SSC CGL 2025 (Held On: 16 Sept, 2025 Shift 1) (Previous Year Paper)

Given sentence: "Reflections are clearer in calm waters."

1. Reflections 2. are 3. clearer 4. in 5. calm 6. waters

6th word is WATERS.

The position and positional values or English Alphabets are:

Rule: If each letter is mirrored using alphabetical symmetry (A↔Z, B↔Y...), then:

2nd letter is A and it's mirroring is Z.

Hence, the correct answer is "Option 4".

Thus, option 2 is exactly the same as the one given.

Hence, the correct answer is "Option 2".

Given: m_u_mg_p_gu_

The pattern followed here is:

1. g,p,u,m,p → mgup / mgup / mgup 2. g,p,u,p,m → mgup / mgup / pgum 3. p,u,u,p,m → mpuu / mgup / pgum 4. p,u,u,p,m → mpuu / mgup / pgum

Clearly, in option 1 "mgup / mgup / mgup" is repeated.

Hence, the correct answer is "Option 1".

The logic followed here is:

Thus, GZNB will replace the question mark (?) in the following series.

Hence, the correct answer is "Option 3".

Given equation: 3.2 Z (0.56 Y 1.4 X 4) W 5 X 5 Y 25 Z 5.

After replacing the letters by their meaning, we get:

= 3.2 + (0.56 ÷ 1.4 × 4) - 5 × 5 ÷ 25 + 5

= 3.2 + (0.4 × 4) - 5 × 5 ÷ 25 + 5

= 3.2 + 1.6 - 5 × 5 ÷ 25 + 5

= 3.2 + 1.6 - 5 × 0.2 + 5

= 3.2 + 1.6 - 1 + 5

= 9.8 - 1

= 8.8

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: First and fourth letter are same, 2nd element is @ and number at 3rd position is the place value of the letter.

1. A@1A → follows the logic
2. B@2B  → follows the logic
3. C@4C  → doesn't follow the logic as the place value of C is 3.
4. D@4D  → follows the logic

Thus, C@4C is the odd one out.

Hence, the correct answer is "Option 3".

Logic: Each number = Previous number × 2 + 1

The logic followed here is:

1

1 × 2 + 1 = 2 + 1 = 3

3 × 2 + 1 = 6 + 1 = 7

7 × 2 + 1 = 14 + 1= 15

15 × 2 + 1 = 30 + 1 = 31

31 × 2 + 1 = 62 + 1 = 63

Thus, 63 will come at the place of question mark.

Hence, the correct answer is "Option 2".

Given:

Statement: Deforestation in tropical rainforests, primarily driven by agricultural expansion and logging, not only contributes to the loss of biodiversity but also exacerbates global warming by reducing the Earth's capacity to absorb carbon dioxide.

Assumptions:

I. Deforestation reduces the Earth's ability to absorb carbon dioxide - True (Because the statement explicitly states that deforestation exacerbates global warming by reducing the Earth's capacity to absorb carbon dioxide. This is directly implied by the statement.)

II. Agricultural expansion and logging are the primary causes of deforestation - True (Because the statement explicitly identifies agricultural expansion and logging as the primary drivers of deforestation in tropical rainforests.)

III. The loss of biodiversity is the most significant consequence of deforestation - False (Because the statement says that deforestation contributes to biodiversity loss and exacerbates global warming, but it does not state or assume that biodiversity loss is the most significant consequence.)

So, only Assumptions I and II are implicit.

Hence, the correct answer is "Option 1".

Given:

Statement: The city experienced its worst rainfall in 50 years

Conclusions:

I. The city's drainage systems may have failed - True ⇒ The phrase “may have failed” indicates a possibility. Extremely heavy rainfall can overwhelm or expose weaknesses in drainage systems, so this is a reasonable inference.

II. Such rainfall occurs regularly - False ⇒ Worst rainfall in 50 years” implies rarity, not regular occurrence. Hence, it contradicts the statement.

So, only Conclusion I follows.

Hence, the correct answer is "Option 1".

Given word: TRAVEL

First, second, third, and fifth letters of the word are: T, R, A, E

The meaningful words are that can be formed using the letters are:

RATE → The speed or measure at which something happens or is calculated.

TEAR → To pull something apart forcefully or a drop of liquid from the eye.

TARE → The weight of a container deducted from the total weight to find the net weight.

Thus, three meaningful four-letter English words can be formed.

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: Letters : Position Value

N x P : 14 x 16

Similarly,

E x I : ?

Thus, 5×9 maintains the same relationship as the first to E x I.

Hence, the correct answer is "Option 1".

The logic followed here is:

Thus, V5 will complete the series.

Hence, the correct answer is "Option 1".

The logic followed here is:

1. Article 14 → Right to Equality
2. Article 19 → Right to Freedom
3. Article 21 → Right to Life and Personal Liberty
4. Article 370 → Provided special status to Jammu & Kashmir (now revoked in 2019). It was not a Fundamental Right and was placed under Part XXI (Temporary, Transitional and Special Provisions).

All options except "Article 370", (14, 19, 21) are Fundamental Rights under Part III of the Indian Constitution.

Thus, Article 370 is the odd one out.

Hence, the correct answer is "Option 4".

Given: A is the daughter of B. B is the son of C. D is the wife of C.

Clearly, D is grandmother of A.

Hence, the correct answer is "Option 3".

Given: If 3 + 4 = 19 and 2 + 5 = 17

The logic followed here is:

Logic: First number × Second number + (First number + Second number)

3 + 4 = 19 → 3 × 4 + (3 + 4)

= 12 + 7

= 19

2 + 5 = 17 → 2 × 5 + (2 + 5)

= 10 + 7

= 17

Similarly,

1 + 6 = 1 × 6 + (1 + 6)

= 6 + 7

= 13

Thus, 13 will come in the place of ?.

Hence, the correct answer is "Option 2".

Correct Option - 3

Given:

SPEED is coded as 83551 

COOL is coded as 9226

then, SCOOP will be coded as:

Hence, the correct answer is "Option 3".

Correct Option - 3

Given: If 5 * 4 = 41 and 6 * 3 = 45

The logic followed here is:

Logic: First number * Second number = (First number)² + (Second number)²

5 * 4 = 41 → 5² + 4² = 25 + 16 = 41

6 * 3 = 45 → 62 + 32 = 36 + 9 = 45

Similarly,

7 * 2? → 72 + 22 = 49 + 4 = 53

Thus, 53 will come in the place of ?.

Hence, the correct answer is "Option 3".

Correct Option - 1

Given:

'READ' is coded as T2@3

'DEAR' is coded as 3@2T

Similarly,

Thus, DARE is coded as 3@T2.

Hence, the correct answer is "Option 1".

Correct Option - 1

5 consecutive odd numbers starting from 13 is 13, 15, 17, 19, 21

Sum = 13 + 15 + 17 + 19 + 21 = 85

Average = Sum of all the number ÷ Number of observations

= 85 ÷ 5

= 17

Hence, the correct answer is "Option 1".

Correct Option - 4

BODMAS Table:

Given equation: 40 x 5 ÷ 8 + 16 - 2=?

Replacing the symbols by their meaning, we get:

⇒ 40 - 5 + 8 ÷ 16 × 2 = ?

⇒ 40 - 5 + 0.5 × 2 = ?

⇒ 40 - 5 + 1 = ?

⇒ 41 - 5 = ?

⇒ 36 = ?

Hence, the correct answer is "Option 4".

Correct Option - 4

Family Chart:

Given: Pointing to a photograph, a man says, "She is the daughter of the only son of my father's wife".

Clearly, the Woman is Daughter of the Man.

Hence, the correct answer is "Option 4".

Correct Option - 2

BODMAS Table:

Now, checking each option one by one:

1. 10 + 2 x 2 - 4 = 10

After replacing the symbols by their meaning, we get:

⇒ 10 − 2 + 2 ÷ 4 = 10

= 10 − 2 + 0.5 = 10

= 10.5 - 2 = 10

= 8.5 ≠ 10

⇒ LHS ≠ RHS

2. 14 ÷ 2 - 2 + 4 = 10

After replacing the symbols by their meaning, we get:

⇒ 14 × 2 ÷ 2 − 4 = 10

⇒ 14 × 1 − 4 = 10

⇒  14 − 4 = 10

⇒ 10 = 10

⇒ LHS = RHS

3. 12 x 2 + 4 - 1 = 27

After replacing the symbols by their meaning, we get:

⇒ 12 + 2 − 4 ÷ 1 = 27

⇒ 12 + 2 − 4 = 27

⇒ 14 − 4 = 27

⇒ 10 ≠ 27

⇒ LHS ≠ RHS

4. 8 + 4 - 2 x 1 = 10

After replacing the symbols by their meaning, we get:

⇒ 8 − 4 ÷ 2 + 1 = 10

⇒ 8 − 2 + 1 = 10

⇒ 9 − 2 = 10

⇒ 7 ≠ 10

⇒ LHS ≠ RHS

Hence, the correct answer is "Option 2".

Statements:

All metals are elements.

Some elements are gases.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some metals are gases. - False (No definite relation given between metal and gases, so it can be possible only)

II. All gases beig elements is a possibility. - True (No definite relation given between metal and gases, so it can be possible only)

Hence, the correct answer is "Option 1".

Correct Option - 4

BODMAS Table:

Given equation:  4 – 2 × 3 + 6 = ?

After replacing the symbols by their meaning, we get:

⇒ 4 ÷ 2 + 3 × 6 = ?

⇒ 2 + 3 × 6 = ?

⇒ 2 + 18 = ?

⇒ 20 = ?

Hence, the correct answer is "Option 4".

Correct Option - 3

Given: The ratio of two numbers is 4:5 and their sum is 180.

Let the numbers be 4a and 5a.

According to question:

⇒ 4a + 5a = 180

⇒ 9a = 180

⇒ a = 180 ÷ 9

⇒ a = 20

Thus, the numbers are:

4a = 4 × 20 = 80

5a = 5 × 20 = 100

Hence, the correct answer is "Option 3".

The correct answer is Steel spikes on doors.

Key Points

• Steel spikes were strategically installed on the gates of Golconda Fort to prevent war elephants from ramming and breaking the gates during attacks.
• The spikes were pointed and sharp, making it impossible for elephants to use their strength to push or damage the gates.
• This feature was a common defensive mechanism in medieval Indian forts to counter the use of war elephants by invading armies.
• The gates of Golconda Fort were also designed with robust wooden structures, reinforced with steel to withstand heavy impacts.
• The strategic use of these spikes showcases the advanced engineering and defensive architecture of the era, ensuring the fort's safety during sieges.

Additional Information

• Deep moat with water channels:
• Moats were commonly used as a defensive feature in forts to prevent enemy forces from approaching the fort walls easily.
• Filled with water, moats acted as a barrier against infantry and cavalry attacks.
• However, Golconda Fort did not primarily rely on moats for defense against war elephants; the steel spikes were the preferred solution.
• Electrified spikes:
• This option is incorrect as the technology for electrified spikes was not available during the time of Golconda Fort's construction.
• Electricity was discovered centuries after the fort was built, making this an anachronistic suggestion.
• Stone cannon towers:
• Stone cannon towers were used for defensive purposes to place cannons for firing at approaching enemies.
• While the fort did have cannon towers, they were not specifically used to defend against war elephants.
• Cannons were primarily aimed at infantry, siege equipment, and other large-scale attackers.

The correct answer is Punha involves symbolic ploughing indoors using willow sticks, reflecting farming hopes during snow..

