The correct answer is option 2,

Key Points

• The 'Sections' feature in MS PowerPoint enables users to organize large presentations into manageable and logical groupings.
• It helps in streamlining navigation during both the editing and presentation phases by dividing the slides into distinct sections.
• Sections allow users to focus on specific parts of the presentation without being overwhelmed by the entire slide deck.
• This feature is particularly useful when dealing with complex or multi-topic presentations, ensuring better structure and clarity.
• Users can rename, collapse, or expand sections for easier navigation, enhancing productivity in managing large presentations.

Additional Information

• Slide Organization in PowerPoint
• PowerPoint includes features like 'Sections,' 'Slide Sorter View,' and 'Outline View' to help users manage and organize content efficiently.
• 'Slide Sorter View' provides a thumbnail view of all slides, enabling easier rearrangement.
• 'Outline View' focuses on text content, offering a structured view of the presentation.
• Benefits of Using Sections
• Enhances collaboration when working on different parts of the presentation.
• Improves workflow by allowing users to concentrate on specific sections.
• Reduces clutter and simplifies navigation in presentations with numerous slides.
• Other Navigation Features in PowerPoint
• Slide Master: Helps maintain consistent design across slides.
• Hyperlinks: Facilitates linking between slides or external resources.
• Slide Zoom: Enables focused navigation on specific slides during presentations.
• PowerPoint for Large Presentations
• For presentations with hundreds of slides, features like 'Sections' and 'Slide Sorter View' are essential for effective management.
• Grouping slides into sections ensures logical flow and improved audience engagement.

Shortcut Trick

Ratio of boys = 15 : 35 → 3 : 7

Assume average height = 160 cm

Deviation in second group = 168 − 160 = +8 cm

Net deviation for total parts (10) = 7 × 8 = +56

Average height = 160 + (56 ÷ 10) = 160 + 5.6 = 165.6 cm

∴ The correct answer is 165.6 cm.

Alternate Method

Given:

Number of boys in first group (n₁) = 15

Average height of first group (A₁) = 160 cm

Total number of boys (N) = 50

Average height of remaining boys (A₂) = 168 cm

Formula Used:

Weighted Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂)

Calculations:

⇒ Total height of first 15 boys = 15 × 160 = 2400 cm

⇒ Number of remaining boys = 50 − 15 = 35 boys

⇒ Total height of 35 boys = 35 × 168 = 5880 cm

⇒ Total height of all 50 boys = 2400 + 5880 = 8280 cm

⇒ Average height of class = 8280 ÷ 50

⇒ Average height = 165.6 cm

∴ The correct answer is 165.6 cm.

Additional Information

Weighted Average Concept

When groups with different averages are combined, the combined average is the total sum divided by the total count: A_avg = Σ (n_i × A_i) ÷ Σ n_i.

Deviation Method (Change in Average)

If one group is considered as the base average, the overall average increases/decreases by the (net deviation) ÷ (total quantity).

Alligation Rule

The ratio of the quantities is inversely proportional to the differences between their averages and the mean average: n₁/n₂ = (A₂ − A_mean) / (A_mean − A₁).

Given: (Left) 103 678 217 111 537 (Right)

According to the question:

So,

Second digit of the highest number → 678 → 7

First digit of the lowest number → 103 → 1

Thus, if the second digit of the highest number is added to the first digit of the lowest number then the resultant will be → 7 + 1 → 8

Hence, the correct answer is "Option 3".

According to the given information,

Thus, 'R' is coded as '7'.

Hence, "Option 3" is the correct answer.

The logic followed here is:

​VBCD is related to MTTV

And,

LVSX is related to CNJP

Similarly,

KXRZ is related to

Hence, "Option 1" is the correct answer.

Given:

When a number is increased by 57 ¹/₂% (57.5%), it becomes 378.

Formula Used:

Final Value = Original Number × (1 + Percentage Increase / 100)

Calculation:

Let the original number be X.

According to the formula:

378 = X × (1 + 57.5 / 100)

378 = X × 1.575

⇒ X = 378 / 1.575

⇒ X = 240

The original number is 240.

The correct answer is option 4,Key Points:

• Pradhan Mantri Awas Yojana - Urban (PMAY-U) 2.0 focuses on providing Affordable Rental Housing Complexes (ARHCs) for urban migrants and workers.
• The ARHC component aims to address housing challenges faced by migrant laborers and urban poor by offering affordable rental housing solutions.
• Under this scheme, ARHCs are developed through Public-Private Partnership (PPP) models or by utilizing existing government-funded housing complexes.
• PMAY-U 2.0 emphasizes the conversion of vacant government-funded housing units into rental housing complexes to maximize utilization.
• Special financial incentives and operational mechanisms are provided to private developers for constructing ARHCs to ensure scalability and sustainability.

Additional Information:

• Pradhan Mantri Awas Yojana - Urban (PMAY-U)
• Launched in June 2015, PMAY-U aims to provide Housing for All by 2022 through affordable housing initiatives.
• The scheme targets urban poor, including economically weaker sections (EWS), low-income groups (LIG), and middle-income groups (MIG).
• It includes various components such as Credit Linked Subsidy Scheme (CLSS), in-situ slum redevelopment, and affordable housing in partnership.
• Affordable Rental Housing Complexes (ARHCs)
• ARHCs aim to provide rental housing for urban migrants and workers, addressing the housing shortage in urban areas.
• These complexes are designed to offer safe and dignified housing at affordable rates near workplaces or industrial hubs.
• ARHCs are developed by leveraging existing housing stock or constructing new units through public-private partnerships.
• Public-Private Partnership (PPP) Model
• PPP is a collaborative framework where private entities and government agencies work together to deliver infrastructure or services.
• In the context of PMAY-U, PPP models are used to construct and operate ARHCs efficiently.
• Credit Linked Subsidy Scheme (CLSS)
• CLSS is a PMAY-U component offering interest subsidies on home loans to eligible beneficiaries.
• It aims to reduce the financial burden on EWS, LIG, and MIG households while purchasing or constructing their homes.

The correct answer is option 4,

Key Points

• The IndiaAI initiative was officially launched by the Union IT Minister to strengthen India’s position as a global hub for artificial intelligence.
• The initiative aims to promote AI research, innovation, and adoption across various sectors such as healthcare, education, agriculture, and governance.
• IndiaAI serves as a national program fostering collaboration between industry, academia, and government entities to accelerate AI-driven solutions.
• The platform has a focus on creating skilled manpower through training programs and enhancing AI literacy at grassroots levels.
• IndiaAI works under the Ministry of Electronics and Information Technology (MeitY) and aligns with the vision of a digitally empowered India.

Additional Information

• Artificial Intelligence (AI):
• AI refers to the simulation of human intelligence in machines programmed to think and learn.
• Key applications include natural language processing, machine learning, and robotics.
• AI is transforming industries by improving efficiency, decision-making, and innovation.
• National AI Strategy:
• The Government of India has laid out a vision for leveraging AI for social and economic growth.
• Focus areas include responsible AI, ethical frameworks, and democratization of AI technologies.
• IndiaAI aligns with this strategy to ensure inclusive development.
• Digital India Program:
• Launched in 2015, the Digital India initiative aims to transform India into a digitally empowered society.
• It focuses on areas such as digital infrastructure, governance, and digital literacy.
• IndiaAI is a major step under this program to enhance technological capabilities.
• AI in Governance:
• AI is being used for policy analysis, decision-making, and service delivery in India.
• Projects like AI-powered chatbots and predictive analytics tools are improving citizen engagement.

The correct answer is option 2,

Key Points

• The Thar Desert, also known as the Great Indian Desert, is the 17^th largest desert in the world and primarily spans the state of Rajasthan, covering approximately 60% of its area.
• It also extends into parts of Gujarat, Punjab, and Haryana, but Rajasthan is the main state covered.
• Rajasthan's districts such as Jaisalmer, Barmer, Bikaner, and Jodhpur are most heavily impacted by the Thar Desert's arid conditions.
• The desert lies in the northwestern part of India and is bordered by the Aravalli mountain range in the southeast.
• The Thar Desert plays a crucial role in shaping the geography, climate, and cultural heritage of Rajasthan.

Additional Information

• Climate: The Thar Desert experiences extreme temperatures, ranging from 50°C during summers to near freezing in winters.
• Vegetation: It is characterized by xerophytic vegetation such as cacti, acacias, and thorny bushes adapted to arid conditions.
• Economy: Agriculture in the Thar region relies heavily on irrigation systems like the Indira Gandhi Canal, while tourism plays a significant role in districts like Jaisalmer.
• Wildlife: The desert is home to unique wildlife such as the Great Indian Bustard, Blackbuck, and Desert Fox, protected under various conservation initiatives.
• Human Settlements: Despite the harsh conditions, the desert supports a dense population, with many villages practicing traditional lifestyles and crafts such as pottery and textiles.

Shortcut Trick

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

Given: The amount becomes 2 times in 11 years.

Target: The amount needs to become 4 times, which is 2² times.

Here, n = 2, T = 11, and k = 2.

∴ Time taken = 2 × 11 = 22 years.

∴ The correct answer is 22 years.

Alternate Method

Given:

Initial Principal = P

Amount after 11 years = 2P

Rate of interest = r% (compounded annually)

Formula Used:

Amount (A) = P(1 + r/100)ⁿ

Calculations:

⇒ 2P = P(1 + r/100)¹¹

⇒ 2 = (1 + r/100)¹¹ --- (Equation 1)

Let the time taken to become 4 times be T years.

⇒ 4P = P(1 + r/100)^T

⇒ 4 = (1 + r/100)^T

⇒ 2² = (1 + r/100)^T

Substitute the value of 2 from Equation 1:

⇒ [(1 + r/100)¹¹]² = (1 + r/100)^T

⇒ (1 + r/100)²² = (1 + r/100)^T

Comparing the exponents:

⇒ T = 22

∴ The correct answer is 22 years.

Additional Information

Exponential Growth in CI

In Compound Interest, the amount grows geometrically. If it doubles in 11 years, it will quadruple (2×2) in 11+11=22 years, and become 8 times (2×2×2) in 11+11+11=33 years.

