 162.

Alternate Method

Given:

First number = 35

Second number = 46

Multiplier = 2

Calculation:

According to the question, we need to find the result of [(35 + 46) × 2].

⇒ Result = (35 + 46) × 2

⇒ Result = 81 × 2

⇒ Result = 162

Alternatively, using the distributive property:

⇒ (35 × 2) + (46 × 2)

⇒ 70 + 92 = 162

∴ The correct answer is 162.

Additional Information

BODMAS Rule

This rule dictates the order of operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction. In this problem, the sum (addition) is performed first because it is implied by the phrasing before the multiplication.

Distributive Property

The property states that a × (b + c) = (a × b) + (a × c). This allows for multiplying each term inside the sum individually before adding them together.

The correct answer is 4) 1, 2 and 3.

Key Points

• Gwalior Fort is located on a hilltop and spans over 2 km in length, making it one of the largest forts in India.
• The southern path of the fort is known for its intricately carved rock-cut Jain temples, dedicated to Jain tirthankars. These sculptures are an example of exquisite craftsmanship.
• The Caturbhuj Mandir (Chaturbhuj Temple) within the fort is a Vaishnavite shrine dedicated to Lord Vishnu.
• Gwalior Fort is renowned for its architectural brilliance and historical importance, being a confluence of various styles and influences from different dynasties.
• It has been a significant cultural and historical landmark, witnessing the reign of the Tomars, Mughals, Marathas, and Scindias.

Additional Information

• Key Examples & Locations of Jain Architecture:
• Dilwara Temples (Mt. Abu, Rajasthan): 11th-13th century masterpieces known for stunning marble carvings, detailed ceilings, and ornate pillars.
• Ranakpur Jain Temple (Rajasthan): Famous for its 1,444 uniquely carved pillars, a 15th-century marvel dedicated to Adinatha.
• Gomateshwara Statue (Shravanabelagola, Karnataka): A towering monolithic granite statue of Bahubali, a key Jain pilgrimage site.
• Palitana & Girnar (Gujarat): Hilltop temple complexes, considered sacred pilgrimage cities.
• Khandagiri & Udayagiri Caves (Odisha): Early rock-cut caves used by Jain monks, featuring simple entrances and carved pillars.

The correct answer is Saguna and Nirguna.

Key Points

• The Bhakti movement in India emphasized devotion to a personal deity and was categorized into two main traditions: Saguna and Nirguna.
• Saguna Bhakti refers to devotion to a deity with form and attributes, such as Lord Krishna, Rama, or Goddess Durga.
• Nirguna Bhakti focuses on the worship of a formless, infinite, and abstract divine reality, often associated with concepts like the ultimate truth or Brahman.
• Famous saints like Meera Bai, Tulsidas, and Surdas were proponents of Saguna Bhakti, while Kabir and Guru Nanak emphasized Nirguna Bhakti.
• The two traditions reflect different approaches to spirituality, offering flexibility for individual beliefs and practices within the Bhakti movement.

Additional Information

• Vedic and Non-Vedic
• The terms "Vedic" and "Non-Vedic" classify traditions based on their alignment with the Vedas.
• Vedic traditions include rituals, sacrifices, and practices outlined in the Vedas, while Non-Vedic traditions involve independent or alternative spiritual practices, such as Jainism and Buddhism.
• However, the Bhakti movement does not strictly align with this classification as it incorporates elements from both.
• Orthodox and Reformist
• The "Orthodox" approach adheres strictly to traditional practices and scriptures, while the "Reformist" approach focuses on modifying or reforming traditional religious practices.
• While the Bhakti movement did involve some reformist ideas, it is not categorized within this binary classification.
• Shaiva and Vaishnava
• Shaiva and Vaishnava traditions refer to sects within Hinduism that worship Lord Shiva and Lord Vishnu, respectively, as the supreme deity.
• These sects are part of the broader Bhakti traditions but are not the two main categories of Bhakti worship.

The correct answer is Wrote over 32,000 songs promoting devotion and equality.

Key Points

• Annamacharya, also known as Tallapaka Annamacharya, was a 15th-century saint and composer hailed as the “Padakavita Pitamaha” (Grandfather of Telugu Songwriting).
• He composed over 32,000 devotional songs (kritis) dedicated primarily to Lord Venkateswara of Tirumala, many of which promote the values of devotion, equality, and social harmony.
• His works were composed in Telugu and Sanskrit, making him one of the earliest composers in Indian music to use vernacular languages for devotional expression.
• He strongly believed in the principle of equality and opposed the caste-based discrimination prevalent during his time, which is often reflected in his compositions.
• Annamacharya’s compositions are not only devotional in nature but also convey profound spiritual and social messages, emphasizing bhakti (devotion) and the importance of human values.

Additional Information

• Contribution of Annamacharya
• Annamacharya was a pioneering figure in Indian devotional music and played a crucial role in enriching Carnatic music with his kritis.
• His works continue to inspire generations of musicians and devotees, and he is celebrated as a reformer who upheld the values of bhakti and equality in society.

The correct answer is Prevents uneven bounce, ensures uniform rebound.

Key Points

• Hockey sticks are designed with a flat left-hand side to ensure uniform rebound and control of the ball during gameplay.
• The flat side prevents uneven bounce, which could occur if the stick had varying surfaces, ensuring consistency in ball handling.
• This uniformity is critical for fair play as it standardizes the performance of all players, regardless of their stick's orientation.
• Using only the flat side helps maintain the game's tempo and predictability, as the ball behaves consistently upon contact.
• Regulations by the International Hockey Federation (FIH) mandate that players use only the left-hand flat side of the stick, ensuring uniformity in gameplay across all levels of hockey.

Additional Information

• General Information about Hockey Sticks:
• The hockey stick's flat side is made of materials like wood, fiberglass, or carbon fiber for durability and performance.
• The curved side is not used for hitting the ball, as it would lead to unpredictable ball movement and uneven rebounds.
• The design and use of hockey sticks are strictly regulated by the International Hockey Federation to ensure fairness and safety in the game.

The correct answer is His role was elevated to Governor-General with a council of four members..

Key Points

• The Regulating Act of 1773 was enacted by the British Parliament to address the administrative and financial issues of the East India Company in India.
• Under this Act, the Governor of Bengal was elevated to the position of Governor-General of Bengal.
• A council of four members was established to assist the Governor-General in decision-making, marking the beginning of centralized administration in India.
• Warren Hastings became the first Governor-General of Bengal under the provisions of this Act.
• The Act also brought the presidencies of Bombay and Madras under the control of the Governor-General of Bengal in matters of war and diplomacy.

Additional Information

• Regulating Act of 1773
• The Regulating Act of 1773 was the first major legislative intervention by the British Parliament in Indian affairs.
• It aimed to reform the administration of the East India Company and ensure better governance in its Indian territories.
• The Act marked the beginning of British parliamentary control over the East India Company's operations in India.
• It also laid the foundation for subsequent acts that gradually centralized British authority in India.

The correct answer is 45.6 seconds.

Key Points

• Milkha Singh, popularly known as the "Flying Sikh," represented India in the 1960 Rome Olympics.
• He participated in the men’s 400 meters race and finished fourth, narrowly missing a medal in one of the most competitive events of the Olympics.
• Milkha Singh clocked a timing of 45.6 seconds, which was an Indian national record at the time.
• This record remained unbroken for 24 years, until 1984, showcasing the exceptional performance of Milkha Singh.
• His fourth-place finish was a remarkable achievement and is considered one of the greatest performances by an Indian athlete in Olympic history.

Additional Information

• Global Icons & Their Records
• Michael Phelps (Swimming, USA): Most decorated Olympian ever with 28 medals, including 23 golds, and the only one to win 8 golds in a single Olympics (2008).
• Usain Bolt (Sprinting, Jamaica): World records in 100m (9.58s) and 200m (19.19s), the fastest man in history.
• Sergei Bubka (Pole Vault, Ukraine): Set 35 world records (17 outdoor, 18 indoor) in the pole vault.
• Jackie Joyner-Kersee (Heptathlon, USA): World record holder in the heptathlon (7291 points)

The correct answer is Option 4: Neither 1 nor 2 is correct.

Key Points

• Statement 1: The India-Uzbekistan Bilateral Investment Treaty (BIT) does not allow unconditional access to international arbitration without exhausting local remedies. Most modern BITs, including India’s revised BIT model, require the exhaustion of local remedies before arbitration.
• The revised Indian Model BIT (2016) emphasizes the protection of national interest and mandates that disputes be attempted to be resolved in domestic courts before proceeding to international arbitration.
• Statement 2: The National Infrastructure Readiness Index (NIRI) does not solely rank Smart Cities, nor does it exclude private investment indicators. It is designed to evaluate the readiness and performance of infrastructure across various sectors, including private investment indicators as a key parameter.
• The NIRI evaluates infrastructure readiness across multiple categories, combining public and private sector contributions, and does not limit its scope to Smart Cities alone.
• Both statements are factually incorrect, making Option 4 the correct answer.

Additional Information

• India-Uzbekistan BIT:
• Bilateral Investment Treaties are agreements between two countries to ensure fair treatment of investments made by investors of one country in the other.
• The revised Indian Model BIT (2016) replaced the earlier version to safeguard the country’s regulatory autonomy and restrict unqualified access to international arbitration.
• India’s Model BIT includes provisions like exhaustion of local remedies, limited protection for investors, and specific definitions of investment and investor.
• National Infrastructure Readiness Index (NIRI):
• The NIRI is a comprehensive framework to assess the readiness of infrastructure projects across sectors, including energy, transportation, water, and social infrastructure.
• It integrates both public and private sector contributions to infrastructure development and considers private investments as a critical performance indicator.
• The index is developed to improve infrastructure planning, execution, and monitoring in alignment with India’s economic growth goals.

The correct answer is New Delhi.

Key Points

• The Dr. S. P. Mookerjee Swimming Stadium is located in New Delhi, India.
• It is a part of the larger sports complex known as the Jawaharlal Nehru Stadium Complex, which was primarily built for the 1982 Asian Games.
• The stadium is equipped with advanced facilities and has hosted various international aquatic events and competitions.
• It was renovated and upgraded for the 2010 Commonwealth Games, which were held in New Delhi.
• It is named after Dr. Syama Prasad Mookerjee, a prominent Indian politician and founder of the Bharatiya Jana Sangh.

Additional Information

• Narendra Modi Stadium (Ahmedabad): World's largest cricket stadium, a major venue for international matches.
• Eden Gardens (Kolkata): One of India's oldest and most iconic cricket grounds, known as the "Mecca of Indian Cricket".
• Wankhede Stadium (Mumbai): Home to the Mumbai Indians (IPL) and India, famous for hosting the 2011 Cricket World Cup Final.
• M. A. Chidambaram Stadium (Chennai): A historic ground known for its passionate crowds.
• M Chinnaswamy Stadium (Bengaluru): Known for its high-scoring matches and central location.

