Shortcut Trick

Time ratio (Initial : New) = 6 : 3 = 2 : 1

Since Distance is constant, Speed ratio is inverse of Time ratio = 1 : 2

1 unit (Initial Speed) = 92 km/hr

Increase in speed required = 2 − 1 = 1 unit = 92 km/hr

∴ The correct answer is 92 km/hr.

Alternate Method

Given: Initial Speed = 92 km/hr, Initial Time = 6 hours, New Time = 3 hours

Formula Used: Distance = Speed × Time

⇒ Total Distance = 92 km/hr × 6 hours

⇒ Total Distance = 552 km

⇒ Required New Speed = Total Distance ÷ New Time

⇒ New Speed = 552 ÷ 3 = 184 km/hr

⇒ Increase in Speed = New Speed − Initial Speed

⇒ Increase in Speed = 184 − 92 = 92 km/hr

∴ The correct answer is 92 km/hr.

Additional Information

Inverse Variation Rule

When distance is constant, speed and time are inversely proportional (S₁ × T₁ = S₂ × T₂).

Average Speed

If a journey is covered in two equal distances at speeds x and y, average speed = 2xy ÷ (x + y).

Relative Speed

Speed of two objects moving in the same direction is (S₁ − S₂) and in opposite directions is (S₁ + S₂).

Given equation:

122 ÷ 4 + 65 × 5 - 41 = ?

Interchanging operators:

When ‘+’ and ‘−’ are interchanged and ‘×’ and ‘÷’ are interchanged, the new equation becomes:

122 × 4 − 65 ÷ 5 + 41

Solution:

Following the order of operations (BODMAS/PEMDAS):

= 122 × 4 - 65 ÷ 5 + 41

= 488 - 13 + 41

= 475 + 41 = 516

Hence, the correct answer is "Option 3".

The Correct answer is Right-click → Select New → Folder

Key Points

• To create a new folder in Windows 11 OS from the desktop, you can use the context menu by right-clicking.
• When you right-click on an empty space on the desktop, a menu appears with various options.
• Selecting the "New" option from the menu expands another sub-menu.
• In the sub-menu, choose "Folder", which automatically creates a new folder on the desktop.
• The newly created folder appears with the default name "New Folder", which can be renamed by typing a desired name immediately or later.
• This method is widely used as it is quick and user-friendly.
• It does not require any additional software or shortcuts, making it ideal for users of all skill levels.
• The functionality is consistent across various versions of Windows, including Windows 10 and Windows 11.

 Additional Information

• Press Alt + F4
• The Alt + F4 shortcut is used to close the active window or application in Windows.
• This shortcut does not create a new folder and serves a different purpose related to window management.
• Drag a file to Desktop
• Dragging a file to the desktop does not create a new folder but moves or copies the selected file to the desktop.
• This action is used for file organization rather than folder creation.
• Press Ctrl + N
• Ctrl + N is commonly used to create a new document or window in applications like Microsoft Word or web browsers.
• This shortcut does not work for creating a folder on the desktop in Windows OS.

The Correct answer is 2014.

Key Points

• The ‘Make in India’ initiative was launched by the Government of India on 25th September 2014.
• The program was aimed at transforming India into a global manufacturing hub.
• It focuses on enhancing skill development, fostering innovation, and protecting intellectual property.
• The initiative encourages both domestic and foreign companies to manufacture their products in India.
• The program is managed by the Department for Promotion of Industry and Internal Trade (DPIIT) under the Ministry of Commerce and Industry.
• ‘Make in India’ emphasizes key sectors such as automobiles, chemicals, IT, pharmaceuticals, textiles, defense manufacturing, and food processing, among others.
• The initiative aims to improve the ease of doing business and strengthen India's position in the global supply chain.
• India's ranking in the World Bank's Ease of Doing Business Index has significantly improved since the launch of this program.
• The ‘Make in India’ initiative aligns with the broader vision of achieving a $5 trillion economy by 2025.

The pattern consists of two parts:

• Letter Shift: The first letter is shifted backward by 4 positions, and the second letter is shifted backward by 3 positions.
• Number Shift: The number is increased by 15.

Applying the logic to the given pairs:

TN 14 is related to PK 29

JD 3 is related to FA 18

Applying the logic to SG -5:

SG -5 is related to what?

Following this logic, SG -5 is related to OD 10.