Key Points

• The Fagli festival, celebrated in the Pattan Valley of Himachal Pradesh, is a traditional winter event celebrated to mark the end of the harsh winter season.
• Punha, one of the rituals, involves symbolic indoor ploughing using willow sticks, symbolizing hope and prosperity for agriculture in the upcoming season.
• This symbolic ritual is significant as it reflects the dependence of the local community on farming and their connection with nature, even during snow-covered winters.
• The festival emphasizes agricultural prosperity, community bonding, and the preparation for the upcoming farming season.
• The use of willow sticks indoors signifies adaptability and reverence toward nature, as agricultural activities halt during heavy snow.

Additional Information

• Some of the key festivals and fairs of Himachal Pradesh:
• Kullu Dussehra (Kullu): A huge, unique Dussehra celebrated with traditional deities brought from villages, unlike other parts of India.
• Mandi Shivratri (Mandi): Features processions of local deities (Devtas) to the main temple, showcasing rich culture.
• Lavi Fair (Rampur Bushahr): An ancient trade fair, famous for trading wool, pashmina, and dry fruits, with significant cultural events.
• Shimla Summer Festival (Shimla): A multi-day cultural extravaganza held in summer, boosting tourism.
• Nalwari Fair (Bilaspur): A prominent cattle fair with wrestling and other activities.

The correct answer is Vaishthika - forced or corvée labour.

Key Points

• In the Gupta administrative and fiscal system, Vaishthika referred to forced labour or corvée labour imposed by the state. Hence, option 4 is correctly matched.
• Corvée labour means compulsory unpaid labour provided by subjects for public works or state purposes.
• Hiranya referred to taxes paid in the form of cash or precious metals, not compulsory labour. Hence, option 1 is incorrect.
• Dhanya referred to taxes paid in the form of grain or agricultural produce, not gold tax. Hence, option 2 is incorrect.
• Bhoga denoted the periodic supply of fruits, flowers, fuel, and other provisions to the king or officials, rather than the king’s share of produce. Hence, option 3 is incorrect.
• Statement-wise Analysis:
• Hiranya - compulsory labour service – Incorrect.
• Dhanya - gold tax – Incorrect.
• Bhoga - king's share of produce – Incorrect.
• Vaishthika - forced or corvée labour – Correct.
• Hence, the correct answer is Vaishthika - forced or corvée labour.

Additional Information

• Gupta Empire: The Gupta period lasted approximately from 320 CE to 550 CE and is often called the “Golden Age of Ancient India”.
• Main Sources: Information about Gupta administration comes from inscriptions, coins, land grants, and accounts of Chinese traveler Faxian.
• Land Revenue: The major source of state income during the Gupta period was land tax, generally collected as a share of agricultural produce.
• Bhaga: The term Bhaga referred to the king’s share of produce, usually around one-sixth of the agricultural yield.
• Economic System: Both cash payments and payments in kind were common during the Gupta age.

Important Points

• Important Gupta Fiscal Terms:
• Hiranya – Cash tax
• Dhanya – Tax in grain
• Bhaga – King’s share of produce
• Vaishthika – Forced labour
• Bhoga – Supplies or provisions to officials
• Questions related to Gupta administration, taxation, and economic terms are frequently asked in history and competitive examinations.

The correct answer is Simikla Koom (Binayo).

Key Points

• Simikla Koom (Binayo) is a traditional musical instrument of Sikkim and is associated with the folklore and culture of the region.
• It is believed to have been crafted for the deity Tetlara Lahadongna, highlighting its mythological and spiritual significance.
• This instrument is made from natural materials, primarily bamboo, which reflects the eco-friendly practices and resourcefulness of the local communities.
• Simikla Koom (Binayo) is classified as a wind instrument, which produces sound through the vibration of air columns.
• The use of this instrument is deeply embedded in Sikkimese culture, often played during festivals, rituals, and traditional ceremonies.

Additional Information

• Niyara Hongsing Ke
• This is not a recognized traditional instrument from Sikkim, but the name may reflect a fictional or lesser-known term.
• In the context of traditional instruments, Sikkim primarily features bamboo-based and folk instruments associated with the Lepcha, Bhutia, and Nepali communities.
• Yema Dhol
• The Yema Dhol is a type of drum and is commonly found in many Indian traditions, but it is not specifically linked to Sikkimese culture.
• Drums like the Dhol are often associated with other regions in India, including Bengal, Assam, and Punjab, and are used in traditional and folk music.
• Nagara of Yeba
• The Nagara is a traditional drum instrument, but it is not particularly associated with Sikkim or the deity Tetlara Lahadongna.
• It is a common percussion instrument used in various parts of India and is often played during religious ceremonies and processions.

The correct answer is Weightlifting.

Key Points

• Karnam Malleswari became the first Indian woman to win an Olympic medal by securing a bronze in Weightlifting at the 2000 Sydney Olympics.
• She competed in the 69 kg Women’s Weightlifting category, lifting a total of 240 kg (110 kg in snatch and 130 kg in clean and jerk).
• Her achievement marked a historic moment for India, as she was the first Indian female athlete to win a medal at the Olympic Games.
• Karnam Malleswari is also a recipient of the prestigious Rajiv Gandhi Khel Ratna Award, India's highest sporting honor.
• She has won several accolades in her career, including multiple medals at the World Weightlifting Championships, solidifying her status as one of India’s greatest athletes.

Additional Information

• Boxing:
• Boxing is a combat sport where two individuals fight using their fists while adhering to a set of rules.
• Indian boxer Mary Kom is one of the most prominent figures in Indian boxing, winning multiple medals at international events, including the Olympics.
• The sport requires exceptional physical fitness, technique, and mental resilience.
• Athletics:
• Athletics encompasses a wide range of sports, including track and field events, road running, and race walking.
• India’s Neeraj Chopra won a historic gold medal in Javelin Throw at the 2020 Tokyo Olympics, becoming the first Indian athlete to win a gold in track and field.
• Athletics is considered one of the most competitive sports globally, requiring high levels of skill and training.
• Long Jump:
• Long Jump is a track and field event where athletes sprint down a runway and jump as far as possible into a sandpit.
• India has produced successful long jumpers, such as Anju Bobby George, who won a bronze medal at the World Athletics Championships in 2003.
• The sport demands explosive strength, speed, and precise technique.
• Weightlifting Overview:
• Weightlifting is a strength-based sport where athletes attempt two lifts: snatch and clean and jerk.
• It is divided into different weight categories for men and women.
• India has a strong presence in weightlifting, with athletes like Mirabai Chanu winning medals at the Olympics and other international competitions.

The correct answer is Option 3: A → 2, B → 1, C → 3, D → 4.

Key Points

• Qutbuddin Aybak (1206–1210): Qutbuddin Aybak was the founder of the Delhi Sultanate and the first ruler of the Slave Dynasty. He ruled from 1206 to 1210 and was known for constructing the Qutub Minar in Delhi.
• Prithviraj Chauhan (1175–1192): Prithviraj Chauhan was a Rajput king of the Chahamana dynasty who ruled during 1175–1192 CE. He is remembered for his valiant battles against Muhammad Ghori.
• Alauddin Khalji (1296–1316): Alauddin Khalji was the second ruler of the Khalji dynasty in India. His reign, from 1296 to 1316, is marked by military expansion and administrative reforms.
• Muhammad bin Tughluq (1324–1351): Muhammad bin Tughluq, the Sultan of Delhi, ruled from 1324 to 1351. He was known for his ambitious and often controversial reforms, such as the attempted introduction of token currency.

Additional Information

• Slave Dynasty (1206–1290): The Slave Dynasty, also known as the Mamluk Dynasty, was the first dynasty of the Delhi Sultanate. It was established by Qutbuddin Aybak, who was originally a slave of Muhammad Ghori. The dynasty laid the foundation of Muslim rule in India.
• Chahamana Dynasty: The Chahamanas (Chauhans) were a prominent Rajput dynasty that ruled over parts of northwestern India, including Delhi and Ajmer. Prithviraj Chauhan is the most celebrated ruler of this dynasty.
• Khalji Dynasty (1290–1320): This dynasty is known for its expansionist policies and economic reforms. Alauddin Khalji implemented price control measures and established a strong central administration.
• Tughluq Dynasty (1320–1414): The Tughluq dynasty was known for its ambitious rulers like Muhammad bin Tughluq, who attempted various reforms, including shifting the capital from Delhi to Daulatabad and the introduction of token currency. These reforms, however, were largely unsuccessful and led to administrative challenges.

The correct answer is One-hundredth.

Key Points

• P. T. Usha, popularly known as the "Payyoli Express," is one of India’s greatest athletes and a legend in track and field.
• At the 1984 Summer Olympics held in Los Angeles, she competed in the women’s 400m hurdles event.
• She narrowly missed the bronze medal, finishing fourth in the final with a timing of 55.42 seconds.
• The difference between P. T. Usha and the bronze medalist was just one-hundredth (0.01) of a second, making it a historic and heartbreaking moment in Indian sports history.
• Despite not winning a medal, her achievement remains significant as she became the first Indian woman to reach the finals of an Olympic event.

Additional Information

• 1984 Summer Olympics:
• The 1984 Summer Olympics were held in Los Angeles, USA, from July 28 to August 12.
• India’s participation in athletics was highlighted by P. T. Usha’s remarkable performance in the 400m hurdles.
• Her near-podium finish brought attention to Indian athletics on the global stage and inspired future generations of athletes.

The correct answer is BRICS Alliance for Green Tourism.

Key Points

• During India’s BRICS Chairship in 2021, one of the major initiatives was the launch of the BRICS Alliance for Green Tourism, focusing on sustainable tourism practices among member nations.
• This initiative aimed to promote collaborative efforts among BRICS countries (Brazil, Russia, India, China, and South Africa) to adopt eco-friendly tourism strategies and reduce the carbon footprint of the tourism sector.
• The alliance emphasized the importance of preserving biodiversity, reducing the environmental impact of tourism, and promoting resource-efficient practices in the sector.
• It provided a platform for sharing best practices, innovations, and policies to strengthen sustainable tourism, benefiting local communities and preserving cultural heritage.
• The BRICS Alliance for Green Tourism also aligned with global sustainability goals, particularly the United Nations Sustainable Development Goals (SDGs), focusing on climate action and responsible consumption and production.

Additional Information

• BRICS Water Security Pact
• This option refers to a hypothetical agreement among BRICS nations to address water security challenges. However, no such pact was launched during India’s BRICS Chairship in 2021.
• Water security is a critical issue for BRICS nations, but it was not a specific focus of the 2021 initiatives.
• BRICS Climate Finance Agreement
• This refers to a potential agreement for financial mechanisms to combat climate change. However, no such specific agreement was introduced during the 2021 BRICS Chairship.
• While climate finance remains a crucial topic for BRICS discussions, it was not a formalized initiative in 2021.
• BRICS Sustainable Energy Forum
• The BRICS Sustainable Energy Forum focuses on energy efficiency and renewable energy cooperation among member nations.
• Although energy sustainability is a significant area of focus for BRICS, it was not one of the key initiatives launched during India’s 2021 Chairship.

The correct answer is Pranay Lal.

Key Points

• Pranay Lal is the author of the critically acclaimed book 'Indica: A Deep Natural History of the Indian Subcontinent'.
• The book presents an in-depth exploration of the geological, biological, and ecological history of the Indian subcontinent.
• 'Indica' is the first comprehensive account of the natural history of India, covering over 4.5 billion years of Earth's evolution.
• The book combines storytelling, scientific research, and illustrations to make complex topics accessible to a broad audience.
• It has been praised for its interdisciplinary approach, blending geology, paleontology, and biodiversity to paint a vivid picture of India's natural history.