Rule of 72

A quick way to estimate doubling time: Time ≈ 72 ÷ Rate of Interest. This is used for mental calculations in finance.

Effective Annual Rate

When interest is compounded more than once a year, the effective rate is higher. Effective Rate = [1 + r/(100×n)]ⁿ − 1, where n is frequency of compounding.

Given Table:

According to the given condition:

Given: 9#@27

Here, (i) and (ii) rules applies i.e. it is to be coded as γ and β respectively.

9 → coded as β

→ coded as I

@ → coded as F

2 → coded as γ

7 → coded as R

So, 9#@27 → Coded as βIFγR

Hence, "Option 4" is the correct answer.

The correct answer is Panaji.

Key Points

• The 55^th International Film Festival of India (IFFI) is scheduled to be held in November 2024 in Panaji, Goa.
• Panaji has been the permanent venue for the International Film Festival of India since 2004.
• The IFFI is organized jointly by the Ministry of Information and Broadcasting, Government of India, and the Entertainment Society of Goa (ESG).
• It is one of the most significant film festivals in Asia, showcasing an extensive range of world cinema and honoring film excellence.
• The festival serves as a platform for fostering global cinematic exchange and offers opportunities for budding filmmakers.

Additional Information

• International Film Festival of India (IFFI):
• Started in 1952, it is the oldest film festival in Asia.
• It aims to promote the art of cinema by bringing together filmmakers and audiences globally.
• IFFI features categories like International Competition, Indian Panorama, Masterclasses, and Special Screenings.
• Goa as a Permanent Venue:
• In 2004, Goa was declared the permanent venue for IFFI due to its infrastructure and scenic appeal.
• The festival is hosted at venues like Dr. Shyama Prasad Mukherjee Indoor Stadium and Inox Multiplex.
• Ministry of Information and Broadcasting:
• The ministry oversees the organization of IFFI in collaboration with state governments and other partners.
• It promotes Indian cinema globally through initiatives like IFFI.
• Significance of IFFI:
• The festival serves as a platform for cultural exchange and dialogue between global filmmakers.
• It also provides a significant boost to the Indian tourism and entertainment industries.

The correct answer is establishing uniform police and judiciary reforms.

Key Points

• The Cornwallis Code, introduced in 1793, marked a significant step in British administrative reforms in India.
• This code established uniformity in police and judiciary systems, laying the foundation for modern legal and administrative systems in India.
• It emphasized the separation of revenue collection and judicial functions to avoid conflicts of interest.
• District judges were made independent of the revenue administration, ensuring impartial judicial processes.
• The code also introduced strict regulations for the conduct of officials, aiming to curb corruption and inefficiency.

Additional Information

• Lord Cornwallis
• He served as the Governor-General of Bengal from 1786 to 1793.
• Known as the "father of civil services in India," he introduced reforms to make administration more efficient and accountable.
• Judicial Reforms under Cornwallis
• The establishment of civil and criminal courts at different levels (district, provincial, and Sadar levels).
• District judges were appointed, and the concept of rule of law was strengthened.
• Police Reforms
• The police system was reorganized, and thanas (police stations) were established to maintain law and order.
• The officers were made accountable, and their powers were clearly defined.
• Legacy of the Cornwallis Code
• It laid the groundwork for the Indian Penal Code and the modern judicial system in India.
• The reforms also influenced subsequent British administrative practices in the country.

Shortcut Trick

Volume of cuboid = Length × Breadth × Height

8832 = 4 × 46 × Height

8832 = 184 × Height

Height = 8832 ÷ 184 = 48 cm

∴ The correct answer is 48 cm.

Alternate Method

Given:

Length (l) = 4 cm

Breadth (b) = 46 cm

Volume (V) = 8832 cm³

Formula Used:

Volume of a Cuboid = Length × Breadth × Height

Calculations:

⇒ 8832 = 4 × 46 × h

⇒ 8832 = 184 × h

⇒ h = 8832 ÷ 184

⇒ h = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Total Surface Area of Cuboid

The total area of all faces is given by 2 × (lb + bh + hl).

Diagonal of a Cuboid

The length of the longest rod that can be placed inside is √(l² + b² + h²).

Lateral Surface Area

The area of the four side walls is given by 2 × Height × (Length + Breadth).

Shortcut Trick

Effective rate of Simple Interest for 2 years = 5% × 2 = 10%

Simple Interest = 10% of Principal

⇒ Interest = 10% of ₹2,200

⇒ Interest = 2200 ÷ 10 = ₹220

∴ The correct answer is ₹220.

Alternate Method

Given:

Principal (P) = ₹2,200

Rate (R) = 5% per annum

Time (T) = 2 years

Formula Used:

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

Calculations:

⇒ SI = (2200 × 5 × 2) ÷ 100

⇒ SI = 2200 × 10 ÷ 100

⇒ SI = 22 × 10

⇒ SI = ₹220

∴ The correct answer is ₹220.

Additional Information

Total Amount Formula

The Amount (A) at the end of the period is the sum of the Principal and the Simple Interest: A = P + SI.

Constant Interest Property

In Simple Interest, the interest earned remains the same every year because it is always calculated on the original Principal (P).

Effective Rate Calculation

For quick calculation, the total Percentage Interest over the entire period is the product of the yearly Rate and the Time duration (R × T).

Shortcut Trick

Total age 3 years ago = 123 years.

Sum of present ages = 123 + (3 × 3) = 132 years.

Let Son's age be x. Then Hitesh = 3x and Daughter = x − 3.

Total: 3x + x + (x − 3) = 132 ⇒ 5x = 135 ⇒ x = 27.

Hitesh's age = 3 × 27 = 81 years.

∴ The correct answer is 81 years.

Alternate Method

Given:

Relation: Hitesh = 3 × Son

Relation: Daughter = Son − 3

Sum of ages 3 years ago = 123

Let Son's present age = x years

Hitesh's present age = 3x years

Daughter's present age = (x − 3) years

Sum of ages 3 years ago:

⇒ (3x − 3) + (x − 3) + (x − 3 − 3) = 123

⇒ 3x + x + x − 3 − 3 − 6 = 123

⇒ 5x − 12 = 123

⇒ 5x = 135

⇒ x = 27

Hitesh's present age = 3 × 27 = 81 years

∴ The correct answer is 81 years.

Additional Information

Sum of Ages Shift

If there are 'n' people, the sum of their ages 't' years ago was (Present Sum − n × t).

Age Ratio Problems

When ages are given in ratios, always represent them as multiples of a variable 'x' to solve linear equations easily.

Constant Age Difference

The difference between the ages of any two people remains constant throughout their lives.

Given:

Equation: 72 ÷ 18 × 32 − 21 x 12 ÷ 3 + 12 = z

Formula Used:

Follow BODMAS (Bracket, Order, Division, Multiplication, Addition, Subtraction) rule to solve the equation.

Calculation:

72 ÷ 18 × 32 − 21 x 12 ÷ 3 + 12 = z

⇒ 4 × 9 − 21 x 4 + 12 = z

⇒ 36 - 84 + 12 = z

⇒ z = 48 - 84

⇒ z = - 36

∴ Option 3 is the correct answer.

Given number: 5123487

According to the question, 

The digits that will remain unchanged as compared to that in the original number → None

Hence, the correct answer is "Option 2".

The correct answer is Kolkata.

Key Points

• Kolkata is one of the metropolitan cities in India and is classified as a Tier I city.
• Tier I cities are those that have a high population density, advanced infrastructure, and are major economic hubs.
• Other Tier I cities in India include Mumbai, Delhi, Bengaluru, Chennai, and Hyderabad.
• Kolkata is the capital of West Bengal and serves as an important cultural, commercial, and educational hub in eastern India.
• The city is home to major industries such as jute, textiles, IT, and finance, contributing significantly to India's economy.

Additional Information

• Tier Classification of Indian Cities:
• Indian cities are classified into Tier I, Tier II, and Tier III based on population, infrastructure, and economic activity.
• Tier I cities are highly urbanized and contribute substantially to India's GDP.
• Tier II cities are mid-sized cities with moderate infrastructure growth.
• Tier III cities are smaller cities with less urbanization and development.
• Population Density:
• Tier I cities generally have a population of over 1 million.
• High population density in these cities leads to increased demand for housing, infrastructure, and services.
• Economic Importance of Tier I Cities:
• Tier I cities are key contributors to India's economy, hosting corporate headquarters, industries, and financial institutions.
• They serve as gateways for international trade and foreign investments.
• Urban Development:
• Tier I cities have advanced transportation networks, healthcare facilities, and educational institutions.
• They are equipped with modern infrastructure, including airports, metro systems, and IT parks.

The correct answer is option 1,

Key Points

• The selection of ministers for the Union Council of Ministers is governed by various considerations, including individual capacity, geographical representation, and representation of backward classes.
• Representation of foreign communities is not a factor considered in the selection process as it is not relevant to the Indian political structure.
• The Prime Minister ensures a balance of representation across regions, social groups, and expertise to reflect the diversity of India.
• Criteria such as individual capability, integrity, and political experience are emphasized during the appointment of ministers.
• Other considerations include maintaining party cohesion, rewarding loyalty, and adhering to constitutional provisions under Articles 74 and 75 of the Indian Constitution.

Additional Information

• Union Council of Ministers:
• The Union Council of Ministers is the executive body of the Indian government, headed by the Prime Minister.
• It consists of Cabinet Ministers, Ministers of State, and Ministers of State (Independent Charge).
• The Council of Ministers is collectively responsible to the Lok Sabha under Article 75 of the Constitution.
• Articles 74 and 75 of the Indian Constitution:
• Article 74 mandates that the Prime Minister leads the Council of Ministers to aid and advise the President.
• Article 75 outlines the appointment of ministers by the President on the advice of the Prime Minister.
• It includes provisions regarding ministerial qualifications and collective responsibility.
• Geographical Representation:
• Geographical representation ensures ministers from different states and regions to reflect India’s federal structure.
• This helps address regional aspirations and maintain political balance.
• Representation of Backward Classes:
• Representation of socially and economically backward classes is crucial to uphold inclusivity and equity in governance.
• This factor aligns with India's commitment to social justice under Articles 15 and 16 of the Constitution.

