SSC CGL 2025 (Held On: 26 Sept, 2025 Shift 3) (Previous Year Paper)

The logic followed here is:

Bengaluru: 5th October

Hyderabad: 15th October

Chennai which is a perfect cube lying between 5 and 15.

The perfect cubes of numbers starting from 1 are:

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64 ...

Among these, the only date between 5 and 15 is 8.

the Chennai event will be on 8th October.

Hence, the correct answer is Option 2.

House No. 18, Sector 17, ChandIgarh - 160017 ('I' instead of 'i' in word Chandigarh)

House No. 18, Sector i7, Chandigarh - 160017  (i7 instead of 17)

House No. 18, Sector 17, Chandigarh - 160017  (Exactly same)

House No. 18, Sector 17, Chandigerh - 160017 (Spelling mistake in word Chandigarh)

Hence, the correct option is Option (3).

The logic followed here is:

169 = 13²

196 = 14²

225 = 15²

256 = 16²

Next

17² = 289

The missing number in the series is 289.

Hence, the correct answer is Option 1.

The logic followed here is:

9 @ 3 = 12

5 @ 2 = 7

8 @ 1 = 9

the pattern follow is:

@ means: first number + second number

9 + 3 = 12

5 + 2 = 7

8 + 1 = 9

7 @ 4:

7 + 4 = 11

7 @ 4 = 11

Hence, the correct answer is Option 1.

The logic followed here is:

H = 17, I = 19, J = 21

The letters are increasing by one (H, I, J) and the numbers are increasing by 2 each time (17, 19, 21)

M:

H → 17 I → 19 J → 21 K → 23 L → 25 M → 27

Hence, the correct answer is Option 4.

The logic followed here is:

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343

216 = 6³ is perfect cube 

225  Not a cube

243 Not a cube

256 Not a cube

Hence, the correct answer is Option 1.

The logic followed here is:

Curtain, Pillow, Brain, Mattress

Curtain, Pillow, and Mattress are items used in a bedroom or for sleeping.

Brain is a part of the human body and not a bedroom item.

Hence, the correct answer is Option 3.

The logic followed here is:

Rearrange the letters to form a meaningful seven-letter English word.

Rearranging the letters, we get the word: STORAGE

S → 6, T → 7, O → 4, R → 5, A → 1, G → 3, E → 2

STORAGE is 6745132.

Therefore, the correct answer is Option 3.

The logic followed here is:

BG : DJ

Analyze the pattern for letters:

B →+2  D ,G →+3 J 

F → H (+2) M → P (+3)

FM : HP

the correct answer is Option 4.

The logic followed here is:

10 : 21

10 × 2 + 1 = 21

Apply the same pattern to 12

12 × 2 + 1 = 25

the missing number is 25.

Hence, the correct answer is 25.

The logic followed here is:

Course of Action 1:

Hiring a cybersecurity team directly addresses the problem of data breaches. It is a practical and effective solution.

Course of Action 2:

Shutting down the entire IT infrastructure is an extreme and impractical step. It would stop company operations instead of solving the issue prope

Hence, the correct answer is Option 1 (Only 1 follows).

Fixed Positions (Common in all valid cases):

• HCL = 7th floor

• Infosys = 1st floor

• Facebook & Google have one floor between them

• Neither Facebook nor Google adjacent to HCL(so not on 6 or 8)

• Airtel even numbered and not adjacent to 7 (so not on 6 or 8)

Case 1A: Flipkart = 4, Jio = 2

Google → 5 → Floors above = 3

Case 1B: Flipkart = 4, Jio = 2

Google → 3 → Floors above = 5

Case 2A: Flipkart = 5, Jio = 3

Google → 4 → Floors above = 4

Case 2B: Flipkart = 5, Jio = 3

Google → 2 → Floors above = 6

Case 3: Flipkart = 6, Jio = 4

Google → 5 → Floors above = 3

Conclusion

Possible number of floors above Google = 3, 4, 5 or 6

Therefore answer is clearly:

Hence the correct option is Option 1.

Given Statements:

Some stars are planets.

No planet is a comet.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some stars are not comets – True

(Because, the stars which are planets cannot be comets, as no planet is a comet.)

II. All comets are planets – False

(Because, the statement clearly says no planet is a comet, so this is not possible.)

So, Only conclusion I follows.

Hence, the correct answer is Option 1

Given Statements:

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some poets are critics – False

(Because, no definite relation is given between Poets and Thinkers. The writers who are thinkers may or may not be poets.)

II. Some writers are critics – True

(Because, some writers are thinkers and all thinkers are critics.)

So, Only conclusion II follows.

Hence, the correct answer is Option 1.

Given words:

Python, Java, C++ and Ruby are programming languages.

HTML is a markup language, not a programming language.

HTML is the odd one out.

Hence, the correct answer is Option 3.

The logical followed here:

1 × 3 = 3

3 × 3 = 9

9 × 3 = 27

The series is multiplied by 3 each time.

27 × 3 = 81

the next number in the series is 81.

Hence, the correct answer is "Option 3".

The logic followed here:

M = 13

N = 14  (+1)

P = 16  (+2)

S = 19  (+3)

W = 23  (+4)

The pattern is increasing by +1, +2, +3, +4.

Next increase should be +5.

23 + 5 = 28

Hence, the correct answer is Option 4.

The logic followed here:

C → B (−1)

H → G (−1)

A → B (+1)

I → H (−1)

R → Q (−1)

The pattern is:

−1, −1, +1, −1, −1

STOOL:

S → R (−1)

T → S (−1)

O → P (+1)

O → N (−1)

L → K (−1)

So, STOOL → RSPNK

Hence, the correct answer is Option 1.

The logic followed here:

First letters:

P → O → N → ?

(Each step −1)

N − 1 = M

Second letters:

L → K → J → ?

(Each step −1)

J − 1 = I

So, the next pair is MI.

Hence, the correct answer is Option 2.

The logic followed here:

MAP → 13116

M = 13

A = 1

P = 16

So, MAP → 13 1 16 

Now for LAP:

L = 12

A = 1

P = 16

LAP → 12 1 16 → 12116

Hence, the correct answer is Option 1.

The logic followed here:

7 × 2 = 14

14 − 2 = 12

12 × 2 = 24

24 − 2 = 22

The pattern is:

×2, −2, ×2, −2, ...

22 × 2 = 44

Hence, the next number is 44.

Hence, the correct answer is Option 1.

Wait for higher authorities to act – Not appropriate (Immediate environmental damage is occurring, and delaying action can worsen the situation.)

Fine the factory after a month – Not appropriate (Delaying enforcement allows the factory to continue polluting and does not address the urgency of the issue.)

Secretly record the act and leak it – Not appropriate (Leaking information is not an official or lawful action and may create legal and ethical complications.)

Take legal action and inform environment department – Appropriate (It ensures immediate lawful intervention, proper documentation, and coordination with the concerned environmental authorities.)

Hence, the correct answer is option 4.

Shortcut Trick

Total cost of 3 pens = 3 × ₹12 = ₹36.

Remaining amount for 2 notebooks = ₹76 − ₹36 = ₹40.

Cost of 1 notebook = ₹40 ÷ 2 = ₹20.

∴ The correct answer is ₹20.

Alternate Method

Given:

Number of pens = 3

Cost of one pen = ₹12

Number of notebooks = 2

Total bill amount = ₹76

Formula Used:

Total Cost = (Number of pens × Price per pen) + (Number of notebooks × Price per notebook)

Calculations:

⇒ Cost of 3 pens = 3 × 12 = ₹36

⇒ Let the cost of one notebook be ₹x

⇒ 36 + 2x = 76

⇒ 2x = 76 − 36

⇒ 2x = 40

⇒ x = 40 ÷ 2

⇒ x = ₹20

∴ The correct answer is ₹20.

Additional Information

Linear Equations

Problems involving fixed prices and quantities are solved using linear equations in the form ax + by = Total, where x and y are the unit prices.

Unitary Method

This method involves finding the value of a single unit (unit rate) to find the value of any number of units.

Verification Rule

Always check the result by substituting the values back: (3 × 12) + (2 × 20) = 36 + 40 = 76, which confirms the answer is correct.

The logic followed here:

F → H (+2)

R → U (+3)

I → M (+4)

E → J (+5)

N → T (+6)

D → K (+7)

The pattern is increasing by +2, +3, +4, +5, +6, +7.

CANDLE:

C → E (+2)

A → D (+3)

N → R (+4)

D → I (+5)

L → R (+6)

E → L (+7)

CANDLE → EDRIRL

Hence, the correct answer is Option 2.

The logic followed here:

Counting from left:

1 → W

2 → %

3 → ^

4 → K

5 → V

6 → P

7 → 1

The 7th element from the left is 1.

Find the second element from the right of 1.

Elements to the right of 1 are:

I (1st), 7 (2nd), E (3rd), ...

The second element from the right of 1 is 7.

Hence, the correct answer is Option 3.

The correct answer is Lionel Robbins.

Key Points

• Lionel Robbins, a renowned British economist, gave the scarcity definition of economics in his famous book, An Essay on the Nature and Significance of Economic Science, published in 1932.
• According to Robbins, economics is the science which studies human behavior as a relationship between ends (wants) and scarce means which have alternative uses.
• He emphasized that resources are limited, and human wants are unlimited, creating a condition of scarcity.
• The scarcity definition highlights the concept of choice and opportunity cost, where individuals must prioritize their needs and decide how to allocate scarce resources efficiently.
• This definition shifted economics from a study of wealth (as proposed by Adam Smith) to a study of human behavior and decision-making under conditions of scarcity.

Additional Information

• Adam Smith:
• Known as the Father of Economics, Adam Smith defined economics as a study of the nature and causes of the wealth of nations.
• His definition focused on the production and accumulation of wealth and was presented in his book, The Wealth of Nations, published in 1776.
• His approach is referred to as the Wealth Definition of Economics.
• P.A. Samuelson:
• Samuelson provided the growth-oriented definition of economics, emphasizing the dynamic aspects of economic growth and development.
• He defined economics as the study of how societies use scarce resources to produce valuable commodities and distribute them among different people.
• He is known for his work in modern economic theory.
• Alfred Marshall:
• Marshall defined economics as the study of human welfare and material well-being, focusing on how people earn income and use it to satisfy their wants.
• His definition is referred to as the Welfare Definition of Economics and was presented in his book, Principles of Economics, published in 1890.
• Marshall's approach emphasized the importance of studying economics in relation to improving human welfare.
• Concept of Scarcity:
• Scarcity occurs when resources are limited and cannot satisfy all human wants.
• This fundamental concept forms the basis of economic problems, such as allocation, production, and distribution.
• Scarcity forces individuals and societies to make choices and prioritize resource utilization effectively.

Important Points

• Key Features of Robbins' Definition:
• Robbins viewed economics as a positive science (descriptive and neutral) rather than a normative one.
• It focuses on the study of individual behavior in making choices based on limited resources.
• Introduces the concept of opportunity cost, where choosing one alternative means forgoing others.
• Emphasizes the universality of economic problems, applicable to all societies irrespective of their stage of development.
• Criticism of Robbins' Definition:
• Critics argue that it overlooks the social and welfare aspects of economics emphasized by Marshall.
• It reduces economics to a study of scarcity and neglects growth and development.
• Robbins' approach is sometimes seen as too abstract and lacking practical application in real-world economic problems.

The correct answer is It will be India's space station in orbit..

Key Points

• The Bharatiya Antariksh Station is envisioned as India's first indigenous space station, which will serve as a permanent platform in orbit for scientific research, technology testing, and other space-related activities.
• The Indian Space Research Organisation (ISRO) announced plans for the space station in 2019, with the goal of achieving operational capability by the 2030s.
• The station will be smaller in scale compared to the International Space Station (ISS), with an expected weight of about 20 tonnes and designed to accommodate astronauts for missions lasting 15-20 days.
• This initiative is part of India's growing ambitions in space exploration, which include the Gaganyaan mission—India's first crewed spaceflight scheduled for the near future.
• The space station will facilitate long-duration human spaceflight, support scientific experiments in microgravity, and enable India to expand its capabilities in space technology.

Additional Information

• Key Features of the Bharatiya Antariksh Station:
• The station will operate at an altitude of approximately 400 km in low Earth orbit (LEO).
• It will be equipped with advanced life support systems to sustain astronauts during missions.
• The station will provide a platform for experiments in areas such as biotechnology, space medicine, and materials science.
• ISRO plans to collaborate with other nations and institutions for research and potential joint missions.
• Gaganyaan Mission:
• The Gaganyaan mission is India's first human spaceflight program, expected to send a crew of three astronauts into orbit for up to seven days.
• The mission is a precursor to the development of the Bharatiya Antariksh Station, as it will provide crucial data on human spaceflight.
• The mission is supported by international collaborations, including agreements with Russia and France for astronaut training and technology sharing.
• Comparison with the International Space Station (ISS):
• The International Space Station is a collaborative project involving NASA, Roscosmos, ESA, JAXA, and CSA. It has been operational since 1998.
• The ISS is significantly larger, with a mass of over 400 tonnes, and supports long-term missions with a rotating crew of astronauts.
• In contrast, India's space station will be a more focused and compact facility, designed for shorter missions and specific research goals.
• India's Space Achievements:
• India has made significant strides in space technology, including successful missions like Chandrayaan (lunar exploration) and Mangalyaan (Mars Orbiter Mission).
• ISRO has also developed a range of launch vehicles, such as the PSLV and GSLV, to support its ambitious space programs.
• The Bharatiya Antariksh Station will elevate India's status as a leading spacefaring nation.