Additional Information

• Sanjeev Sanyal
• Sanjeev Sanyal is an Indian economist and author known for works like 'The Ocean of Churn' and 'The Indian Renaissance'.
• His books primarily focus on history, economics, and the cultural heritage of India.
• Rohan Murty
• Rohan Murty is a computer scientist, entrepreneur, and founder of the Murty Classical Library of India.
• The Murty Classical Library of India publishes English translations of classical Indian texts to promote India's literary heritage.
• Tony Joseph
• Tony Joseph is the author of the popular book 'Early Indians: The Story of Our Ancestors and Where We Came From'.
• The book delves into the genetic and archaeological history of the Indian subcontinent, focusing on human migration and settlement patterns.

The correct answer is Option 1: Both 1 and 2 are correct.

Key Points

• The men's hockey event at the 38th National Games 2025 witnessed intense competition among several states, with Karnataka emerging as the winner by defeating Uttar Pradesh 3-2 in the final.
• Karnataka and Maharashtra had a closely contested semifinal match, which ended in a 2-2 draw and was ultimately decided by a shootout.
• These results highlight Karnataka's dominance in the tournament and their ability to perform under pressure, especially during the shootout in the semifinals.
• The National Games are a multi-sport event held in India every few years, and hockey remains one of the most competitive sports in the event.
• Option 1 is correct as both statements 1 and 2 about the men's hockey event align with the verified outcomes reported during the 38th National Games.

Additional Information

• National Games
• The National Games are organized by the Indian Olympic Association and are held in different states to promote sports culture in India.
• The event features competitions in various sports, including hockey, athletics, swimming, and more.
• It serves as a platform for athletes to showcase their talent and prepare for international tournaments like the Asian Games and Olympics.

The correct answer is e-Civil List 2025 e-book.

Key Points

• The e-Civil List 2025 e-book was launched by Dr. Jitendra Singh in May 2025 in his capacity as Minister of State (MoS) PMO and MoS Personnel.
• This initiative is part of the government’s effort to digitize official records and make them accessible to stakeholders in a user-friendly manner.
• The e-Civil List provides comprehensive information about IAS officers across India, including their postings, cadre, and career details.
• It aims to enhance transparency, accountability, and ease of access to the details of civil servants for administrative and public use.
• The platform leverages technology to ensure up-to-date record maintenance and helps in better governance and human resource management of IAS officers.

Additional Information

• National Grievance Redress Portal
• This is an online platform designed to address public grievances efficiently and transparently.
• It allows citizens to register complaints related to government services and schemes, ensuring timely resolution.
• The portal is part of the government’s ongoing efforts to improve governance and citizen-centric service delivery.
• While significant, this initiative was not launched in May 2025.
• Central Police Commemoration Day
• Observed annually on October 21, this day is dedicated to honoring police personnel who have sacrificed their lives in the line of duty.
• It was first commemorated in 1959 following the tragic loss of police forces during a skirmish with Chinese forces at Hot Springs, Ladakh.
• While it is an important event, it is not related to the initiative launched by Dr. Jitendra Singh in May 2025.
• Digital Land Records Survey
• This initiative focuses on digitizing land records to streamline land ownership documentation and minimize disputes.
• It aims to create a transparent and accessible database of land records for citizens and government authorities.
• It is a separate program from the e-Civil List 2025 e-book and was not launched in May 2025.

The correct answer is 16%.

Key Points

• As per the 2011 Census of India, Uttar Pradesh accounted for approximately 16% (or 0.16) of the total population of India.
• The total population of India in 2011 was around 1.21 billion, and Uttar Pradesh contributed approximately 199.81 million to this figure.
• Uttar Pradesh is the most populous state in India, followed by Maharashtra and Bihar.
• The state’s high population is attributed to factors such as a high fertility rate, large geographical area, and significant rural population.
• Uttar Pradesh alone has a population larger than many countries, ranking as the fifth most populous administrative unit globally, after China, India, the United States, and Indonesia.

Additional Information

• Population Dynamics of India 
• The Census of India, conducted every 10 years, is the most reliable source of demographic data for the country.
• India has a highly uneven distribution of population, with Uttar Pradesh being the most populous state, while states like Sikkim and Mizoram have much smaller populations.
• Population distribution in India is influenced by factors like fertility rates, migration, urbanization, and socioeconomic conditions.

The correct answer is Human LVM3.

Key Points

• The designation HLVM3 stands for Human LVM3, which refers to the modified version of the Launch Vehicle Mark-3 (LVM3) developed by the Indian Space Research Organisation (ISRO) for the Gaganyaan mission.
• LVM3, also known as GSLV Mk-III, is India’s heaviest and most advanced rocket, capable of launching payloads up to 8 tons into low Earth orbit.
• For the Gaganyaan mission, LVM3 has been customized to carry humans safely into space, integrating advanced safety features such as an abort system and life-support mechanisms.
• The HLVM3 is equipped with a crew module and service module designed to house astronauts and ensure their safety during the mission.
• The Gaganyaan program aims to send a crew of three astronauts into low Earth orbit for a duration of up to seven days using the HLVM3 rocket.

Additional Information

• Gaganyaan Mission:
• The Gaganyaan mission is India’s first human spaceflight program, aimed at demonstrating the capability of sending astronauts into space and bringing them back safely.
• It involves critical technologies such as crew safety systems, life-support systems, space medicine, and crew training.
• The program is a significant milestone in India’s space exploration journey and aims to position India as a key player in human spaceflight capabilities.

The correct answer is Iron ore.

Key Points

• Iron ore is one of the most abundant and economically significant minerals in India, making the country the fourth-largest producer in the world as of 2022.
• The majority of iron ore production in India comes from states like Odisha, Chhattisgarh, Karnataka, and Jharkhand, with Odisha being the largest producer.
• India's iron ore is predominantly used in the production of steel, which is critical to the construction, automobile, and infrastructure sectors.
• The two main types of iron ore found in India are hematite and magnetite, with hematite being the more dominant type.
• India exports significant quantities of iron ore to countries like China and Japan, contributing to its foreign exchange reserves and economic growth.

Additional Information

• Copper
• Copper is an important metal used in electrical equipment, construction, and industrial machinery due to its excellent conductivity and malleability.
• India has limited copper reserves, with key mining areas located in Madhya Pradesh, Rajasthan, and Jharkhand.
• India imports a significant amount of its copper requirements to meet domestic demand.
• Petroleum
• Petroleum is a crucial energy resource and raw material for industries like petrochemicals, plastics, and fertilizers.
• India is the third-largest consumer of crude oil globally but relies heavily on imports to meet around 85% of its demand.
• Major petroleum-producing regions in India include Assam, Gujarat, Rajasthan, and offshore fields in the Mumbai High basin.
• Tin
• Tin is primarily used in soldering, tin plating, and the production of alloys like bronze.
• India has very limited tin reserves, with some deposits found in Chhattisgarh and Jharkhand, making it dependent on imports to meet domestic demand.

The correct answer is Only 1.

Key Points

• Statement 1: Both the Prime Minister (PM) and the Chief Minister (CM) can advise the dissolution of their respective Lower Houses. Correct.
• The Prime Minister, along with the Council of Ministers, can advise the President to dissolve the Lok Sabha under Article 85 of the Constitution.
• Similarly, the Chief Minister, along with the Council of Ministers, can advise the Governor to dissolve the Legislative Assembly (Vidhan Sabha) of the state.
• The President and Governor generally act on the aid and advice of their respective Councils of Ministers.

Hence, Statement 1 is correct.

• Statement 2: Only the PM can recommend President's Rule under Article 356 of the Constitution. Incorrect.
• President's Rule is imposed by the President of India under Article 356 when a state's constitutional machinery fails.
• The recommendation is generally made by the Union Council of Ministers, headed by the Prime Minister, and is often based on a report from the Governor of the concerned state.
• The Constitution does not provide that the Prime Minister alone can recommend President's Rule.

Hence, Statement 2 is incorrect.

• Therefore, only Statement 1 is correct.

Additional Information

• Article 74 provides that the President shall act in accordance with the aid and advice of the Council of Ministers headed by the Prime Minister.
• Article 163 provides that the Governor acts on the aid and advice of the Council of Ministers headed by the Chief Minister, except in matters where the Constitution grants discretionary powers.
• Article 356 deals with the imposition of President's Rule in a state if the constitutional machinery fails.
• During President's Rule, the powers of the state government are assumed by the President, and the state legislature may be suspended or dissolved.
• The proclamation of President's Rule must be approved by both Houses of Parliament within two months from the date of its issuance.

Important Points

• The landmark S.R. Bommai vs Union of India (1994) judgment placed significant restrictions on the arbitrary use of Article 356.
• The Lok Sabha is the Lower House at the Union level, while the Legislative Assembly (Vidhan Sabha) is the Lower House at the state level.
• The power to dissolve the Lower House is exercised formally by the President or Governor, but usually on the advice of the elected executive.

The correct answer is Micronutrient - Boron.

Key Points

• Boron is classified as a micronutrient because plants require it in very small amounts for proper growth and development.
• It plays a critical role in cell wall formation, membrane stability, and pollen tube growth in plants.
• Deficiency of boron leads to symptoms like stunted root growth, hollow stems, or flower abortion in plants.
• Boron is absorbed by plants from the soil in the form of borate ions (BO3^3-).
• Common sources of boron in agriculture include fertilizers like borax and boric acid.

Additional Information

• Micronutrient - Calcium (Option 1):
• Calcium is actually a macronutrient, not a micronutrient, as it is required by plants in relatively large amounts.
• It is essential for cell wall integrity and acts as a secondary messenger in various cellular processes.
• Calcium deficiency can lead to problems like blossom end rot in tomatoes and weak cell walls in plants.
• Macronutrient - Zinc (Option 2):
• Zinc is a micronutrient, not a macronutrient, as it is needed in small quantities by plants.
• It is essential for enzyme activation, protein synthesis, and hormone production in plants.
• Zinc deficiency often manifests as chlorosis (yellowing of leaves) and stunted growth.
• Macronutrient - Iron (Option 4):
• Iron is also a micronutrient, not a macronutrient.
• It plays a key role in chlorophyll synthesis and acts as a cofactor for various enzymatic reactions.
• Iron deficiency is characterized by interveinal chlorosis in young leaves.
• Macronutrients vs Micronutrients:
• Macronutrients (e.g., nitrogen, phosphorus, potassium) are required in large amounts by plants, while micronutrients (e.g., zinc, boron, iron) are needed in trace amounts.
• Both types of nutrients are vital for plant health, growth, and productivity.

The correct answer is Electric Bell: Uses the magnetic effect of electric current.

Key Points

• An electric bell operates on the principle of the magnetic effect of electric current, where an electromagnet is created by passing an electric current through a coil of wire.
• The magnetic field generated attracts a metal armature, causing it to strike the bell and produce sound.
• The electric bell consists of components like an electromagnet, armature, contact screw, and hammer, all working in synchronization to create the ringing sound.
• When the circuit is complete, the current flows through the coil, magnetizing the core, and pulling the armature, which simultaneously breaks the circuit and causes the hammer to strike the bell.
• This cycle repeats rapidly as long as the switch is pressed, producing a continuous ringing sound.