The correct answer is Non-fiction memoir, ineligible for Edgar Award.

Key Points

• 'Heart: A History' by Sandeep Jauhar is a non-fiction memoir that explores the historical and scientific journey of cardiology.
• The Edgar Award is primarily awarded to works in the genre of mystery, crime, and suspense fiction, making non-fiction works like 'Heart: A History' ineligible.
• The book chronicles the evolution of heart treatments and delves into personal and emotional experiences with heart diseases.
• As a memoir focused on medical history, science, and the personal reflections of the author, it does not align with the criteria for the Edgar Award.

Additional Information

• Theme of 'Heart: A History'
• The book provides an insightful look into the history of cardiology, exploring how scientific advancements have revolutionized heart treatments.
• It also shares the author Sandeep Jauhar's personal experiences with heart disease in his family, making it a blend of science, history, and memoir.
• The book emphasizes the emotional and psychological connections people have with their hearts, making it a compelling narrative in the field of medical literature.

The correct answer is Hard rocks.

Key Points

• Gorges are deep valleys with steep, almost vertical walls, commonly formed by the erosive action of rivers over geological time.
• They are most commonly developed in hard rock formations because these rocks resist erosion better than softer materials, allowing the river to cut deep, narrow valleys rather than wide ones.
• Examples of hard rocks where gorges are formed include granite, basalt, and other igneous or metamorphic rocks.
• Famous gorges like the Grand Canyon in the USA and Kali Gandaki Gorge in Nepal were carved through hard rock formations by persistent river erosion.
• The steep walls of gorges are created as the hard rock resists weathering and collapsing, maintaining the narrow and deep profile of the valley.

Additional Information

• Soft sediment:
• Soft sediment consists of loose, unconsolidated materials such as sand, silt, and clay, which are easily eroded by water and wind.
• In areas dominated by soft sediment, rivers typically form wide valleys or floodplains rather than narrow gorges.
• These areas are more prone to erosion and deposition processes, leading to the formation of features like meanders and oxbow lakes.
• Glacial till:
• Glacial till is an unsorted mixture of sediments deposited directly by glacier ice.
• It does not form gorges, as it lacks the structural integrity required to create steep walls.
• Instead, glacial till can form moraines, drumlins, and other glacial landforms.
• Volcanic ash:
• Volcanic ash consists of fine particles of volcanic glass and minerals ejected during volcanic eruptions.
• It is highly susceptible to erosion and compaction, making it unsuitable for gorge formation.
• However, volcanic activity can create other landforms, such as calderas, lava plateaus, and volcanic cones.
• Gorge Formation:
• Gorges are typically formed by the long-term erosive action of rivers cutting through resistant rock layers over millions of years.
• Other factors, such as tectonic uplift and changes in sea level, can enhance the formation of gorges by increasing river gradient and erosive power.
• Gorges can also be formed by glacial activity, where a glacier carves out a U-shaped valley that later becomes a gorge after glacial retreat.

The correct answer is Rajeev Chandrasekhar, Minister of State for Electronics and Information Technology.

Key Points

• India's first lithium-ion cell manufacturing plant was inaugurated in Tirupati, Andhra Pradesh, in September 2022.
• Rajeev Chandrasekhar, the Minister of State for Electronics and Information Technology, inaugurated this significant facility.
• The plant is part of India's push towards promoting self-reliance in the manufacturing of advanced energy storage systems and reducing dependency on imports.
• The facility was established by Munoth Industries Limited, marking a key milestone in India's journey toward building a local ecosystem for lithium-ion battery production.
• This project aligns with the government’s Make in India initiative, aiming to bolster domestic manufacturing and support the clean energy transition.

Additional Information

• Prime Minister Narendra Modi
• Prime Minister Narendra Modi has been at the forefront of launching several initiatives like Make in India, Atmanirbhar Bharat, and the National Electric Mobility Mission Plan to promote domestic production and clean energy adoption in India.
• Nirmala Sitharaman, Minister of Finance
• Nirmala Sitharaman has played a crucial role in facilitating financial incentives for the electric vehicle (EV) industry, including schemes like Faster Adoption and Manufacturing of Electric Vehicles (FAME).
• Lithium-Ion Battery Technology
• Lithium-ion batteries are rechargeable batteries widely used in electronics, electric vehicles, and renewable energy storage systems due to their high energy density, lightweight, and long cycle life.
• The production of lithium-ion batteries aligns with India's National Electric Mobility Mission Plan (NEMMP) and its commitment to achieving net-zero carbon emissions by 2070.

The correct answer is Nagaland.

Key Points

• According to the Census of India 2011, Nagaland was the only state to record a negative population growth rate of -0.58% during the period 2001-2011.
• The negative growth rate was attributed to migration, as many people left Nagaland for better opportunities outside the state.
• Another contributing factor was an overestimation of the population in the 2001 Census, which led to a decrease in the reported population for 2011.
• The population of Nagaland decreased from 1,988,636 in 2001 to 1,978,502 in 2011, reflecting the negative growth rate.
• Such trends highlight demographic challenges in the northeastern region, including migration and socio-economic factors.

Additional Information

• Total Population: 1,210,854,977 (approx. 1.21 billion).
• Decadal Growth (2001-2011): 17.7%, a decrease from the previous decade's 21.5%.
• Sex Ratio: 940 females per 1000 males (up from 933 in 2001).
• Child Sex Ratio (0-6 years): 914 females per 1000 males (down from 927).
• Density: 382 persons per square kilometer.

The correct answer is It is the difference between the birth rate and the death rate.

Key Points

• The rate of natural increase (RNI) of a population refers to the difference between the crude birth rate (CBR) and the crude death rate (CDR).
• It measures the population growth occurring solely due to natural factors (births and deaths), excluding migration.
• The rate of natural increase is typically expressed as a percentage or per 1,000 individuals per year.
• For example, if the crude birth rate is 20 per 1,000 and the crude death rate is 8 per 1,000, the RNI is 12 per 1,000 (or 1.2%).
• The RNI is a critical indicator in demography, as it provides insights into the natural population growth and helps in policy planning, resource allocation, and infrastructure development.

Additional Information

• Crude Birth Rate (CBR)
• Definition: The number of live births per 1,000 people in a year.
• Formula: (Total Live Births / Total Population) * 1,000.
• Significance: Indicates fertility levels and contributes to population growth.
• Crude Death Rate (CDR)
• Definition: The number of deaths per 1,000 people in a year.
• Formula: (Total Deaths / Total Population) * 1,000.
• Significance: Reflects mortality conditions and public health, with declines often due to better healthcare and sanitation.

The correct answer is Option 2.

Key Points

• The SSLV-D2 mission demonstrated rapid deployment capability, characterized by a reduced turnaround time and minimal ground infrastructure requirements.
• The mission is specifically designed for launching small satellites into low Earth orbit, catering to increasing demand in the satellite market.
• The SSLV (Small Satellite Launch Vehicle) is equipped with a three-stage propulsion system, featuring solid propulsion for the first two stages and a liquid propulsion system for the velocity trimming module (VTM).
• On February 10, 2023, SSLV-D2 successfully deployed EOS-07, Janus-1, and AzaadiSAT-2 into a 450 km circular orbit, marking a significant milestone in India's space program.
• The mission reinforces the Indian Space Research Organisation's (ISRO) capability to facilitate cost-effective, timely launches for small satellite payloads.

Additional Information

• Statement 1 is incorrect: SSLV-D2 uses a three-stage propulsion system with solid propulsion for the first two stages and a velocity trimming module (VTM) powered by liquid propulsion. The claim that all stages are powered entirely by liquid propulsion is inaccurate.
• EOS-07: It is an Earth observation satellite developed by ISRO to meet the demands of various applications, such as environmental monitoring, agriculture, and disaster management.
• Janus-1: A satellite built by Antaris, a US-based company, featuring significant contributions from Indian space industry partners. It is designed for commercial applications and showcases international collaboration.
• AzaadiSAT-2: Developed by Indian students under the guidance of Space Kidz India, this satellite aims to encourage female participation in STEM fields and includes payloads designed for communication and data collection.

The correct answer is Telomere lengthening.

Key Points

• Telomerase is an enzyme responsible for maintaining the length of telomeres, which are repetitive nucleotide sequences at the ends of chromosomes.
• Telomeres protect chromosomes from degradation, fusion with neighboring chromosomes, and incomplete replication during cell division.
• The enzyme adds DNA sequence repeats to the telomere regions, compensating for the shortening that occurs during DNA replication.
• Telomerase activity is particularly high in germ cells, stem cells, and certain cancer cells, ensuring their proliferative capacity.
• In most somatic cells, telomerase activity is low or absent, leading to progressive telomere shortening and cellular aging.

Additional Information

• DNA replication
• DNA replication is the process by which a cell duplicates its DNA, resulting in two identical DNA molecules.
• This process involves enzymes like DNA polymerase, helicase, and ligase, but telomerase is not directly involved in the core replication process.
• While telomerase addresses the replication of telomeres, it is not responsible for the replication of the entire DNA strand.
• DNA repair
• DNA repair refers to the mechanisms by which cells identify and correct damage to the DNA molecules that encode their genome.
• It involves processes like base excision repair, nucleotide excision repair, and mismatch repair.
• Telomerase does not play a role in DNA repair but focuses on preventing the loss of genetic information at chromosome ends.
• RNA transcription
• RNA transcription is the process of synthesizing RNA from a DNA template, carried out by RNA polymerase enzymes.
• It is a critical step in gene expression but is unrelated to telomere lengthening or the activity of telomerase.

The correct answer is Sarkaria Commission.

Key Points

• The Sarkaria Commission was established by the Government of India in June 1983 to review the Centre-State relations.
• It recommended the establishment of a Permanent Secretariat for the Inter-State Council to institutionalize Centre-State cooperation.
• The Inter-State Council is a constitutional body under Article 263 of the Indian Constitution to investigate and discuss subjects of common interest between the Union and States.
• The Sarkaria Commission emphasized the need for better communication and coordination between the Centre and States to strengthen federalism in India.
• Its recommendations led to the formation of the Inter-State Council in 1990, which serves as an important platform for resolving disputes and promoting cooperative federalism.