The correct answer is option 1,

Key Points

• Nucleated settlements in the Bundelkhand region primarily developed due to security concerns and invasions, especially during historical periods marked by frequent conflicts and raids.
• The settlements were designed as compact clusters to facilitate defense mechanisms and collective safety against external threats.
• Bundelkhand's geographical location, surrounded by rugged terrain, made it vulnerable to invasions, necessitating closely-knit communities for protection.
• Historical records indicate that fortifications and defensive structures were often built in conjunction with these settlements to ensure security.
• The prevalence of such settlements during medieval times was a strategic response to safeguard inhabitants from both human threats and environmental adversities.

Additional Information

• Nucleated Settlements:
• Nucleated settlements are characterized by houses grouped closely together, often around a central feature like a fort, market, or temple.
• These settlements are common in regions where people need to stay united for social, economic, or security reasons.
• Historical Invasions in Bundelkhand:
• Bundelkhand faced invasions from various rulers, including Mughals, Afghans, and Marathas, leading to the need for defensive settlements.
• Fortifications like the Kalinjar Fort were built to protect inhabitants from external threats.
• Geographical Influence:
• The rugged terrain of Bundelkhand provided natural barriers but also isolated communities, making compact settlements a necessity.
• Limited agricultural potential in the region further emphasized the need for cooperative living in nucleated clusters.
• Defensive Architecture:
• Structures like forts, watchtowers, and walls were integral to nucleated settlements, ensuring collective security.
• These architectural features are still evident in Bundelkhand's historical sites.

The correct answer is option 3,

Key Points

• The Perform, Achieve, and Trade (PAT) scheme is a market-based mechanism introduced by the Bureau of Energy Efficiency (BEE) under the National Mission on Enhanced Energy Efficiency (NMEEE).
• It aims to improve energy efficiency in energy-intensive industries by setting specific energy consumption reduction targets.
• Industries that achieve energy efficiency targets earn tradable Energy Saving Certificates (ESCerts), which can be sold to entities that fail to meet their targets.
• The PAT scheme covers sectors like cement, power, steel, pulp and paper, textiles, and others, making it a key component of India's energy efficiency strategy.
• The scheme helps India balance industrial growth with energy efficiency and reduction in greenhouse gas emissions, aligning with sustainable development goals.

Additional Information

• Energy Saving Certificates (ESCerts):
• These certificates are awarded to industries that exceed their energy savings targets under the PAT scheme.
• ESCerts can be traded on power exchanges, incentivizing industries to adopt energy-efficient technologies.
• National Mission on Enhanced Energy Efficiency (NMEEE):
• NMEEE is one of the eight missions under India's National Action Plan on Climate Change (NAPCC).
• Its goal is to promote energy efficiency and reduce the energy intensity of India's economy.
• Market-based Mechanism:
• The PAT scheme is an example of a market-based approach to address environmental challenges.
• It incentivizes industries to achieve energy efficiency through tradable certificates rather than penalties.
• Impact of PAT Scheme:
• Since its inception in 2012, the PAT scheme has led to significant energy savings and reduction in carbon emissions.
• It has helped industries adopt advanced technologies and improve competitiveness in global markets.

Given number: 7521432

According to the question, 

The sum of digits which are second from left and third from right in the new number thus formed → 4 + 5 = 9

Hence, the correct answer is "Option 1".

The logic followed here is:  

Option 1) LSO

Option 2) JQM

​

Option 3) GNI

Option 4) ELH

Thus, out of all 'GNI' is the odd one out.

Hence, "Option 3" is the correct answer.

Given: (Left) 307 485 194 237 937 (Right)

According to the question:

So,

Third digit of the highest number → 937 → 7

First digit of the lowest number → 194 → 1

Thus, if the third digit of the highest number is subtracted from the first digit of the lowest number then the resultant will be → 1 - 7 → -6

Hence, the correct answer is "Option 2".

The correct answer is option 3,

The correct answer is F2.

Key Points

• F2 is the default keyboard shortcut in MS PowerPoint 365 to start editing text inside a selected shape or text box.
• When a shape or text box is selected, pressing F2 directly activates the editing mode, allowing users to modify the text content.
• This shortcut is specifically designed for convenience and saves time compared to manually double-clicking on the shape or text box.
• F2 is widely recognized across different versions of PowerPoint, making it a universal shortcut for text editing.
• Using keyboard shortcuts like F2 significantly enhances productivity and workflow efficiency while working with presentations.

Additional Information

• Keyboard Shortcuts in MS PowerPoint:
• Ctrl + M: Adds a new slide to the presentation.
• Ctrl + D: Duplicates the selected slide.
• Alt + Shift + Up/Down Arrow: Moves slides up or down in the slide order.
• Ctrl + P: Activates the pen tool while presenting in slideshow mode.
• Ctrl + Z: Undoes the last action.
• Text Editing in PowerPoint:
• Text boxes and shapes are primary tools for adding content to slides.
• Double-clicking or using F2 allows users to quickly access the text-editing mode.
• Formatting options like font size, color, and alignment can be adjusted using the toolbar or ribbon.
• Keyboard shortcuts like Ctrl + B (bold) and Ctrl + I (italic) enhance text formatting efficiency.
• Productivity Benefits of Shortcuts:
• Keyboard shortcuts streamline workflows by reducing dependency on mouse navigation.
• They help users perform repetitive tasks faster, enhancing overall productivity.
• Memorizing key shortcuts can significantly improve presentation creation time.
• PowerPoint Versions:
• MS PowerPoint 365 is the latest subscription-based version, offering regular updates and cloud-based features.
• Older versions like PowerPoint 2016 and PowerPoint 2019 support many of the same shortcuts.
• Shortcuts remain consistent across versions, ensuring ease of use for long-time users.

Shortcut Trick

Total Work (LCM of 9, 21, 6) = 126 units.

Efficiency: Mohan = 14, Pramod = 6, All three (Mohan+Pramod+Akshay) = 21.

Akshay's Efficiency = 21 − (14 + 6) = 1 unit.

Time for Akshay alone to do twice the work = (126 × 2) ÷ 1 = 252 hours.

∴ The correct answer is 252 hours.

Alternate Method

Given: Mohan = 9 hours, Pramod = 21 hours, Mohan + Pramod + Akshay = 6 hours.

Formula Used: Total Work = Time × Efficiency

⇒ Total efficiency of M + P + A = 21

⇒ Akshay efficiency = 21 − (14 + 6) = 1

⇒ Total work to be done = 2 × 126 = 252 units

⇒ Time taken by Akshay = 252 ÷ 1 = 252 hours

∴ The correct answer is 252 hours.

Additional Information

Individual Efficiency Rule

The efficiency of an individual is found by subtracting the sum of efficiencies of known members from the combined efficiency of the group.

Work and Time Relation

Time is inversely proportional to Efficiency (Time = Work ÷ Efficiency). If work is doubled, time taken at the same efficiency also doubles.

The correct answer is Summing values from a specific cell across a sequence of identically structured worksheets representing monthly data for a year.

Key Points

• A three-dimensional (3D) reference in MS Excel is used to reference the same cell or range of cells across multiple worksheets within the same workbook.
• It is particularly useful for performing calculations (e.g., summing, averaging) that involve data stored in the same cell position across identically structured worksheets.
• The correct scenario here—"Summing values from a specific cell across a sequence of identically structured worksheets representing monthly data for a year"—requires referencing the same cell (e.g., B2) across worksheets such as "Jan," "Feb," "Mar," etc.
• An example of a 3D reference formula for summing cell B2 across worksheets "Jan" to "Dec" would be: =SUM(Jan:Dec!B2).
• This method simplifies complex calculations and avoids manually navigating to each worksheet, making it efficient for managing large datasets.

Additional Information

• 3D Reference in MS Excel
• A 3D reference allows users to refer to the same range or cell across multiple worksheets.
• It is denoted by the syntax: Sheet1:SheetN!CellReference.
• Common functions used with 3D references include SUM, AVERAGE, COUNT, MAX, and MIN.
• Conditional Formatting vs. 3D References
• Conditional formatting applies visual formatting rules to data within a single worksheet or across specific ranges, but it does not use 3D references.
• 3D references are designed for computations rather than formatting tasks.
• Identically Structured Worksheets
• These are worksheets that share the same layout, such as having the same rows, columns, and headings.
• Identical structure is a prerequisite for effectively using 3D references.
• Advantages of 3D References
• Enables efficient data analysis across multiple worksheets.
• Reduces redundancy by avoiding repetitive formulas for each worksheet.
• Improves consistency and minimizes human error in calculations.
• Limitations of 3D References
• 3D references cannot be used in array formulas or with certain Excel features like Data Tables.
• Deleting or rearranging worksheets within the referenced range may lead to formula errors.

The correct answer is option 4,On the eve of independence, India was characterized by poor healthcare and literacy outcomes due to colonial exploitation. Below are key points and additional information to understand this further:

Key Points

• Life expectancy at birth during India’s independence (1947) was estimated to be around 32 years, which is considered extremely low compared to global standards.
• Birth rates and death rates were both high, indicative of poor healthcare infrastructure and lack of access to medical facilities.
• Infant mortality rates were alarming, with nearly 150-200 infant deaths per 1,000 live births.
• Literacy rates were abysmally low, with only around 12% of the population being literate, reflecting widespread educational deprivation.
• The colonial government focused primarily on revenue generation and military needs, leaving public health and education severely neglected.