Important Points

• Budget and Funding:
• The development of the space station is expected to require substantial investment, with funding sourced from the Government of India and potential international partnerships.
• Strategic Importance:
• The space station will bolster India's national security, provide technological independence, and enhance its influence in global space governance.

The correct answer is Bidar.

Key Points

• Lahore: Lahore was one of the prominent provincial capitals of the Mughal Empire. It served as the capital of the Punjab Subah (province) and was a center of administration and culture during the Mughal period. Hence, this option is incorrect.
• Allahabad: Allahabad (modern-day Prayagraj) held significance during the Mughal era and was designated as a provincial capital. It played a crucial role in administration and strategy. Hence, this option is incorrect.
• Ajmer: Ajmer was a key provincial capital in the Mughal Empire, especially under Akbar's reign. It was also an important center for trade and Sufi culture. Hence, this option is incorrect.
• Bidar: Bidar was historically significant, especially during the Bahmani and later Deccan Sultanates, but it was not a Mughal provincial capital. The Mughal influence in the Deccan did not elevate Bidar to the status of a provincial capital. Hence, this option is correct.

Additional Information

• Provincial Capitals of the Mughal Empire:
• The Mughal Empire was divided into several administrative units called Subahs, governed by Subahdars.
• Prominent provincial capitals included Delhi, Agra, Lahore, Ajmer, and Allahabad.
• These capitals served as administrative hubs, centers of trade, and locations for imperial courts.
• Bidar's Historical Context:
• Bidar was the capital of the Bahmani Sultanate and later the Barid Shahi dynasty.
• It is located in present-day Karnataka and is renowned for its architectural landmarks, such as the Bidar Fort.
• Although the Mughals had influence in the Deccan, Bidar remained a regional center rather than a provincial capital under Mughal rule.
• Lahore as a Mughal Capital:
• Lahore gained prominence under Emperor Akbar, who made it one of his main residences.
• The city flourished as a cultural and architectural hub, featuring landmarks like the Shalimar Gardens and Lahore Fort.
• Ajmer's Role in the Mughal Empire:
• Ajmer was strategically and spiritually significant, housing the shrine of Khwaja Moinuddin Chishti, a revered Sufi saint.
• It served as a provincial capital and a center for Mughal diplomacy in Rajasthan.
• Allahabad's Significance:
• Allahabad became an important administrative and military center under the Mughals, particularly during Akbar's reign.
• The city is also historically significant for the construction of the Allahabad Fort by Akbar.

Important Points

• Administrative Structure of the Mughal Empire:
• The empire was divided into Subahs (provinces), Sarkars (districts), Parganas (sub-districts), and villages.
• The head of each Subah was the Subahdar, responsible for administration, revenue collection, and military affairs.
• The Deccan and Mughal Influence:
• The Mughals extended their control over the Deccan during the reigns of Akbar, Jahangir, and Aurangzeb.
• Despite this, cities like Bidar, which were part of the Deccan Sultanates, never became central administrative hubs for the Mughals.

The correct answer is Deendayal Antyodaya Yojana-National Rural Livelihoods Mission (DAY-NRLM).

Key Points

• Deendayal Antyodaya Yojana-National Rural Livelihoods Mission (DAY-NRLM) was launched by the Ministry of Rural Development, Government of India, in June 2011.
• The scheme aims to reduce rural poverty by enabling poor households to access gainful self-employment and skilled wage employment opportunities.
• It focuses on building strong institutional platforms for the rural poor, particularly women Self-Help Groups (SHGs), to enable them to achieve sustainable livelihoods.
• The program provides financial assistance and capacity building to women SHGs, helping them start micro-enterprises and other income-generating activities.
• DAY-NRLM also provides interest subvention on loans to SHGs to ensure affordable credit and promotes financial inclusion for rural women.
• It focuses on empowering rural women by engaging them in economic activities, ensuring their socio-economic development.

Additional Information

• Rashtriya Mahila Kosh (RMK):
• Launched in 1993, RMK is a government initiative aimed at providing micro-credit to poor and underprivileged women.
• It acts as an apex financing agency to promote income-generating activities among women.
• However, it primarily acts as a financial intermediary and does not directly assist Self-Help Groups (SHGs).
• Pradhan Mantri Jan Dhan Yojana (PMJDY):
• Launched in 2014, PMJDY is a national mission aimed at ensuring financial inclusion by providing access to banking facilities to every household.
• It focuses on opening zero-balance bank accounts, issuing RuPay debit cards, and providing accidental and life insurance cover.
• While it improves financial literacy and inclusion, it is not directly aimed at providing financial assistance to SHGs.
• Mahila E-Haat:
• Launched in 2016, Mahila E-Haat is an online marketing platform under the Ministry of Women and Child Development.
• It enables women entrepreneurs and SHGs to showcase and sell their products to a wider audience.
• This initiative enhances market accessibility but does not provide direct financial assistance to SHGs.
• Important Features of DAY-NRLM:
• Targets the poorest households in rural areas and aims to uplift them through social and economic empowerment.
• Promotes the formation of Self-Help Groups (SHGs) and provides them with training, credit linkages, and technical support.
• Emphasizes women's participation and leadership in SHGs, ensuring their active role in decision-making processes.

Important Points

• Eligibility: Beneficiaries are selected through a participatory identification process targeting the poorest of the poor households.
• Financial Support: Includes revolving funds, community investment funds, and interest subvention on loans.
• Implementation: The program is implemented in partnership with state governments and various NGOs, ensuring the holistic development of SHGs.
• Livelihood Promotion: DAY-NRLM focuses on enhancing livelihood opportunities through agricultural and non-agricultural activities, skill development, and entrepreneurship.

The correct answer is Speed increases, wavelength increases..

Key Points

• When a sound wave travels from air to water, its speed increases. This is because the speed of sound depends on the medium's density and elasticity. Water, being denser and more elastic than air, allows sound to travel faster.
• The wavelength of the sound wave also increases when it enters water. This is because the wavelength is proportional to the speed of the wave when the frequency remains constant.
• Frequency of the sound wave does not change during the transition between two media. It remains constant because frequency is determined by the source of the sound and not by the medium.
• The relationship between the speed, wavelength, and frequency is given by the formula: v = f × λ, where v is the speed, f is the frequency, and λ is the wavelength. When speed (v) increases and frequency (f) remains constant, the wavelength (λ) must also increase.
• As water is a denser and more elastic medium compared to air, sound travels approximately 4.3 times faster in water than in air.

Additional Information

• Speed of Sound in Different Media:
• The speed of sound in air (at 20°C) is approximately 343 m/s.
• In water, the speed of sound is about 1482 m/s (at 25°C), which is much faster than in air.
• In solids like steel, sound travels even faster, at around 5000 m/s.
• Factors Affecting Speed of Sound:
• Density: Higher density generally decreases speed; however, in liquids and solids, elasticity plays a more significant role.
• Elasticity: Greater elasticity increases the speed of sound.
• Temperature: In gases, an increase in temperature increases the speed of sound.
• Why Frequency Remains Constant:
• The frequency of a wave is determined by its source and does not change when the wave moves between different media.
• As a result, changes in the wave's speed directly affect its wavelength in the new medium.
• Applications of Sound Wave Properties:
• Sonar systems use the increased speed of sound in water for underwater navigation and communication.
• Medical ultrasonography leverages the varying speeds of sound in different tissues for imaging.

Important Points

• Formula for Wave Behavior: The equation v = f × λ is critical for understanding how speed, frequency, and wavelength are interrelated.
• Why Sound Travels Faster in Water: Water molecules are more tightly packed compared to air molecules, allowing sound waves to transfer energy more efficiently.
• Effect of Medium on Amplitude: While speed and wavelength may change, the amplitude of the sound wave can also vary, affecting the wave's intensity.
• Real-Life Phenomena: This principle explains why sounds underwater (like a dolphin's echolocation clicks) can be heard over long distances.

The correct answer is Swish.

Key Points

• A swish in basketball refers to a successful shot where the ball passes cleanly through the hoop without touching the rim or backboard.
• It is considered a perfect shot, demonstrating precision and accuracy in shooting.
• The swish sound occurs due to the ball brushing against the net, creating an audible "swish" sound that players and spectators recognize as a sign of skillful shooting.
• Swish shots are often celebrated in basketball as they symbolize a high level of shooting proficiency and are aesthetically pleasing.
• Players practice extensively to achieve consistent swish shots, as these contribute significantly to scoring efficiency and overall performance in the game.

Additional Information

• Slam Dunk:
• A slam dunk is a basketball move where a player jumps and forcefully scores by directly putting the ball into the hoop, often with one or both hands.
• It is an emphatic and high-energy play that is often used to showcase athleticism and skill.
• Slam dunks are popular in competitive games and highlight reels but are different from a swish as they involve physical contact with the hoop.
• Layup:
• A layup is a close-range shot where a player moves towards the hoop and gently places the ball into the basket using one hand, often off the backboard.
• It is a fundamental scoring move and is commonly used during fast breaks or close-range plays.
• Alley-oop:
• An alley-oop is a play where one player passes the ball near the basket, and another player catches it mid-air and scores, often with a dunk or layup.
• This move is a collaborative and visually impressive scoring method, requiring coordination and timing between teammates.
• Importance of Shooting Precision:
• Effective shooting contributes significantly to a team's success in basketball, with swish shots being a prime example of precise scoring.
• Players focus on improving their shooting techniques, such as grip, release angle, and follow-through, to achieve consistent accuracy.

Important Points

• Historical Context:
• The term "swish" became popular in basketball due to the distinct sound made when the ball cleanly goes through the net.
• Players like Larry Bird and Steph Curry are renowned for their ability to consistently produce swish shots, demonstrating exceptional shooting skills.
• Training Techniques:
• To achieve swish shots, players often practice using shooting drills that focus on form, release, and trajectory.
• Basketball coaches emphasize the importance of muscle memory and proper foot positioning during shooting practice.

The correct answer is Shah Jahan.

Key Points

• Shah Jahan, the fifth Mughal emperor, was responsible for constructing the iconic Red Fort in Delhi.
• The Red Fort, also known as Lal Qila, was built in 1639 when Shah Jahan decided to shift his capital from Agra to Delhi.
• The fort was designed by architect Ustad Ahmad Lahauri, who is also credited with designing the Taj Mahal.
• The Red Fort served as the main residence of the Mughal emperors for nearly 200 years, until the British colonial period.
• The fort is constructed primarily using red sandstone, giving it its distinctive appearance and name.
• The Red Fort is a UNESCO World Heritage Site, recognized for its historical and architectural significance.

Additional Information

• Architecture of the Red Fort:
• The fort complex includes structures such as the Diwan-i-Aam (Hall of Public Audience), Diwan-i-Khas (Hall of Private Audience), Moti Masjid, and Rang Mahal.
• It features a blend of Persian, Timurid, and Indian architectural styles.
• The fort's walls stretch for approximately 2.5 kilometers and are about 18 meters high.
• Historical Significance:
• The Red Fort became the symbol of sovereignty after India gained independence in 1947.
• Every year, on Independence Day (15th August), the Prime Minister hoists the national flag and delivers a speech from the Red Fort.
• Shah Jahan’s Legacy:
• Shah Jahan is remembered for his patronage of art and architecture, evident in monuments like the Taj Mahal, Jama Masjid, and Shalimar Gardens.
• His reign is often considered the golden age of Mughal architecture.
• Other Mughal Contributions:
• Babur: Laid the foundation of the Mughal Empire and introduced Persian-style gardens, known as Charbagh.
• Akbar: Known for constructing Fatehpur Sikri and the Agra Fort, showcasing his architectural innovation.
• Aurangzeb: Built structures such as the Bibi Ka Maqbara, but his reign focused more on military campaigns than architecture.

Important Points

• The Red Fort was commissioned by Shah Jahan in 1638 and took 10 years to complete.
• The fort's design reflected the Mughal vision of paradise on Earth, with extensive gardens and water channels.
• During colonial rule, the British destroyed many of the fort’s internal structures and used it as a military barracks.
• The fort remains one of India's most visited tourist sites and serves as a symbol of India’s rich heritage.
• Its preservation is managed by the Archaeological Survey of India (ASI).

The correct answer is cytoskeleton.

Key Points

• The cytoskeleton is an internal network of protein filaments and tubules present in all eukaryotic cells.
• It provides structural support to the cell, maintaining its shape and organization.
• The cytoskeleton is composed of three primary components: microfilaments (actin filaments), microtubules, and intermediate filaments.
• This network is involved in various cellular processes, including intracellular transport, cell division (mitosis and meiosis), and cell signaling.
• The cytoskeleton plays a crucial role in cell motility, enabling cells to move and change shape during processes like wound healing and immune responses.