Additional Information

• Rajamannar Committee
• It was set up by the Tamil Nadu government in 1969 under the chairmanship of P.V. Rajamannar.
• The committee recommended more autonomy to states and a redefinition of Centre-State relations.
• It suggested the abolition of the Planning Commission, the discontinuation of All India Services, and the transfer of subjects from the Union List to the State List.
• However, its recommendations were not implemented at the national level.
• Punchhi Commission
• The Punchhi Commission was set up in April 2007 under the chairmanship of Justice M.M. Punchhi to look into Centre-State relations.
• It recommended a fixed term for Governors, a comprehensive framework for disaster management, and changes to the role of the Governor in state politics.
• It focused on addressing challenges to federalism in the context of globalization and socio-economic changes.
• Its report is considered significant for improving Centre-State relations but is yet to be fully implemented.
• Venkatachelliah Commission
• Also known as the National Commission to Review the Working of the Constitution (NCRWC), it was established in 2000 under Justice M.N. Venkatachaliah.
• The commission aimed to review the Constitution to make it more effective and relevant in contemporary times.
• It suggested reforms in electoral processes, judicial accountability, and the functioning of parliamentary institutions.
• It did not specifically focus on Centre-State relations or the Inter-State Council.

The correct answer is Copper is a good conductor of heat, ensuring even heating.

Key Points

• Copper is an excellent conductor of heat, meaning it allows heat to spread quickly and evenly across its surface.
• This property ensures uniform cooking, preventing hotspots that could cause unevenly cooked food.
• Copper’s high thermal conductivity allows it to respond quickly to temperature changes, offering better control over cooking processes.
• Many cooking utensils have copper bottoms because it improves energy efficiency, reducing cooking time and saving fuel or electricity.
• Due to its efficient heat transfer, copper is commonly used in combination with other materials like stainless steel, which adds durability and prevents direct contact with food.

Additional Information

• Properties of Copper
• Copper is widely used in cookware due to its unique thermal properties, making it ideal for efficient and precise cooking.
• Its ability to conduct heat efficiently makes it a preferred choice for professional chefs and cooking enthusiasts.
• However, due to its reactivity with certain foods, it is usually paired with other materials to ensure safety.

The correct answer is To provide a common set of personal laws for all citizens, irrespective of religion.

Key Points

• The Uniform Civil Code (UCC) aims to replace personal laws based on the customs and scriptures of various religious communities with a common set of laws governing marriage, divorce, inheritance, and adoption for all citizens of India.
• The UCC is enshrined as a directive principle under Article 44 of the Indian Constitution, which calls for the State to endeavor to secure a uniform civil code for all citizens.
• It seeks to ensure equality and secularism by eliminating discriminatory practices embedded in personal laws based on religion or gender.
• The implementation of UCC is viewed as a significant step toward achieving gender justice and promoting national integration by ensuring uniformity in legal matters across all communities.
• The Supreme Court of India has, on several occasions, emphasized the need for a Uniform Civil Code to ensure equality and uniformity in personal laws.

Additional Information

• Classification of DPSPs (Philosophical Basis):
• Socialistic Principles: Aim for socio-economic justice, reducing inequality (e.g., equal pay, right to work).
• Gandhian Principles: Promote Gandhian ideals (e.g., village panchayats, cow protection, cottage industries).
• Liberal-Intellectual Principles: Advocate for individual freedoms and state action (e.g., uniform civil code, separation of judiciary from executive).

The correct answer is Agri-logistics and cold chains.

Key Points

• The Agriculture Infrastructure Fund (AIF) is a financing facility launched by the Government of India in 2020 to strengthen agricultural infrastructure across the country.
• It provides medium to long-term debt financing for investment in viable projects for post-harvest management infrastructure and community farming assets.
• Key projects funded under the AIF include cold storage, warehousing, sorting & grading units, packhouses, ripening chambers, and other agri-logistics infrastructure.
• Through this initiative, farmers and Farmer Producer Organizations (FPOs) can access loans with interest subvention and credit guarantee support.
• The fund also aims to reduce post-harvest losses and improve the efficiency of agricultural supply chains, ensuring better price realization for farmers.

Additional Information

• Free Fertilizers
• The government of India provides subsidies on fertilizers to ensure affordability for farmers, but there is no provision for free fertilizers under the AIF.
• Subsidized fertilizers such as urea, DAP, and MOP are made available through schemes like the Nutrient Based Subsidy (NBS) scheme.
• The subsidy is directly transferred to fertilizer companies, which reduces the retail price for farmers.
• Cash Transfers
• Cash transfers are direct benefit transfers (DBT) under schemes like PM-KISAN, where eligible farmers receive Rs. 6,000 annually in three installments.
• The objective of cash transfers is to provide income support to farmers for agricultural and allied activities.
• However, AIF does not involve cash transfers and focuses on infrastructure development.
• MGNREGA Work
• The Mahatma Gandhi National Rural Employment Guarantee Act (MGNREGA) provides wage employment for rural households through unskilled manual work.
• It aims to enhance livelihood security and create durable assets in rural areas.
• While MGNREGA contributes to rural development, it is unrelated to the objectives of AIF, which targets agricultural infrastructure.

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

Key Points

• India Industrial Land Bank (IILB) is a GIS-based portal that provides detailed information about industrial land and resources in India.
• The platform integrates data from various states and union territories to offer a one-stop repository for industrial information.
• It is part of the Indian government’s effort to improve the ease of doing business and attract investments by providing transparent and accessible data.
• The IILB is managed by the Department for Promotion of Industry and Internal Trade (DPIIT), under the Ministry of Commerce and Industry.
• The platform allows investors to identify and select industrial land based on their requirements, such as connectivity, infrastructure, and natural resources.

Additional Information

• GIS-Based Information
• GIS (Geographic Information System)-based platforms are used to map and analyze data related to geography and infrastructure.
• The IILB platform provides details such as the availability of land, location-specific infrastructure, and connectivity through GIS mapping.
• Such platforms are instrumental in improving transparency and aiding investors in making informed decisions.
• Ease of Doing Business
• The India Industrial Land Bank initiative is a step towards improving the ease of doing business in India, a key government priority.
• It aims to attract both domestic and foreign investments by providing a seamless and transparent platform for industrial information.

The correct answer is Ongoing Conflict and Safety & Security have deteriorated, while Militarisation has improved.

Key Points

• The Global Peace Index (GPI) measures the relative peacefulness of nations using three primary domains: Ongoing Conflict, Safety & Security, and Militarisation.
• Since 2008, there has been a deterioration in the domains of Ongoing Conflict and Safety & Security globally, primarily due to an increase in conflicts, violence, and political instability.
• However, the Militarisation domain has shown improvement, reflecting reduced military expenditure and fewer armed forces in some regions.
• Global peace has deteriorated over the years, with 2024 marking the continuation of this trend, driven by factors such as geopolitical tensions, terrorism, and internal conflicts.
• The GPI provides a comprehensive overview of global trends in peace, helping policymakers address issues related to conflict and security.

Additional Information

• Human Development Report (UNDP): Measures health, education, and living standards.
• Global Gender Gap Report (WEF): Ranks countries on gender equality.
• Global Innovation Index (WIPO): Ranks innovation performance.
• Global Hunger Index (Concern Worldwide/Welthungerhilfe): Assesses hunger levels.
• World Economic Outlook (IMF): Forecasts global economic growth, ranking economies.
• Corruption Perceptions Index (Transparency International): Ranks perceived levels of public sector corruption.

The correct answer is Agriculture and Water Resources.

Key Points

• Under India's updated State Action Plan on Climate Change (SAPCC) for 2025, the highest budgetary allocation for climate adaptation is given to the Agriculture and Water Resources sector.
• Agriculture is highly vulnerable to climate change as it depends heavily on climatic conditions such as rainfall, temperature, and seasonal variations.
• Water resources management is critical in addressing climate adaptation challenges, such as droughts, floods, and water scarcity exacerbated by climate change.
• The allocation aims to enhance irrigation infrastructure, promote sustainable agricultural practices, and improve water conservation efforts to adapt to shifting climate patterns.
• This step aligns with India's National Action Plan on Climate Change (NAPCC), which emphasizes resource efficiency, climate resilience, and sustainable development in agriculture and water management.

Additional Information

• Renewable Energy
• Renewable energy is a key area in mitigating climate change by reducing dependency on fossil fuels and cutting carbon emissions.
• India has launched ambitious initiatives, including the National Solar Mission, to transition towards cleaner energy sources.
• While important for mitigation, renewable energy is often considered less relevant to immediate adaptation needs, leading to lower allocation under the SAPCC for adaptation purposes.

The correct answer is It is self-reliant across the full value chain.

Key Points

• India's textile industry is one of the most self-reliant sectors in the country, encompassing the entire value chain from raw materials to finished goods. This includes cotton production, spinning, weaving, dyeing, printing, and garment manufacturing.
• The Indian textile sector is among the largest globally, contributing approximately 2.3% to India's GDP, 7% to the industrial output, and 12% to the country’s export earnings.
• India is the largest producer of cotton and jute globally and ranks second in silk and polyester production, ensuring a strong foundation for raw material availability.
• With a balanced integration of traditional handloom techniques and advanced machinery, the textile industry caters to both domestic and international markets.
• Government initiatives such as the Production Linked Incentive (PLI) Scheme for textiles and the establishment of Mega Textile Parks aim to further boost the self-reliance of the sector across the value chain.

Additional Information

• Post-Independence (1948 & 1956): Focused on public sector dominance, planned development, and import substitution, with state control over strategic industries (IPR 1956).
• 1991 New Industrial Policy (LPG Reforms): Marked a major shift towards liberalization, privatization, and globalization, dismantling the License Raj, opening up sectors to private and foreign investment, and encouraging exports.
• National Manufacturing Policy (NMP 2011): Aimed to increase manufacturing's share in GDP, create jobs, promote SEZs, and develop National Investment and Manufacturing Zones (NIMZs).
• Make in India (2014): Aims to make India a global manufacturing hub, improve ease of doing business, attract FDI, and boost key sectors like defense, aerospace, and electronics.
• Current Focus (Post-2020): Emphasizes Production Linked Incentive (PLI) Schemes for high-growth sectors, further deregulation, skill development, and infrastructure, seeking self-reliance (Atmanirbhar Bharat).

The correct answer is Article 263.

Key Points

• Article 263 of the Indian Constitution empowers the President of India to establish an Inter-State Council.
• The primary function of the Inter-State Council is to inquire into and advise upon disputes that may arise between states.
• It also aims to investigate and discuss subjects of common interest between states or between the Union and one or more states.
• The Inter-State Council promotes coordination and cooperation among the states and the Union of India, ensuring smooth governance and resolving inter-state disputes.
• The council may also be assigned the responsibility to formulate policies and improve the implementation of laws and schemes for national integration.