Additional Information

• Life Expectancy
• Life expectancy serves as an indicator of the general health and living conditions in a country.
• India’s low life expectancy at independence reflected widespread poverty, malnutrition, and limited access to healthcare.
• High Birth and Death Rates
• High birth rates were driven by social and economic factors such as lack of contraception and the need for large families in agrarian societies.
• High death rates were caused by rampant diseases like malaria, cholera, and tuberculosis, compounded by poor sanitation and healthcare.
• Infant Mortality
• Infant mortality refers to the death of children under one year of age per 1,000 live births.
• India’s high infant mortality rate at independence underscored the lack of maternal and child healthcare services.
• Colonial Impact on Education
• Colonial policies largely ignored mass education, focusing instead on creating a small class of elites for administrative purposes.
• The absence of widespread educational initiatives led to the low literacy rates observed at independence.
• Post-Independence Reforms
• After independence, India embarked on initiatives to improve healthcare and education, such as the establishment of primary schools and public health programs.
• These efforts contributed to gradual improvements in life expectancy and literacy rates over the decades.

The correct answer is option 2,

Key Points

• Forest laws and land settlements introduced by the British led to the alienation of tribal lands and disruption of traditional tribal practices.
• The British declared vast areas of forests as reserved or protected, prohibiting tribal communities from practicing shifting cultivation, hunting, and gathering.
• Land settlements imposed by colonial authorities forced tribals into exploitative relationships with moneylenders and landlords, further marginalizing them.
• These laws deprived tribes of their access to resources, leading to widespread discontent, loss of livelihood, and eventual uprisings such as the Santhal Rebellion (1855-56) and Munda Rebellion (1899-1900).
• Tribal revolts were fueled by oppressive policies that disregarded their traditional rights and cultural practices, making British forest laws a major factor behind tribal unrest.

Additional Information

• Santhal Rebellion (1855-56): A significant tribal uprising against exploitative landlords and British policies affecting tribal lands and resources.
• Munda Rebellion (1899-1900): Led by Birsa Munda, this revolt was against British land policies and the exploitation of tribal populations.
• Shifting Cultivation: A traditional agricultural practice of many tribal communities, which was banned under British forest laws.
• Impact of Forest Laws: These laws restricted access to forests for hunting, gathering, and cultivation, leading to economic hardships for tribal communities.
• Doctrine of Lapse: Although unrelated to tribal unrest, this British policy annexed princely states, further consolidating colonial control in India.

The logic followed here is:  

Option 1) HJD

Option 2) PLO

​

Option 3) PRL

Option 4) IKE

Thus, out of all 'PLO' is the odd one out.

Hence, "Option 2" is the correct answer.

According to the given information,

Thus, 'K' is coded as '8'.

Hence, "Option 1" is the correct answer.

The correct answer is option 1.

Key Points

• The Dandi March was a non-violent protest led by Mahatma Gandhi as part of the Civil Disobedience Movement.
• The march began on 12^th March 1930 from Sabarmati Ashram near Ahmedabad and ended at Dandi, a coastal village in Gujarat, covering a distance of 241 miles.
• The primary objective of the march was to protest against the British salt monopoly and the imposition of tax on salt.
• The Dandi March is considered a pivotal event in India's struggle for independence as it mobilized mass participation.
• Statement (b) regarding the Poona Pact is incorrect because the Poona Pact was signed in 1932, unrelated to the Dandi March.

Additional Information

• Civil Disobedience Movement:
• Launched by Mahatma Gandhi in 1930 as a non-violent campaign against British rule.
• Focused on breaking unjust laws, such as the salt tax, and boycotting British goods and institutions.
• The movement significantly contributed to raising national consciousness and mass participation.
• Salt Tax:
• Imposed by the British colonial government, forcing Indians to purchase salt from British-controlled sources.
• Seen as exploitative and symbolized the broader economic injustices of colonial rule.
• Poona Pact:
• An agreement signed in 1932 between Mahatma Gandhi and Dr. B.R. Ambedkar regarding the reservation of seats for depressed classes in the legislature.
• It is unrelated to the Dandi March and occurred two years later.
• Impact of Dandi March:
• Inspired civil disobedience campaigns across India against various colonial laws.
• Highlighted the power of non-violent protest in challenging oppressive regimes.

Given number: 7415385

According to the question, 

The digits that will remain unchanged as compared to that in the original number → None

Hence, the correct answer is "Option 4".

The correct answer is the Union Council of Ministers.

Key Points

• The President of India, as the head of the state, exercises executive powers, including the removal of officials like Governors and Ambassadors, based on the advice of the Union Council of Ministers.
• This advisory role of the Union Council of Ministers is mandated by Article 74(1) of the Indian Constitution, which states that the President must act in accordance with their advice.
• The Union Council of Ministers is headed by the Prime Minister, who communicates decisions and advice to the President on behalf of the council.
• The removal or recall of Governors and Ambassadors is an executive action that reflects the collective will of the council in alignment with constitutional provisions.
• The President acts as a constitutional authority, and their actions are bound by the advice provided by the democratically elected government, ensuring parliamentary democracy.

Additional Information

• Article 74 of the Indian Constitution:
• It mandates the President to act on the advice of the Council of Ministers with the Prime Minister as its head.
• The advice tendered by the council is binding on the President.
• This ensures the functioning of the parliamentary system where the real executive authority lies with the council.
• Role of Governors in India:
• Governors serve as the constitutional head of a state, appointed by the President.
• They hold office during the pleasure of the President and can be removed based on the Union Council of Ministers' advice.
• Role of Ambassadors in India:
• Ambassadors represent India in foreign countries and act as the official diplomatic channel.
• They are appointed and can be recalled by the President based on the advice of the Union government.
• Parliamentary Democracy in India:
• India follows a parliamentary system where the President acts as the nominal executive and the Prime Minister-led council functions as the real executive.
• The system ensures accountability and collective decision-making in governance.

Shortcut Trick

Use the direct percentage reduction formula: [r / (100 + r)] × 100

Here, r = 12%

Reduction = [12 / (100 + 12)] × 100

Reduction = [12 / 112] × 100 ≈ 10.71%

∴ The correct answer is 10.71%.

Alternate Method

Given: A’s salary is 12% more than B’s salary.

Formula Used: Percentage Less = (Difference / Higher Value) × 100

⇒ Let B’s salary = 100

⇒ A’s salary = 100 + 12% of 100 = 112

⇒ Difference = 112 − 100 = 12

⇒ Percentage by which B is less than A = (12 / 112) × 100

⇒ 3/28 × 100 = 300 / 28 = 75 / 7

⇒ 10.7142...

∴ The correct answer is 10.71%.

Additional Information

Percentage Increase/Decrease

If A is R% more than B, then B is [R / (100 + R)] × 100% less than A.

Inverse Case

If A is R% less than B, then B is [R / (100 − R)] × 100% more than A.

Fraction Equivalents

12% = 3/25. If B is 25, A is 25+3=28. B is 3/28 less than A. 1/28 ≈ 3.57%, so 3/28 ≈ 10.71%.

Shortcut Trick

Total distance = 2742 m, Time = 457 seconds.

Relative speed = Distance ÷ Time = 2742 ÷ 457 = 6 m/s.

In opposite directions, Relative speed = Speed₁ + Speed₂.

6 = 4.6 + X ⇒ X = 6 − 4.6 = 1.4 m/s.

∴ The correct answer is 1.4 m/s.

Alternate Method

Given:

Length of circular path (D) = 2742 m

Speed of Riya (S₁) = 4.6 m/s

Speed of Trisha (S₂) = X m/s

Time taken to meet (T) = 457 s

Formula Used:

Relative Speed (opposite direction) = S₁ + S₂

Time = Distance ÷ Relative Speed

Calculations:

⇒ 457 = 2742 ÷ (4.6 + X)

⇒ 4.6 + X = 2742 ÷ 457

⇒ 4.6 + X = 6

⇒ X = 6 − 4.6

⇒ X = 1.4 m/s

∴ The correct answer is 1.4 m/s.

Additional Information

Relative Speed in Circular Motion

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂). If they move in the same direction, it is the difference |S₁ − S₂|.

First Meeting on a Circular Track

The time taken for the first meeting is given by T = L ÷ (S₁ ± S₂), where L is the length of the track.

Meeting at the Starting Point

The time taken to meet at the starting point for the first time is the LCM of the times taken by each person to complete one full lap: LCM(L/S₁, L/S₂).

Shortcut Trick

Calculate Total Frequency (N) = 11 + 16 + 14 + 29 + 40 + 21 + 17 = 148.

Find Median Position = N/2 = 148/2 = 74.

Identify Median Class: Cumulative frequencies are 11, 27, 41, 70, 110... Since 74 falls in the 110 group, the class is 60-70.

Apply Interpolation: Median = 60 + [(74 − 70) / 40] × 10 = 60 + [4 / 40] × 10 = 60 + 1 = 61.

∴ The correct answer is 61 years.

Alternate Method

Given: Frequency distribution of patients' ages.

Formula Used: Median = l + [((N/2) − cf_p) / f] × h

Where: l = lower limit of median class, N = total frequency, cf_p = cumulative frequency of preceding class, f = frequency of median class, h = class width.

Calculations:

⇒ Cumulative Frequencies (cf):

20-30: 11

30-40: 11 + 16 = 27

40-50: 27 + 14 = 41

50-60: 41 + 29 = 70 (cf_p)

60-70: 70 + 40 = 110 (Median Class)

70-80: 110 + 21 = 131

80-90: 131 + 17 = 148 (N)

⇒ N/2 = 148 / 2 = 74

⇒ Median Class is 60-70 because its cf (110) is just greater than 74.

⇒ l = 60, cf_p = 70, f = 40, h = 10

⇒ Median = 60 + [(74 − 70) / 40] × 10

⇒ Median = 60 + [4 / 40] × 10

⇒ Median = 60 + [1 / 10] × 10

⇒ Median = 60 + 1 = 61

∴ The correct answer is 61 years.

Additional Information

Empirical Relationship

The relationship between Mean, Median, and Mode is given by: Mode = 3 Median − 2 Mean.

Median for Ungrouped Data

If n is odd, Median = ((n+1)/2)^th term. If n is even, Median = Average of (n/2)^th and (n/2 + 1)^th terms.

Class Mark

The Class Mark (x_i) is the mid-value of a class interval: (Upper Limit + Lower Limit) ÷ 2.