Additional Information

• Electric Generator: Converts electrical energy to mechanical energy
• This statement is incorrect. An electric generator actually converts mechanical energy into electrical energy.
• It operates on the principle of electromagnetic induction, as discovered by Michael Faraday, where a conductor moving through a magnetic field induces an electric current.
• Generators are widely used in power plants to produce electricity for homes, industries, and businesses.
• Transformer: Operates on DC current to change voltage
• This statement is incorrect because transformers only work with alternating current (AC), not direct current (DC).
• Transformers are used to step up (increase) or step down (decrease) the voltage levels in an AC circuit.
• They function based on the principle of mutual induction between two coils of wire.
• Electromagnet: A permanent magnet
• This statement is incorrect because an electromagnet is a temporary magnet that only exhibits magnetic properties when an electric current flows through its coil.
• Once the current is stopped, the magnetic field disappears, unlike a permanent magnet that retains its magnetism.
• Electromagnets are widely used in devices like motors, relays, and magnetic cranes.

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

Key Points

• Mission Amrit Sarovar is a flagship initiative by the Government of India, launched to promote water conservation and sustainable water resource management.
• The objective of Mission Amrit Sarovar is to develop and rejuvenate 75 water bodies in every district across the country by August 15, 2023, as part of the Azadi Ka Amrit Mahotsav celebrations.
• The project aims to strengthen water security and ensure environmental sustainability by constructing ponds and water reservoirs to increase water storage and recharge groundwater levels.
• The Reason (R) correctly explains the Assertion (A) since the construction of 75 ponds in each district directly contributes to promoting water conservation efforts.
• Mission Amrit Sarovar seeks to involve multiple stakeholders, including rural communities, local bodies, and government agencies, to revive traditional water conservation practices.

Additional Information

• Water Conservation
• Water conservation refers to the strategies, policies, and activities aimed at managing the use of fresh water to meet current and future demands sustainably.
• It includes measures such as rainwater harvesting, groundwater recharge, and the construction of water bodies like ponds and reservoirs.
• Programs like Mission Amrit Sarovar align with the United Nations Sustainable Development Goals (SDG), particularly Goal 6: Clean Water and Sanitation.
• Importance of Groundwater Recharge
• Groundwater recharge is essential for maintaining aquifer levels, which are a critical source of drinking water and irrigation in India.
• Constructing ponds and water reservoirs helps in reducing surface runoff and ensures that water infiltrates into the ground, replenishing groundwater levels.
• Community Participation in Water Conservation
• Mission Amrit Sarovar emphasizes the active involvement of local communities in creating and maintaining water bodies.
• Such participation fosters awareness about water conservation and ensures the sustainability of the initiative in the long term.

The correct answer is Option 3: Both 1 & 2 correct.

Key Points

• Rauf is a traditional dance form of Kashmir, primarily performed by women during festive occasions such as Eid.
• The dance is performed in groups, with women aligning themselves into two facing rows.
• Rauf is characterized by intricate toe-heel footwork, which is locally referred to as ras.
• The dance reflects the vibrant culture and traditions of Kashmir and is a significant representation of its rich heritage.
• It is a popular folk dance that also serves as a medium of social interaction and communal harmony among women in the region.

Additional Information

• Key Kashmiri Dances:
• Rouf (or Rauf):
• Who: Performed by women in colorful dresses.
• When: Celebrates spring and festivals like Eid and Ramzan.
• How: Simple, rhythmic footwork, with dancers facing each other, mimicking bees.
• Dumhal:
• Who: Exclusively performed by men of the Wattal tribe.
• When: Performed on special, auspicious occasions.
• How: Dancers wear vibrant robes and conical caps, plant a banner, and dance around it, singing folk songs in chorus.
• Hikat:
• A simple dance by two girls holding hands, moving clockwise and anti-clockwise, often involving weeding/harvesting themes.
• Bhand Pather:
• A traditional folk theatre form that combines dance, music, and satire, performed by wandering minstrels (Bhand).

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

Key Points

• India's rank in the Global Happiness Index has been consistently low, with neighboring South Asian countries like Nepal, Bhutan, and Bangladesh often ranking higher.
• As per the 2023 World Happiness Report, India ranked 126th out of 137 countries, while countries like Bhutan ranked significantly higher.
• The Global Happiness Index considers factors such as social support, life expectancy, freedom to make life choices, and perceptions of corruption—not just economic indicators like GDP per capita.
• India's GDP per capita is not the lowest in South Asia. For example, countries like Afghanistan have a lower GDP per capita than India.
• Hence, while the assertion (A) is correct that India's rank is lower than most South Asian neighbors, the reason (R) is incorrect since the ranking is based on several factors beyond GDP per capita.

Additional Information

• Factors influencing the Global Happiness Index:
• The ranking is based on six key metrics: GDP per capita, social support, healthy life expectancy, freedom to make life choices, generosity, and perceptions of corruption.
• While GDP per capita is an important factor, other social and psychological well-being aspects weigh heavily in the index.
• Countries like Bhutan rank higher because of their focus on Gross National Happiness, despite having lower GDP per capita.
• India's performance in the Global Happiness Index:
• India's lower ranking is attributed to factors such as weak social support, perceived corruption, and lower levels of individual freedom.
• Efforts to improve quality of life through government initiatives like Digital India, Swachh Bharat Abhiyan, and Ayushman Bharat are aimed at addressing these issues.
• GDP per capita in South Asia:
• India's GDP per capita is higher than several South Asian nations, including Afghanistan and Nepal.
• However, countries like Bhutan and Maldives have higher GDP per capita due to their smaller populations and focused economic strategies.

The correct answer is Musi.

Key Points

• The Musi River, located in Telangana, was added to the National River Conservation Plan (NRCP) in 2024.
• The NRCP is an initiative by the Ministry of Environment, Forest and Climate Change (MoEFCC) to conserve and rejuvenate polluted rivers across India.
• The Musi River flows through Hyderabad and has been heavily polluted due to untreated sewage, industrial waste, and urban runoff.
• Under the NRCP, measures such as sewage treatment plants, bio-remediation, riverbank restoration, and public awareness campaigns are implemented to revive polluted rivers.
• The inclusion of the Musi River in the NRCP aims to address environmental concerns and improve water quality for better ecological and human health.

Additional Information

• Yamuna River:
• The Yamuna River is the largest tributary of the Ganga and flows through several states, including Uttarakhand, Haryana, and Uttar Pradesh.
• It has been part of the NRCP since 1993 due to high pollution levels in cities like Delhi and Agra.
• Under the NRCP, projects such as sewage treatment, riverfront development, and waste management have been undertaken.
• Krishna River:
• The Krishna River is one of India's major rivers, flowing through Maharashtra, Karnataka, Telangana, and Andhra Pradesh.
• It is known for its importance in irrigation, hydroelectric power generation, and water supply.
• The Krishna is not part of the NRCP, as pollution levels are comparatively lower than rivers like the Yamuna or Musi.
• Gomti River:
• The Gomti River flows through Uttar Pradesh, including the city of Lucknow.
• It has been part of the NRCP since the late 1990s due to pollution from untreated sewage and industrial effluents.
• Efforts under the NRCP include constructing sewage treatment plants and reducing pollution sources.

The correct answer is Directive Principles of State Policy.

Key Points

• Article 50 of the Indian Constitution falls under the Directive Principles of State Policy (DPSP), which are outlined in Part IV of the Constitution (Articles 36-51).
• It specifically states, "The State shall take steps to separate the judiciary from the executive in the public services of the State."
• The purpose of Article 50 is to maintain the independence of the judiciary, a vital component of a democratic system, and to ensure the impartiality of justice.
• DPSPs, including Article 50, are non-justiciable, meaning they are not enforceable by the courts, but they act as guiding principles for the State to establish policies and laws.
• Article 50 has played a crucial role in the Indian judicial system by emphasizing the doctrine of separation of powers between the judiciary and the executive.

Additional Information

• Fundamental Duties:
• Fundamental Duties are listed under Part IVA of the Indian Constitution (Article 51A).
• They were added by the 42nd Amendment Act, 1976, based on the recommendations of the Swaran Singh Committee.
• These duties serve as moral obligations for citizens to promote a spirit of patriotism and uphold the unity of the nation.
• Examples include abiding by the Constitution, respecting the national flag and anthem, protecting the environment, and promoting harmony.
• Fundamental Rights:
• Fundamental Rights are enshrined in Part III of the Indian Constitution (Articles 12-35).
• They guarantee basic rights to all citizens, essential for their development and protection against state actions.
• Key Fundamental Rights include the Right to Equality (Article 14-18), Right to Freedom (Article 19-22), and Right to Constitutional Remedies (Article 32).
• These rights are justiciable, meaning individuals can approach the courts if their rights are violated.
• Preamble:
• The Preamble serves as the introductory statement of the Indian Constitution, outlining its objectives and philosophy.
• It declares India as a Sovereign, Socialist, Secular, and Democratic Republic and ensures Justice, Liberty, Equality, and Fraternity to all its citizens.
• Although not enforceable in a court of law, the Preamble is considered a guiding light for interpreting the Constitution.

The correct answer is 2-1-3.

Key Points

• The Goods and Services Tax (GST) was implemented in India on July 1, 2017, marking the beginning of a unified indirect tax structure across the nation.
• Before its launch, the Constitution (101st Amendment) Act was enacted on September 8, 2016, to provide the legal framework for GST implementation.
• The GST Council, which is responsible for making recommendations on GST-related matters, was constituted on September 15, 2016.
• Hence, the correct chronological order is as follows:
• Constitution (101st Amendment) Act enacted (September 8, 2016)
• GST Council constituted (September 15, 2016)
• GST launched (July 1, 2017)
• This sequence aligns with the preparatory steps taken by the Indian government to roll out GST effectively.

Additional Information

• Constitution (101st Amendment) Act, 2016:
• The 101st Constitutional Amendment was enacted to introduce GST in India.
• It empowered both the Centre and the States to levy and collect GST on goods and services.
• The amendment subsumed several indirect taxes such as excise duty, service tax, and VAT into a single tax structure.
• GST Council:
• The GST Council is a constitutional body established under Article 279A of the Indian Constitution.
• It is chaired by the Union Finance Minister and includes State Finance Ministers as members.
• The council is responsible for making recommendations on tax rates, exemptions, and other policy matters related to GST.

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

Key Points

• The Industrial Policy Resolution (IPR) of 1956 was a significant policy document that aimed to classify industries into three categories: Schedule A, Schedule B, and Schedule C.
• Schedule A included industries that were exclusively under state control and were regarded as essential for national defense and economic development (e.g., railways, atomic energy).
• Schedule B covered industries that were to be developed under joint sector arrangements, i.e., both state and private enterprises could participate in these industries.
• Schedule C comprised industries that were to be left for private sector operations, allowing individual entrepreneurship to thrive.
• The classification was designed to promote a mixed economy model and was rooted in the philosophy of socialist planning, which sought to balance private and state ownership.

Additional Information

• Post-Independence (IPR 1948, 1956): Focused on public sector dominance and planned development.
• Liberalization (IPR 1991): Shifted towards opening the economy and global integration.
• Recent Initiatives:
• Make in India (2014): To make India a global manufacturing hub.
• Atmanirbhar Bharat Abhiyan (2020): Aimed at self-reliance and reducing import dependency.
• Production-Linked Incentive (PLI) Scheme (2020): Offers incentives for increased domestic production.

The correct answer is Option 3: Both 1 and 2.