Additional Information

• Components of the Cytoskeleton:
• Microfilaments: These are composed of actin proteins. They are involved in maintaining the cell's shape, enabling cell motility, and facilitating intracellular transport.
• Microtubules: Made up of tubulin, they provide structural support and serve as tracks for the movement of organelles and vesicles. They are also essential in forming the mitotic spindle during cell division.
• Intermediate Filaments: These filaments provide mechanical strength to cells, helping them withstand tension. Examples include keratin in skin cells.
• Functions of the Cytoskeleton:
• Maintains the overall cell structure and integrity.
• Assists in cellular movements such as crawling, contraction, and division.
• Plays a role in intracellular transport of organelles and molecules using motor proteins like kinesin and dynein.
• Facilitates communication between the cell surface and its internal organelles.
• Comparison with Other Organelles:
• Nucleus: It is the control center of the cell, housing DNA and regulating gene expression. However, it does not provide structural support like the cytoskeleton.
• Mitochondria: Known as the powerhouse of the cell, it generates energy in the form of ATP but has no role in structural support.
• Ribosomes: These are involved in protein synthesis and are not connected to structural functions.
• Applications and Research:
• Understanding the cytoskeleton is critical in studying diseases such as cancer, as changes in cell shape and motility are hallmarks of metastasis.
• Cytoskeletal proteins are targeted in therapies for neurodegenerative disorders like Alzheimer's disease.
• Research into cytoskeletal dynamics has applications in tissue engineering and regenerative medicine.

Important Points

• The cytoskeleton is dynamic, meaning it can rapidly reorganize to adapt to the cell's needs, such as during cell division or migration.
• It interacts with motor proteins like myosin, kinesin, and dynein, which help in the movement of vesicles and organelles within the cell.
• Mutations in cytoskeletal proteins can lead to diseases such as cardiomyopathy, muscular dystrophy, and neurodegenerative disorders.
• Cytoskeleton research has greatly advanced with the development of advanced microscopy techniques like fluorescent and electron microscopy.

The correct answer is 2005.

Key Points

• National Anti-Doping Agency (NADA) was established in 2005 under the aegis of the Ministry of Youth Affairs and Sports, Government of India.
• The primary purpose of NADA is to promote clean sport and ensure the integrity of athletes by preventing the use of performance-enhancing drugs.
• NADA operates in accordance with the World Anti-Doping Code, which is a set of global standards for anti-doping measures created by the World Anti-Doping Agency (WADA).
• The agency is responsible for conducting dope testing, educating athletes about anti-doping rules, and implementing awareness programs.
• NADA's jurisdiction includes domestic athletes and events, ensuring compliance with anti-doping regulations at the national level.

Additional Information

• Functions of NADA:
• Testing and Investigation: NADA conducts doping tests during competitions and outside of competitions to ensure adherence to anti-doping rules.
• Education and Awareness: The agency organizes workshops, seminars, and campaigns to educate athletes, coaches, and support staff about the consequences of doping.
• Results Management: NADA handles the process of analyzing test results and taking necessary disciplinary actions against those found guilty of doping violations.
• Coordination with WADA: NADA ensures compliance with international anti-doping standards and collaborates with WADA for global anti-doping efforts.
• World Anti-Doping Agency (WADA):
• WADA was established in 1999 as an independent agency to combat doping in sports worldwide.
• The agency introduced the World Anti-Doping Code to unify anti-doping policies globally.
• WADA's headquarters are located in Montreal, Canada, and it oversees compliance for all national anti-doping organizations.
• Significant Achievements of NADA:
• NADA has successfully conducted thousands of doping tests and implemented stringent measures for ensuring fair play in sports.
• The agency actively collaborates with international bodies like WADA and regional anti-doping organizations to enhance its capabilities.
• It has played a critical role in preventing doping scandals and maintaining India's reputation in the global sports arena.
• Anti-Doping Rules in India:
• The Anti-Doping Rules, 2021 were introduced to align with WADA's 2021 Code and improve the enforcement of anti-doping measures.
• These rules cover various aspects, including testing protocols, prohibited substances, therapeutic use exemptions (TUEs), and penalties for violations.
• They emphasize the importance of athlete education and aim to create a doping-free sports culture.

Important Points

• Prohibited Substances:
• NADA publishes a list of prohibited substances and methods annually, which is based on WADA's guidelines.
• Substances like anabolic steroids, stimulants, narcotics, and masking agents are banned for athletes.
• Consequences of Doping:
• Athletes found guilty of doping face sanctions, including disqualification, suspension, fines, and lifetime bans depending on the severity of the violation.
• Repeated violations can lead to harsher penalties and loss of reputation in the sports community.
• Role of Athletes:
• Athletes are required to stay updated on anti-doping regulations and avoid using prohibited substances.
• They must ensure compliance with testing protocols and cooperate with NADA during investigations.

The correct answer is The reorganization of the police and the establishment of the Criminal Investigation Department (CID).

Key Points

• Lord Curzon's tenure as Viceroy of India (1899-1905) was marked by several administrative reforms aimed at strengthening British control over India.
• The reorganization of the police force and the establishment of the Criminal Investigation Department (CID) were intended to enhance law enforcement capabilities and improve the surveillance network for maintaining British authority.
• The CID was introduced as a specialized department to handle criminal investigations, intelligence gathering, and monitoring revolutionary activities, which were on the rise during British rule.
• This reform was primarily aimed at consolidating British strategic interests by curbing dissent and ensuring the stability of colonial administration.
• While the CID contributed to improving policing infrastructure, its emphasis on intelligence and surveillance highlighted its role in safeguarding British dominance rather than addressing local needs.

Additional Information

• Lord Curzon's reforms:
• Imperial Department of Agriculture: Established to promote agricultural research and education, this initiative aimed at improving the agricultural economy but wasn't directly tied to British strategic interests.
• Universities Act of 1904: Curzon introduced this act to regulate higher education in India, enhance academic standards, and centralize control over universities. This was more focused on educational reforms rather than administrative control.
• Preservation of ancient monuments: Curzon took steps to protect India's heritage by establishing the Archaeological Survey of India. This was more cultural and conservation-oriented rather than a measure to consolidate British power.
• The police reorganization and CID establishment were directly linked to improving governance and ensuring the suppression of anti-colonial movements, making it a key strategy for consolidating British control.
• Functions of the CID:
• The CID focused on investigating criminal cases, collecting intelligence, and tracking revolutionary activities.
• It played a pivotal role in monitoring political dissent and nationalist movements that posed a threat to British rule.
• The establishment of CID was critical in maintaining colonial law and order during a period of growing unrest and dissatisfaction among Indians.
• Historical Context:
• During Curzon's tenure, nationalist movements such as the Swadeshi movement and calls for self-rule were gaining momentum.
• The CID served as a mechanism to counteract revolutionary activities and suppress freedom movements.
• Legacy:
• The CID continues to exist as an integral part of India's police structure, playing a vital role in modern criminal investigations and intelligence operations.
• The reorganization of the police force under Curzon laid the foundation for a systematic and centralized policing framework in India.

Important Points

• Lord Curzon: Served as Viceroy of India from 1899 to 1905.
• Reorganization of Police: Focused on improving law enforcement and intelligence gathering during British rule.
• CID Establishment: Specialized department created to monitor revolutionary activities and ensure British administrative control.
• Role in Nationalist Movements: CID was instrumental in suppressing emerging freedom movements, reflecting its strategic importance to colonial governance.
• Impact: Laid the groundwork for modern police infrastructure in India, with lasting effects on intelligence and investigation processes.

The correct answer is Ensure admissibility of modern evidence.

Key Points

• The Bharatiya Sakshya Adhiniyam, also known as the Indian Evidence Act, was enacted in 1872 and has undergone amendments to incorporate modern developments in evidence law.
• The main objective of the Act is to regulate the admissibility of evidence in Indian courts to ensure justice.
• With advancements in technology, the Act has been updated to include provisions for the admissibility of modern evidence, such as electronic records and digital evidence.
• The admissibility of modern evidence ensures that contemporary forms of proof, like emails, CCTV footage, digital signatures, and electronic communications, are legally recognized.
• This objective ensures fairness in trials by allowing relevant evidence in both civil and criminal cases, aligning the judicial process with technological progress.

Additional Information

• Indian Evidence Act (1872):
• The Act defines the rules and guidelines for the admissibility of evidence in courts of law in India.
• It includes provisions for the relevance of facts, oral and documentary evidence, and burden of proof.
• The Act is divided into 11 chapters and 167 sections, covering a wide range of evidentiary aspects.
• Modern Evidence:
• Modern evidence includes electronic evidence such as emails, SMS, electronic records, and digital images.
• Section 65B of the Indian Evidence Act governs the admissibility of electronic records, provided certain conditions are met.
• This ensures that evidence derived from technological sources is treated on par with traditional forms of evidence.
• Amendments for Technological Advancements:
• In 2000, the Indian Evidence Act was amended to include provisions from the Information Technology Act, introducing the admissibility of electronic evidence.
• These amendments ensure that courts can adapt to the changing nature of evidence in the digital age.
• Importance of Admissibility:
• Admissibility ensures that only relevant and authentic evidence is considered in judicial proceedings, maintaining the integrity of the legal system.
• It helps in preventing manipulation and ensures transparency in trials.

Important Points

• Options Analysis:
• Option 1: Reduce the role of evidence in trials - Incorrect. Evidence is the cornerstone of judicial proceedings, and its role cannot be diminished.
• Option 2: Ensure admissibility of modern evidence - Correct. This aligns with the Act's objective to adapt to technological advancements.
• Option 3: Make only physical evidence valid - Incorrect. Limiting evidence to physical forms would disregard the importance of digital evidence.
• Option 4: Allow anonymous evidence without checks - Incorrect. Evidence must be verified for authenticity and reliability before being admitted in court.

The correct answer is Tushar Mehta.

Key Points

• Tushar Mehta was reappointed as the Solicitor General of India for a term of three years in 2023. He continued in this role in 2024, making him the incumbent Solicitor General during that year.
• The Solicitor General is the second-highest law officer in India, assisting the Attorney General in legal matters related to the government.
• Tushar Mehta has been serving as Solicitor General of India since October 2018. His tenure has been marked by handling several high-profile cases.
• The Solicitor General is appointed by the President of India on the advice of the Union Cabinet. The appointment is typically for a term of three years.
• Mehta has represented the Union Government in various cases concerning constitutional, civil, and criminal matters, including landmark hearings in the Supreme Court.

Additional Information

• Solicitor General of India:
• The Solicitor General is responsible for advising the government on legal matters and representing it in courts, particularly in the Supreme Court.
• The position is a statutory one, but not explicitly mentioned in the Indian Constitution. It is governed by statutory rules under the Law Officers (Conditions of Service) Rules, 1987.
• The Solicitor General assists the Attorney General of India, who is the highest law officer of the country.
• The office of the Solicitor General is supported by several Additional Solicitors General, who also represent the Union Government in various courts.
• Role and Functions:
• The Solicitor General does not have independent executive authority and works under the direction of the Attorney General.
• They provide advice on legal issues and appear on behalf of the government in high-profile cases.
• Prominent cases handled by the Solicitor General include disputes involving constitutional law, taxation, and criminal law.
• Historical Context:
• Previous Solicitor Generals have included eminent legal personalities like K.K. Venugopal, Mukul Rohatgi, and Ranjit Kumar.
• The position is considered prestigious and crucial in maintaining the legal framework of the government.
• Important Developments:
• Tushar Mehta has represented the government in cases like the abrogation of Article 370, the Citizenship Amendment Act (CAA), and various economic and criminal matters.
• His tenure has been extended multiple times, reflecting the confidence of the government in his legal expertise.

Important Points

• Appointment Process:
• The Solicitor General is appointed by the President of India on the recommendation of the Union Cabinet.
• The term of appointment is typically three years, but extensions are possible based on performance and the government’s discretion.
• Eligibility and Qualifications:
• The Solicitor General is typically an eminent lawyer with extensive experience in constitutional and civil law.
• They must possess a strong understanding of judicial processes and have a proven track record in handling complex legal cases.
• Difference from Attorney General:
• While the Attorney General is the chief legal advisor to the government and represents it in crucial cases, the Solicitor General assists in these responsibilities and handles specific cases as directed.
• The Solicitor General does not directly advise the government but focuses on litigation and courtroom representation.

The correct answer is Karnataka.

Key Points

• Yakshagana is a traditional folk dance-drama form that originated in the state of Karnataka, India.
• It is a unique blend of dance, music, dialogue, costumes, and stage techniques with a distinct style and form.
• The performance is primarily based on mythological stories and epics such as the Ramayana and Mahabharata.
• It is traditionally performed in an open-air theatre, usually during the night, and can last for several hours, sometimes extending till dawn.
• The musical instruments used in Yakshagana include the chenda, maddale (a percussion instrument), and the harmonium, which accompany the actors' performances.