The logic followed here is:

Hence, "Option 4" is the correct answer.

Given:

1. S is the father of W.

2. D is the only sister of W.

3) R is the mother's mother of D. 

4) L, the mother of P, is married to S.

Thus, P is W's brother.

Hence, "Option 3" is the correct answer.

Shortcut Trick

Let Cost Price (CP) be 100.

Marked Price (MP) = 100 + 70% of 100 = 170.

Selling Price (SP) = 100 + 53% of 100 = 153.

Discount = MP − SP = 170 − 153 = 17.

Discount % (X) = (Discount ÷ MP) × 100 = (17 ÷ 170) × 100 = 10%.

∴ The correct answer is 10.

Alternate Method

Given → Markup % = 70%, Profit % = 53%.

Formulas: 100 + Profit% = (100 + Markup%) × (100 − Discount%) ÷ 100

⇒ 100 + 53 = (100 + 70) × (100 − X) ÷ 100

⇒ 153 = 170 × (100 − X) ÷ 100

⇒ 1530 = 17 × (100 − X)

⇒ 90 = 100 − X

⇒ X = 10

∴ The correct answer is 10.

Additional Information

Relationship between MP and CP

The ratio of Marked Price to Cost Price is given by MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Successive Percentage Change

Profit% can be calculated using successive formula: Profit% = Markup% − Discount% − (Markup × Discount) ÷ 100.

The correct answer is option 3,

Key Points

• Prabhakaravardhana was the first notable king of the Pushyabhuti dynasty and ruled between 580-605 CE.
• He is remembered for consolidating the dynasty's power and expanding its territorial influence in northern India.
• Prabhakaravardhana was the father of Harsha Vardhana, who later became one of the most celebrated rulers of ancient India.
• Under Prabhakaravardhana's reign, the Pushyabhuti dynasty established its capital at Thanesar (modern-day Haryana).
• Prabhakaravardhana strengthened the military and administration of the dynasty, laying the foundation for its future prominence.

Additional Information

• Pushyabhuti Dynasty:
• The Pushyabhuti dynasty was an ancient Indian dynasty that ruled in northern India during the early medieval period.
• Its capital was located at Thanesar, a prominent city in modern-day Haryana.
• The dynasty gained prominence under Prabhakaravardhana and reached its zenith under Harsha Vardhana.
• Harsha Vardhana:
• Harsha Vardhana, son of Prabhakaravardhana, succeeded him and ruled from 606-647 CE.
• Harsha expanded the empire significantly and is known for his patronage of arts and Buddhism.
• He authored notable works like ""Nagananda,"" a Sanskrit drama.
• Thanesar:
• Thanesar was a historic city and the capital of the Pushyabhuti dynasty.
• It was an important center of politics and religion in ancient India.
• Thanesar is associated with various historical and archaeological sites today.
• Administration under Prabhakaravardhana:
• Prabhakaravardhana focused on strengthening the military and administrative systems of the dynasty.
• He laid the groundwork for the expansion of the dynasty under his successors.

Shortcut Trick

Sum of interior angle (I) and exterior angle (E) is always 180°.

Given: I − E = 140° and we know I + E = 180°.

Subtracting the equations: (I + E) − (I − E) = 180° − 140°.

2E = 40° ⇒ E = 20°.

Number of sides (n) = 360° ÷ Exterior Angle = 360° ÷ 20° = 18.

∴ The correct answer is 18.

Alternate Method

Given: The difference between an interior angle and an exterior angle of a regular polygon is 140°.

Formula Used: Interior angle (I) + Exterior angle (E) = 180°; Number of sides (n) = 360° ÷ E.

Calculations:

⇒ I − E = 140° ... (Equation 1)

⇒ I + E = 180° ... (Equation 2)

⇒ Subtracting Eq. 1 from Eq. 2:

⇒ (I + E) − (I − E) = 180° − 140°

⇒ 2E = 40°

⇒ E = 20°

⇒ n = 360° ÷ 20°

⇒ n = 18

∴ The correct answer is 18.

Given: In a row of 22 students facing north, Suman is 14^th from the left end. If Dipa is 2^nd to the right of Suman.

Suman's position from the left end = 14^th

Dipa's position from the left end = 14 + 2 → 16^th

Dipa's position from the right end = Total students - Dipa's position from the left end + 1

= 22 - 16 + 1 → 7^th

Hence, "Option 4" is the correct answer.

Shortcut Trick

Let initial Cost of Production be 100.

Initial Selling Price = 100 × 1.30 = 130.

New Cost of Production = 100 × 1.30 = 130.

New Selling Price = 130 × 1.70 = 221.

Profit = 221 − 130 = 91. Profit% = (91 ÷ 130) × 100 = 70%.

∴ The correct answer is 70%.

Alternate Method

Given → Initial CP = 100, Initial Profit = 30%, Increase in CP = 30%, Increase in SP = 70%. Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100 | SP = CP × (100 + P%) ÷ 100.

⇒ Assume initial Cost Price (CP₁) = 100. ⇒ Selling Price (SP₁) = 100 + 30% of 100 = 130. ⇒ New Cost Price (CP₂) = 100 + 30% of 100 = 130. ⇒ New Selling Price (SP₂) = 130 + 70% of 130 = 130 + 91 = 221. ⇒ New Profit = SP₂ − CP₂ = 221 − 130 = 91. ⇒ Profit Percentage = (91 ÷ 130) × 100 = 70%.

∴ The correct answer is 70%.

Additional Information

Profit Percentage on Cost Price

The Profit Percentage is always calculated on the Cost Price unless stated otherwise. Formula: P% = (P ÷ CP) × 100.

Effect of Multiplier

When a value increases by x%, it becomes (100 + x)% of its original value, which can be represented by the multiplier (1 + x÷100).

The correct answer is option 3,

Key Points

• Sound waves are mechanical waves, meaning they require a medium to travel through and cannot propagate in a vacuum.
• An elastic medium is necessary for the transmission of sound, as it allows the particles within the medium to oscillate and transfer energy between them.
• Vibration is the primary mechanism for the generation of sound waves, as oscillating objects create disturbances in the surrounding elastic medium.
• Examples of elastic media include air, water, and solid materials, which facilitate the propagation of sound waves.
• Without both vibration and an elastic medium, sound waves cannot be generated or transmitted effectively.

Additional Information

• Mechanical Waves: Sound waves fall under the category of mechanical waves, which require a material medium to propagate.
• Elastic Medium: An elastic medium is a material that returns to its original shape after being deformed, allowing energy transfer through particle oscillation.
• Vibration: Vibration refers to the periodic motion of particles about an equilibrium position, which generates disturbances that propagate as sound waves.
• Speed of Sound: The speed of sound varies depending on the medium; it is fastest in solids, slower in liquids, and slowest in gases.
• Vacuum: Sound cannot propagate in a vacuum because there are no particles to oscillate and transfer the energy of the wave.

Shortcut Trick

LCM of 40, 78, and X = 7800.

Prime factorization: 7800 = 2³ × 3¹ × 5² × 13¹.

Check options for prime factors: 200 = 2³ × 5². These factors are already within the LCM prime factorization.

Testing other options: 180 has 3², which exceeds 3¹ in the LCM; 127 and 152 have prime factors (127, 19) not present in 7800.

∴ The correct answer is 200.

Alternate Method

Given: Numbers 40, 78 and X. LCM = 7800.

Formula Used: LCM is the product of the highest power of all prime factors present in the numbers.

Calculations:

⇒ Prime factorization of 40 = 2³ × 5¹

⇒ Prime factorization of 78 = 2¹ × 3¹ × 13¹

⇒ Prime factorization of LCM (7800) = 2³ × 3¹ × 5² × 13¹

⇒ Comparing LCM factors with 40 and 78, we see that the factor 5² is not fully satisfied (40 only has 5¹).

⇒ Therefore, X must contain 5² and its other prime factors must not exceed the powers in the LCM.

⇒ Checking Option (4): 200 = 2³ × 5². This fits perfectly as 5² is present and 2³ is within limits.

∴ The correct answer is 200.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is divisible by all the numbers in a set. It is found by taking the highest power of each prime factor.

HCF and LCM Relation

For two numbers a and b, Product of numbers = HCF(a, b) × LCM(a, b).

Prime Factorization Method

Express each number as a product of primes. LCM includes every prime factor at its maximum observed exponent.

The logic followed here is:

SAND - DSNA - SDAN

And,

GUNT - TGNU - GTUN

Similarly,

Option 4) JUNK - KJNU - JKUN

​

​

Hence, "Option 4" is the correct answer.

Calculation:

$\frac{(0.06)^2 + (0.66)^2 + (0.666)^2}{(0.006)^2 + (0.066)^2 + (0.0666)^2}$

⇒ $\frac{(0.06)^2 + (0.66)^2 + (0.666)^2}{(\frac{0.06}{10})^2 + (\frac{0.66}{10})^{2} + (\frac{0.666}{10})^2}$

⇒ $\frac{(0.06)^2 + (0.66)^2 + (0.666)^2}{\frac{1}{100}[(0.06)^2 + (0.66)^2 + (0.666)^2]}$

⇒ 100

∴ Option 4 is the correct answer.

Shortcut Trick

Income ratio (Sohit : Rohit) = 17 : 14

Expenditure ratio (Sohit : Rohit) = 12 : 8 → 3 : 2

Since savings are equal (₹3,392), make the parts difference equal: (17−14) = 3 and (3−2) = 1.

Multiply expenditure ratio by 3: 17 : 14 (Income) and 9 : 6 (Expenditure). Difference = 17 − 9 = 8 units.

8 units = ₹3,392 ⇒ 1 unit = ₹424. Sohit's money = 17 × 424 = ₹7,208.

∴ The correct answer is ₹7,208.

Alternate Method

Given: Initial ratio = 17 : 14, Expenditure ratio = 12 : 8 = 3 : 2, Saving = ₹3,392 each.

Formula Used: Income − Expenditure = Savings

⇒ Let Sohit's initial money be 17x and Rohit's be 14x.