Additional Information

• Article 264:
• This article deals with the provisions related to the "definition" of the territory of India for taxation purposes.
• It defines the jurisdiction of the Union and States in terms of taxes and duties, but it does not relate to the Inter-State Council.
• Article 312:
• Article 312 provides for the creation of All India Services, such as the Indian Administrative Service (IAS) and Indian Police Service (IPS).
• It allows Parliament to create new All India Services by passing a law, if deemed necessary in the national interest.
• Article 262:
• Article 262 grants Parliament the power to adjudicate disputes related to inter-state rivers or river valleys.
• It also provides that Parliament may, by law, bar the jurisdiction of the Supreme Court or any other court in such disputes.

The correct answer is Option 4: Neither 1 nor 2.

Key Points

• Statement 1 is incorrect: Most weather phenomena, such as clouds, rain, storms, and winds, occur in the troposphere, the lowest atmospheric layer, not the stratosphere.
• Statement 2 is incorrect: The ozone layer is primarily found in the stratosphere, not the troposphere. It is located at an altitude of approximately 10 to 50 km above Earth's surface.
• The troposphere is the closest layer to Earth's surface and extends up to about 8-15 km, depending on geographic location. It contains most of the atmospheric water vapor, making it the layer where weather occurs.
• The stratosphere lies above the troposphere and extends from about 15 km to 50 km. It is characterized by the presence of the ozone layer, which absorbs harmful ultraviolet radiation from the Sun.
• Both statements in the question are factually incorrect, and therefore, the correct option is "Neither 1 nor 2."

Additional Information

• Troposphere:
• It is the lowest atmospheric layer and contains approximately 75% of the atmosphere's mass and nearly all its water vapor.
• Weather phenomena such as cloud formation, precipitation, storms, and winds occur in this layer.
• The temperature decreases with altitude in this layer due to the Earth's heat radiation.
• Stratosphere:
• It lies above the troposphere and extends from about 15 km to 50 km above the Earth's surface.
• The ozone layer is located within the stratosphere, helping to absorb harmful UV radiation from the Sun.
• The temperature increases with altitude in the stratosphere due to the absorption of UV radiation by ozone.
• This layer is relatively stable and free from weather disturbances, making it ideal for high-altitude flights.
• Ozone Layer:
• It is a region within the stratosphere containing high concentrations of ozone (O₃).
• The ozone layer plays a critical role in protecting life on Earth by absorbing the majority of the Sun's harmful ultraviolet radiation.
• Depletion of the ozone layer, often caused by human-made chemicals like chlorofluorocarbons (CFCs), leads to increased UV radiation exposure, causing health and environmental risks.
• Atmospheric Layers:
• The Earth's atmosphere is divided into five principal layers: troposphere, stratosphere, mesosphere, thermosphere, and exosphere.
• Each layer has unique characteristics, such as composition, temperature variation, and phenomena occurring within it.

Shortcut Trick Substitute approximate decimal values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 ⇒ x ≈ 1.414 + 1.732 = 3.146 ⇒ y ≈ 2.236 + 1 = 3.236 ⇒ z ≈ 2 × 1.414 + 1 = 3.828 ⇒ Comparing values: 3.146 < 3.236 < 3.828 ⇒ x < y < z ∴ The correct answer is x < y < z. Alternate Method Given: x = √2 + √3 y = √5 + √1 z = 2√2 + 1 Calculations: ⇒ Squaring all the given terms to compare their magnitudes: ⇒ x2 = (√2 + √3)2 = 2 + 3 + 2√6 = 5 + √24 ⇒ y2 = (√5 + 1)2 = 5 + 1 + 2√5 = 6 + √20 ⇒ z2 = (2√2 + 1)2 = 8 + 1 + 4√2 = 9 + √32 ⇒ Using approximations for easier comparison: √20 ≈ 4.47, √24 ≈ 4.89, √32 ≈ 5.65 ⇒ x2 ≈ 5 + 4.89 = 9.89 ⇒ y2 ≈ 6 + 4.47 = 10.47 ⇒ z2 ≈ 9 + 5.65 = 14.65 ⇒ Comparing the calculated squares gives: 9.89 < 10.47 < 14.65 ⇒ x2 < y2 < z2 ⇒ Since x, y, and z are all positive numerical values, their order remains the same: x < y < z. ∴ The correct answer is x < y < z. Additional Information Comparing Surds To compare irrational numbers (surds), squaring both sides is a standard method. It removes the square roots and helps compare the resulting rational and irrational parts more easily. Important Square Roots Memorising common square roots helps quickly estimate and simplify values in exams: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449. Inequality Property of Squares For any positive real numbers a and b, if a2 > b2, it directly implies a > b. This fundamental property allows us to determine the order of numbers by ordering their squares.

Shortcut Trick

Net percentage change can be calculated directly using the weighted average of the given ratios.

Net % Change = [(Ratio₁ × %₁) + (Ratio₂ × %₂) + (Ratio₃ × %₃)] ÷ Total Ratio parts

Net % Change = [5 × 5 + 3 × 10 + 2 × (−20)] ÷ (5 + 3 + 2)

Net % Change = (25 + 30 − 40) ÷ 10 = 15 ÷ 10

Net % Change = +1.5% (A positive value denotes an increase)

∴ The correct answer is 1.5% increase.

Alternate Method

Given:

Ratio of cost of land acquisition : building materials : labor = 5 : 3 : 2

Change in land acquisition cost = +5%

Change in building materials cost = +10%

Change in labor cost = −20%

Formula Used:

Percentage change = (Net Change ÷ Initial Total) × 100

Land (500)+5% (→ +25)Materials (300)+10% (→ +30)Labor (200)−20% (→ −40)Total Base = 1000Net Change = +25 + 30 − 40 = +15

Calculation:

⇒ Let the initial cost of land acquisition, building materials, and labor be 500, 300, and 200 respectively.

⇒ Total initial project cost = 500 + 300 + 200 = 1000

⇒ Change in land acquisition cost = 5% of 500 = +25

⇒ Change in building materials cost = 10% of 300 = +30

⇒ Change in labor cost = −20% of 200 = −40

⇒ Net change in total cost = 25 + 30 − 40 = +15

⇒ Percentage change = (15 ÷ 1000) × 100

⇒ Percentage change = 1.5%

⇒ Since the net change is positive, there is an overall increase in the total cost.

∴ The correct answer is 1.5% increase.

Additional Information

Percentage Change

Percentage change is calculated as [(New Value − Old Value) ÷ Old Value] × 100. A positive result denotes an increase, and a negative result denotes a decrease.

Weighted Average Method

When components of a total are subjected to different percentage changes, the net percentage change can be found using the weighted average formula: Net % = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n).

Successive Percentage Change

If a single quantity changes by a% and then further changes by b%, the effective percentage change is given by: Net Change = a + b + (a × b)÷100 %.

Shortcut Trick

According to BODMAS, solve inner brackets first: (1 − 1/3) = 2/3

Multiply inner terms: 1/2 × 2/3 = 1/3

Solve curly brackets: 1/6 + 1/3 = 1/2

Solve square brackets: 2/3 ÷ 1/2 = 2/3 × 2 = 4/3

Final subtraction: 6 − 4/3 = 14/3

∴ The correct answer is 14/3.

Alternate Method

Given:

Expression: 6 − [2/3 ÷ {1/6 + (1/2 × (1 − 1/3))}]

Formula Used:

BODMAS Rule: Brackets → Of → Division → Multiplication → Addition → Subtraction

Calculations:

⇒ Step 1: Solve the innermost round bracket ( )

⇒ 1 − 1/3 = (3 − 1)/3 = 2/3

⇒ The expression becomes: 6 − [2/3 ÷ {1/6 + (1/2 × 2/3)}]

⇒ Step 2: Solve the multiplication inside the remaining round bracket

⇒ 1/2 × 2/3 = 2/6 = 1/3

⇒ The expression becomes: 6 − [2/3 ÷ {1/6 + 1/3}]

⇒ Step 3: Solve the addition inside the curly bracket { }

⇒ 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2

⇒ The expression becomes: 6 − [2/3 ÷ 1/2]

⇒ Step 4: Solve the division inside the square bracket [ ]

⇒ 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3

⇒ The expression becomes: 6 − 4/3

⇒ Step 5: Perform the final subtraction

⇒ 6 − 4/3 = (18 − 4)/3 = 14/3

∴ The correct answer is 14/3.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction. It sets the standard order of operations to solve mathematical expressions correctly.

Types of Brackets

Brackets must be solved in a specific order: Line bracket or Vinculum first, followed by Round brackets ( ), Curly brackets { }, and finally Square brackets [ ].

Fraction Division Rule

When dividing two fractions, you must multiply the first fraction by the reciprocal of the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c).

Shortcut Trick

Since the time period is identical for all, the profit ratio equals the investment ratio.

Investment Ratio of X : Y : Z = 25000 : 35000 : 40000 = 5 : 7 : 8

Total profit parts = 5 + 7 + 8 = 20 parts

Y's share = (7 ÷ 20) × 30000 = 7 × 1500 = 10500

∴ The correct answer is ₹10,500.

Alternate Method

Given:

Investment of X = ₹25,000

Investment of Y = ₹35,000

Investment of Z = ₹40,000

Total profit at the end of the year = ₹30,000

Formula Used:

Profit Ratio = Capital Investment × Time Period

Partnership Investment & Profit Ratio

₹25,000 5 parts Partner X

₹35,000 7 parts Partner Y

₹40,000 8 parts Partner Z

Total Profit = ₹30,000 (20 parts)

Calculations:

⇒ Profit Ratio of X : Y : Z = (25000 × 1) : (35000 × 1) : (40000 × 1)

⇒ X : Y : Z = 25 : 35 : 40

⇒ X : Y : Z = 5 : 7 : 8

⇒ Sum of profit ratios = 5 + 7 + 8 = 20 parts

⇒ 20 parts = ₹30,000

⇒ 1 part = 30000 ÷ 20 = ₹1,500

⇒ Y's share = 7 parts = 7 × 1500 = ₹10,500

∴ The correct answer is ₹10,500.

Additional Information

Simple Partnership

When all partners invest their money for the exact same duration, the profit or loss is distributed purely in the ratio of their original investment amounts.

Compound Partnership

When partners invest their capital for different time periods, the profit ratio is calculated as the product of their investment and their respective time duration (Profit = Capital × Time).

Active vs Sleeping Partner

An active partner manages the business operations and generally receives a fixed percentage or salary from the profit before the remaining profit is distributed among all partners (including sleeping partners) based on their capital ratio.