Additional Information

• Characteristics of Yakshagana:
• Yakshagana is known for its elaborate costumes, rich facial makeup, and colorful headgear, which add grandeur to the performance.
• The dance is performed by a group of artists, where one plays the role of the narrator (called Bhagavata) and the others enact the story.
• The dialogues are often improvised, making each performance unique.
• The language used is mainly Tulu or Kannada, depending on the region of the performance.
• Regional Variations:
• Yakshagana has two prominent styles: Tenkutittu (southern style) and Badagutittu (northern style).
• The Tenkutittu style is more focused on dance, while Badagutittu emphasizes dialogue delivery and dramatic performance.
• Historical Context:
• Yakshagana is believed to have originated around the 11th to 16th centuries, evolving from ancient forms of dance and drama in South India.
• It was influenced by temple arts and folk traditions, making it a significant cultural heritage of Karnataka.
• Performances and Audience:
• The performances are typically organized during festivals and special occasions, especially in rural areas of Karnataka.
• Yakshagana has also gained international recognition and is performed in various countries to showcase India's rich cultural diversity.

Important Points

• Other Folk Arts of Karnataka:
• Dollu Kunitha: A drum dance performed by men in groups, showcasing synchronized drumbeats and movements.
• Krishna Parijatha: A folk theatre form focusing on the life and stories of Lord Krishna.
• Puppetry: Karnataka is renowned for its leather puppetry shows, which narrate mythological tales.
• Yakshagana in Modern Times:
• Efforts are being made to preserve and promote Yakshagana through cultural organizations and government initiatives.
• Educational institutions in Karnataka also include Yakshagana in their cultural programs to teach younger generations about this art form.

The correct answer is Rishabhadev.

Key Points

• Rishabhadev, also known as Adinath, is considered the first Tirthankara of Jainism. The Tirthankaras are revered spiritual teachers in Jain philosophy who have achieved enlightenment.
• He is mentioned in Jain scriptures as the one who inaugurated the tradition of Jain teachings and established the core tenets of Jainism.
• Rishabhadev is also depicted as the one who taught people various fundamental skills such as agriculture, trade, and governance, contributing significantly to the development of society.
• He is believed to have lived in the ancient period, and his teachings are considered timeless, forming the foundation of Jain philosophy.
• The Jain tradition holds that there are 24 Tirthankaras, and Rishabhadev was the first among them. He is often symbolized with the bull (his emblem), which represents strength and determination.

Additional Information

• Jain Tirthankaras:
• Jainism recognizes 24 Tirthankaras or spiritual teachers who have attained liberation and guide others on the path to enlightenment.
• The last Tirthankara, Mahavira, was a contemporary of Buddha and lived around the 6th century BCE.
• Symbolism of Rishabhadev:
• The bull is the symbol associated with Rishabhadev, representing power, patience, and work ethic.
• In Jain art and iconography, he is depicted as a serene figure with the bull as his identifying emblem.
• Key Teachings of Rishabhadev:
• He is believed to have introduced the concept of non-violence (Ahimsa), which is one of the core principles of Jainism.
• He emphasized the importance of renunciation, self-discipline, and ethical living.
• Comparison with Other Tirthankaras:
• Unlike Mahavira, who focused on organizing and spreading Jain teachings, Rishabhadev is credited with laying the foundational principles of Jainism.
• He is often considered a figure of mytho-historical importance and is revered both in Jainism and other Indic traditions.
• Historical Evidence:
• Although Jain texts provide a detailed account of Rishabhadev, historical evidence about his existence is sparse and mostly symbolic.
• References to Rishabhadev can also be found in other ancient Indian scriptures like the Bhagavata Purana, which mentions him as a significant figure.

The correct answer is 2) B only.

Key Points

• Statement A: Level of economic development and terrain are the determinants of road density in a region.
• This statement is correct. Economic development influences the availability of resources for infrastructure development, while terrain impacts construction costs and feasibility. For example, plains have higher road density due to easier construction, whereas hilly or forested terrains face challenges.
• Statement B: Road density is higher in the North-Eastern states than in the Southern states of India.
• This statement is incorrect. The Southern states like Kerala and Tamil Nadu have much higher road density compared to the North-Eastern states. Factors such as better economic development, higher population density, and favorable terrain contribute to this disparity.
• Statement C: Road density of Kerala is higher than the average national road density.
• This statement is correct. Kerala has one of the highest road densities in India due to its well-developed infrastructure and high population density. The state's road density exceeds the national average significantly.

Additional Information

• Road Density in India:
• The term "road density" refers to the length of road per 100 square kilometers of area.
• India's average road density is approximately 1.94 km/sq km as of recent data.
• States like Kerala, West Bengal, and Tamil Nadu have road densities higher than the national average.
• North-Eastern states have lower road density due to challenging terrain, lower population density, and limited infrastructure development.
• Factors Affecting Road Density:
• Economic Development: Wealthier regions can allocate more resources for road construction and maintenance.
• Terrain: Plains are favorable for road construction, while mountainous and forested terrains are less accessible.
• Population Density: Higher population density leads to greater demand for transportation infrastructure.
• Urbanization: Urbanized areas tend to have more extensive road networks compared to rural areas.
• North-Eastern States:
• These states include Assam, Meghalaya, Nagaland, Manipur, Mizoram, Tripura, Arunachal Pradesh, and Sikkim.
• The region is known for its rugged terrain, heavy rainfall, and forest cover, which pose challenges for infrastructure development.
• Efforts have been made to improve connectivity, such as through the Pradhan Mantri Gram Sadak Yojana (PMGSY).
• Kerala's Road Network:
• Kerala has a well-developed road network with a density of over 4.62 km/sq km, one of the highest in India.
• The state's infrastructure development is driven by its high population density and urbanization.
• Roads in Kerala play a crucial role in connecting remote villages to urban centers, boosting trade and tourism.

Important Points

• National Highway Development Project (NHDP):
• This project aims to upgrade, rehabilitate, and widen major highways to international standards.
• It includes initiatives like the Golden Quadrilateral and the North-South and East-West corridors.
• Pradhan Mantri Gram Sadak Yojana (PMGSY):
• Launched in 2000, this scheme aims to provide all-weather road connectivity to unconnected villages.
• It has significantly improved rural road density in India.

The correct answer is 30.76 % of total population.

Key Points

• As per the 2011 Census of India, the proportion of the 'Juvenile Population' (individuals aged 0-14 years) was recorded at 30.76% of the total population.
• The juvenile population is an essential demographic segment as it represents the country's future workforce and potential contributors to economic and social development.
• The total population of India as per the 2011 Census was approximately 1.21 billion, and out of this, about 373 million people belonged to the 0-14 years age group.
• This percentage highlights the significance of investing in healthcare, education, and child welfare to ensure the holistic development of this segment.
• In comparison to global standards, India had a relatively high proportion of its population in the juvenile category, reflecting a young population profile.

Additional Information

• Significance of the Juvenile Population:
• The juvenile population forms the foundation of a nation's human resource capital.
• It emphasizes the need for policy interventions focusing on child nutrition, education, and healthcare.
• Challenges of a Young Population:
• High juvenile population can lead to a burden on education and healthcare infrastructure.
• It necessitates investments in schools, hospitals, and skill development programs to meet the growing needs.
• Comparison with Global Trends:
• Globally, the juvenile population percentage varies. In developed countries, it is generally lower due to lower birth rates.
• In contrast, many developing nations, including India, have a higher proportion of juvenile populations due to higher fertility rates.
• Government Initiatives for Juvenile Welfare:
• Integrated Child Development Services (ICDS): Aims to provide food, preschool education, and healthcare to children below 6 years and their mothers.
• Mid-Day Meal Scheme: Provides nutritious meals to school children to improve enrollment, retention, and nutritional levels.
• Right to Education Act (2009): Ensures free and compulsory education for children aged 6-14 years.

Important Points

• Census 2011 Highlights:
• The census is conducted every 10 years and serves as the primary source of demographic data in India.
• The 2011 Census was the 15th National Census since 1872 and the 7th after India’s independence.
• The overall literacy rate in India rose to 74.04% in 2011.
• Juvenile Dependency Ratio:
• It is the ratio of individuals aged 0-14 years to the working-age population (15-64 years).
• India's high dependency ratio highlights the importance of creating employment opportunities for the working-age population to support the dependents.
• Future Implications:
• India's high juvenile population provides a demographic dividend, but only if investments are made in education, health, and skill development.
• If not addressed, this demographic advantage can turn into a demographic liability due to unemployment and poverty.

The correct answer is Indus River.

Key Points

• The Indus River, known as the "Lion's Mouth" or Singi Khamban in Tibet, originates near the Mansarovar Lake in Tibet.
• It flows in a northwest direction through the regions of Ladakh and Zaskar, carving deep gorges as it progresses. Hence, this statement is correct.
• Near Gilgit, the river cuts a spectacular gorge, which is one of the remarkable geographical features of the Indus River. This statement is correct.
• The Indus River enters Pakistan near the towering peak of Nanga Parbat, marking a significant turn in its flow. This statement is correct.
• The Indus is one of the world's longest rivers, covering a total length of approximately 3,180 kilometers, passing through China, India, and Pakistan.

Additional Information

• Shayok River:
• The Shayok River is a tributary of the Indus River and flows through the Ladakh region.
• It originates from the Rimo Glacier in the Karakoram Range.
• Although it merges with the Indus, it does not cut through regions like Zaskar or Gilgit.
• Sutlej River:
• The Sutlej is one of the major tributaries of the Indus River and originates from Rakshastal Lake near Mansarovar in Tibet.
• It flows through Himachal Pradesh and Punjab but does not pass through the regions mentioned in the question.
• Galwan River:
• The Galwan River is a small tributary of the Shyok River, located in the Ladakh region.
• It became internationally significant due to the India-China clashes in the Galwan Valley.
• However, this river is not the correct answer as it is not connected to the regions like Zaskar or Gilgit.
• Indus River Facts:
• The Indus River is the primary water source for Pakistan and supports its agriculture and economy.
• It is also the source of the name of the country India, derived from the term "Sindhu," the Sanskrit name for the Indus.
• The river has a large basin spanning over 1.1 million square kilometers, encompassing parts of China, India, Afghanistan, and Pakistan.

Important Points

• Indus Waters Treaty (1960):
• This treaty, signed between India and Pakistan, governs the sharing of the waters of the Indus and its tributaries.
• It was brokered by the World Bank and is considered one of the most successful water-sharing agreements globally.
• Historical Significance:
• The Indus Valley Civilization, one of the world's oldest urban civilizations, developed along the banks of the Indus River.
• Major cities like Harappa and Mohenjo-Daro flourished near its basin.
• Hydroelectric Projects:
• Several hydroelectric projects have been developed along the Indus River, especially in Pakistan, such as the Tarbela Dam.
• India is also utilizing its share of water through projects like the Baglihar Dam on the Chenab River, a tributary of the Indus.

The correct answer is Ishtiaq Ahmed.

Key Points

• Dr. Ishtiaq Ahmed is the author of the book titled "Jinnah: His Successes, Failures and Role in History." The book provides an in-depth analysis of Muhammad Ali Jinnah, the founder of Pakistan.
• The book was published in 2020 and critically examines Jinnah's political journey, his successes, failures, and his role in shaping the subcontinent's history.
• Ishtiaq Ahmed is a renowned political scientist and historian of Pakistani origin. He is an Emeritus Professor of Political Science at Stockholm University, Sweden.
• The book highlights Jinnah's transformation from a staunch advocate of Hindu-Muslim unity to the leader of the Muslim League, advocating for the creation of Pakistan.
• It delves into key historical events such as the Lucknow Pact (1916), Pakistan Resolution (1940), and the eventual Partition of India (1947), analyzing Jinnah's role in each.

Additional Information

• About the Author:
• Dr. Ishtiaq Ahmed has written extensively on South Asian politics, history, and the partition of India. His works are known for their neutral and factual approach.
• He has contributed significantly to the understanding of the complex relationship between India and Pakistan through his academic and research work.
• Key Themes of the Book:
• The book explores how Jinnah's political ideology evolved over time, from being a liberal secularist to a leader advocating for a separate nation for Muslims.
• It critically examines Jinnah's failures, such as his inability to prevent communal violence during the partition.
• The book also sheds light on the internal workings of the Muslim League and its strategy to achieve the goal of Pakistan.
• Important Historical Context:
• The partition of India in 1947 led to the creation of two nations, India and Pakistan. Jinnah played a pivotal role in this process as the leader of the Muslim League.
• The book discusses how Jinnah's vision of Pakistan as a secular state was later overshadowed by religious and political factors.
• Other Works on Jinnah:
• Books like "Jinnah of Pakistan" by Stanley Wolpert and "The Sole Spokesman" by Ayesha Jalal also provide significant insights into Jinnah's life and politics.
• However, Dr. Ishtiaq Ahmed's work stands out for its comprehensive analysis and critical approach.

Important Points

• Muhammad Ali Jinnah:
• Born on 25 December 1876 in Karachi, Jinnah was a barrister by profession and became the leader of the All-India Muslim League.
• He is often referred to as the "Quaid-e-Azam" (Great Leader) in Pakistan.
• Jinnah's leadership played a crucial role in the passage of the Pakistan Resolution in 1940.
• Partition of India:
• The partition in 1947 was one of the largest mass migrations in history, resulting in significant loss of life and property.
• Jinnah's demand for a separate homeland for Muslims was based on the Two-Nation Theory, which argued that Hindus and Muslims were distinct nations with their own customs, religion, and traditions.
• Legacy of Jinnah:
• Jinnah's vision for Pakistan remains a subject of debate, with differing interpretations of his secular and Islamic ideologies.
• He passed away on 11 September 1948, just over a year after Pakistan's independence.