⇒ (17x − 3392) / (14x − 3392) = 12 / 8

⇒ (17x − 3392) / (14x − 3392) = 3 / 2

⇒ 2 × (17x − 3392) = 3 × (14x − 3392)

⇒ 34x − 6784 = 42x − 10176

⇒ 42x − 34x = 10176 − 6784

⇒ 8x = 3392

⇒ x = 424

⇒ Sohit's initial money = 17 × 424 = ₹7,208

∴ The correct answer is ₹7,208.

Additional Information

Income and Savings Relationship

Income = Expenditure + Savings. If two people have equal savings, the difference in their incomes is proportional to the difference in their expenditures.

Ratio Balancing Method

When a constant value is added or subtracted from both terms of a ratio, the difference between the ratio parts must be made equal to maintain proportionality.

Shortcut Trick

Isolate the trigonometric ratios: m/a = cos³β and n/b = sin³β.

Raise both sides to the power 2/3: (m/a)^2/3 = cos²β and (n/b)^2/3 = sin²β.

Apply the Pythagorean Identity: sin²β + cos²β = 1.

∴ The correct answer is 1.

Alternate Method

Given:

m = a cos³β

n = b sin³β

Formula Used:

sin²θ + cos²θ = 1

Calculations:

⇒ m/a = cos³β

⇒ (m/a)^1/3 = cosβ

⇒ [(m/a)^1/3]² = cos²β

⇒ (m/a)^2/3 = cos²β ---- (i)

⇒ n/b = sin³β

⇒ (n/b)^1/3 = sinβ

⇒ (n/b)^2/3 = sin²β ---- (ii)

⇒ Adding (i) and (ii): (m/a)^2/3 + (n/b)^2/3 = cos²β + sin²β

⇒ Since sin²β + cos²β = 1, the result is 1.

∴ The correct answer is 1.

Additional Information

Trigonometric Identities

Fundamental identities include sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Laws of Exponents

For any base x, (x^a)^b = x^a×b. This property is used to eliminate cube powers by raising to 1/3 or 2/3.

Substitution Method

In competitive exams, you can substitute β = 0° or 90° to quickly find the value: if β = 0°, m/a = 1 and n/b = 0, result = 1 + 0 = 1.

Given: Seven boxes D, E, F, G, I, J and K are kept one over the other but not necessarily in the same order. 

1) Only three boxes are kept above G.

2) Only one box is kept between J and G. 

3) Only three boxes are kept between J and K.

4) K is kept at some place above G.

5) E is kept immediately below K.

6) F is kept at one of the places above D.

7) I is not kept immediately above or below J.

Thus, D box is kept at the lowermost position.

Hence, "Option 1" is the correct answer.

The correct answer is Members of the Lok Sabha.

Key Points

• The Estimates Committee is constituted every year by the Parliament of India.
• Members of this committee are elected exclusively by the Members of the Lok Sabha (House of the People).
• The committee consists of 30 members, all of whom are drawn from the Lok Sabha.
• Rajya Sabha (Council of States) members are not involved in the election or composition of this committee.
• The function of the Estimates Committee is to examine the estimates included in the budget and suggest economies in public expenditure.

Additional Information

• Origin of the Committee
• The Estimates Committee was first established in 1950 under the Government of India Act, 1935.
• The provisions for its formation were later incorporated into the Constitution of India.
• Key Responsibilities
• The committee examines whether the proposed expenditure is necessary and whether funds are being utilized effectively.
• It also suggests measures to ensure efficiency in administration.
• Chairperson
• The chairperson of the Estimates Committee is appointed by the Speaker of the Lok Sabha from amongst its members.
• Traditionally, the chairperson is selected from the ruling party or coalition.
• Limitations
• The committee does not have the authority to examine policy issues or demand grants.
• Its focus is restricted to scrutinizing the technical and financial aspects of the estimates.

The correct answer is Option 1) “DRAM is much faster and doesn’t need to refresh.”

Key Points

• Why Option 1 is invalid: DRAM is slower than SRAM and it does require periodic refresh to retain data. The statement reverses both facts.
• Actual DRAM cell: Typically 1 transistor + 1 capacitor (1T1C) per bit → high density, low cost, but needs refresh → higher latency than SRAM.
• Actual SRAM cell: Typically multiple transistors (≈6T) forming a bistable latch → no refresh needed, faster access, but lower density and higher cost.
• Common usage: SRAM → CPU caches (L1/L2/L3). DRAM → system main memory (DIMMs).

Additional Information

• Option 2 (Correct statement): DRAM uses 1T1C (compact, dense), while SRAM uses multiple transistors per bit (less dense).
• Option 3 (Correct statement): SRAM is faster and more expensive → used for cache; DRAM is slower and cheaper → used for main memory.
• Option 4 (Correct statement): SRAM’s multi-transistor latch enables faster access than DRAM’s capacitor-based cell.
• Performance note: Refresh operations and sense-amp charging in DRAM introduce extra delay; SRAM reads/writes directly from a stable latch.

The Correct answer is 44^th Constitutional Amendment 1978.

Key Points

• The 44^th Amendment Act was enacted to reverse many provisions of the 42^nd Amendment introduced during the Emergency.
• The 42^nd Constitutional Amendment Act 1976 is often called the “Mini Constitution” due to the large number of changes it introduced in the Indian Constitution.
• This amendment made significant changes to the Directive Principles of State Policy (DPSPs), particularly by inserting new directives under Article 39 of the Constitution.
• It added a directive for the State to secure the minimization of inequalities in income, property, facilities, and opportunities among citizens.
• The amendment emphasized social, economic, and political justice, aligning with the goals of a welfare state.
• It also added provisions related to the protection and improvement of the environment, safeguarding of forests and wildlife, and promotion of equal justice.
• The 42^nd Amendment was passed during the period of Emergency (1975–1977) under the leadership of then Prime Minister Indira Gandhi.
• Through this amendment, the words “Socialist” and “Secular” were also inserted into the Preamble of the Constitution.

Additional Information

• 97^th Constitutional Amendment 2011
• The 97^th Amendment Act dealt with matters related to co-operative societies.
• It gave constitutional status and protection to co-operatives by inserting Part IXB in the Constitution.
• The amendment also added the words “co-operative societies” in Article 19(1)(c) and inserted Article 43B as part of DPSPs.
• 86^th Constitutional Amendment 2002
• The 86^th Amendment Act made free and compulsory education a Fundamental Right for children aged 6 to 14 years.
• It inserted Article 21A into the Constitution and amended Article 45 and Article 51A.
• This was a crucial step in achieving universal elementary education in India.
• 44^th Constitutional Amendment 1978
• It restored the power of judicial review and strengthened the position of Fundamental Rights.
• It also safeguarded Article 352 by making the proclamation of Emergency more difficult.

The correct answer is option 3,Key Points:

• The Secondary sector is responsible for transforming raw materials or natural products into finished goods through manufacturing and industrial processes.
• This sector includes industries such as textiles, automobile manufacturing, steel production, cement production, and chemical manufacturing.
• The secondary sector contributes significantly to India's Gross Domestic Product (GDP), driving economic growth and industrial development.
• Key activities within this sector include construction, factories, and production units that create value-added goods.
• It plays a crucial role in employment generation, especially in urban areas, by providing jobs in manufacturing and industrial operations.

Additional Information:

• Primary Sector:
• Involves the extraction and harvesting of natural resources like agriculture, fishing, forestry, and mining.
• Forms the base of the economy and supplies raw materials to the secondary sector.
• Tertiary Sector:
• Also known as the service sector, it provides services like banking, education, healthcare, and retail.
• It supports the primary and secondary sectors by facilitating trade and offering essential services.
• Role of the Secondary Sector in India:
• Contributes to industrialization and urbanization, leading to infrastructure development.
• Promotes export-oriented industries, enhancing foreign exchange reserves.
• Economic Impact:
• The secondary sector is vital for creating a balanced economy by reducing dependency on agriculture.
• It helps in improving living standards through the production of essential and luxury goods.

The logic followed here is:

Given: # : RMN :: TOP : %  

Option 1) #=MOU, % : YLU → MOU : RMN :: TOP : YLU  

Option 2) # NHI, % : YOK → NHI : RMN :: TOP : YOK

​

Option 3) # : KJI, % = YMU → KJI : RMN :: TOP : YMU

Option 4) #= MHI, % = YTU → MHI : RMN :: TOP : YTU

Hence, "Option 4" is the correct answer.

Shortcut Trick

Divide the given number 22938 by 7 to find the remainder.

22938 ÷ 7 leaves a remainder of 6.

The smallest number to be added = 7 − Remainder = 7 − 6 = 1.

∴ The correct answer is 1.

Alternate Method

Given:

The number to be made divisible is 22938.

The divisor is 7.

Formula Used:

Required Number to be added = Divisor − Remainder

Calculations:

⇒ 22938 ÷ 7

⇒ 22938 = 7 × 3276 + 6

⇒ Remainder = 6

⇒ Required smallest number to be added = 7 − 6

⇒ Required smallest number to be added = 1

∴ The correct answer is 1.

Additional Information

Divisibility Rule of 7

Double the last digit of the number and subtract it from the remaining number. If the result is divisible by 7, then the original number is also divisible by 7.

Smallest Number to Subtract

To make a number completely divisible by a divisor, the smallest number that should be subtracted is equal to the Remainder itself.

Smallest Number to Add

To make a number completely divisible by a divisor, the smallest number that should be added is equal to (Divisor − Remainder).

The correct answer is Hitesh Gulia.

Key Points

• Hitesh Gulia became the first Indian boxer to win a gold medal at the World Boxing Cup 2025.
• He achieved this historic milestone in the 75 kg middleweight category.
• The World Boxing Cup 2025 was held in Belgrade, Serbia, showcasing elite boxing talents from around the globe.
• Hitesh defeated Sergey Ivanov of Russia in the final match with a unanimous decision from the judges.
• This victory marked a significant achievement for Indian boxing on the international stage, reinforcing India’s growing prominence in the sport.