Shortcut Trick

Ratio of Profit = (Capital_A × Time_A) : (Capital_B × Time_B)

Profit Ratio = [(48 × 6) + (24 × 6)] : [(72 × 6) + (96 × 6)]

Profit Ratio = [6 × (48 + 24)] : [6 × (72 + 96)] = 72 : 168 = 3 : 7

B's share = (7 ÷ 10) × 55000 = ₹38,500

∴ The correct answer is ₹38,500.

Alternate Method

Given:

Initial capital of A = ₹48,000

Initial capital of B = ₹72,000

Total time of investment = 1 year = 12 months

Total profit earned at the end of the year = ₹55,000

A

₹48k × 6m

₹24k × 6m = 432k

B

₹72k × 6m

₹96k × 6m = 1008k

Profit Ratio (A : B) = 432 : 1008 = 3 : 7

Formula Used:

Profit Ratio = Product of Capital and Time Period = C₁T₁ : C₂T₂

Calculations:

⇒ Equivalent capital of A = (₹48,000 × 6 months) + (₹24,000 × 6 months)

⇒ Equivalent capital of A = ₹2,88,000 + ₹1,44,000 = ₹4,32,000

⇒ Equivalent capital of B = (₹72,000 × 6 months) + (₹96,000 × 6 months)

⇒ Equivalent capital of B = ₹4,32,000 + ₹5,76,000 = ₹10,08,000

⇒ Ratio of profit (A : B) = 4,32,000 : 10,08,000

⇒ Ratio of profit (A : B) = 432 : 1008 = 3 : 7

⇒ Sum of ratio parts = 3 + 7 = 10 parts

⇒ B's share in profit = (7 ÷ 10) × Total Profit

⇒ B's share = (7 ÷ 10) × 55,000

⇒ B's share = 7 × 5,500 = ₹38,500

∴ The correct answer is ₹38,500.

Additional Information

Partnership Profit Ratio

When investments are made for unequal time periods, the profit is distributed in the ratio of the product of capital and corresponding time period: C₁T₁ : C₂T₂ : C₃T₃.

Sleeping vs Working Partner

A working partner manages the business and usually receives a fixed salary or a certain percentage of the total profit before the remaining profit is distributed according to capital ratios.

Joining Mid-Way

If a partner joins x months after the start of a year, their investment time period is taken as (12 − x) months when calculating the effective capital for profit distribution.

Shortcut Trick

Using the deviation method to find the average easily:

Let the assumed average be 220.

Deviations of Worker C's daily wages (220, 230, 215, 225) from 220 are: 0, +10, −5, +5.

Net deviation = 0 + 10 − 5 + 5 = +10

Average deviation = +10 ÷ 4 = +2.5

Actual average = 220 + 2.5 = 222.5

∴ The correct answer is 222.5.

Alternate Method

Given:

Daily earnings of Worker C extracted from the table:

Day 1 = 220, Day 2 = 230, Day 3 = 215, Day 4 = 225

Total number of days = 4

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Total earnings of Worker C over 4 days = 220 + 230 + 215 + 225

⇒ Total earnings of Worker C = 890

⇒ Average daily earning = 890 ÷ 4

⇒ Average daily earning = 222.5

∴ The correct answer is 222.5.

Additional Information

Arithmetic Mean (Average)

It is the simplest and most widely used measure of central tendency. The formula is simply the sum of elements ÷ number of elements.

Deviation Method for Averages

Assume a base average value. Find the deviation (difference) of each number from this base. The actual average is the assumed base plus the net average of these deviations. This greatly reduces calculation time for large numbers.

Data Interpretation (Tables)

Tabular data presents numerical figures in rows and columns. Always read the row and column headers carefully to extract the precise data points needed before initiating any calculation.

Shortcut Trick

Net increase in population = 25250 − 25000 = 250.

Overall percentage change = (250 ÷ 25000) × 100 = +1%.

Using the Alligation method on the percentage change of Males (−5%) and Females (+10%):

Ratio of Males to Females = (10 − 1) : (1 − (−5)) = 9 : 6 = 3 : 2.

Present number of literate males = (3 ÷ 5) × 25000 = 15000.

∴ The correct answer is 15,000 males.

Alternate Method

Given:

Total current literate population = 25,000

New total literate population = 25,250

Net increase = 25,250 − 25,000 = 250

Total: 25,000Males (M)Females (F)−5%+10%Net Change: +250

Calculations:

Let the current number of literate males be M and females be F.

⇒ M + F = 25,000 --- (Equation 1)

The number of males decreases by 5% and females increases by 10%, resulting in a net increase of 250.

⇒ 10% of F − 5% of M = 250

⇒ (10 ÷ 100) × F − (5 ÷ 100) × M = 250

Multiplying the entire equation by 100:

⇒ 10F − 5M = 25,000

Dividing by 5 to simplify:

⇒ 2F − M = 5,000 --- (Equation 2)

Adding Equation 1 and Equation 2:

⇒ (M + F) + (2F − M) = 25,000 + 5,000

⇒ 3F = 30,000

⇒ F = 10,000

Substituting the value of F back into Equation 1:

⇒ M + 10,000 = 25,000

⇒ M = 15,000

∴ The correct answer is 15,000 males.

Additional Information

Alligation Rule

It is a powerful technique used to quickly find the ratio in which two components are combined to achieve a given overall change. The shortcut structure dictates: Ratio = (Higher % − Net %) : (Net % − Lower %).

Percentage Shift Method (Alternate Mental Trick)

If both groups decreased by 5%, the total decrease would be 5% of 25,000 = 1,250. The actual change is an increase of 250, creating a numerical gap of 1,500. This gap is entirely filled by the 15% difference in the females' group (from −5% to +10%). Hence, 15% of Females = 1,500 ⇒ Females = 10,000.

Linear Equations in Two Variables

When solving questions with two interconnected parts (like males and females in a total population), setting up two equations (Sum of parts = Total, and Sum of their changes = Net change) reliably yields an exact algebraic solution without getting confused by ratios.

Shortcut Trick

Let the price of standard flour = ₹1/kg and premium flour = ₹2/kg.

Original cost = 10 × 2 + x × 1 = 20 + x

New cost = x × 2 + 10 × 1 = 2x + 10

As cost decreased by 25%, New Cost = 75% of Original Cost ⇒ 3/4

⇒ (2x + 10) / (20 + x) = 3/4 ⇒ 8x + 40 = 60 + 3x ⇒ 5x = 20 ⇒ x = 4 kg.

∴ The correct answer is 5:2.

Alternate Method

Original Order Prem: 10 kg @ 2P Std: x kg @ 1P Cost = 20P + xP

New Order Prem: x kg @ 2P Std: 10 kg @ 1P Cost = 2xP + 10P

−25% Cost

Given:

⇒ Premium flour quantity = 10 kg

⇒ Let standard flour quantity = x kg

⇒ Let standard flour price = P per kg, then premium flour price = 2P per kg

Calculations:

⇒ Original Total Cost = (10 × 2P) + (x × P) = P(20 + x)

⇒ After interchange, New Total Cost = (x × 2P) + (10 × P) = P(2x + 10)

⇒ The new cost is 25% less than the original cost.

⇒ New Cost = (100% − 25%) of Original Cost = 75% of Original Cost

⇒ P(2x + 10) = 0.75 × P(20 + x)

⇒ 2x + 10 = 0.75(20) + 0.75x

⇒ 2x + 10 = 15 + 0.75x

⇒ 2x − 0.75x = 15 − 10

⇒ 1.25x = 5

⇒ x = 5 ÷ 1.25 = 4

⇒ The original quantity of standard flour is 4 kg.

⇒ Ratio of premium to standard flour = 10 : 4 = 5 : 2

∴ The correct answer is 5:2.

Additional Information

Cost-Quantity Relationship

The total cost or value of a commodity is the product of its unit price and the total quantity purchased: Total Cost = Price per unit × Quantity.

Percentage Change

A decrease of R% in a value means the new value becomes (100 − R)% of the original value. For example, a 25% decrease results in a multiplier of 0.75 or 3/4.

Ratio and Proportion

A ratio compares two quantities of the same unit, representing how many times one value contains or is contained within the other. Ratios should always be expressed in their simplest form.

Shortcut Trick

Using the combined ratio method: Amount₁ ÷ (Quantity₁ × SP₁%) = Amount₂ ÷ (Quantity₂ × SP₂%)

⇒ 3000 ÷ (15 × 125%) = 2816 ÷ (N × 110%)

⇒ 200 ÷ 125 = 2816 ÷ 110N

⇒ 8 ÷ 5 = 25.6 ÷ N ⇒ N = 25.6 × 5 ÷ 8 = 16

∴ The correct answer is 16 toys.

Alternate Method

Given:

Selling price of 15 toys = ₹3000

Initial profit percentage = 25%

Required profit percentage = 10%

Target selling amount = ₹2816

CP: ₹160

SP&sub1;: ₹200

SP&sub2;: ₹176

+25% Profit

+10% Profit

Calculations:

⇒ Selling price of 1 toy (SP₁) = 3000 ÷ 15 = ₹200

⇒ Cost price of 1 toy (CP) = SP₁ × [100 ÷ (100 + P₁%)]

⇒ CP = 200 × (100 ÷ 125) = ₹160

⇒ New selling price of 1 toy for 10% profit (SP₂) = CP × [100 + P₂%] ÷ 100

⇒ SP₂ = 160 × (110 ÷ 100) = ₹176

⇒ Number of toys to sell = Target amount ÷ SP₂

⇒ Number of toys = 2816 ÷ 176 = 16

∴ The correct answer is 16 toys.

Additional Information

Cost Price Formula

Cost Price = Selling Price × [100 ÷ (100 + Profit %)]. It acts as the fundamental base for calculating any subsequent profit or loss margins.

Successive Changes in Selling Price

When the SP varies due to a different profit or loss percentage on the same object, use the direct ratio: SP₁ ÷ (100 + P₁) = SP₂ ÷ (100 + P₂).

Unitary Method for Quantities

Total Value = Price per unit × Total Quantity. Applying this helps compare different transaction scenarios independently of the absolute Cost Price.

Shortcut Trick

CP of Brand A = 150 × (100 / 75) = ₹200/kg

CP of Brand B = 200 × (100 / 80) = ₹250/kg

For an equal mixture (1 kg each), Total CP = 200 + 250 = ₹450

Total SP for 2 kg = 175 × 2 = ₹350

Loss % = ((450 − 350) / 450) × 100 = (100 / 450) × 100 = 22.22%

∴ The correct answer is Loss of 22.22%.