The correct answer is Both statements 1 and 2 are correct.

Key Points

• Statement 1: The 36th National Games of India were held in 2022 in the state of Gujarat. This is correct because Gujarat hosted the event, making it the first time the state organized the National Games.
• Statement 2: The events were hosted across six cities in Gujarat, namely Ahmedabad, Gandhinagar, Surat, Vadodara, Rajkot, and Bhavnagar. This statement is also correct as these cities were the venues for various sporting events.
• The 36th National Games were initially scheduled for 2020 but were postponed to 2022 due to the COVID-19 pandemic.
• It was one of the largest sporting events in India, with participation from athletes across the country competing in various disciplines.
• Gujarat made significant infrastructure advancements to successfully host the games, including setting up modern sports complexes in multiple cities.

Additional Information

• Historical Context:
• The National Games of India is a multi-sport event held every few years to showcase the best talent from different states and union territories.
• The first edition of the National Games was held in Lahore in 1924 (then part of British India).
• Key Features of the 36th National Games:
• The event featured over 36 sports disciplines, including traditional games like kabaddi, kho-kho, and mallakhamb, alongside modern sports.
• Approximately 7,000 athletes from 28 states and 8 union territories participated.
• The motto of the 36th National Games was "Celebrating Unity Through Sports."
• It was inaugurated by Prime Minister Narendra Modi on 29th September 2022 in Ahmedabad.
• Role of Gujarat:
• Gujarat became the first state to complete the National Games within a record time of just three months after being awarded the hosting rights.
• Significant investments were made in upgrading sports facilities, such as the Sardar Patel Sports Complex in Ahmedabad and the Ekana Sports City in other cities.
• Previous Editions:
• The 35th National Games were held in Kerala in 2015, with over 11,000 participants.
• The National Games are organized under the aegis of the Indian Olympic Association (IOA).

Important Points

• Significance of the National Games:
• The National Games serve as a platform for athletes to qualify for international events like the Asian Games, Commonwealth Games, and the Olympics.
• They promote sports culture and infrastructure development in the host state, leaving a lasting legacy.
• Challenges of Hosting:
• Logistical challenges include accommodating and managing thousands of athletes, officials, and spectators.
• Infrastructure upgrades and compliance with international standards are essential for successful hosting.
• Future Editions:
• The 37th National Games are scheduled to be held in Goa, with a focus on eco-friendly and sustainable practices.

The correct answer is Union.

Key Points

• Taxes on railway fares and freights are a part of the revenue collected by the Union Government in India. This is because railways fall under the Union List of the Seventh Schedule of the Indian Constitution.
• The Union List enumerates subjects on which only the Parliament has the power to legislate, and railways are one of these subjects (Entry 22 of the Union List).
• The revenue generated from taxes on railway fares and freight charges is used by the Union Government to manage and maintain the Indian Railways, which is one of the largest railway networks in the world.
• State governments, municipal corporations, and panchayats do not have any jurisdiction over railway taxation as it is exclusively managed by the Union Government.
• This centralized system ensures uniformity in railway operations and taxation across the country.

Additional Information

• Seventh Schedule of the Indian Constitution:
• It defines the division of powers between the Union Government and State Governments through three lists: Union List, State List, and Concurrent List.
• The Union List contains subjects of national importance such as defense, foreign affairs, atomic energy, and railways.
• Only the Parliament can legislate on matters in the Union List.
• Indian Railways:
• Indian Railways is a centralized organization under the Ministry of Railways.
• It operates one of the largest rail networks in the world, with over 68,000 kilometers of track.
• The revenue generated from freight and passenger services contributes significantly to the Union Government's budget.
• Role of the Union Government:
• The Union Government is responsible for the operation, maintenance, and expansion of the railway network in India.
• It also determines the fares, freight rates, and taxation policies for railways.
• Importance of Railways:
• Railways are the lifeline of India's economy, providing affordable and efficient transportation for goods and passengers.
• They play a critical role in national integration and economic development.

Important Points

• Entry 22 of the Union List: Railways are explicitly mentioned in Entry 22 of the Union List, giving the Union Government exclusive power to legislate on matters related to railways, including taxation.
• Railway Budget: Until 2017, a separate railway budget was presented in the Parliament. However, it has now been merged with the general Union budget.
• Revenue Sources: The major sources of revenue for Indian Railways include passenger fares, freight charges, and other taxes collected by the Union Government.
• Centralized Governance: The centralized governance of railways ensures a unified and consistent approach to railway management, safety, and development across the country.

The correct answer is Manipur (Naga Hills).

Key Points

• Rani Gaidinliu was a prominent Naga spiritual and political leader from the Zeliangrong tribe of the Naga community.
• She led a fierce armed rebellion against the British colonial rule in the Naga Hills of Manipur in the 1930s, aiming to secure the independence of her people.
• She was deeply influenced by the Heraka religious movement, which sought to revive Naga tribal culture and traditions, and used it as a platform to unite her community against colonial oppression.
• At the young age of 16, she was captured by the British in 1932 and sentenced to life imprisonment but was later released in 1947 after India gained independence.
• Rani Gaidinliu is honored in Indian history as a freedom fighter and is remembered for her efforts to protect the cultural and political autonomy of her people.

Additional Information

• About Rani Gaidinliu:
• Born on 26 January 1915, in the village of Nungkao (now in Manipur), she belonged to the Zeliangrong community of Nagas.
• She joined the Heraka religious movement under the guidance of her cousin Jadonang, who was also a spiritual and political leader.
• After Jadonang's execution by the British in 1931, she took over the leadership of the movement and expanded its scope to include political resistance against colonial rule.
• Heraka Movement:
• The Heraka movement was initially a spiritual reform movement but evolved into a political movement under Gaidinliu's leadership.
• It sought to preserve the Naga culture and identity while resisting British interference in their socio-political life.
• Recognition and Legacy:
• She was given the title of "Rani" (Queen) by Pandit Jawaharlal Nehru, India's first Prime Minister, in recognition of her bravery and contribution to India's freedom struggle.
• The Government of India posthumously awarded her the Padma Bhushan in 1982.
• Several institutions, roads, and awards have been named after her to honor her legacy.
• Other Related Regions:
• Arunachal Pradesh (Naga Hills): While part of the larger northeastern region, this area was not the central location of Gaidinliu's rebellion.
• Meghalaya (Khasi Hills): The Khasi Hills are known for their own unique tribal identity but were not associated with Rani Gaidinliu's movement.
• Mizoram (Lushai Hills): This region is distinct from the Naga Hills and was not a part of the resistance led by Gaidinliu.

Important Points

• Rani Gaidinliu spent 14 years in British prisons, enduring harsh conditions before being released in 1947.
• She advocated for the autonomy of the Naga people and opposed the conversion of Nagas to Christianity, emphasizing the importance of preserving their indigenous traditions.
• A commemorative Rani Gaidinliu Tribal Freedom Fighters Museum was inaugurated in Manipur to honor her contributions.
• The Indian government released a postal stamp in her honor in 1996.
• Her life and struggle remain a symbol of the resistance of tribal communities against colonial and external forces.

The correct answer is N₂ & O₂.

Key Points

• Earth's atmosphere is primarily composed of nitrogen (N₂) and oxygen (O₂), which together account for nearly 99% of the atmosphere by volume.
• Nitrogen (N₂) constitutes approximately 78% of Earth's atmosphere, making it the most abundant gas.
• Oxygen (O₂) is the second most abundant gas, constituting roughly 21% of the atmosphere.
• Other gases like argon (Ar), carbon dioxide (CO₂), and trace amounts of neon (Ne), helium (He), and others make up the remaining 1%.
• The dominance of N₂ and O₂ plays a crucial role in supporting life on Earth, with oxygen being essential for respiration and nitrogen being integral to biological processes like protein synthesis.

Additional Information

• Nitrogen (N₂):
• Nitrogen is a chemically inert gas under normal conditions, which helps stabilize the atmosphere and prevent rapid combustion.
• It is essential for life as it is a fundamental component of amino acids, DNA, and RNA, and is utilized in biological processes like the nitrogen cycle.
• Nitrogen is widely used in industries, such as in the production of fertilizers, explosives, and refrigerants.
• Oxygen (O₂):
• Oxygen is vital for cellular respiration, which is the process by which organisms convert glucose into energy.
• It is highly reactive and combines with other elements to form oxides, which are crucial for various chemical reactions.
• Oxygen also supports combustion and is used in industries like welding, metallurgy, and medical applications.
• Other Gases:
• Argon (Ar): Constitutes about 0.93% of the atmosphere. It is inert and is primarily used in welding and lighting applications.
• Carbon Dioxide (CO₂): Although present in small amounts (~0.04%), it plays a critical role in the greenhouse effect and is essential for photosynthesis in plants.
• Neon (Ne): A trace gas used in lighting and high-voltage indicators.
• Atmospheric Layers:
• Earth's atmosphere is divided into layers: Troposphere, Stratosphere, Mesosphere, Thermosphere, and Exosphere.
• The gases are most concentrated in the troposphere, where weather phenomena occur.

Important Points

• Historical Context: Earth's atmosphere has evolved over billions of years, transitioning from a primordial atmosphere rich in carbon dioxide and methane to the current nitrogen-oxygen-rich composition.
• Impact on Life: The presence of oxygen in the atmosphere increased significantly during the Great Oxidation Event, approximately 2.5 billion years ago, enabling the evolution of complex life forms.
• Industrial Applications: Atmospheric gases are extracted for various uses, such as liquid nitrogen for cryopreservation and oxygen tanks for medical purposes.
• Climate Change: The increase in greenhouse gases like CO₂ due to human activities is altering the atmospheric composition, impacting global temperatures and ecosystems.

The correct answer is Poverty alleviation and modernization of technology.

Key Points

• The Sixth Five-Year Plan (1980-1985) of India aimed at achieving a balance between growth and social justice. Its primary focus was on poverty alleviation through employment generation and modernization of industries.
• It introduced programs like the Integrated Rural Development Programme (IRDP) and National Rural Employment Programme (NREP), which were key strategies for reducing poverty by providing employment opportunities in rural areas.
• The plan emphasized the modernization of technology to boost productivity in both agriculture and industry. This included promoting mechanization in agriculture and upgrading industrial infrastructure.
• Special attention was given to sectors like energy, irrigation, and transport to support growth in key industries and improve rural connectivity.
• The Sixth Plan also marked a shift towards a more technology-driven economy, focusing on introducing modern technologies in various sectors to strengthen India's industrial and economic base.

Additional Information

• Integrated Rural Development Programme (IRDP):
• Introduced in 1980, this program aimed at providing employment and alleviating poverty in rural areas.
• It provided financial and technical assistance to poor families, enabling them to engage in productive activities.
• National Rural Employment Programme (NREP):
• Launched in 1980, the NREP aimed at generating wage employment in rural areas through the construction of durable community assets.
• Examples include roads, irrigation works, and drinking water facilities.
• Modernization of Technology:
• The Sixth Plan emphasized introducing modern agricultural machinery like tractors and harvesters to improve farm productivity.
• It also focused on upgrading industrial technology to increase efficiency and reduce production costs.
• Energy Sector Growth:
• The plan prioritized the development of energy infrastructure, including hydroelectric, thermal, and nuclear power plants.
• This was crucial for meeting the growing energy demands of industries and households.
• Transport and Connectivity:
• Improved road and rail networks were a significant focus to ensure better market access for rural and urban areas.
• This also supported industrial growth by reducing logistics costs.

Important Points

• The Sixth Plan marked a departure from the earlier "growth-first" approach, focusing instead on employment generation and poverty reduction.
• Agriculture was given equal importance along with industrial modernization to ensure food security and rural development.
• The plan also stressed the importance of achieving self-reliance in critical technologies, particularly in defense and nuclear energy.
• For the first time, issues like population control and environmental conservation were included as part of the development agenda.
• The Sixth Plan's success laid the groundwork for India's subsequent economic reforms and modernization in the 1990s.

The correct answer is Globalisation.

Key Points

• Foreign Direct Investment (FDI) is a key component of globalisation, as it involves the investment of capital by a foreign entity into the economy of another country.
• FDI facilitates the integration of domestic economies with the global market, enabling the exchange of resources, technology, and expertise across borders.
• The concept of globalisation involves the removal of barriers to international trade and investment, including policies that encourage FDI, such as easing trade restrictions and reducing tariffs.
• FDI is critical for fostering economic growth and development in countries by bringing in foreign capital, creating jobs, and improving infrastructure and technology.
• Globalisation promotes an interconnected world where FDI plays a pivotal role in enabling businesses to operate internationally, establish global supply chains, and enhance competitiveness.