Key Points

• Statement 1: Tropical zones experience minimal seasonal temperature variation because they are located near the equator and receive consistent solar radiation throughout the year. This results in relatively stable temperatures.
• Statement 2: Temperate zones are characterized by four distinct seasons: spring, summer, autumn, and winter. These zones experience significant temperature variations due to their location between the tropical and polar zones.
• The Earth's major climatic zones are divided into tropical, temperate, and polar zones based on latitude, solar radiation, and prevailing weather patterns.
• The tropical zone spans from the Tropic of Cancer to the Tropic of Capricorn and is known for its hot and humid climate, while the temperate zones lie between the tropics and polar regions and experience moderate climates.
• The distinction between these zones is important for understanding global weather patterns, biodiversity, and human activities adapted to these climates.

Additional Information

• Earth's Climatic Zones
• The Earth is divided into three major climatic zones based on latitude: tropical, temperate, and polar.
• Tropical Zone: Located between 23.5°N and 23.5°S, it experiences high temperatures and rainfall year-round.
• Temperate Zone: Found between 23.5° and 66.5° in both hemispheres, it has distinct seasonal changes and moderate climates.
• Polar Zone: Situated beyond 66.5° in both hemispheres, it experiences extremely cold temperatures and long winters.
• These climatic zones influence global weather patterns, ecosystem diversity, and human adaptation to different environments.

Shortcut Trick

Use the column multiplication method for continuous ratios:

P = 6 × 5 × 9 = 270

Q = 11 × 5 × 9 = 495

R = 11 × 4 × 9 = 396

S = 11 × 4 × 7 = 308

∴ The correct answer is 270 : 495 : 396 : 308.

Alternate Method

Given:

⇒ P : Q = 6 : 11

⇒ Q : R = 5 : 4

⇒ R : S = 9 : 7

Visual Method: Fill empty spaces with adjacent numbers

P Q R S

6

11

11

11

5

5

4

4

9

9

9

7

× Multiply columns vertically

270

495

396

308

Ratio → 270 : 495 : 396 : 308

Calculations:

⇒ Equalize Q in the first two ratios (LCM of 11 and 5 is 55).

⇒ P : Q = (6 × 5) : (11 × 5) = 30 : 55

⇒ Q : R = (5 × 11) : (4 × 11) = 55 : 44

⇒ Combining them, P : Q : R = 30 : 55 : 44

⇒ Now, equalize R in P : Q : R and R : S (LCM of 44 and 9 is 396).

⇒ P : Q : R = (30 × 9) : (55 × 9) : (44 × 9) = 270 : 495 : 396

⇒ R : S = (9 × 44) : (7 × 44) = 396 : 308

⇒ Combine all to get the final ratio:

⇒ P : Q : R : S = 270 : 495 : 396 : 308

∴ The correct answer is 270 : 495 : 396 : 308.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, it is called a compound ratio. For a:b and c:d, the compound ratio is (a × c) : (b × d).

Duplicate & Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b². The sub-duplicate ratio is √a : √b.

Componendo and Dividendo

If a/b = c/d, then (a + b) / (a − b) = (c + d) / (c − d). This rule is heavily used to simplify complex algebraic fraction statements.

Shortcut Trick

Total value ratio = (10 × 1) + (5 × 2) + (2 × 3) = 10 + 10 + 6 = 26 units.

Total given money = ₹390.

Value of 1 ratio unit = 390 ÷ 26 = 15.

Multiply the initial coin ratio (1 : 2 : 3) by 15.

Number of coins = (1 × 15), (2 × 15), and (3 × 15) = 15, 30, and 45.

∴ The correct answer is 15 coins of ₹10, 30 of ₹5, and 45 of ₹2.

Alternate Method

Given:

Denominations of coins = ₹10, ₹5, and ₹2

Ratio of number of coins = 1 : 2 : 3

Total amount in the bag = ₹390

₹10

₹5

₹2

Ratio: x Ratio: 2x Ratio: 3x

Value: 10x

Value: 10x

Value: 6x

Total Value = 10x + 10x + 6x = 26x = ₹390

Calculation:

⇒ Let the number of ₹10, ₹5, and ₹2 coins be x, 2x, and 3x respectively.

⇒ The values of these coins will be ₹(10 × x), ₹(5 × 2x), and ₹(2 × 3x) respectively.

⇒ Total value in terms of x = 10x + 10x + 6x = 26x

⇒ We are given that the total amount is ₹390.

⇒ 26x = 390

⇒ x = 390 ÷ 26 = 15

⇒ Number of ₹10 coins = x = 15

⇒ Number of ₹5 coins = 2x = 2 × 15 = 30

⇒ Number of ₹2 coins = 3x = 3 × 15 = 45

∴ The correct answer is 15 coins of ₹10, 30 of ₹5, and 45 of ₹2.

Additional Information

Total Value of Coins

The total value of any specific coin denomination is found by multiplying the face value of the coin by the number of coins of that denomination.

Coin Ratio vs Value Ratio

If the ratio of the number of coins is given, multiply it by their respective denominations to get the ratio of their values. Conversely, if the ratio of values is given, divide by their denominations to get the ratio of the number of coins.

Proportion Rule

When values are given in a ratio a : b : c, they can be represented as ax, bx, and cx where x is a common multiplier.

Shortcut Trick

Convert the fractions into decimals to make calculation faster.

Total capacity = 5¾ = 5.75 L

Total drained = 2.375 × 2 = 4.75 L

Remaining water = 5.75 − 4.75 = 1 L

∴ The correct answer is 1 liter.

Alternate Method

Given:

Total water = 5¾ L

Water drained each time = 2.375 L

Number of times drained = 2

Drained: 4.75 LLeft: 1 LTotal Water: 5.75 L

Calculations:

⇒ Total water = 23/4 L

⇒ Total water drained = 2 × (19/8) = 19/4 L

⇒ Remaining water = Total water − Total water drained

⇒ Remaining water = 23/4 − 19/4 = 4/4 = 1 L

∴ The correct answer is 1 liter.

Additional Information

Converting Mixed Fractions

To convert a mixed fraction a(b/c) to an improper fraction, multiply the whole number by the denominator and add the numerator: (a × c + b) / c.

Fraction to Decimal Conversion

Common standard conversions: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/8 = 0.125, 3/8 = 0.375. Memorizing these saves time in exams.

Operations on Fractions

When adding or subtracting fractions, always take the Least Common Multiple (LCM) of the denominators to equate them before computing the numerators.

Shortcut Trick

Solve Numerator: 2/3 + (4/5 × 7/2) = 2/3 + 14/5 = 52/15

Solve Denominator: 3/2 − (15/24) = 3/2 − 5/8 = 7/8

Divide them: 52/15 ÷ 7/8 = 52/15 × 8/7 = 416/105

∴ The correct answer is 416/105.

Alternate Method

Given:

Expression: [(2/3) + (4/5 ÷ 2/7)] ÷ [(3/2) − (5/6 × 3/4)]

Formula Used:

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

Calculations:

⇒ Numerator = (2/3) + (4/5 ÷ 2/7)

⇒ Numerator = (2/3) + (4/5 × 7/2)

⇒ Numerator = (2/3) + (14/5)

⇒ Numerator = (10/15) + (42/15) = 52/15

⇒ Denominator = (3/2) − (5/6 × 3/4)

⇒ Denominator = (3/2) − (15/24)

⇒ Denominator = (3/2) − (5/8)

⇒ Denominator = (12/8) − (5/8) = 7/8

⇒ Result = (52/15) ÷ (7/8)

⇒ Result = (52/15) × (8/7)

⇒ Result = (52 × 8) / (15 × 7)

⇒ Result = 416/105

∴ The correct answer is 416/105.

Additional Information

BODMAS Rule

Dictates the correct sequence of operations in mathematical expressions: Brackets first, then Orders (roots/powers), Division, Multiplication, Addition, and finally Subtraction.

Fraction Division

To divide by a fraction, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Fraction Addition and Subtraction

Find the Least Common Multiple (LCM) of the denominators to ensure fractions are equivalent before proceeding to add or subtract their respective numerators.

Shortcut Trick

Convert decimals to fractions for faster calculation.

⇒ (5/2 + 18/5) ÷ 3/4 + (5/4 × 4/5)

⇒ (61/10) × 4/3 + 1

⇒ 122/15 + 1 = 137/15

∴ The correct answer is 137/15.

Alternate Method

Given:

(2 1/2 + 3.6) ÷ 0.75 + (1.25 × 4/5)

Method Used:

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

Calculations:

⇒ Step 1: Solve the brackets first.

⇒ First bracket: 2 1/2 + 3.6 = 2.5 + 3.6 = 6.1

⇒ Second bracket: 1.25 × 4/5 = 1.25 × 0.8 = 1

⇒ Step 2: Substitute values back into the expression.

⇒ 6.1 ÷ 0.75 + 1

⇒ Step 3: Perform division.

⇒ 6.1 ÷ (3/4) = 6.1 × 4/3

⇒ 24.4 / 3 = 244 / 30 = 122 / 15

⇒ Step 4: Perform addition.

⇒ 122 / 15 + 1 = (122 + 15) / 15

⇒ 137 / 15

∴ The correct answer is 137/15.

Additional Information

BODMAS Rule

Always strictly follow the sequence of operations: Brackets → Orders → Division → Multiplication → Addition → Subtraction to ensure the final result is accurate.

Fraction to Decimal Conversion

Converting common decimals into fractions (e.g., 0.75 to 3/4 and 1.25 to 5/4) allows division to become simple multiplication by the reciprocal.

Adding Fractions

To add a whole number to a fraction, express the whole number as a fraction with the same denominator: a/b + c = (a + b × c) / b.

Shortcut Trick

Time Ratio = Profit Ratio ÷ Investment Ratio

Time Ratio of A : B = (10 ÷ 5) : (9 ÷ 6) = 2 : 1.5 = 4 : 3

Here, B invested for 12 months, which corresponds to 3 parts in the ratio.

So, A's investment time (4 parts) = (12 ÷ 3) × 4 = 16 months.

∴ The correct answer is 16 months.

Alternate Method

Given:

Ratio of investments of A and B = 5 : 6

Ratio of profits of A and B = 10 : 9

Time period of investment for B = 12 months

Formula Used:

Profit = Investment × Time

⇒ Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Partnership: Capital × Time = Profit

Partner A

Capital = 5 units Time = t months Profit Ratio = 10

Partner B

Capital = 6 units Time = 12 months Profit Ratio = 9 VS

Calculations:

Let the time period for A's investment be T_A months.

⇒ P_A / P_B = (I_A × T_A) / (I_B × T_B)

⇒ 10 / 9 = (5 × T_A) / (6 × 12)

⇒ 10 / 9 = 5T_A / 72

⇒ 10 × 72 = 9 × 5T_A

⇒ 720 = 45T_A

⇒ T_A = 720 ÷ 45 = 16

∴ The correct answer is 16 months.

Additional Information

Simple Partnership

When investments of all the partners are for the same time period, the profit or loss is distributed purely in the ratio of their original capital investments.

Compound Partnership

When investments of the partners are for different time periods, the profit or loss is distributed in the ratio of the product of their capital and respective time periods (Profit Ratio = Capital × Time).

Working and Sleeping Partner

A working partner actively manages the business and usually receives a fixed salary or a specific percentage of the profit before the final division. A sleeping partner simply invests money and only receives a share of the remaining profit based on their capital and time invested.