Additional Information

• World Boxing Cup
• It is an international boxing competition organized by the International Boxing Association (IBA).
• The tournament features top boxers from various countries competing across different weight categories.
• It serves as a platform for nations to showcase their boxing talent and gain international exposure.
• Indian Boxing Achievements
• India has a rich history in boxing, with notable achievements in the Olympics, Asian Games, and Commonwealth Games.
• Prominent Indian boxers include Mary Kom, Vijender Singh, and Lovelina Borgohain.
• Hitesh Gulia’s win adds to India’s growing legacy in international boxing events.
• International Boxing Association (IBA)
• The IBA is the world governing body for amateur boxing, formerly known as AIBA.
• It was founded in 1946 and is headquartered in Lausanne, Switzerland.
• The IBA is responsible for organizing major boxing events and maintaining rules and standards for the sport.
• Training and Preparation
• Indian boxers undergo rigorous training at facilities like the National Boxing Academy in Haryana and other Sports Authority of India (SAI) centers.
• Coaching and support from experienced trainers, alongside exposure to international competitions, play a crucial role in their success.

The correct answer is option 1,

Key Points

• Salim Chishti’s tomb is located in Fatehpur Sikri, Uttar Pradesh, India.
• Fatehpur Sikri was built by Mughal Emperor Akbar in the late 16^th century and served as his capital for a short period.
• The tomb honors Sheikh Salim Chishti, a revered Sufi saint whose blessings were believed to have led to the birth of Akbar's son, Jahangir.
• Salim Chishti’s tomb is an exquisite example of Mughal architecture, featuring intricate marble work and a highly decorative design.
• The tomb is located within the sacred complex of the Jama Masjid in Fatehpur Sikri and is considered a significant pilgrimage site.

Additional Information

• Fatehpur Sikri:
• Built between 1571 and 1585, Fatehpur Sikri is a UNESCO World Heritage Site.
• The city features several architectural marvels, including the Buland Darwaza, Jama Masjid, and Panch Mahal.
• It was abandoned due to water scarcity but remains an iconic representation of Mughal grandeur.
• Sheikh Salim Chishti:
• He was a 16^th-century Sufi saint of the Chishti order.
• Akbar sought his blessings for a male heir, leading to the birth of Jahangir.
• His tomb is a revered site attracting pilgrims from across the globe.
• Mughal Architecture:
• Mughal architecture is characterized by symmetrical designs, intricate carvings, and use of red sandstone and marble.
• Key features include domes, arches, minarets, and large courtyards.
• Prominent examples include the Taj Mahal, Red Fort, and Fatehpur Sikri.
• Jama Masjid in Fatehpur Sikri:
• One of the largest mosques in India, built by Akbar in 1571.
• It serves as a central structure within the Fatehpur Sikri complex.
• Its design integrates Persian and Indian architectural styles.

Shortcut Trick

Mean proportional between a and b is √(a × b).

Product = (9 + 3√7) × (18 − 6√7) = (9 + 3√7) × 2(9 − 3√7).

Using (x + y)(x − y) = x² − y² ⇒ 2 × (9² − (3√7)²) = 2 × (81 − 63) = 2 × 18 = 36.

Mean proportional = √36 = 6.

∴ The correct answer is 6.

Alternate Method

Given: First value (a) = 9 + 3√7, Second value (b) = 18 − 6√7.

Formula Used: Mean Proportional (M.P.) = √(a × b)

⇒ M.P. = √[(9 + 3√7) × (18 − 6√7)]

⇒ M.P. = √[(9 + 3√7) × 6(3 − √7)]

⇒ M.P. = √[3(3 + √7) × 6(3 − √7)]

⇒ M.P. = √[18 × (3² − (√7)²)]

⇒ M.P. = √[18 × (9 − 7)]

⇒ M.P. = √[18 × 2]

⇒ M.P. = √36

⇒ M.P. = 6

∴ The correct answer is 6.

Additional Information

Third Proportional

For two numbers a and b, the third proportional is calculated as c = b² ÷ a.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d satisfies the ratio a:b = c:d, so d = (b × c) ÷ a.

Mean Proportional Concept

If x is the mean proportional between a and b, then a:x = x:b, which results in x² = ab.

The correct answer is option 3,"

Key Points

• Windows operating systems, including Windows 11, prohibit the use of certain special characters in file and folder names.
• Characters such as asterisks (*), question marks (?), colons (:), slashes (/ \), angle brackets (< >), pipes (|), quotation marks (""), and others are reserved for system use and cannot be included in file or folder names.
• These restrictions are implemented to avoid conflicts with the file system and ensure proper functioning of the operating system.
• Hyphens (-), underscores (_), letters, and numbers are allowed in file and folder names, as they do not interfere with system processes.
• Attempting to use restricted characters will result in an error message from Windows, preventing the creation or renaming of the file or folder.

Additional Information

• File System in Windows:
• Windows primarily uses the NTFS (New Technology File System), which governs how data is stored and retrieved on a disk.
• NTFS supports features like file permissions, encryption, and large file sizes.
• It enforces character restrictions to maintain system integrity and compatibility.
• Reserved Characters:
• Characters like asterisks (*) and question marks (?) are used as wildcards in search and command-line operations.
• Other reserved characters include angle brackets (< >) for redirection and pipes (|) for command chaining.
• File Naming Best Practices:
• Use descriptive names to make files easier to identify.
• Avoid excessively long file names, as some applications may face issues handling them.
• Stick to alphanumeric characters, underscores (_), and hyphens (-) for compatibility across different systems.
• System Reserved Names:
• Windows prohibits file and folder names like CON, PRN, AUX, NUL, and others, as they are reserved for system devices.
• These restrictions apply regardless of file extension.

"

Shortcut Trick

Identify half of the coefficients of x and y: a = 10/2 = 5 and b = 26/2 = 13.

Check if a² + b² equals the constant: 5² + 13² = 25 + 169 = 194.

Since the sum matches, the equation is (x − 5)² + (y + 13)² = 0.

This directly gives x = 5 and y = −13.

Calculating x² + y² = 5² + (−13)² = 194.

∴ The correct answer is 194.

Alternate Method

Given: x² + y² − 10x + 26y + 194 = 0

Formula Used: (a − b)² = a² − 2ab + b² and (a + b)² = a² + 2ab + b²

⇒ Group the terms: (x² − 10x) + (y² + 26y) + 194 = 0

⇒ Complete the square for x: (x² − 2 × 5 × x + 5²) − 25

⇒ Complete the square for y: (y² + 2 × 13 × y + 13²) − 169

⇒ Substitute back: (x − 5)² − 25 + (y + 13)² − 169 + 194 = 0

⇒ (x − 5)² + (y + 13)² − 194 + 194 = 0

⇒ (x − 5)² + (y + 13)² = 0

⇒ Since the sum of squares of real numbers is zero, each term must be zero.

⇒ x − 5 = 0 ⇒ x = 5

⇒ y + 13 = 0 ⇒ y = −13

⇒ Value of x² + y² = 5² + (−13)²

⇒ 25 + 169 = 194

∴ The correct answer is 194.

Additional Information

Sum of Squares Property

If the sum of squares of multiple real expressions is zero, i.e., A² + B² + C² = 0, then A = 0, B = 0, and C = 0 individually.

Perfect Square Trinomial

A quadratic expression ax² + bx + c is a perfect square if its discriminant D = b² − 4ac = 0.

Completing the Square Technique

To make x² + bx a perfect square, we add and subtract (b/2)², resulting in (x + b/2)² − (b/2)².

The correct answer is ICAR (Indian Council of Agricultural Research).

Key Points

• ICAR (Indian Council of Agricultural Research) is an autonomous organization under the Department of Agricultural Research and Education (DARE), Ministry of Agriculture and Farmers Welfare, Government of India.
• It played a pivotal role in the success of the Green Revolution by spearheading agricultural research and promoting the use of High Yielding Variety (HYV) seeds in India.
• ICAR was instrumental in collaborating with international organizations like the International Rice Research Institute (IRRI) and CIMMYT (International Maize and Wheat Improvement Center) for the transfer and adaptation of HYV seeds.
• The organization established a strong network of agricultural research institutions and state agricultural universities (SAUs) to disseminate new technologies and practices to farmers.
• ICAR's coordinated efforts helped achieve self-sufficiency in food grain production during the 1960s and 1970s, a hallmark of the Green Revolution in India.

Additional Information

• Green Revolution:
• It refers to a period of agricultural transformation in India during the 1960s-70s, marked by increased food grain production due to the adoption of HYV seeds, chemical fertilizers, and irrigation techniques.
• Dr. M.S. Swaminathan, often referred to as the "Father of the Green Revolution in India," played a key role in introducing these agricultural reforms.
• The Green Revolution primarily focused on crops such as wheat and rice, with significant success in states like Punjab, Haryana, and Uttar Pradesh.
• High Yielding Variety (HYV) Seeds:
• These seeds are genetically engineered to produce higher yields compared to traditional varieties.
• They require proper inputs, including adequate irrigation, fertilizers, and pesticides, to achieve their full potential.
• Role of International Organizations:
• The International Rice Research Institute (IRRI) in the Philippines developed the semi-dwarf rice variety IR8, which contributed to increased rice production in India.
• CIMMYT in Mexico developed high-yielding and disease-resistant wheat varieties that were adapted for Indian conditions.
• Challenges of the Green Revolution:
• While it boosted food grain production, it led to environmental issues such as soil degradation, water scarcity, and overuse of chemical fertilizers and pesticides.
• The benefits were unevenly distributed, with some regions and crops receiving more focus than others.
• Current Role of ICAR:
• ICAR continues to contribute to agricultural research, focusing on sustainable practices, climate-resilient crops, and biotechnology to meet modern challenges.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

(I) Some figs are not eagles. → Follow (As all tanks are figs and no eagles are tanks. As whole tanks comes in figs and cannot be eagles, therefore is true)

(II) Some mugs are not tanks → Follow (As no eagles are tanks and some mugs are eagles. As no tanks can be eagles which has some part common with mugs, therefore it is true.)