Additional Information

• Definition of Globalisation:
• Globalisation refers to the process of increasing interdependence and interconnectedness among countries, particularly in terms of trade, investment, and cultural exchange.
• It involves the removal of trade barriers, increased mobility of capital and labor, and the spread of technology and innovation.
• FDI and Economic Growth:
• FDI is a significant driver of economic development as it brings in foreign capital, creates employment opportunities, and improves productivity.
• Countries often compete to attract FDI by offering incentives such as tax breaks, subsidies, and liberalised trade policies.
• For example, India has attracted significant FDI in sectors like information technology, manufacturing, and retail due to its liberal economic policies and large consumer market.
• Globalisation and Technology Transfer:
• FDI facilitates the transfer of advanced technologies and management practices from developed to developing economies.
• This transfer enhances the capabilities of domestic industries, making them more competitive in the global market.
• Privatisation vs Globalisation:
• While privatisation involves transferring ownership of public sector enterprises to private entities, globalisation focuses on integrating economies with the global market.
• FDI aligns more closely with globalisation, as it involves cross-border investments and fosters international economic integration.
• Examples of FDI in India:
• India has witnessed substantial FDI in sectors like e-commerce (Amazon, Walmart-owned Flipkart), automobile (Toyota, Hyundai), and telecommunication (Vodafone, Airtel).
• As per the Department for Promotion of Industry and Internal Trade (DPIIT), India received a record FDI inflow of USD 83.57 billion during the financial year 2021-22.

Important Points

• FDI Policy in India:
• The Indian government has liberalised its FDI policy over the years to attract more foreign investment. Sectors like defence, insurance, and retail have been opened up for higher FDI limits.
• Currently, India permits 100% FDI in several sectors such as automobiles, infrastructure, and telecom under the automatic route.
• Some sectors, like defence and insurance, require approval from the Foreign Investment Promotion Board (FIPB) or the Ministry of Finance.
• Global Institutions Promoting FDI:
• Organisations like the World Bank, International Monetary Fund (IMF), and World Trade Organization (WTO) play a crucial role in promoting globalisation and facilitating FDI flows.
• These institutions provide financial assistance, technical expertise, and policy guidance to countries to enhance their global competitiveness.

The correct answer is Article 360.

Key Points

• Article 360 of the Indian Constitution empowers the President of India to declare a Financial Emergency in the country if he/she is satisfied that the financial stability or credit of India or any part of its territory is threatened.
• During a Financial Emergency, the executive authority of the Union extends over all financial matters in the country.
• The President can issue directions to states to ensure financial propriety, including reducing the salaries and allowances of government employees, including those of judges of the Supreme Court and High Courts.
• Any money bill or financial legislation passed by the Parliament during this time must also be approved by the President.
• No Financial Emergency has been declared in India so far, making it a provision that has never been invoked in Indian history.

Additional Information

• Other Types of Emergencies Under the Indian Constitution:
• National Emergency (Article 352): This can be declared by the President when there is a threat to the security of India or any part of its territory due to war, external aggression, or armed rebellion.
• State Emergency (Article 356): Also called President's Rule, it can be proclaimed if the President is satisfied that the governance of a state cannot be carried out as per the Constitution.
• Process for Declaring a Financial Emergency:
• The proclamation must be approved by both Houses of Parliament within two months of its issuance.
• Once approved, the Financial Emergency remains in force until it is revoked by the President.
• Impact of Financial Emergency:
• The Union Government gains absolute control over state financial matters.
• The salaries of government officials, including judges, can be reduced.
• All money bills passed by state legislatures require the President's approval.
• Comparison with Other Countries:
• Similar emergency provisions exist in other countries, like the United States, where the President can declare a financial emergency, but the specifics of implementation vary.

Important Points

• Historical Context:
• Though no Financial Emergency has ever been declared in India, the provisions were included to address extreme financial crises that could potentially destabilize the country.
• Checks and Balances:
• The requirement for Parliamentary approval within two months ensures that the use of this provision is not arbitrary or misused.
• Significance:
• Article 360 acts as a safeguard for the financial stability and credit of the nation, ensuring that the government can take necessary actions during a fiscal crisis.

Shortcut Trick

Recognize that 7396 = 86² and use the algebraic identity a² − b² = (a − b)(a + b).

⇒ (34.12)² − (86)²

⇒ (34.12 − 86) × (34.12 + 86)

⇒ (−51.88) × 120.12 = −6231.8256

∴ The correct answer is −6231.8256.

Alternate Method

Given:

Expression = (34.12)² − 7396

Calculations:

⇒ First, find the exact square value of 34.12.

⇒ (34.12)² = 34.12 × 34.12 = 1164.1744

⇒ Substitute this calculated value back into the original expression.

⇒ 1164.1744 − 7396

⇒ −6231.8256

∴ The correct answer is −6231.8256.

Additional Information

Algebraic Identities

The difference of squares formula, a² − b² = (a − b)(a + b), is very helpful for quickly solving expressions involving squared numbers.

Perfect Squares Near 100

Estimating squares helps in quick identification: since 80² = 6400 and 90² = 8100, a perfect square ending in 6 between them must be 84² or 86². Checking 86 gives exactly 7396.

BODMAS Rule

Always follow the operational order for simplification: Brackets, Orders (Powers and Roots), Division and Multiplication, Addition and Subtraction.

Shortcut Trick

Rate of Growth = [ √(Final Value ÷ Initial Value) − 1 ] × 100

Rate = [ √(26000 ÷ 16000) − 1 ] × 100

Rate = [ √1.625 − 1 ] × 100

Rate ≈ [ 1.2747 − 1 ] × 100 ≈ 27.47% ≈ 27.5%

∴ The correct answer is 27.5%.

Alternate Method

Given:

Initial production (P) = 16000

Final production (A) = 26000

Time (n) = 2 years

Formula Used:

A = P × (1 + R/100)ⁿ

Time (n) = 2 Years

Initial (P) 16000

Final (A) 26000

Rate (R) = [ √(A ÷ P) − 1 ] × 100

Calculations:

⇒ 26000 = 16000 × (1 + R/100)²

⇒ (1 + R/100)² = 26000 ÷ 16000

⇒ (1 + R/100)² = 26 ÷ 16

⇒ (1 + R/100)² = 13 ÷ 8 = 1.625

⇒ 1 + R/100 = √1.625

⇒ 1 + R/100 ≈ 1.2747

⇒ R/100 ≈ 1.2747 − 1 = 0.2747

⇒ R ≈ 27.47%

⇒ R ≈ 27.5%

∴ The correct answer is 27.5%.

Additional Information

Compound Annual Growth Rate (CAGR)

It represents the constant rate at which a quantity grows over a period, assuming growth is reinvested. Formula: CAGR = [ (Final Value ÷ Initial Value)^1/n − 1 ] × 100.

Successive Percentage Change

If a population or production value grows by x% in year 1 and y% in year 2, the effective net growth over two years is calculated as: Net % Change = x + y + (x × y) ÷ 100.

Rule of 72 Shortcut

To quickly estimate the number of years required to double an investment or quantity at a constant annual growth rate (R), use the approximate formula: Years ≈ 72 ÷ R.

Shortcut Trick

Effective Simple Interest rate for 3 years = 6% × 3 = 18%

Total Amount = Principal + Interest = 100% + 18% = 118%

Total Amount = 118% of 1500

⇒ Amount = (118 × 1500) ÷ 100 = 1770

∴ The correct answer is Rs. 1770.

Alternate Method

Given:

Principal (P) = Rs. 1500

Rate (R) = 6% per annum

Time (T) = 3 years

Formula Used:

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

Amount (A) = P + I

SI = 18% of Principal (6% × 3 years)

Principal Rs. 1500

Interest

• Rs. 270

Amount Rs. 1770

Calculations:

⇒ I = (1500 × 6 × 3) ÷ 100

⇒ I = 15 × 18

⇒ I = 270

⇒ Amount = 1500 + 270 = 1770

∴ The correct answer is Rs. 1770.

Additional Information

Simple vs Compound Interest

Simple Interest is calculated only on the principal amount, whereas Compound Interest is calculated on the principal and also on the accumulated interest of previous periods.

Amount Formula for Simple Interest

A = P(1 + (RT ÷ 100)), where A is Amount, P is Principal, R is Rate per annum, and T is Time in years.

Rule of 72

A quick mental math trick used to estimate the number of years required to double your invested money at a given annual interest rate (Time ≈ 72 ÷ Rate).

Shortcut Trick

Convert speed directly to m/s and multiply by time to find the distance.

Distance = Speed × Time

Distance = 30 × (5/18) × 72

Distance = (25/3) × 72 = 600 m

∴ The correct answer is 600 m.

Alternate Method

Start End

Speed = 30 kmph = 25/3 m/s Time = 72 seconds

Distance = ?

Given:

Speed of Suraj = 30 kmph

Time taken = 72 seconds

Formula Used:

Distance = Speed × Time

1 kmph = 5/18 m/s

Calculations:

⇒ First, convert the speed into m/s.

⇒ Speed = 30 × (5/18) m/s

⇒ Speed = 150/18 m/s = 25/3 m/s

⇒ Now, substitute the values into the distance formula.

⇒ Distance = (25/3) × 72

⇒ Distance = 25 × 24

⇒ Distance = 600 m

∴ The correct answer is 600 m.

Additional Information

Unit Conversion

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

Average Speed

When traveling equal distances at different speeds S₁ and S₂, the average speed formula is (2 × S₁ × S₂) ÷ (S₁ + S₂).

Relative Speed

If two objects are moving in the same direction, relative speed is |S₁ − S₂|. In opposite directions, it is S₁ + S₂.

Shortcut Trick

Let the dimensions in Arithmetic Progression (AP) be (a − d), a, and (a + d).

Sum of dimensions = 3a = 21 ⇒ a = 7 cm. So, the middle dimension must be exactly 7 cm.

Check the given options for a middle value of 7 cm and a product (volume) of 168 cm³.

Only Option 4 has dimensions 2 cm, 7 cm, and 12 cm.

Sum = 2 + 7 + 12 = 21 cm | Volume = 2 × 7 × 12 = 168 cm³ (Both conditions satisfied).

∴ The correct answer is 2 cm, 7 cm, 12 cm.

Alternate Method

Given:

Sum of the three dimensions = 21 cm

Volume of the parallelepiped = 168 cm³

Formula Used:

Volume of a cuboid (parallelepiped) = Length × Breadth × Height

Rectangular Parallelepiped Volume = 168 cm³, Sum of dimensions = 21 cm Length (a + d) Height (a) Width (a − d)

Calculation:

⇒ Let the dimensions in AP be (a − d), a, and (a + d).

⇒ Sum of dimensions = (a − d) + a + (a + d) = 21

⇒ 3a = 21

⇒ a = 7 cm

⇒ Volume = (a − d) × a × (a + d) = 168

⇒ a(a² − d²) = 168

⇒ 7(7² − d²) = 168

⇒ 49 − d² = 168 ÷ 7

⇒ 49 − d² = 24

⇒ d² = 49 − 24 = 25

⇒ d = 5 cm

⇒ The required dimensions are: (7 − 5) cm, 7 cm, and (7 + 5) cm.

⇒ The dimensions are 2 cm, 7 cm, and 12 cm.

∴ The correct answer is 2 cm, 7 cm, 12 cm.

Additional Information

Arithmetic Progression (AP)

When taking 3 unknown terms in an AP, always assume them as (a − d), a, (a + d). This instantly eliminates the common difference (d) when their sum is provided.

Rectangular Parallelepiped

A rectangular parallelepiped is simply a 3D box or a cuboid. Its volume is given by Length × Breadth × Height.

Total Surface Area of Cuboid

The total surface area is 2(lb + bh + hl) where l, b, and h are its side dimensions.

Shortcut Trick

The radius of a circle can be calculated directly using the arc length (l) and central angle (θ): r = (l × 360°) ÷ (θ × 2π)

⇒ r = (22 × 360) ÷ (72 × 2 × 22/7)

⇒ r = (360 × 7) ÷ (72 × 2)

⇒ r = (5 × 7) ÷ 2 = 35 ÷ 2

⇒ r = 17.5 cm

∴ The correct answer is 17.5 cm.

Alternate Method

72° Arc length (l) = 22 cm r r

O

Given:

Arc length (l) = 22 cm

Central angle (θ) = 72°

Formula Used:

Length of an arc = (θ ÷ 360°) × 2πr

Calculations:

⇒ 22 = (72° ÷ 360°) × 2 × (22 ÷ 7) × r

⇒ 22 = (1 ÷ 5) × (44 ÷ 7) × r

Dividing both sides by 22:

⇒ 1 = (1 ÷ 5) × (2 ÷ 7) × r

⇒ 1 = (2 ÷ 35) × r

⇒ r = 35 ÷ 2

⇒ r = 17.5 cm

∴ The correct answer is 17.5 cm.

Additional Information

Area of a Sector

The area of a sector of a circle with central angle θ is given by A = (θ ÷ 360°) × πr². Alternatively, if the arc length (l) is known, the area can be calculated as A = (1 ÷ 2) × l × r.

Radian Measure

When the central angle θ is measured in radians, the formula for the arc length simplifies to l = r × θ. Note that 180° is equal to π radians.

Circumference and Area of Circle

The total boundary length (circumference) of a complete circle is 2πr, corresponding to an angle of 360°. The total area enclosed by the circle is πr².

Shortcut Trick

Total expenditure = 30% + 20% + 25% = 75%.

Remaining savings = 100% − 75% = 25%.

Given that 25% of the total salary corresponds to Rs. 9,000.