Alternate Method

Given:

SP of Brand A = ₹150/kg, Loss = 25%

SP of Brand B = ₹200/kg, Loss = 20%

Mixture proportion = 1 : 1

SP of the mixture = ₹175/kg

Formula Used:

CP = SP × [100 / (100 − Loss %)]

Loss % = (Loss / Total CP) × 100

Brand A (1 kg) CP = ₹200

Brand B (1 kg) CP = ₹250

Mixture (2 kg) Total CP = ₹450

Total SP = ₹350 Overall Loss = ₹100

Calculations:

⇒ CP of Brand A = 150 × [100 / (100 − 25)]

⇒ CP of Brand A = 150 × (100 / 75) = ₹200/kg

⇒ CP of Brand B = 200 × [100 / (100 − 20)]

⇒ CP of Brand B = 200 × (100 / 80) = ₹250/kg

⇒ Assuming he mixes exactly 1 kg of each brand, Total Quantity = 2 kg

⇒ Total CP for 2 kg = 200 + 250 = ₹450

⇒ Total SP for 2 kg = 175 × 2 = ₹350

⇒ Overall Loss = Total CP − Total SP = 450 − 350 = ₹100

⇒ Overall Loss % = (Loss / Total CP) × 100

⇒ Overall Loss % = (100 / 450) × 100 = (2 / 9) × 100 ≈ 22.22%

∴ The correct answer is Loss of 22.22%.

Additional Information

Weighted Average Cost Price

When mixing two quantities in ratio n₁ : n₂, the average CP = (n₁ × CP₁ + n₂ × CP₂) / (n₁ + n₂).

Alligation Method

The ratio of quantities mixed is inversely proportional to the difference between their unit prices and the mean price: (Quantity 1) / (Quantity 2) = (CP₂ − Mean Price) / (Mean Price − CP₁).

Profit and Loss on Overall Transaction

To find the exact profit or loss percentage for mixed articles, always calculate the total Selling Price (SP) of the final mixture and compare it against the total Cost Price (CP) of the combined ingredients.

Shortcut Trick

Total work = LCM of 12, 15 and 20 = 60 units.

Combined efficiencies: (A + B) = 60 ÷ 12 = 5, (B + C) = 60 ÷ 15 = 4, (C + A) = 60 ÷ 20 = 3.

Total efficiency of 2(A + B + C) = 5 + 4 + 3 = 12 ⇒ Efficiency of (A + B + C) = 6.

Efficiency of A = Total Efficiency − Efficiency of (B + C) = 6 − 4 = 2 units/day.

Time taken by A alone = Total work ÷ Efficiency of A = 60 ÷ 2 = 30 days.

∴ The correct answer is 30 days.

Alternate Method

Given:

Time taken by A and B together = 12 days

Time taken by B and C together = 15 days

Time taken by C and A together = 20 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Adding all the combined efficiencies: (A + B) + (B + C) + (C + A) = 5 + 4 + 3

⇒ 2(A + B + C) = 12

⇒ Efficiency of A + B + C = 12 ÷ 2 = 6 units/day

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

⇒ Efficiency of A = 6 − 4 = 2 units/day

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

⇒ Time taken by A alone = 60 ÷ 2 = 30 days

∴ The correct answer is 30 days.

Additional Information

Pipes & Cisterns

This concept works exactly like Time & Work. Inlet pipes have positive efficiency (filling) and outlet pipes have negative efficiency (emptying).

Work & Wages

Wages are always distributed in the direct ratio of the total work done by each person. If the working time is equal for all, wages are distributed in the ratio of their efficiencies.

Alternate Day Work

When people work on alternate days, calculate the combined work done in one complete cycle (e.g., 2 days for 2 people) and use it to find the number of cycles required to complete the total work.

Shortcut Trick

Direct Replacement Formula: Remaining Spirit = Initial Quantity × (1 − Replaced Quantity / Initial Quantity)ⁿ

⇒ Remaining Quantity = 64 × (1 − 8 / 64)³

⇒ Remaining Quantity = 64 × (7 / 8)³ = 64 × (343 / 512)

⇒ Remaining Quantity = 343 / 8 = 42.875 ≈ 42.88 liters

∴ The correct answer is 42.88 liters.

Alternate Method

Repeated Replacement Process

Initial 64L Pure Spirit

• 8L Mixture

• 8L Water (Repeated 3 Times)

Final 42.88L Spirit

Given:

Initial volume of pure spirit (V) = 64 liters

Volume of mixture replaced with water each time (x) = 8 liters

Total number of operations (n) = 3

Formula Used:

Final Quantity of Spirit = V × (1 − x / V)ⁿ

Calculations:

⇒ Final Quantity = 64 × (1 − 8 / 64)³

⇒ Final Quantity = 64 × (1 − 1 / 8)³

⇒ Final Quantity = 64 × (7 / 8)³

⇒ Final Quantity = 64 × (343 / 512)

⇒ Final Quantity = 343 / 8

⇒ Final Quantity = 42.875 ≈ 42.88 liters

∴ The correct answer is 42.88 liters.

Additional Information

Repeated Replacement Formula

If a container holds V units of pure liquid, and x units are replaced by water n times, the final quantity is given by V × (1 − x/V)ⁿ.

Successive Replacement

If varying amounts a, b, and c are replaced successively, the remaining pure liquid is V × (1 − a/V) × (1 − b/V) × (1 − c/V).

Rule of Alligation

Used to find the ratio in which two ingredients of different prices or concentration are mixed. The ratio is calculated as (Dearer Quantity − Mean Quantity) : (Mean Quantity − Cheaper Quantity).

Shortcut Trick

When a mixture is sold at the cost price of the pure liquid, the profit earned is entirely due to the free liquid (water) added.

Ratio of Water : Pure Liquid = Profit % : 100%

Ratio of Water : Pure Orange Juice = 20 : 100

Ratio of Water : Pure Orange Juice = 1 : 5

∴ The correct answer is 1 : 5.

Alternate Method

Given:

Profit earned = 20%

Selling price of the mixture = Cost price of pure orange juice

Formula Used:

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

Mean Price (CP of Mixture) = (SP × 100) ÷ (100 + Profit %)

Calculation:

Water CP = 0

Juice CP = 100

Mean = 500/6

100 − 500/6 100/6

500/6 − 0 500/6

Ratio = 1 : 5

⇒ Let the cost price (CP) of pure orange juice be ₹100 per unit.

⇒ Since the mixture is sold at the CP of pure juice, Selling Price (SP) of the mixture = ₹100.

⇒ The given profit is 20%, so CP of the mixture = (SP × 100) ÷ (100 + 20)

⇒ CP of the mixture = (100 × 100) ÷ 120 = 500/6.

⇒ The cost price of water is ₹0 per unit.

⇒ Applying the rule of alligation as shown in the diagram above:

⇒ Quantity of water = 100 − 500/6 = 100/6

⇒ Quantity of pure juice = 500/6 − 0 = 500/6

⇒ Ratio of water to pure juice = 100/6 : 500/6

⇒ Ratio = 1 : 5

∴ The correct answer is 1 : 5.

Additional Information

Rule of Alligation

It states that if two ingredients at given prices are mixed to produce a mixture at a given mean price, the ratio of their quantities is inversely proportional to the differences of their prices from the mean price. Ratio = (Price of Dearer − Mean Price) : (Mean Price − Price of Cheaper).

Dishonest Dealer Concept

When a dealer professes to sell his goods at cost price but uses a false weight or adds impurities, the profit percentage is calculated as: Profit % = (Error ÷ True Value) × 100. This is conceptually identical to adding free water to pure goods.

Replacement Formula in Mixtures

If from a vessel containing 'x' units of pure liquid, 'y' units are removed and replaced with water, and this operation is repeated 'n' times, the quantity of pure liquid left = x × (1 − y/x)ⁿ.

Shortcut Trick

Let the total distance be assumed as 100 km.

First part of the distance = 40 km, covered at 20 km/h.

Time taken = 40 ÷ 20 = 2 hours.

Remaining distance = 60 km, covered at 30 km/h.

Time taken = 60 ÷ 30 = 2 hours.

Average Speed = Total Distance ÷ Total Time = 100 ÷ (2 + 2) = 25 km/h.

∴ The correct answer is 25 km/h.

Alternate Method

Given:

First part of distance (D₁) = 40% of total distance

Speed for first part (S₁) = 20 km/h

Remaining distance (D₂) = 60% of total distance

Speed for remaining part (S₂) = 30 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

40% Distance 60% Distance

20 km/h 30 km/h

Total Distance Journey Average Speed = ?

Calculations:

⇒ Let the total distance be D km.

⇒ Time taken for the first part = 0.4D ÷ 20 = 4D ÷ 200 = D ÷ 50 hours

⇒ Time taken for the remaining part = 0.6D ÷ 30 = 6D ÷ 300 = D ÷ 50 hours

⇒ Total Time taken = (D ÷ 50) + (D ÷ 50) = 2D ÷ 50 = D ÷ 25 hours

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = D ÷ (D ÷ 25)

⇒ Average Speed = 25 km/h

∴ The correct answer is 25 km/h.

Additional Information

Average Speed for Equal Distances

When a certain distance is covered at speed 'x' and an equal distance at speed 'y', the Average Speed = (2xy) ÷ (x + y).

Average Speed for Equal Time Intervals

When travelling for equal time periods at speeds 'x' and 'y', the Average Speed = (x + y) ÷ 2.

Relative Speed

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

Shortcut Trick

Relative speed (opposite directions) = 5 + 7 = 12 km/h.

Time for first meeting = Distance ÷ Relative speed = 1.2 ÷ 12 = 0.1 hours = 6 minutes.

Time for 5th meeting = 5 × 6 = 30 minutes.

Meeting Time = 6:00 AM + 30 minutes = 6:30 AM.

∴ The correct answer is 6:30 AM.

Alternate Method

Start Point (6:00 AM)

A: 5 km/h

B: 7 km/h

Perimeter = 1.2 km Opposite Directions

Given:

Perimeter of track (Distance) = 1.2 km

Speed of A (S₁) = 5 km/h

Speed of B (S₂) = 7 km/h

Formula Used:

Time = Distance ÷ Relative Speed

Relative Speed (Opposite Directions) = S₁ + S₂

Calculations:

⇒ Relative Speed = 5 + 7 = 12 km/h

⇒ Time for 1st meeting = 1.2 ÷ 12 = 0.1 hours

⇒ Converting hours into minutes: 0.1 × 60 = 6 minutes

⇒ Time taken to meet for the 5th time = 5 × 6 = 30 minutes

⇒ Meeting Time = 6:00 AM + 30 minutes = 6:30 AM

∴ The correct answer is 6:30 AM.

Additional Information

Relative Speed in Same Direction

When two bodies move in the same direction, their relative speed is the difference between their speeds: |S₁ − S₂|.