Shortcut Trick

Removed Number = Old Average + Remaining Numbers × Decrease in Average

⇒ Removed Number = 56 + 8 × (56 − 54)

⇒ Removed Number = 56 + 8 × 2

⇒ Removed Number = 56 + 16

⇒ Removed Number = 72

∴ The correct answer is 72.

Alternate Method

Given:

Average of 9 numbers = 56

Average of remaining 8 numbers (after one is removed) = 54

Formula Used:

Total Sum = Average × Number of items

Total Sum of 9 numbers = 9 × 56 = 504

Sum of 8 numbers = 8 × 54 = 432

72

Removed

Calculation:

⇒ Initial total sum of 9 numbers = 9 × 56 = 504

⇒ New total sum of remaining 8 numbers = 8 × 54 = 432

⇒ The removed number = Initial total sum − New total sum

⇒ The removed number = 504 − 432 = 72

∴ The correct answer is 72.

Additional Information

Averages

The average is the central value of a set of numbers, calculated as the sum of all values divided by the number of values: Average = Sum of observations ÷ Total number of observations.

Deviation Method

The net deviation from the average across all terms is always zero. This property helps in quickly finding missing numbers, replacing numbers, or discovering new averages without multiplying large values.

Weighted Average

When combining groups, the combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n represents the number of items and A is the average of each respective group.

Shortcut Trick

Average of Truck 1 = (18 + 20 + 22 + 21 + 19) ÷ 5 = 100 ÷ 5 = 20 litres

Average of Truck 2 = (25 + 24 + 26 + 23 + 24) ÷ 5 = 122 ÷ 5 = 24.4 litres

Overall Average = (20 + 24.4) ÷ 2 = 44.4 ÷ 2 = 22.2 litres

∴ The correct answer is 22.2 litres.

Alternate Method

Given:

Fuel usage of Truck 1 for 5 days: 18, 20, 22, 21, 19

Fuel usage of Truck 2 for 5 days: 25, 24, 26, 23, 24

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Calculations:

⇒ Total fuel used by Truck 1 = 18 + 20 + 22 + 21 + 19 = 100 litres

⇒ Total fuel used by Truck 2 = 25 + 24 + 26 + 23 + 24 = 122 litres

⇒ Total fuel used by both trucks = 100 + 122 = 222 litres

⇒ Total number of observations = 5 days × 2 trucks = 10

⇒ Overall average = 222 ÷ 10 = 22.2 litres

∴ The correct answer is 22.2 litres.

Additional Information

Combined Average

If two groups have sizes n₁ and n₂ with averages A₁ and A₂, the combined average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Data Interpretation (Tabulation)

In tabular data interpretation, it is often faster to find sums of individual rows or columns first before computing total averages, avoiding multiple complex calculations.

Percentage Change

Frequently tested in DI questions. Percentage change is mathematically given as ((New Value − Old Value) ÷ Old Value) × 100.

Shortcut Trick

Rate = 10% = 1 / 10. Multiplying factor = 11 / 10.

Ratio of Principal to Amount for 2 years = (10)² : (11)² = 100 : 121.

Here, 121 units correspond to ₹12100.

1 unit = ₹12100 ÷ 121 = ₹100.

Principal = 100 units = 100 × ₹100 = ₹10000.

∴ The correct answer is ₹10000.

Alternate Method

Given:

Amount (A) = ₹12100

Time (T) = 2 years

Rate (R) = 10% per annum

Formula Used:

A = P × (1 + R / 100)^T

Where A = Amount, P = Principal, R = Rate, and T = Time.

+10% CI +10% CI

Year 0 Year 1 Year 2

P = ? A = ₹12100

Calculations:

⇒ 12100 = P × (1 + 10 / 100)²

⇒ 12100 = P × (1 + 1 / 10)²

⇒ 12100 = P × (11 / 10)²

⇒ 12100 = P × (121 / 100)

⇒ P = (12100 × 100) ÷ 121

⇒ P = 100 × 100 = 10000

∴ The correct answer is ₹10000.

Additional Information

Effective Compound Interest Rate

For 2 years, the effective CI rate can be found using the successive percentage formula: R + R + (R × R) / 100. For 10%, the effective rate is 10 + 10 + (10 × 10) / 100 = 21%.

Difference Between SI and CI (2 Years)

The difference between Compound Interest and Simple Interest for exactly 2 years can be directly calculated using the formula: Difference = P × (R / 100)².

Half-Yearly Compounding

When interest is compounded half-yearly, the rate of interest is halved (R / 2) and the given time period is doubled (2T) because interest is computed every 6 months.

Shortcut Trick

Find the profit or loss per kg for each variety relative to the selling price.

Profit on 50 kg rice = (48 − 40) × 50 = +8 × 50 = +₹400

Loss on 20 kg rice = (48 − 55) × 20 = −7 × 20 = −₹140

Net Profit = 400 − 140 = ₹260

∴ The correct answer is Profit of ₹260.

Alternate Method

Given:

Quantity 1 = 50 kg, Rate 1 = ₹40/kg

Quantity 2 = 20 kg, Rate 2 = ₹55/kg

Selling rate of mixture = ₹48/kg

Formula Used:

Total CP = (Q₁ × R₁) + (Q₂ × R₂)

Profit = Total SP − Total CP

50 kg @ ₹40 CP: ₹2000

20 kg @ ₹55 CP: ₹1100

Total CP ₹3100

70 kg @ ₹48 SP: ₹3360

Profit = SP − CP = 3360 − 3100 = ₹260

Calculations:

⇒ Cost Price of 50 kg of rice = 50 × 40 = ₹2000

⇒ Cost Price of 20 kg of rice = 20 × 55 = ₹1100

⇒ Total Cost Price (CP) = 2000 + 1100 = ₹3100

⇒ Total quantity of mixture = 50 + 20 = 70 kg

⇒ Total Selling Price (SP) of the mixture = 70 × 48 = ₹3360

⇒ Since SP > CP, there is a profit.

⇒ Profit = Total SP − Total CP = 3360 − 3100 = ₹260

∴ The correct answer is Profit of ₹260.

Additional Information

Profit and Loss Basic Formula

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

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100. Loss % = (Loss ÷ CP) × 100.

Weighted Average Concept

Average Cost Price per unit of a mixture is calculated as (Q₁ × C₁ + Q₂ × C₂) ÷ (Q₁ + Q₂).

Shortcut Trick

Intermediate Price after 1st discount =  ₹20,000 × 85/100 = ₹17,000

Value of 2nd discount = ₹17,000 − ₹15,300 = ₹1,700

Percentage of 2nd discount = (1700 ÷ 17000) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Marked Price (MP) = ₹20,000

First discount = 15%

Final Selling Price (SP) = ₹15,300

Formula Used:

Price after discount = MP × (100 − D%)/100

Discount % = (Discount Amount ÷ Base Price) × 100

₹20,000 Marked Price

− 15%

₹17,000

2nd Discount (−₹1,700)

₹15,300 Selling Price

Calculations:

⇒ Price after 1st discount = 20000 × (100 − 15)/100

⇒ Price after 1st discount = 20000 × 85/100 = ₹17,000

⇒ 2nd discount amount = Price after 1st discount − Final SP

⇒ 2nd discount amount = 17000 − 15300 = ₹1,700

⇒ 2nd discount percentage = (1700 ÷ 17000) × 100

⇒ 2nd discount percentage = 1/10 × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Equivalent Successive Discount

For two successive discounts of a% and b%, the single equivalent discount formula is (a + b − ab/100)%.

Marked Price and Cost Price Relation

MP/CP = (100 + Profit%) ÷ (100 − Discount%). This directly gives the ratio of Marked Price to Cost Price without knowing the Selling Price.

Multiplier Method for Successive Discounts

Final Selling Price = Marked Price × (1 − d₁/100) × (1 − d₂/100) × ...

Shortcut Trick

Let the Cost Price (CP) be 100.

Since the product is marked 30% above CP, Marked Price (MP) = 130.

With a profit of 17%, Selling Price (SP) = 117.

Discount = MP − SP = 130 − 117 = 13.

Discount % = (13 ÷ 130) × 100 = 10%.

∴ The correct answer is 10%.

Alternate Method

Given:

Mark-up percentage = 30%

Profit percentage = 17%

Formula Used:

MP = CP × (1 + Mark-up% ÷ 100)

SP = CP × (1 + Profit% ÷ 100)

Discount % = [(MP − SP) ÷ MP] × 100

CP: 100

SP: 117

MP: 130

Mark Up: +30%

Profit: +17%

Discount: ?

Calculations:

Let the Cost Price (CP) of the product be 100.

⇒ Marked Price (MP) = 100 × (1 + 30 ÷ 100) = 130

⇒ Selling Price (SP) = 100 × (1 + 17 ÷ 100) = 117

⇒ Discount given = MP − SP = 130 − 117 = 13

⇒ Rate of Discount = (Discount ÷ MP) × 100

⇒ Rate of Discount = (13 ÷ 130) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Direct Relation Between CP and MP

There is a direct ratio relationship connecting Cost Price, Marked Price, Profit% and Discount%: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Successive Discounts

If two successive discounts of a% and b% are offered, the single equivalent discount is calculated as: (a + b − a × b ÷ 100)%.

Dishonest Dealer Concept

When a dealer cheats by using false weights while selling at CP, the profit percentage is calculated as: (Error ÷ True Value) × 100.

Shortcut Trick

Alcohol from Container A = (4 ÷ 5) × 10 = 8 L and Water from A = 10 − 8 = 2 L

Alcohol from Container B = (5 ÷ 7) × 14 = 10 L and Water from B = 14 − 10 = 4 L

Total Alcohol in new mixture = 8 + 10 = 18 L

Total Water in new mixture = 2 + 4 = 6 L

Required Ratio of Alcohol to Water = 18 : 6 = 3 : 1

∴ The correct answer is 3:1.

Alternate Method

Given:

Ratio of alcohol to water in Container A = 4:1

Ratio of alcohol to water in Container B = 5:2

Mixture drawn from Container A = 10 liters

Mixture drawn from Container B = 14 liters

Formula Used:

Quantity of component = (Ratio of component ÷ Total ratio parts) × Total quantity

Container A 10 L (Ratio 4:1) Alc: 8L | Wat: 2L

Container B 14 L (Ratio 5:2) Alc: 10L | Wat: 4L

New Mixture Alc: 18L | Wat: 6L Ratio → 3:1

Calculations:

⇒ In Container A, the sum of ratio parts = 4 + 1 = 5

⇒ Quantity of alcohol drawn from A = (4 ÷ 5) × 10 = 8 liters

⇒ Quantity of water drawn from A = (1 ÷ 5) × 10 = 2 liters

⇒ In Container B, the sum of ratio parts = 5 + 2 = 7

⇒ Quantity of alcohol drawn from B = (5 ÷ 7) × 14 = 10 liters

⇒ Quantity of water drawn from B = (2 ÷ 7) × 14 = 4 liters

⇒ Total alcohol in the third container = 8 + 10 = 18 liters

⇒ Total water in the third container = 2 + 4 = 6 liters

⇒ Ratio of alcohol to water in the new mixture = 18 : 6

⇒ Simplifying the ratio by dividing both sides by 6, we get 3 : 1

∴ The correct answer is 3:1.

Additional Information

Multiple Mixtures Concept

When components from multiple different mixtures are mixed together, individually calculate the absolute quantities of each component from every source. Sum these respective quantities up to find the final total amount before constructing the new ratio.