Total Salary (100%) = Rs. 9,000 × 4 = Rs. 36,000.

∴ The correct answer is Rs. 36,000.

Alternate Method

Given:

Percentage spent on rent = 30%

Percentage spent on groceries = 20%

Percentage spent on other expenses = 25%

Total savings = Rs. 9,000

Salary Distribution

Rent 30%

Grocery 20%

Other 25%

Savings 25%

= Rs. 9,000

Total Salary = 100% = ?

Calculations:

Let the total salary of the man be x.

⇒ Total percentage spent = 30% + 20% + 25%

⇒ Total percentage spent = 75%

⇒ Percentage of salary saved = 100% − 75% = 25%

⇒ 25% of x = Rs. 9,000

⇒ (25 ÷ 100) × x = 9000

⇒ x ÷ 4 = 9000

⇒ x = 9000 × 4

⇒ x = 36000

∴ The correct answer is Rs. 36,000.

Additional Information

Successive Percentage

When a person spends a percentage of the "remaining" amount rather than the total amount, the final value is calculated using the successive discount principle: Final = Initial × (1 − r₁/100) × (1 − r₂/100).

Percentage to Fraction Conversion

Memorizing common percentage-to-fraction values speeds up calculations significantly. Common examples: 20% = 1/5, 25% = 1/4, 30% = 3/10, and 33.33% = 1/3.

Base Change Rule

If value A is x% more than value B, then B is (x ÷ (100 + x)) × 100% less than A.

Shortcut Trick

Direct multiplying factor method for successive percentage changes:

Final Amount = MP × (100 − Discount)% × (100 + VAT)%

Final Amount = 1600 × 75% × 105%

Final Amount = 1600 × (3 ÷ 4) × (21 ÷ 20)

Final Amount = 1600 × (63 ÷ 80) = 20 × 63 = 1260

∴ The correct answer is ₹1260.

Alternate Method

Given:

Marked Price (MP) = ₹1600

Discount = 25%

VAT = 5%

Formula Used:

Selling Price (SP) = MP − Discount Amount

Final Amount = SP + VAT Amount

−25%

+5% VAT

MP ₹1600

SP ₹1200

Final ₹1260

Calculations:

⇒ Discount = 25% of 1600 = (25 ÷ 100) × 1600 = ₹400

⇒ Selling Price (SP) = 1600 − 400 = ₹1200

⇒ VAT = 5% of 1200 = (5 ÷ 100) × 1200 = ₹60

⇒ Final Amount = SP + VAT = 1200 + 60 = ₹1260

∴ The correct answer is ₹1260.

Additional Information

Marked Price (MP) & Discount

Discount is always offered directly on the Marked Price (or MRP) of an article. Selling Price (SP) = MP − Discount.

Value Added Tax (VAT) / GST

Tax is always calculated on the discounted Selling Price, not on the Marked Price. Customer Final Amount = SP + Tax Amount.

Successive Percentage Changes

When two percentage changes occur sequentially, Net Change % = a + b + (a × b) ÷ 100. Here, Net Change % = −25 + 5 + (−25 × 5) ÷ 100 = −21.25% overall reduction.

Shortcut Trick

Using the LCM method for equal distances:

LCM of given speeds (30, 45, 90) = 90. Let each equal distance segment be 90 km.

Total Distance = 3 × 90 = 270 km.

Total Time = (90 ÷ 30) + (90 ÷ 45) + (90 ÷ 90) = 3 + 2 + 1 = 6 hours.

Average Speed = Total Distance ÷ Total Time = 270 ÷ 6 = 45 km/h.

∴ The correct answer is 45 km/h.

Alternate Method

Given:

Three equal distances are covered at speeds of 30 km/h, 45 km/h, and 90 km/h.

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

30 km/h Dist = d

45 km/h Dist = d

90 km/h Dist = d

Total Distance = 3d Result: Average Speed = 45 km/h

Calculations:

⇒ Let each equal distance part be d km.

⇒ Total Distance = d + d + d = 3d km.

⇒ Time taken for the first part = d ÷ 30 hours.

⇒ Time taken for the second part = d ÷ 45 hours.

⇒ Time taken for the third part = d ÷ 90 hours.

⇒ Total Time = (d ÷ 30) + (d ÷ 45) + (d ÷ 90)

⇒ Taking the LCM of denominators (30, 45, and 90), which is 90:

⇒ Total Time = (3d + 2d + d) ÷ 90 = 6d ÷ 90 = d ÷ 15 hours.

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = 3d ÷ (d ÷ 15)

⇒ Average Speed = 3 × 15 = 45 km/h.

∴ The correct answer is 45 km/h.

Additional Information

Average Speed for Two Equal Distances

When two equal distances are covered at speeds x and y, the average speed can directly be calculated using the formula: 2xy ÷ (x + y).

Average Speed for Three Equal Distances

When three equal distances are covered at speeds x, y, and z, the average speed formula expands to: 3xyz ÷ (xy + yz + zx).

General Average Speed Formula

For unequal distance splits, the direct formulas do not apply. Always use the fundamental definition: Average Speed = Total Distance Covered ÷ Total Time Taken.

Shortcut Trick

Remaining mixture = 40 − 8 = 32 L. Acid = 32 × (3/4) = 24 L, Water = 32 − 24 = 8 L.

Added mixture = 8 L. Acid = 8 × (1/4) = 2 L, Water = 8 − 2 = 6 L.

New Acid = 24 + 2 = 26 L

New Water = 8 + 6 = 14 L

Final ratio = 26 : 14 = 13 : 7

∴ The correct answer is 13:7.

Alternate Method

Given:

Initial mixture = 40 L

Initial ratio of Acid : Water = 3 : 1

Mixture removed = 8 L

Mixture added = 8 L (Acid : Water = 1 : 3)

Remaining 32 Litres Acid: 24L Water: 8L + Added 8 Litres Acid: 2L Water: 6L = Final 40L Acid: 26L Water: 14L Final Ratio = 26 : 14 = 13 : 7

Calculations:

⇒ Volume of the mixture remaining = 40 − 8 = 32 L

⇒ In 32 L mixture, amount of Acid = 32 × (3/4) = 24 L

⇒ In 32 L mixture, amount of Water = 32 × (1/4) = 8 L

⇒ In 8 L added solution, amount of Acid = 8 × (1/4) = 2 L

⇒ In 8 L added solution, amount of Water = 8 × (3/4) = 6 L

⇒ Final amount of Acid = 24 + 2 = 26 L

⇒ Final amount of Water = 8 + 6 = 14 L

⇒ Final ratio of Acid to Water = 26 : 14 = 13 : 7

∴ The correct answer is 13:7.

Additional Information

Alligation Rule

When mixing two quantities of concentrations C and D to get a mixture of mean concentration M, the ratio is (D − M) : (M − C).

Replacement Formula

When y units are repeatedly removed from x units of pure liquid and replaced with water n times, final pure liquid = x × (1 − y/x)ⁿ.

Multiple Mixtures

When multiple mixtures are combined, the total quantity of each individual component is directly summed up to find the resulting final ratio.

Shortcut Trick

Total work = LCM(15, 25) = 75 units.

Lalit's efficiency = 75 ÷ 15 = 5 units/day | Laxman's efficiency = 75 ÷ 25 = 3 units/day.

Work completed by both in 5 days = 5 × (5 + 3) = 40 units.

Remaining work = 75 − 40 = 35 units, which Lalit and Deven do in 4 days.

Deven's efficiency = (35 ÷ 4) − 5 = 8.75 − 5 = 3.75 units/day.

∴ The correct answer is 20 days.

Alternate Method

Given:

Time taken by Lalit to finish the work = 15 days

Time taken by Laxman to finish the work = 25 days

They work together for 5 days, and then Lalit and Deven finish the rest in 4 days.

Calculations:

⇒ Efficiency of Lalit + Laxman = 5 + 3 = 8 units/day

⇒ Work done by Lalit and Laxman in 5 days = 8 × 5 = 40 units

⇒ Remaining work to be done = 75 − 40 = 35 units

⇒ Lalit and Deven complete these 35 units in 4 days.

⇒ Combined efficiency of Lalit and Deven = 35 ÷ 4 = 8.75 units/day

⇒ Let Deven's efficiency be D.

⇒ 5 + D = 8.75

⇒ D = 8.75 − 5 = 3.75 units/day

⇒ Time taken by Deven alone to finish the whole work = Total Work ÷ D

⇒ Time taken = 75 ÷ 3.75 = 20 days

∴ The correct answer is 20 days.

Additional Information

Pipes and Cisterns

Works on the same mathematical principle as time and work, where inlet pipes have a positive efficiency and outlet (leak) pipes possess a negative efficiency.

Work and Wages

Total wages are distributed among workers in the direct ratio of the total amount of work physically done by each individual, not simply by their daily efficiency unless they work for the same number of days.

Alternate Day Work

Calculate the total work done in one complete cycle of days, divide the total work by this cycle work to find the number of full cycles, and assign the remaining work to the correct person based on their starting turn.

Shortcut Trick

Since Volume ∝ r² × h, we have height h ∝ V / r².

Ratio of heights (h₁/h₂) = (V₁/V₂) × (r₂/r₁)²

⇒ h₁/h₂ = (2/3) × (2/1)²

⇒ h₁/h₂ = (2/3) × 4 = 8/3

∴ The correct answer is 8:3.

Alternate Method

Given:

Ratio of volumes (V₁ : V₂) = 2 : 3

Ratio of radii (r₁ : r₂) = 1 : 2

Formula Used:

Volume of a cylinder (V) = πr²h

1x

h₁ V₁ = 2 units

2x

h₂ V₂ = 3 units

Calculations:

⇒ Volume of the first cylinder, V₁ = π × (r₁)² × h₁

⇒ Volume of the second cylinder, V₂ = π × (r₂)² × h₂

⇒ Ratio V₁ / V₂ = [π × (1x)² × h₁] / [π × (2x)² × h₂]

⇒ 2 / 3 = (x² × h₁) / (4x² × h₂)

⇒ 2 / 3 = (1 / 4) × (h₁ / h₂)

⇒ h₁ / h₂ = (2 × 4) / 3

⇒ h₁ / h₂ = 8 / 3

∴ The correct answer is 8:3.

Additional Information

Curved Surface Area of a Cylinder

The lateral or curved surface area is calculated as CSA = 2πrh, where 'r' is the base radius and 'h' is the height.

Total Surface Area of a Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πr(r + h).

Proportionality in Cylinders

If the radius is doubled while keeping the height constant, the volume increases by a factor of 4 (since Volume ∝ r²).

Shortcut Trick

The volume of any prism is given by the product of its base area and height.

Volume = Base Area × Height.

When the base area is kept constant, the volume becomes directly proportional to the height (Volume ∝ Height).

Therefore, if the height is multiplied by 3, the volume is also multiplied by 3.

∴ The correct answer is Tripled.

Alternate Method

Area = A

h Volume = V

Height × 3 →

Area = A

3h Volume = 3V

Given:

Initial Base Area = A

Initial Height = h

New Height = 3h

Formula Used:

Volume of a prism (V) = Base Area × Height

Calculations:

⇒ Let the initial volume be V₁.

⇒ V₁ = A × h

⇒ Let the new volume be V₂.

⇒ V₂ = A × (3h)

⇒ V₂ = 3 × (A × h)

⇒ V₂ = 3 × V₁

∴ The correct answer is Tripled.

Additional Information

Volume of a Pyramid

A pyramid's volume is exactly one-third of the volume of a prism that has the identical base area and vertical height. Formula: V = 1/3 × Base Area × Height.

Scaling 3D Shapes

When all dimensions of a perfectly similar 3D shape are multiplied by a uniform scale factor 'k', its total surface area is scaled by k², and its overall volume is scaled by k³.

Volume of a Cylinder

A cylinder functions identically to a prism but features a circular base rather than a polygon. Its volume is directly calculated as Area of Circular Base × Height, which evaluates to π × r² × h.

Shortcut Trick

When a solid is melted and recast, its total volume remains constant.

The volume of a cone is directly proportional to r² × h.

If the radius is halved, its square (r²) becomes 1/4 of the original.

To keep the overall volume constant, the new height must become 4 times the original.

New height = 4 × 12 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Original height of cone (h₁) = 12 cm

Original radius (r₁) = r

New radius (r₂) = r / 2

Let the new height be h₂.

Formula Used:

Volume of a cone = 1/3 × π × r² × h

r h = 12 Original Cone

→ Melted

r/2 h = ? New Cone

Calculations:

⇒ Volume of original cone = Volume of new cone

⇒ 1/3 × π × (r₁)² × h₁ = 1/3 × π × (r₂)² × h₂

⇒ 1/3 × π × (r)² × 12 = 1/3 × π × (r/2)² × h₂

⇒ r² × 12 = (r² / 4) × h₂

Canceling r² from both sides:

⇒ 12 = h₂ / 4

⇒ h₂ = 12 × 4 = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Volume of a Cone

V = 1/3 × π × r² × h, where r is the base radius and h is the vertical height.

Curved Surface Area (CSA)

CSA = π × r × l, where l is the slant height given by l = √(r² + h²).

Melting and Recasting Property

When any 3D object is melted and recast into a different shape, its volume remains conserved while the surface area may change.