Meeting on a Circular Track

Time taken to meet for the first time anywhere on the track = Track Length ÷ Relative Speed.

Meeting at Starting Point

Time taken to meet for the first time exactly at the starting point = LCM of individual times taken by each person to complete one full round.

Shortcut Trick

A central angle of 90° means the sector is exactly a quarter of the full circle.

Area of a full circle = π × r²

⇒ Full Area = (22/7) × 7 × 7 = 154 cm²

Area of the sector = 154 ÷ 4 = 38.5 cm²

∴ The correct answer is 38.5 cm².

Alternate Method

r = 7 cm 90° Area of Sector

Given:

Radius of the circle (r) = 7 cm

Central angle (θ) = 90°

Formula Used:

Area of a sector = (θ ÷ 360°) × π × r²

Calculations:

⇒ Area = (90° ÷ 360°) × (22/7) × 7²

⇒ Area = (1/4) × (22/7) × 49

⇒ Area = (1/4) × 154

⇒ Area = 38.5 cm²

∴ The correct answer is 38.5 cm².

Additional Information

Length of an Arc

The length of the arc bounding the sector is given by: l = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The perimeter of a sector includes the arc length and the two radii forming the sector: Perimeter = l + 2r.

Area of a Segment

A segment is the region bounded by a chord and an arc. Its area is: Area of segment = Area of sector − Area of the corresponding triangle.

Shortcut Trick

To convert any angle from degrees to radians, multiply the degree value by π / 180°.

⇒ Radian measure = 150° × (π / 180°)

⇒ Radian measure = 15π / 18

⇒ Radian measure = 5π / 6

∴ The correct answer is 5π/6.

Alternate Method

Given:

Angle measure = 150°

Formula Used:

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

150° 0° 150° = 5π/6

Calculations:

⇒ Radian measure = 150° × (π / 180°)

⇒ Radian measure = (150 / 180) × π

⇒ Radian measure = (15 / 18) × π

⇒ Radian measure = 5π / 6

∴ The correct answer is 5π/6.

Additional Information

Degrees to Radians Conversion

The formula to convert degrees to radians is: Radians = Degrees × (π / 180°).

Radians to Degrees Conversion

The formula to convert an angle from radians back to degrees is: Degrees = Radians × (180° / π).

Standard Angles in Radians

Important values to remember: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π.

Shortcut Trick

Since the central angle is 120°, the sector represents exactly 120° ÷ 360° = 1/3 of the full circle.

Area of the full circle = π × r² = π × 8² = 64π cm².

Area of the sector = (1 / 3) × 64π = 64π/3 cm².

∴ The correct answer is 64π/3 cm².

Alternate Method

120° r = 8 cm r = 8 cm

Centre O Area of Sector

Given:

⇒ Radius of the sector (r) = 8 cm

⇒ Central angle (θ) = 120°

Formula Used:

⇒ Area of Sector = (θ ÷ 360°) × πr²

Calculations:

⇒ Area = (120° ÷ 360°) × π × (8)²

⇒ Area = (1 / 3) × π × 64

⇒ Area = 64π/3 cm²

∴ The correct answer is 64π/3 cm².

Additional Information

Area of a Circle

The total region enclosed by a circle is given by πr², which corresponds to a full central angle of 360°.

Length of an Arc

The length of the curved boundary of a sector is proportional to its central angle, given by the formula L = (θ ÷ 360°) × 2πr.

Perimeter of a Sector

The total boundary length of a sector is obtained by adding the arc length to twice the radius: Perimeter = L + 2r.

Area of a Segment

The region bounded by a chord and its corresponding arc. It is calculated by subtracting the area of the corresponding triangle from the total area of the sector.

Shortcut Trick

For any linear equation ax + by = c, all lines parallel to it are of the identical form ax + by = k (where k ≠ c).

The given line is 3x + 2y = 5.

By comparing the options, only the equation 3x + 2y = 7 matches this exact a and b coefficient format.

∴ The correct answer is 3x + 2y = 7.

Alternate Method

Given:

The equation of the original line is 3x + 2y = 5.

Concept Used:

Two lines are strictly parallel if their slopes are equal (m₁ = m₂).

For a line represented as ax + by = c, the slope m is determined by −a/b.

X Y O

3x + 2y = 5 3x + 2y = 7

Calculations:

⇒ Slope of the given line 3x + 2y = 5 is m = −3/2.

⇒ Let us calculate the slopes of all given options to find the identical one:

⇒ 1) 3x + 2y = 7 → Slope = −3/2 (Parallel to given line)

⇒ 2) 2x + 3y = 5 → Slope = −2/3 (Not parallel)

⇒ 3) 3x − 2y = 5 → Slope = 3/2 (Not parallel)

⇒ 4) x + y = 5 → Slope = −1 (Not parallel)

⇒ Only the first equation has the same slope as the original given line.

∴ The correct answer is 3x + 2y = 7.

Additional Information

Parallel Lines Condition

Two general lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are strictly parallel and do not touch when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Perpendicular Lines Condition

Two lines cross each other at exactly 90 degrees if the mathematical product of their slopes equals −1 (m₁ × m₂ = −1). The perpendicular inverse format is bx − ay + k = 0.

Intersecting Lines Condition

Linear graphs will always intersect at exactly one unique point if they possess structurally different slopes, proven when a₁/a₂ ≠ b₁/b₂.

Coincident Lines Condition

Two drawn lines represent the exact same path and hold infinite solutions universally if their constants are fully proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.

Shortcut Trick

Using the identity sin(90° − θ) = cos(θ), we get cos(x) = cos(2x).

By comparing the angles directly for the principal value, we can write:

x = 2x ⇒ x = 0°

Checking the options, 0° satisfies the equation: sin(90°) = cos(0°) ⇒ 1 = 1.

∴ The correct answer is 0°.

Alternate Method

Given:

sin(90° − x) = cos(2x)

Formula Used:

sin(90° − θ) = cos(θ)

cos(2θ) = 2cos²(θ) − 1

Calculation:

⇒ sin(90° − x) = cos(2x)

⇒ cos(x) = cos(2x)

⇒ cos(x) = 2cos²(x) − 1

⇒ 2cos²(x) − cos(x) − 1 = 0

⇒ 2cos²(x) − 2cos(x) + cos(x) − 1 = 0

⇒ 2cos(x)(cos(x) − 1) + 1(cos(x) − 1) = 0

⇒ (2cos(x) + 1)(cos(x) − 1) = 0

⇒ cos(x) = 1 or cos(x) = −1/2

For x in the first quadrant (as per options), cos(x) = 1.

⇒ x = 0°

∴ The correct answer is 0°.

Additional Information

Complementary Angle Identities

For any angle θ, sin(90° − θ) = cos(θ), cos(90° − θ) = sin(θ), and tan(90° − θ) = cot(θ).

General Solution of Cosine

If cos(x) = cos(y), then the general solution is x = 360° × n ± y, where n is any integer.

Double Angle Formula

cos(2θ) can be expressed as cos²(θ) − sin²(θ), or 2cos²(θ) − 1, or 1 − 2sin²(θ).

Shortcut Trick

Given x = √5 + 2

Square both sides directly: x² = 5 + 4 + 4√5 = 9 + 4√5

We need to find the value of x² − 4√5

Substitute x²: 9 + 4√5 − 4√5 = 9

∴ The correct answer is 9.

Alternate Method

Given:

x = √5 + 2

Formula Used:

(a + b)² = a² + b² + 2ab

Calculation:

⇒ x² = (√5 + 2)²

⇒ x² = (√5)² + (2)² + 2 × 2 × √5

⇒ x² = 5 + 4 + 4√5

⇒ x² = 9 + 4√5

Now, the required expression is x² − 4√5.

Substituting the value of x²:

⇒ (9 + 4√5) − 4√5

⇒ 9

∴ The correct answer is 9.

Additional Information

Algebraic Identities

Basic identities like (a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab are fundamental in simplification questions involving surds and radicals.

Simplification of Surds

When squaring a binomial containing a surd, the rational parts and irrational parts are combined separately. For example, (√x + y)² = x + y² + 2y√x.

Conjugate Surds

The expressions (a + √b) and (a − √b) are conjugates. Their product (a + √b)(a − √b) = a² − b is always a rational number.

Shortcut Trick

The congruence of triangles is determined by matching their corresponding sides and angles.

Since the problem states that all three corresponding sides are exactly equal (AB = XY, BC = YZ, CA = ZX).

This perfectly matches the defining condition for the Side-Side-Side (SSS) congruence rule.

∴ The correct answer is SSS.

Alternate Method

Given:

⇒ In Triangle ABC and Triangle XYZ:

⇒ AB = XY

⇒ BC = YZ

⇒ CA = ZX

A B C

X Y Z

Concept Used:

⇒ SSS Congruence Rule: If three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.

Calculations:

⇒ We map the sides of both triangles to verify their relation.

⇒ AB = XY (First pair of sides is equal)

⇒ BC = YZ (Second pair of sides is equal)

⇒ CA = ZX (Third pair of sides is equal)

⇒ Since all three corresponding sides are exactly equal, Triangle ABC is identically congruent to Triangle XYZ under the Side-Side-Side property.

∴ The correct answer is SSS.

Additional Information

SAS (Side-Angle-Side) Congruence

Two triangles are congruent if two sides and the included angle of one triangle are exactly equal to two sides and the included angle of the other.

ASA (Angle-Side-Angle) Congruence

Two triangles are congruent if two angles and the included side of one triangle are exactly equal to two angles and the included side of the other.

AAS (Angle-Angle-Side) Congruence

Two triangles are congruent if any two angles and a non-included side of one triangle are equal to the corresponding angles and side of the other.

RHS (Right Angle-Hypotenuse-Side) Congruence

Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other.

Shortcut Trick

A and B are complementary angles ⇒ A + B = 90° ⇒ B = 90° − A

tanB = tan(90° − A) = cotA

Given sinA = 3/5 = Perpendicular / Hypotenuse ⇒ Base = √(5² − 3²) = 4

cotA = Base / Perpendicular = 4/3

∴ The correct answer is 4/3.

Alternate Method

Given:

sinA = 3/5

A and B are complementary angles (A + B = 90°).

Formula Used:

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

tanθ = Perpendicular ÷ Base

ABBase = 4Perpendicular = 3Hypotenuse = 5

Calculations:

⇒ In a right-angled triangle, sinA = Perpendicular ÷ Hypotenuse = 3 ÷ 5.

⇒ By Pythagoras theorem: Base = √(Hypotenuse² − Perpendicular²)

⇒ Base = √(5² − 3²) = √(25 − 9) = √16 = 4.