∴ Here, Both conclusions (I) and (II) follow.​

Hence, the correct answer is "Option 3".

The correct answer is Trade, defense, clean energy, and connectivity.

Key Points

• In April 2025, India and Italy advanced their strategic partnership, focusing on trade, defense, clean energy, and connectivity.
• The collaboration emphasizes strengthening bilateral trade, with both nations aiming to boost economic exchanges and enhance market access.
• Defense cooperation includes joint military exercises, technology sharing, and defense manufacturing under India’s ‘Make in India’ initiative.
• Clean energy collaboration focuses on renewable energy, particularly in the areas of solar, wind, and hydrogen energy, to combat climate change.
• Connectivity initiatives highlight projects like transport infrastructure, smart cities, and digital innovation to improve economic integration between the two countries.

Additional Information

• India-Italy Bilateral Relations:
• India and Italy enjoy a robust relationship based on shared democratic values, cultural ties, and economic interests.
• Italy is one of India’s largest trading partners in the European Union, with bilateral trade crossing significant milestones in recent years.
• Defense Cooperation:
• The partnership includes joint military exercises, defense R&D, and technology-sharing agreements.
• Italy has expressed interest in collaborating on India’s defense manufacturing programs, aligning with the ‘Make in India’ initiative.
• Clean Energy Initiatives:
• India and Italy are committed to achieving sustainable development through renewable energy projects.
• Both nations are part of international frameworks like the International Solar Alliance (ISA) to promote clean energy adoption.
• Connectivity and Digital Innovation:
• Connectivity efforts include enhancing infrastructure in sectors like transport and smart cities.
• Digital innovation focuses on AI, IoT, and blockchain technologies to boost economic and technological integration.

Given: Anu, Beena, Deepa, Heena, Kareena and Reena sit in a straight line, facing north.

1) Kareena sits third to the right of Beena, who sits second to the right of Deepa.

2) Reena sits to the immediate left of Leena.

3) Anu sits second to the left of Heena.

Thus, according to the final arrangement:

Option 1) Leena and Kareena → Immediate neighbours 

Option 2) Anu and Deepa → Immediate neighbours

Option 3) Beena and Reena → Immediate neighbours

Option 4) Heena and Reena → Heena sits second to the left of Renna

Hence, "Option 4" is the correct answer.

Shortcut Trick

Since A varies directly as B, the ratio A/B remains constant.

Direct proportion logic: A₁ ÷ B₁ = A₂ ÷ B₂

210 ÷ 51 = A₂ ÷ 136

A₂ = (210 × 136) ÷ 51 = 210 × (8 ÷ 3) × 1 = 70 × 8 = 560

∴ The correct answer is 560.

Alternate Method

Given: A ∝ B, A₁ = 210, B₁ = 51, B₂ = 136.

Formula Used: A = k × B, where k is the constant of proportionality.

⇒ 210 = k × 51

⇒ k = 210 ÷ 51 = 70 ÷ 17

⇒ A = (70 ÷ 17) × 136

⇒ A = 70 × 8

⇒ A = 560

∴ The correct answer is 560.

Additional Information

Direct Variation

When one quantity increases, the other increases in the same ratio. Formula: y = kx or y₁/x₁ = y₂/x₂.

Inverse Variation

When one quantity increases, the other decreases in the same ratio. Formula: xy = k or x₁y₁ = x₂y₂.

Constant of Proportionality

The fixed ratio 'k' between two proportional quantities that defines their relationship.

Given:

Mr. RST starts from Point A and drives 16 km towards the south.

He then takes a left turn, drives 6 km, turns left and drives 23 km.

He then takes a left turn and drives 8 km, turns left and drives 3 km.

He then takes a right turn and drives 4 km.

He takes a final left turn, drives 4 km and stops at Point A. 

Thus, Mr. RST should drive 6 km (4 km + 8 km - 6 km) in the east direction in order to reach Point A again.​

Hence, "Option 2" is the correct answer.

The logic followed here is:

Hence, "Option 4" is the correct answer.

The correct answer is option 1,"Key Points: - The Prayagraj Prashasti, also known as the Allahabad Pillar Inscription, is the primary source of information about Samudragupta's reign. - It was composed by Harisena, a court poet of Samudragupta, in classical Sanskrit. - The inscription highlights Samudragupta's military campaigns, administrative achievements, and his conquests over various regions. - The Prashasti refers to Samudragupta as the 'Napoleon of India' due to his extensive military expeditions and victories. - It provides insights into the political and cultural landscape of India during the Gupta Empire. Additional Information: - Samudragupta was a ruler of the Gupta dynasty who reigned from approximately 335-380 CE and is considered one of India's greatest emperors. - The Prayagraj Prashasti is inscribed on the Ashokan pillar located in Prayagraj (Allahabad) and represents an important piece of historical evidence for the Gupta period. - Harisena, the composer of the Prashasti, was not only a poet but also served as a minister and military officer under Samudragupta. - The inscription follows the tradition of eulogizing the king, a common practice during ancient Indian dynasties. - The Gupta Empire under Samudragupta witnessed advancements in arts, science, and literature, often referred to as the 'Golden Age of India'."

The correct answer is Azathof CSI Lier.

Key Points

• The Azathof CSI Lier is an international equestrian competition held in Lier, Belgium, renowned for showcasing top-tier talent in showjumping events.
• The 2025 edition of the Azathof CSI Lier ran from March 25 to March 30, featuring competitive events across multiple categories.
• Witharika Singhania, representing India, secured the first position in one of the prestigious showjumping events during this competition.
• The Azathof CSI Lier is recognized for attracting elite riders and horses from across the globe, making it a significant event on the international equestrian calendar.
• The competition adheres to the regulations set by the International Federation for Equestrian Sports (FEI), ensuring high standards of integrity and professionalism.

Additional Information

• Equestrian Sports:
• Equestrian sports include disciplines like showjumping, dressage, and eventing, governed internationally by the FEI.
• Showjumping involves riders and their horses navigating a course of fences within a set time, testing speed, agility, and precision.
• Dressage focuses on the horse's ability to perform predefined movements, emphasizing harmony between rider and horse.
• International Federation for Equestrian Sports (FEI):
• The FEI is the global governing body for equestrian sports, founded in 1921 and headquartered in Lausanne, Switzerland.
• It oversees major competitions like the Olympics, World Equestrian Games, and other international events.
• Witharika Singhania:
• Witharika Singhania is an emerging talent in the equestrian world and a prominent Indian rider excelling in international competitions.
• Her success at the Azathof CSI Lier 2025 highlights India’s growing presence in equestrian sports on the global stage.
• Belgium’s Role in Equestrian Sports:
• Belgium is known for its rich equestrian tradition, hosting numerous international events and producing world-class riders and horses.
• Facilities like the Azathof equestrian center offer state-of-the-art amenities for training and competitions.

Given: Seven people, A, B, C, D, S, T, and U are sitting in a row, facing north. 

1) No one sits to the right of D. 

2) Only three people sit between D and S.

3) Only two people sit between S and T.

4) C sits third to the left of U.

5) A sits to the immediate right of U.

After positioning C, U and A only one position is left which will be occupied by the only person left i.e. B.

Thus, three people sit between B and T.

Hence, "Option 1" is the correct answer.

Shortcut Trick

For any quadratic equation of the form x² + bx + K = 0, the condition for equal roots is K = (b/2)².

In the given equation x² + 4x + K = 0, the value of b is 4.

Substituting the value: K = (4 ÷ 2)² = 2² = 4.

∴ The correct answer is 4.

Alternate Method

Given: The quadratic equation is x² + 4x + K = 0.

Formula Used: For a standard quadratic equation ax² + bx + c = 0, the roots are equal if and only if the discriminant (D) is zero.

The formula for the discriminant is D = b² − 4ac.

⇒ Comparing x² + 4x + K = 0 with ax² + bx + c = 0, we get:

⇒ a = 1, b = 4, c = K

⇒ For equal roots, D = 0

⇒ (4)² − 4 × 1 × K = 0

⇒ 16 − 4K = 0

⇒ 4K = 16

⇒ K = 16 ÷ 4

⇒ K = 4

∴ The correct answer is 4.

Additional Information

Nature of Roots

The value of D = b² − 4ac determines the nature of roots: D > 0 means real and distinct roots, D = 0 means real and equal roots, and D < 0 means imaginary roots.

Sum and Product of Roots

For the equation ax² + bx + c = 0, the Sum of roots (α + β) = −b/a and the Product of roots (αβ) = c/a.

Quadratic Formula

The roots of a quadratic equation are calculated using x = (−b ± √D) ÷ 2a.

Shortcut Trick

Mass = Density × Volume

Volume of shell = 4/3 × π × (R³ − r³)

Mass = 7 × (4/3 × 22/7 × (3³ − 0³)) = 4/3 × 22 × 27 = 88 × 9 = 792 g

Weight in kg = 792 ÷ 1000 = 0.792 kg

∴ The correct answer is 0.792 kg.

Alternate Method

Given:

External radius (R) = 3 cm

Internal radius (r) = 0 cm

Density (ρ) = 7 g/cm³

Formula Used:

Volume of Sphere = 4/3 × π × R³

Mass = Density × Volume

Calculations:

⇒ Volume = 4/3 × 22/7 × (3)³

⇒ Volume = 4/3 × 22/7 × 27

⇒ Volume = 4 × 22/7 × 9 = 792/7 cm³

⇒ Mass = Density × Volume

⇒ Mass = 7 × (792/7) = 792 g

⇒ Weight in kg = 792 ÷ 1000

⇒ Weight = 0.792 kg

∴ The correct answer is 0.792 kg.

Additional Information

Volume of Hollow Sphere

The volume of a hollow spherical shell is given by V = 4/3 × π × (R³ − r³), where R is the outer radius and r is the inner radius.

Mass and Density Relation

The relationship is Mass = Density × Volume. Ensure units are consistent (e.g., g/cm³ with cm³ results in grams).

Unit Conversion

To convert grams to kilograms, divide the value by 1000 since 1 kg = 1000 g.