Shortcut Trick

In a right-angled ▵ABC at C, the sum of the remaining two angles is 90°.

⇒ ∠A + ∠B = 90° → B = 90° − A

Using the complementary angle property: cos(90° − θ) = sin θ

⇒ cos B = cos(90° − A) = sin A = 5/13

∴ The correct answer is 5/13.

Alternate Method

A C B 13 5 12

Given:

⇒ In ▵ABC, ∠C = 90°

⇒ sin A = 5/13

Formula Used:

⇒ sin θ = Perpendicular / Hypotenuse

⇒ cos θ = Base / Hypotenuse

⇒ Pythagoras Theorem: H² = P² + B²

Calculation:

For angle A, Perpendicular = BC and Hypotenuse = AB.

⇒ sin A = BC / AB = 5 / 13

Let BC = 5 units and AB = 13 units.

Applying Pythagoras theorem in ▵ABC:

⇒ AB² = BC² + AC²

⇒ 13² = 5² + AC²

⇒ 169 = 25 + AC²

⇒ AC² = 169 − 25 = 144

⇒ AC = 12 units

Now, for angle B, Base = BC and Hypotenuse = AB.

⇒ cos B = Base / Hypotenuse = BC / AB

⇒ cos B = 5 / 13

∴ The correct answer is 5/13.

Additional Information

Complementary Angles

Two angles are complementary if their sum is 90°. Key trigonometric identities include sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the base and perpendicular: H² = P² + B².

Common Pythagorean Triplets

Integers that satisfy the Pythagoras theorem are called Pythagorean triplets. Common exam triplets are (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Shortcut Trick

To convert from degrees to radians, simply multiply the given angle by π ÷ 180.

Angle in radians = 3 × (π ÷ 180)

Angle in radians = π ÷ 60

∴ The correct answer is π/60 radians.

Alternate Method

Given:

Angle = 3°

Formula Used:

Angle in radians = Angle in degrees × (π ÷ 180)

180° = π rad 0° Center

3° π/60

Degree to Radian Conversion

Calculations:

⇒ We know that a half circle measures 180° and is equivalent to π radians.

⇒ 1° = π ÷ 180 radians

⇒ 3° = 3 × (π ÷ 180) radians

⇒ 3° = (3 × π) ÷ 180 radians

⇒ Simplifying the fraction by dividing the numerator and denominator by 3:

⇒ 3° = π ÷ 60 radians

∴ The correct answer is π/60 radians.

Additional Information

Radian to Degree Conversion

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

Relationship between Arc Length and Radius

Angle in radians (θ) = Arc length (s) ÷ Radius (r). This is the fundamental geometrical definition of a radian.

Common Angles in Radians

30° = π/6 | 45° = π/4 | 60° = π/3 | 90° = π/2 | 360° = 2π

Shortcut Trick

If the sum of two angles is 90°, they are complementary angles.

For any two complementary angles A and B (where A + B = 90°), we have the standard trigonometric identity: cos B = sin A.

Since we are given that sin A = 5/13, we can substitute this value directly.

Therefore, cos B = 5/13.

∴ The correct answer is 5/13.

Alternate Method

Given:

A + B = 90°

sin A = 5/13

Formula Used:

In a right-angled triangle:

⇒ sin θ = Opposite Side ÷ Hypotenuse

⇒ cos θ = Adjacent Side ÷ Hypotenuse

A

B

A C B

5 12 13

Right Triangle ABC

Calculations:

⇒ Let us consider a right-angled triangle ABC, right-angled at C.

⇒ Since A + B = 90°, A and B are the two acute angles of this right-angled triangle.

⇒ We are given sin A = 5/13, which means the ratio of the side Opposite to A (BC) to the Hypotenuse (AB) is 5/13.

⇒ Let BC = 5 units and AB = 13 units.

⇒ Using Pythagoras theorem: AC² + BC² = AB²

⇒ AC² + 5² = 13²

⇒ AC² + 25 = 169

⇒ AC² = 144 ⇒ AC = 12 units

⇒ We need to find the value of cos B.

⇒ For angle B, the adjacent side is BC.

⇒ cos B = Adjacent Side ÷ Hypotenuse = BC / AB

⇒ cos B = 5 / 13

∴ The correct answer is 5/13.

Additional Information

Complementary Angles in Trigonometry

If two angles A and B sum up to 90°, they are known as complementary angles. Important conversion identities include sin A = cos B, tan A = cot B, and sec A = cosec B.

Basic Trigonometric Ratios

In any right-angled triangle, the sine function (sin θ) represents the ratio of the Perpendicular to the Hypotenuse, and the cosine function (cos θ) represents the ratio of the Base to the Hypotenuse.

Pythagorean Triplets

A set of three positive integers that strictly satisfy the condition a² + b² = c² is called a Pythagorean triplet. Frequently tested triplets in exams include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Shortcut Trick

When two straight lines intersect, the vertically opposite angles are always equal in measure.

Since one of the angles is given as 45°, its vertically opposite angle must exactly be 45°.

∴ The correct answer is 45°.

Alternate Method

Given:

One angle at the intersection of two lines = 45°.

45° ?

A B C D O

Concept Used:

For any two intersecting lines, the vertically opposite angles are equal.

Calculations:

⇒ Let the two intersecting straight lines be AB and CD, intersecting at a common point O.

⇒ Let the given angle ∠BOC be 45°.

⇒ The angle that is vertically opposite to ∠BOC is ∠AOD.

⇒ According to the geometrical property of intersecting lines, opposite angles are identical.

⇒ ∴ ∠AOD = ∠BOC

⇒ ∠AOD = 45°

∴ The correct answer is 45°.

Additional Information

Vertically Opposite Angles

When two straight lines intersect, the pairs of non-adjacent angles directly opposite to each other are equal. In the diagram, ∠AOD = ∠BOC and ∠AOC = ∠BOD.

Linear Pair (Adjacent Angles)

The adjacent angles resting on the same straight line always sum up to 180°. In this case, 45° + 135° = 180°.

Supplementary Angles

Two angles are considered supplementary if their total sum equals exactly 180°.

Shortcut Trick

The incenter of a triangle is exactly the center of its inscribed circle.

By definition, the perpendicular distance (altitude) from the incenter to any side of the triangle is equal to the inradius.

Given the altitude to side AB is 2 units, the inradius directly equals 2 units.

The coordinates of the incenter (3, 4) are provided merely as a distractor and are not needed.

∴ The correct answer is 2 units.

Alternate Method

Incenter I (3, 4) r = 2 C A B

Given:

Incenter coordinates = (3, 4)

Perpendicular distance (altitude) from the incenter to side AB = 2 units

Concept Used:

The incenter is the point where the internal angle bisectors of a triangle intersect.

It forms the center of the incircle, which perfectly touches all three sides of the triangle.

The perpendicular distance from the center of the incircle to any tangent (the side of the triangle) is its radius.

Calculation:

⇒ Inradius (r) = Perpendicular distance from incenter to side AB

⇒ r = 2 units

⇒ The given coordinate (3, 4) is unneeded data intended to consume calculation time.

∴ The correct answer is 2 units.

Additional Information

Incenter and Area Relation

The area of any triangle can be calculated directly using its semi-perimeter (s) and inradius (r) with the formula: Area = r × s.

Circumcenter

The point of intersection of the perpendicular bisectors of the triangle's sides. Its distance to all three vertices is strictly equal (known as the circumradius).

Orthocenter

The point where the three perpendicular altitudes drawn from the vertices to the opposite sides intersect.

Centroid

The intersection of the medians. It divides each median in a distinct 2:1 ratio and represents the exact geometric center of gravity of the triangle.

Shortcut Trick

Use the standard algebraic identity: (a + b)² + (a − b)² = 2(a² + b²)

Substitute x for a and 1 for b: Expression = 2(x² + 1²)

Given x = √3, its square is x² = 3.

Expression = 2(3 + 1) = 2 × 4 = 8.

∴ The correct answer is 8.

Alternate Method

Given:

x = √3

Formula Used:

(a + b)² = a² + b² + 2ab

(a − b)² = a² + b² − 2ab

Calculations:

⇒ Expand the first term: (x + 1)² = x² + 1 + 2x

⇒ Expand the second term: (x − 1)² = x² + 1 − 2x

⇒ Add both expressions together: (x² + 1 + 2x) + (x² + 1 − 2x)

⇒ Group the like terms: x² + x² + 1 + 1 + 2x − 2x

⇒ The 2x and −2x terms cancel each other out.

⇒ Sum = 2x² + 2

⇒ Substitute the value of x = √3 into the simplified expression.

⇒ Sum = 2(√3)² + 2

⇒ Sum = 2 × 3 + 2

⇒ Sum = 6 + 2 = 8

∴ The correct answer is 8.

Additional Information

Related Algebraic Identity

When subtracting the squares of sums and differences, use the counterpart formula: (a + b)² − (a − b)² = 4ab.

Difference of Squares

Another highly useful algebraic property in simplification problems is: a² − b² = (a − b)(a + b).

Properties of Surds

The square of a square root returns the original number value: (√a)² = a, assuming a ≥ 0.

Shortcut Trick

By the Basic Proportionality Theorem (Thales Theorem), if DE || BC, then AD / DB = AE / EC.

Given AD / DB = 1 / 2 and AE = 5 cm.

Substitute directly: 1 / 2 = 5 / EC ⇒ EC = 5 × 2 = 10 cm.

∴ The correct answer is 10 cm.

Alternate Method

Given:

⇒ In ΔABC, DE || BC

⇒ AD / DB = 1 / 2

⇒ AE = 5 cm

Formula Used:

Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

A B C D E 1 part 2 parts AE = 5 cm EC = ?

Calculation:

⇒ According to the theorem, AD / DB = AE / EC

⇒ Substituting the given values: 1 / 2 = 5 / EC

⇒ EC = 5 × 2

⇒ EC = 10 cm

∴ The correct answer is 10 cm.

Additional Information

Basic Proportionality Theorem (BPT)

Also known as Thales Theorem. If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally: AD / DB = AE / EC.

Converse of BPT

If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

Similar Triangles (AA Similarity)

Since DE || BC, ΔADE ∼ ΔABC. The ratio of their corresponding sides is equal: AD / AB = AE / AC = DE / BC.

Mid-Point Theorem

A special case of BPT where D and E are mid-points. The line segment connecting the mid-points is parallel to the third side and half of its length (DE = BC / 2).

Shortcut Trick

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

Since chord AB = chord CD, the angles subtended by them must be exactly equal.

∠AOB = ∠COD ⇒ 50° = x° ⇒ x = 50.

∴ The correct answer is 50°.

Alternate Method

Given:

⇒ Chord AB = Chord CD

⇒ Angle subtended by AB at center (∠AOB) = 50°

⇒ Angle subtended by CD at center (∠COD) = x°

O

A B C D

50° x°

Calculations:

⇒ In ΔAOB and ΔCOD:

⇒ OA = OC (Radii of the same circle)

⇒ OB = OD (Radii of the same circle)

⇒ AB = CD (Given chords are equal)

⇒ ∴ ΔAOB ≅ ΔCOD (By SSS congruence criterion)

⇒ ∠AOB = ∠COD (Corresponding parts of congruent triangles are equal)

⇒ 50° = x°

⇒ x = 50

∴ The correct answer is 50°.

Additional Information

Angles Subtended by Equal Chords

Equal chords of a circle subtend equal angles at the center. Conversely, if the angles subtended by the chords are equal, then the chords are equal in length.

Perpendicular from the Center

A perpendicular drawn from the center of a circle to a chord perfectly bisects the chord. Conversely, the line joining the center to the midpoint of a chord is perpendicular to it.

Equidistant Chords Property

Equal chords of a circle are always equidistant from the center. Conversely, chords that are equidistant from the center are identically equal in length.

Shortcut Trick

For a segment subtending 90° at the center, we can directly use the formula: Area = r² × (π/4 − 1/2)

⇒ Area = 10² × (π/4 − 1/2)

⇒ Area = 100 × (π/4 − 1/2)

⇒ Area = 25π − 50 cm²

∴ The correct answer is 25π − 50 cm².

Alternate Method

O

A B

90°

10 cm 10 cm

Segment of the Circle

Given:

Radius of the circle (r) = 10 cm

Angle subtended at the center (θ) = 90°

Formula Used:

Area of sector = (θ ÷ 360°) × πr²

Area of right-angled triangle = (1/2) × base × height

Area of segment = Area of sector − Area of triangle

Calculations:

⇒ Area of sector = (90° ÷ 360°) × π × (10)²

⇒ Area of sector = (1/4) × 100π = 25π cm²

⇒ The triangle OAB formed by the radii is a right-angled triangle with base = 10 cm and height = 10 cm.

⇒ Area of triangle = (1/2) × 10 × 10 = 50 cm²

⇒ Area of segment = 25π − 50 cm²

∴ The correct answer is 25π − 50 cm².

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and central angle θ is given by the formula: Area = (θ ÷ 360°) × πr².

Area of a Segment

The minor segment area is found by subtracting the area of the corresponding triangle from the sector's area: Area = (θ ÷ 360°)πr² − (1/2)r²sin(θ).