⇒ Since A and B are complementary, they form the two acute angles of the same right-angled triangle.

⇒ For angle B, the perpendicular and base sides are swapped relative to angle A.

⇒ Perpendicular to B = Base to A = 4.

⇒ Base to B = Perpendicular to A = 3.

⇒ tanB = (Perpendicular to B) ÷ (Base to B)

⇒ tanB = 4/3

∴ The correct answer is 4/3.

Additional Information

Trigonometric Ratios of Complementary Angles

sin(90° − θ) = cosθ and tan(90° − θ) = cotθ. Complementary angles sum to exactly 90°.

Pythagorean Triplets

A set of positive integers satisfying a² + b² = c². Standard base triplets include (3, 4, 5) and (5, 12, 13).

Basic Trigonometric Ratios

sinθ = P/H, cosθ = B/H, tanθ = P/B, where P is Perpendicular, B is Base, and H is Hypotenuse.

Shortcut Trick

The sum of opposite angles in a cyclic quadrilateral is always 180°.

Angle A and Angle C are opposite angles in the cyclic quadrilateral ABCD.

∠C = 180° − ∠A

∠C = 180° − 100° = 80°

∴ The correct answer is 80°.

Alternate Method

A B C D

100° 80°

∠A + ∠C = 180°

Given:

∠A = 100°

ABCD is a cyclic quadrilateral.

Formula Used:

In a cyclic quadrilateral, the sum of a pair of opposite interior angles is equal to 180°.

Calculation:

⇒ ∠A + ∠C = 180°

⇒ 100° + ∠C = 180°

⇒ ∠C = 180° − 100°

⇒ ∠C = 80°

∴ The correct answer is 80°.

Additional Information

Cyclic Quadrilateral

A quadrilateral is called cyclic if all its four vertices lie on the circumference of a single circle. The primary property of a cyclic quadrilateral is that the sum of its opposite interior angles is always 180°.

Exterior Angle Property

If any one side of a cyclic quadrilateral is extended outwardly, the exterior angle formed is strictly equal to the interior opposite angle.

Ptolemy's Theorem

For a cyclic quadrilateral, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of its opposite sides (AC × BD = AB × CD + AD × BC).

Shortcut Trick

Using the fundamental trigonometric identity: sin²A + cos²A = 1

Subtract the given equation (sin²A − cos²A = 1/4) from this identity:

(sin²A + cos²A) − (sin²A − cos²A) = 1 − 1/4

2cos²A = 3/4

cos²A = 3/8

∴ The correct answer is 3/8.

Alternate Method

Given:

sin²A − cos²A = 1/4

Formula Used:

sin²A + cos²A = 1 ⇒ sin²A = 1 − cos²A

Calculations:

⇒ Substitute the value of sin²A in the given equation.

⇒ (1 − cos²A) − cos²A = 1/4

⇒ 1 − 2cos²A = 1/4

⇒ 2cos²A = 1 − 1/4

⇒ 2cos²A = 3/4

⇒ cos²A = 3/4 × 1/2

⇒ cos²A = 3/8

∴ The correct answer is 3/8.

Additional Information

Trigonometric Square Identities

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

Double Angle Formula for Cosine

The double angle formula can be expressed in three different forms:
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

The correct answer is: Sarcastic.

Key Points

• The word "Caustic" means sarcastic in a scathing and bitter way, or capable of burning, corroding, or destroying organic tissue by chemical action. (तीखा, कटु, या जलाने वाला)
• Example: Her caustic comments about his work left him feeling insulted.
• "Sarcastic" refers to using irony to mock or convey contempt. (व्यंग्यात्मक)
• Example: He made a sarcastic remark about her punctuality, knowing she was always late.
• Hence, the word "caustic" aligns closely with "sarcastic" in terms of its meaning.

Therefore, the correct answer is: Sarcastic.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Kind (दयालु): Having or showing a friendly, generous, and considerate nature.
• Example: She is always kind to animals and takes care of stray dogs.
• Mild (हल्का): Not severe, serious, or harsh.
• Example: The weather today is mild, perfect for a picnic.
• Gentle (कोमल): Having or showing a mild, kind, or tender temperament.
• Example: He gave her a gentle push to signal that it was her turn to speak.

The correct answer is: Option 3.

Key Points

• The idiom "Make no bones about it" means to be direct, clear, and honest about something without hesitation or ambiguity. (स्पष्ट और ईमानदार होना)
• Example: He made no bones about his dislike for the new policy and spoke openly against it.
• It is used when someone expresses their thoughts or opinions in a straightforward manner without sugar-coating their words. (बिना झिझक या संकोच के अपनी बात रखना)

Therefore, the correct answer is: Option 3.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Option 1: To hesitate before speaking (बोलने से पहले झिझकना): Refers to being uncertain or pausing before expressing thoughts.
• Example: He hesitated before speaking in the meeting, unsure of how his ideas would be received.
• Option 2: To remain neutral (निष्पक्ष रहना): Refers to not taking sides in a conflict or argument.
• Example: The mediator remained neutral throughout the discussion to resolve the dispute.
• Option 4: To confuse the listener (श्रोता को भ्रमित करना): Refers to speaking in a way that creates misunderstanding or lack of clarity.
• Example: His vague explanation confused the listener and led to further questions.

The correct answer is: Enduring.

Key Points

• The word "Evanescent" means something that is fleeting, vanishing, or lasting only for a very short time. (क्षणिक/अस्थायी)
• Example: The evanescent beauty of a sunset leaves us in awe, even though it lasts only a few moments.
• "Enduring" refers to something that is lasting, permanent, or continuing over time despite challenges or changes. (स्थायी)
• Example: His love for music is enduring, having lasted throughout his entire life.
• Hence, we can conclude that the opposite of 'Evanescent' is 'Enduring'.

Therefore, the correct answer is: Enduring.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Ephemeral (क्षणभंगुर): Something that lasts for a very short period of time.
• Example: The joy of winning the lottery was ephemeral, as money brought its own challenges.
• Transient (क्षणिक): Something that is temporary or short-lived.
• Example: The transient nature of the spring bloom makes it even more precious.
• Fleeting (क्षणिक): Something that passes quickly or is momentary.
• Example: She caught a fleeting glimpse of the bird before it disappeared into the forest.

The correct answer is option 4.

Key Points

The misspelled word is 'Inadvertant'.

• The correct spelling would be 'Inadvertent'.
• The word 'Inadvertent' is an adjective which means 'not resulting from or achieved through deliberate planning'. (असावधान)
• For Example -
• An inadvertent administrative error caused confusion.

Therefore, the correct answer is 'Inadvertant'.

Additional Information

Let's understand the meanings of the other marked words - 

• Avaricious - having or showing an extreme greed for wealth or material gain (लोभी)
• Temerarious - recklessly or presumptuously daring (अधीर)
• Penitent - feeling or showing sorrow and regret for having done wrong (पश्चातापी)

The correct answer is option 2.

Key Points

The misspelled word is 'Prophyroblastic'.

• The correct spelling would be 'Porphyroblastic'.
• The word 'Porphyroblastic' refers to a metamorphic texture characterized by large crystals (called porphyroblasts) within a finer-grained matrix.
• Porphyroblasts are formed during the metamorphism of rocks and can include minerals like garnet, staurolite, or kyanite.

Therefore, the correct answer would be 'Porphyroblastic'.

Additional Information

Other options and their details - 

• Option 1: Prophyroblastic - This is an incorrect spelling and does not refer to any known geological term.
• Option 3: Porfyrblastic - This is also an incorrect spelling.
• Option 4: Porphyroblastik - This is a misspelling and not a recognized term.

The correct answer is: Option 3 i.e. Plutocracy.

Key Points

• The word "Plutocracy" refers to a political system or society that is ruled or controlled by the wealthy. (धनपतियों का शासन)
• Example: Some critics argue that modern democracies are evolving into plutocracies where the rich have disproportionate influence over political decisions.
• "Plutocracy" is derived from the Greek words "ploutos" (wealth) and "kratos" (power), which signify "the rule of the wealthy."

Therefore, the correct answer is: Option 3 i.e. Plutocracy.

Additional Information

Here are the other options explained along with their Hindi meanings and example sentences:

• Oligarchy (गुटतंत्र): A form of government in which power is held by a small group of people.
• Example: The country was run as an oligarchy by a small elite group of political leaders.
• Aristocracy (कुलीनतंत्र): A form of government in which power is held by the nobility.
• Example: The aristocracy of the old European societies wielded significant power over the common people.
• Meritocracy (योग्यता आधारित शासन): A system in which advancement is based on individual ability or achievement.
• Example: The company promoted employees based on meritocracy, rewarding hard work and talent.

The correct answer is 'did she'.

Key Points

• The correct structure for inversion in the sentence is "did she" because the sentence starts with "Only after" requiring the auxiliary verb to precede the subject.
• Option 1 ("she had") is incorrect because it does not follow the grammatical rule of inversion.
• Option 3 ("had she") is incorrect because it suggests past perfect tense, which does not fit the context.
• Option 4 ("she did") is incorrect because it does not follow the correct inversion structure.

Therefore, the correct answer is Option 2.

Complete Sentence:"Only after a decade of fieldwork did she fully grasp the dialect's nuances."

Additional Information

• Option 1: "she had" is incorrect because it does not follow inversion rules after "Only after."
• Option 3: "had she" is incorrect because it implies past perfect tense, which is not required here.
• Option 4: "she did" is incorrect because it does not follow the correct inversion structure.

The correct answer is 'opacity'.

Key Points

• The term "opacity" refers to the quality of being difficult to understand or lacking clarity, which fits the context of the professor's jargon-heavy remarks.
• Option 2 (opaque) is an adjective, not a noun, and doesn't fit grammatically in the sentence.
• Option 3 (opacus) refers to a type of cloud and is irrelevant in this context.
• Option 4 (opalescent) means displaying iridescent colors, which is unrelated to the context of comprehension.

Therefore, the correct answer is Option 1.

Complete Sentence:"The professor's remarks were so dense with jargon that they bordered on opacity."

Additional Information

• Option 2: "opaque" is an adjective and would not fit as a noun in this sentence structure.
• Option 3: "opacus" is unrelated to the context of the sentence as it pertains to meteorological terms.
• Option 4: "opalescent" refers to something displaying a play of colors and does not align with the intended meaning of complexity or lack of clarity.

The correct answer is 'address'.

Key Points

• The word "address" means to deal with or discuss a particular issue or topic.
• The sentence suggests that although the rebuttal was eloquent, it failed to directly deal with the central issue raised by the opposition.
• Options 2, 3, and 4 are incorrect as they imply deviating from or avoiding the issue, which is contrary to the intended meaning of the sentence.