Alligation Method

Used to find the ratio in which two mixtures must be mixed to obtain a desired concentration. The fundamental rule relates the quantities: (Quantity of Cheaper) ÷ (Quantity of Dearer) = (Price of Dearer − Mean Price) ÷ (Mean Price − Price of Cheaper).

Replacement Method

If a certain volume 'y' is replaced 'n' times from a total pure volume 'x', the final quantity of the original pure substance left is given by the formula: Final Quantity = Initial Quantity × (1 − y/x)ⁿ.

Shortcut Trick

For any 3D shape, if its linear dimensions (like radius) are multiplied by a factor 'n', its surface area becomes n² times and volume becomes n³ times.

Here, the radius is doubled, meaning the scaling factor n = 2.

Ratio of New Surface Area to Original = 2² : 1 = 4 : 1

Ratio of New Volume to Original = 2³ : 1 = 8 : 1

∴ The correct answer is 4:1 and 8:1.

Alternate Method

r Original Sphere SA = 4×π×r²

2r New Sphere SA = 16×π×r²

Given:

Let the original radius of the sphere be r.

The new radius of the sphere is doubled, so new radius = 2r.

Formula Used:

Surface Area of a Sphere = 4 × π × radius²

Volume of a Sphere = 4/3 × π × radius³

Calculations:

⇒ Original Surface Area = 4 × π × r²

⇒ New Surface Area = 4 × π × (2r)² = 16 × π × r²

⇒ Ratio of New Surface Area to Original = (16 × π × r²) : (4 × π × r²) = 4 : 1

⇒ Original Volume = 4/3 × π × r³

⇒ New Volume = 4/3 × π × (2r)³ = 4/3 × π × 8r³ = 8 × (4/3 × π × r³)

⇒ Ratio of New Volume to Original = 8 × (4/3 × π × r³) : (4/3 × π × r³) = 8 : 1

∴ The correct answer is 4:1 and 8:1.

Additional Information

Scaling Principle

For any 3D geometric solid, if all its linear dimensions are scaled by a factor 'k', its total surface area directly increases by a factor of k², and its volume increases by a factor of k³.

Hemisphere Volume

When a sphere is cut exactly in half, it forms a hemisphere. The volume of a solid hemisphere is given by 2/3 × π × r³.

Hemisphere Surface Area

The Curved Surface Area (CSA) of a hemisphere is 2 × π × r², whereas the Total Surface Area (TSA), which includes the flat circular base, is 3 × π × r².

Shortcut Trick

Since the hemisphere and the cone have equal volumes and share the same base radius R, we directly equate their volume formulas.

⇒ (2/3) × πR³ = (1/3) × πR²h

⇒ 2R = h

∴ The correct answer is 2R.

Alternate Method

Given:

Volume of Hemisphere = Volume of Cone

Radius of Hemisphere = Radius of Cone = R

R Hemisphere V = (2/3)×π×R³

R

h Cone V = (1/3)×π×R²h

=

Formula Used:

Volume of a Hemisphere = (2/3) × πr³

Volume of a Cone = (1/3) × πr²h

Calculation:

Equating the volume of the hemisphere and the cone:

⇒ (2/3) × πR³ = (1/3) × πR²h

Multiplying both sides by 3 to remove the fraction:

⇒ 2 × πR³ = πR²h

Dividing both sides by πR²:

⇒ 2R = h

⇒ h = 2R

∴ The correct answer is 2R.

Additional Information

Cylinder vs Cone Relationship

A cylinder with base radius r and height h has a volume of V = πr²h. Its volume is exactly 3 times that of a cone with the identical base and height.

Sphere vs Hemisphere Volume

A complete sphere has a volume of V = (4/3)πr³, while a hemisphere (which is half a sphere) has a volume of V = (2/3)πr³.

Surface Area of a Cone

The total surface area of a cone is calculated as TSA = πr(r + l), where l is the slant height given by l = √(r² + h²).

Shortcut Trick

When a hemisphere is melted into 2 equal cones, their total volumes remain equal.

Volume of Hemisphere = Volume of 2 Cones

2/3 × π × R³ = 2 × (1/3 × π × r² × h)

⇒ R³ = r² × h

⇒ 10³ = 5² × h ⇒ 1000 = 25 × h ⇒ h = 40

∴ The correct answer is 40 cm.

Alternate Method

Given:

Radius of the solid hemisphere (R) = 10 cm

Radius of each newly formed cone (r) = 5 cm

Number of cones formed = 2

Formula Used:

Volume of a hemisphere = 2/3 × π × R³

Volume of a cone = 1/3 × π × r² × h

R = 10 Hemisphere

Melted

r = 5 h = ?

r = 5 h = ?

Two Equal Cones

Calculations:

⇒ According to the problem, Total Volume of 2 Cones = Volume of the Hemisphere

⇒ 2 × (1/3 × π × r² × h) = 2/3 × π × R³

⇒ 2/3 × π × (5)² × h = 2/3 × π × (10)³

⇒ Canceling (2/3 × π) from both sides:

⇒ 25 × h = 1000

⇒ h = 1000 ÷ 25

⇒ h = 40 cm

∴ The correct answer is 40 cm.

Additional Information

Principle of Recasting

When a solid object is melted and recast into new shapes, the total volume always remains constant regardless of the new shape forms.

Hemisphere Formulas

Volume = 2/3 × π × r³ | Curved Surface Area = 2 × π × r² | Total Surface Area = 3 × π × r²

Cone Formulas

Volume = 1/3 × π × r² × h | Slant height (l) = √(r² + h²) | Curved Surface Area = π × r × l

Shortcut Trick

For any regular pyramid, the lateral surface area can be calculated directly using its base perimeter and slant height.

Lateral Surface Area = ½ × Base Perimeter × Slant Height

⇒ LSA = ½ × 40 × 10

⇒ LSA = 200 cm²

∴ The correct answer is 200 cm².

Alternate Method

Given:

Base perimeter = 40 cm

Slant height (l) = 10 cm

l = 10 cm a = 10 cm Base Perimeter = 40 cm

Formula Used:

Area of one lateral triangle = ½ × Base × Slant height

Side of square base (a) = Perimeter ÷ 4

Calculation:

⇒ Side of the base (a) = 40 ÷ 4 = 10 cm

⇒ A square pyramid has 4 identical lateral triangular faces.

⇒ Area of one triangular face = ½ × 10 × 10

⇒ Area of one triangular face = 50 cm²

⇒ Total Lateral Surface Area = 4 × Area of one face

⇒ Total LSA = 4 × 50 = 200 cm²

∴ The correct answer is 200 cm².

Additional Information

Total Surface Area (TSA)

The TSA of a regular pyramid is the sum of its Lateral Surface Area and Base Area. For a square pyramid of base side 'a': TSA = LSA + a².

Volume of a Pyramid

The volume of any pyramid is one-third of the product of its base area and true vertical height. V = ⅓ × Base Area × h.

Relation between Heights

For a regular square pyramid, the true vertical height (h), slant height (l), and base side (a) form a right-angled triangle. By Pythagoras theorem: l² = h² + (a/2)².

Shortcut Trick

Ratio of areas = (Ratio of radii)²

Area ratio = 2² : 3² = 4 : 9

Given 4 units = 100π cm² ⇒ 1 unit = 25π cm²

Area of larger pizza = 9 units = 9 × 25π = 225π cm²

∴ The correct answer is 225π cm².

Alternate Method

Given:

Ratio of radii of two circular pizzas, r₁ : r₂ = 2 : 3

Area of smaller pizza = 100π cm²

Formula Used:

Area of a circle = πr²

r₁ = 2x A₁ = 100π

r₂ = 3x A₂ = ?

Calculations:

Let the radii of the smaller and larger pizzas be 2x and 3x respectively.

⇒ Area of smaller pizza = π(2x)²

⇒ 4πx² = 100π

⇒ 4x² = 100

⇒ x² = 25

⇒ Area of larger pizza = π(3x)²

⇒ 9πx²

Substituting the value of x²:

⇒ Area of larger pizza = 9π × 25

⇒ Area of larger pizza = 225π cm²

∴ The correct answer is 225π 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 A = (θ ÷ 360) × πr².

Circumference of a Circle

The total boundary length of a circle is its circumference, calculated as C = 2πr.

Similar Figures Property

For any two similar geometric figures, the ratio of their areas is strictly equal to the square of the ratio of their corresponding linear dimensions (e.g., radii, diameters, or sides).

Shortcut Trick

The intersection of medians of a triangle is called its centroid.

The centroid always divides a median in the ratio 2 : 1 (from vertex to base).

Length of AG = (2/3) × Total length of median AD

AG = (2/3) × 12 cm = 8 cm

∴ The correct answer is 8 cm.

Alternate Method

Given:

Length of median AD = 12 cm

Intersection of medians AD and BE is G.

Concept Used:

The point where the medians of a triangle intersect is called the centroid.

The centroid divides each median in the ratio 2 : 1, measuring from the vertex to the midpoint of the opposite side.

 A B C D E G 2 Units 1 Unit

Calculations:

⇒ AG : GD = 2 : 1

⇒ Total parts on median AD = 2 + 1 = 3 parts

⇒ 3 parts = 12 cm

⇒ 1 part = 12 ÷ 3 = 4 cm

⇒ Length of AG = 2 parts = 2 × 4 cm

⇒ AG = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Centroid

The point of intersection of the three medians of a triangle. It divides each median in the ratio 2 : 1. The centroid always lies strictly inside the triangle.

Incenter

The point of intersection of the internal angle bisectors of the triangle. It is equidistant from all three sides and forms the center of the inscribed circle (incircle).

Circumcenter

The point of intersection of the perpendicular bisectors of the sides. It is equidistant from all three vertices and forms the center of the circumscribed circle (circumcircle).

Orthocenter

The point of intersection of the three altitudes (perpendiculars dropped from vertices to opposite sides) of the triangle.

Shortcut Trick

In the given right-angled triangles ABC and DEF, we are given:

∠B = ∠E = 90° (Right angles)

AC = DF (Hypotenuses are equal)

AB = DE (One pair of corresponding sides are equal)

Since we have matching right angles, equal hypotenuses, and one pair of equal sides, the RHS (Right angle − Hypotenuse − Side) congruence criterion is perfectly satisfied.

∴ The correct answer is Yes, by RHS.

Alternate Method

Given:

⇒ Two right-angled triangles, ΔABC and ΔDEF.

⇒ ∠B = ∠E = 90°

⇒ Hypotenuse AC = Hypotenuse DF

⇒ Side AB = Side DE

A B C

D E F

Proof:

⇒ In right-angled ΔABC and right-angled ΔDEF:

⇒ ∠B = ∠E = 90° (Given right angles)

⇒ AC = DF (Given equal hypotenuses)

⇒ AB = DE (Given equal corresponding sides)

⇒ According to the RHS (Right angle − Hypotenuse − Side) congruence rule, if the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the two triangles are congruent.

⇒ ∴ ΔABC ≅ ΔDEF directly by the RHS congruence criterion.

∴ The correct answer is Yes, by RHS.

Additional Information

SSS (Side-Side-Side) Congruence

If all three sides of one triangle are exactly equal to the corresponding three sides of another triangle, the two triangles are congruent.

SAS (Side-Angle-Side) Congruence

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, they are congruent.

ASA (Angle-Side-Angle) Congruence

If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, they are congruent.

AAS (Angle-Angle-Side) Congruence

If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the two triangles are congruent.

Shortcut Trick

The hypotenuse of the right triangle is the diameter of the inscribed circle, so AC = 10 cm.

By the right triangle altitude theorem: AB² = AD × AC

⇒ 6² = AD × 10

⇒ 36 = 10 × AD

⇒ AD = 3.6 cm

∴ The correct answer is 3.6 cm.

Alternate Method

Given:

⇒ Diameter (Hypotenuse AC) = 10 cm

⇒ Leg AB = 6 cm

O A C B D

6 cm

AD = ?

AC = 10 cm

Formula Used:

⇒ Angle in a semicircle is always a right angle, so ∠ABC = 90°.

⇒ In a right-angled triangle, the altitude drawn to the hypotenuse creates similar triangles: ΔADB ∼ ΔABC.

Calculations:

⇒ From the similarity of ΔADB and ΔABC: AB / AC = AD / AB

⇒ By cross-multiplying: AB² = AD × AC

⇒ Substituting the given values: 6² = AD × 10

⇒ 36 = 10 × AD

⇒ AD = 36 ÷ 10

⇒ AD = 3.6 cm

∴ The correct answer is 3.6 cm.

Additional Information

Right Triangle Similarity Theorem

When an altitude is drawn to the hypotenuse of a right triangle, it divides the triangle into two smaller triangles that are similar to the original triangle and to each other.

Geometric Mean Theorem

The length of the altitude drawn to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse: BD² = AD × CD.

Angle in a Semicircle

Any angle subtended by a diameter at the circumference of a circle is always exactly a right angle (90°).

Shortcut Trick When two circles intersect, they overlap each other, meaning the distance between their centers is less than the sum of their radii. Any internal (transverse) common tangent must pass between their centers, which is impossible without intersecting the overlapping area. Thus, no internal common tangents can exist for intersecting circles. Only external (direct) tangents can be drawn to touch the outer boundaries of both circles simultaneously. Consequently, exactly 2 direct common tangents can be drawn. ∴ The correct answer is Only two common tangents exist.

Alternate Method Given: Two circles intersect at two distinct points. Concept Used: The possible number of common tangents depends entirely on the distance between the circle centers (d) and their radii (r1 and r2).

C₁ C₂ Direct Common Tangent 1 Direct Common Tangent 2

Calculations: ⇒ For intersecting circles, the distance between their centers follows the condition: |r1 − r2| < d < r1 + r2. ⇒ Because d < r1 + r2, the Transverse (internal) Common Tangents are restricted as they would cut through the overlapping region of the circles instead of behaving as tangents. ⇒ Only Direct (external) Common Tangents can be drawn, touching both circles on their outer boundaries without crossing the central intersecting zone. ⇒ Therefore, exactly 2 Direct Common Tangents are geometrically possible. ∴ The correct answer is Only two common tangents exist.

Additional Information Number of Common Tangents Rule If d > r1 + r2 → 4 tangents. If d = r1 + r2 → 3 tangents.If d = |r1 − r2| → 1 tangent. If d < |r1 − r2| → 0 tangents.

Length of Direct Common Tangent (DCT) The formula to find the length of a direct tangent when circles are separated by distance d is: DCT = √[d2 − (r1 − r2)2]

Length of Transverse Common Tangent (TCT) The formula to find the length of an internal cross tangent when circles are non-intersecting is: TCT = √[d2 − (r1 + r2)2]

Shortcut Trick

When two circles with radii r₁ and r₂ touch externally, the distance between their centers is exactly d = r₁ + r₂.

By applying this to the standard Direct Common Tangent formula, it reduces to a direct standard result.

The length of the direct common tangent is the geometric mean of their diameters.

Direct Formula: T = √(4 × r₁ × r₂)

This simplifies directly to T = 2√(r₁r₂).

∴ The correct answer is 2√(r₁r₂).

Alternate Method

r₁ r₂ T (Tangent)

C₁ C₂

r₁ − r₂ T r₁ + r₂

Given:

Radius of the larger circle = r₁

Radius of the smaller circle = r₂

Since the circles touch each other externally, the distance between their centers (d) = r₁ + r₂.

Length of the direct common tangent = T

Formula Used:

In the right-angled triangle formed by drawing a horizontal line from C₂ to the radius of C₁:

⇒ Base² + Perpendicular² = Hypotenuse²

Calculations:

⇒ T² + (r₁ − r₂)² = (r₁ + r₂)²

⇒ T² = (r₁ + r₂)² − (r₁ − r₂)²

Applying the standard algebraic identity (a + b)² − (a − b)² = 4ab:

⇒ T² = 4 × r₁ × r₂

⇒ T = √(4r₁r₂)

⇒ T = 2√(r₁r₂)

∴ The correct answer is 2√(r₁r₂).

Additional Information

General Direct Common Tangent (DCT)

DCT = √[d² − (r₁ − r₂)²], where d is the straight line distance between the centers of the two circles.

Transverse Common Tangent (TCT)

TCT = √[d² − (r₁ + r₂)²]. The transverse tangent diagonally crosses the line joining the centers.

Maximum Common Tangents

When two circles touch each other externally, exactly 3 common tangents can be drawn (2 direct common tangents and 1 transverse common tangent touching at the intersection point).

Internally Touching Circles

If two circles touch each other internally, the distance between their centers is d = |r₁ − r₂| and only 1 common tangent can be drawn.

Shortcut Trick

The angle at the center is twice the angle at the circumference → 2 × 30° = 60°.

The triangle formed by the center and the chord is an isosceles triangle with a 60° vertex, making it an equilateral triangle.

Radius of the circle = Chord length = 12 cm.

The distance from the center to the chord is simply the altitude of this equilateral triangle.

Distance = (√3 ÷ 2) × Side = (√3 ÷ 2) × 12 = 6√3 cm.

∴ The correct answer is 6√3 cm.

Alternate Method

Given:

Length of the chord (AB) = 12 cm

Angle subtended at the circumference (∠ACB) = 30°

Formula Used:

Angle at the center = 2 × Angle at the circumference

Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²

60° 30°

O A B C M

AB = 12 cm d = ?

Calculation:

⇒ Angle subtended at the center (∠AOB) = 2 × 30° = 60°

⇒ In ▵AOB, OA = OB (radii of the same circle), making it an isosceles triangle.

⇒ Base angles ∠OAB = ∠OBA = (180° − 60°) ÷ 2 = 60°

⇒ Since all internal angles are exactly 60°, ▵AOB is an equilateral triangle.

⇒ Radius (OA) = AB = 12 cm

⇒ Let OM be the perpendicular drawn from the center O to the chord AB.

⇒ The perpendicular from the center bisects the chord perfectly, so AM = AB ÷ 2 = 12 ÷ 2 = 6 cm.

⇒ In right-angled ▵OMA, applying Pythagoras Theorem:

⇒ OA² = OM² + AM²

⇒ 12² = OM² + 6²

⇒ 144 = OM² + 36

⇒ OM² = 144 − 36 = 108

⇒ OM = √108 = √(36 × 3) = 6√3 cm

∴ The correct answer is 6√3 cm.

Additional Information

Angle Subtended by an Arc

The angle subtended by an arc at the center is always double the angle subtended by it at any remaining part of the circle's circumference.

Perpendicular from Center to Chord

A perpendicular drawn from the center of a circle to any chord will perfectly bisect the chord into two equal halves.

Equilateral Triangle Altitude

In an equilateral triangle where all sides are equal, the height or altitude can be calculated directly using the standard formula h = (√3 ÷ 2) × side.

Shortcut Trick

Area of segment = Area of sector − Area of triangle

Area = [(θ ÷ 360°) × πr²] − [(1 ÷ 2) × r² × sin(θ)]

Substitute r = 8 cm and θ = 60°:

Area = [(60 ÷ 360) × π × 8²] − [(1 ÷ 2) × 8² × sin(60°)]

Area = (64π ÷ 6) − (32 × √3 ÷ 2)

Area = 32π/3 − 16√3 cm²

∴ The correct answer is 32π/3 − 16√3 cm².

Alternate Method

O A B 8 cm 8 cm 60°

Segment Area = Sector OAB − Triangle OAB

Given:

Radius of the circle, r = 8 cm

Angle subtended at the center by the chord, θ = 60°

Formula Used:

Area of Segment = Area of Sector − Area of Triangle

Area of Sector = (θ ÷ 360°) × πr²

Since the two sides bounding the angle 60° are the radii of the same circle, they are equal in length, forming an isosceles triangle. With the included angle being 60°, it becomes an equilateral triangle.

Area of Equilateral Triangle = (√3 ÷ 4) × a² (where a = r)

Calculations:

⇒ Area of Sector = (60° ÷ 360°) × π × 8²

⇒ Area of Sector = (1 ÷ 6) × 64π = 32π/3 cm²

⇒ Area of Triangle = (√3 ÷ 4) × 8²

⇒ Area of Triangle = (√3 ÷ 4) × 64 = 16√3 cm²

⇒ Area of Segment = Area of Sector − Area of Triangle

⇒ Area of Segment = 32π/3 − 16√3 cm²

∴ The correct answer is 32π/3 − 16√3 cm².

Additional Information

Area of a Sector

The region bounded by two radii and an arc is called a sector. Its area is calculated as (θ ÷ 360°) × πr².

Length of an Arc

The distance along the curved line making up the arc is given by the formula (θ ÷ 360°) × 2πr.

Area of Triangle in a Circle

For a triangle formed by two circle radii and a connecting chord, the general area can be found using (1 ÷ 2) × r² × sin(θ).

Perimeter of a Sector

The perimeter of a circular sector is the sum of its arc length and the two boundary radii: Length of Arc + 2r.

The correct answer is '1' i.e. Explain .

Key Points

• Illuminate means to make something clear or understandable.
• The word is often used metaphorically, meaning to explain or shed light on a topic.
• Among the options provided:
• Explain: Synonym for illuminate, as it clarifies or makes something clear.
• Darken: Opposite in meaning, as it refers to making things less clear.
• Destroy: Unrelated meaning; refers to ruining or breaking something.
• Hide: Opposite of illuminate; means to conceal or obscure something.

Correct synonym: Explain

Additional Information

• Usage in Sentence: The professor used diagrams to illuminate the complex concept for the students.
• Other Synonyms: Clarify, enlighten, elucidate.

The correct answer is '1' i.e. Agree .

Key Points

• The word "Demur" means to raise objections or show reluctance.
• Its antonym would be a word that conveys agreement or acceptance.
• From the given options:
• Agree: Correct, as it is the opposite of showing objection.
• Oppose: Incorrect, as it is similar to the meaning of "Demur".
• Object: Incorrect, as it is similar to "Demur".
• Resist: Incorrect, as it conveys a meaning similar to "Demur".

Correct sentence: The correct antonym of "Demur" is Agree, as it signifies a lack of objection or reluctance.

The correct answer is '3' i.e. To accept insult or defeat gracefully .

Key Points

• The idiom "Eat humble pie" is used to describe a situation where one has to accept defeat or admit their mistakes gracefully.
• This idiom originates from historical usage, where "humble pie" referred to a dish made from the less desirable parts of an animal, symbolizing humility and submission.
• Among the given options:
• Option 1: "To demand an apology" is incorrect as the idiom implies accepting, not demanding.
• Option 2: "To celebrate one's success" is incorrect as the idiom refers to humility, not celebration.
• Option 4: "To prepare for failure" is incorrect as the idiom signifies the acceptance of failure, not preparation.

Correct meaning: To accept insult or defeat gracefully.

Additional Information

• Examples of usage:
• After realizing his mistake, he had to eat humble pie and apologize to his colleagues.
• She boasted about her success, but when the truth came out, she had to eat humble pie.