Hence, the correct answer is "Option 2".

Shortcut Trick

Property: If each observation in a data set is increased by a constant k, the mean average of the set also increases by exactly k.

New Average = Old Average + Increase Value

⇒ New Average = 18 + 5 = 23

∴ The correct answer is 23.

Alternate Method

Given: Number of observations = 6 | Original average = 18 | Increase per number = 5

Formula Used: Average = Sum of observations ÷ Total number of observations

⇒ Sum of original 6 numbers = 18 × 6 = 108

⇒ Total increase after adding 5 to each number = 5 × 6 = 30

⇒ New Total Sum = 108 + 30 = 138

⇒ New Average = 138 ÷ 6 = 23

∴ The correct answer is 23.

Additional Information

Effect of Operations on Average

If every term is multiplied by k, the average is multiplied by k. If every term is divided by k, the average is divided by k.

Average of Consecutive Numbers

The average of n consecutive numbers is the middle term if n is odd, or the mean of the two middle terms if n is even.

Weighted Average Formula

Weighted Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), used when combining two groups with different averages.

The Correct answer is Junagadh caves.

Key Points

• The Junagadh caves are a group of Buddhist caves located in the Junagadh district of Gujarat, India.
• These caves are famous for their high citadel and architectural significance.
• The caves are divided into three main groups: Uparkot caves, Khapra Kodiya caves, and the Baba Pyare caves.
• They were primarily used as monasteries and meditation centers by Buddhist monks during the 2nd to 4th centuries CE.
• The caves are adorned with pillars, water tanks, and prayer halls, reflecting the influence of Buddhist rock-cut architecture.
• The Uparkot caves, in particular, are known for their location within a high citadel, offering panoramic views of the surroundings.
• The intricate carvings and structural designs of the caves showcase the advanced techniques of ancient Indian craftsmen.

 Additional Information

• Sudama caves
• The Sudama caves are part of the Barabar caves, located in Bihar, India.
• These caves are the earliest examples of rock-cut architecture in India and were used by the Ajivika sect.
• The caves were carved during the reign of Emperor Ashoka in the 3rd century BCE and are known for their polished interior surfaces.
• Ellora caves
• The Ellora caves are a UNESCO World Heritage Site located in Maharashtra, India.
• They represent a mix of Buddhist, Hindu, and Jain religious traditions.
• The caves are renowned for the Kailasa temple, a monolithic rock-cut structure dedicated to Lord Shiva.
• Bagh caves
• The Bagh caves are located in the Dhar district of Madhya Pradesh, India.
• These caves are famous for their ancient mural paintings, which reflect Indian art from the Gupta period.
• They are associated with Buddhist monks who used them as viharas (monasteries).

Shortcut Trick

Use the relationship: MP ÷ CP = (100 + P%) ÷ (100 − D%).

Given: MP = 152 (if CP = 100) and Profit = 14%.

152 ÷ 100 = (100 + 14) ÷ (100 − D).

152 ÷ 100 = 114 ÷ (100 − D) ⇒ 100 − D = (114 × 100) ÷ 152.

100 − D = 75 ⇒ D = 25%.

∴ The correct answer is 25%.

Alternate Method

Given: Marked Price (MP) is 52% above Cost Price (CP), Profit = 14%.

Formulas: SP = CP × (100 + P%) ÷ 100 | Discount% = (MP − SP) ÷ MP × 100

⇒ Let Cost Price (CP) = 100.

⇒ Marked Price (MP) = 100 + 52% of 100 = 152.

⇒ Desired Profit = 14%, so Selling Price (SP) = 100 + 14 = 114.

⇒ Discount Amount = MP − SP = 152 − 114 = 38.

⇒ Discount% = (38 ÷ 152) × 100 = 1/4 × 100 = 25%.

∴ The correct answer is 25%.

Additional Information

Relation between CP, MP, Profit and Discount

The ratio of MP to CP is always constant for a given profit and discount: MP/CP = (100 + Gain%) / (100 − Discount%).

Marked Price

It is the price displayed on the label of an item, also known as the List Price.

Shortcut Trick

Perform Prime Factorisation of 6627.

6627 = 3 × 13 × 13 × 13 = 3¹ × 13³.

To make it a perfect square, all prime factors must have even powers.

Required multipliers = 3¹ × 13¹ = 39. (Wait, let us re-check the division).

6627 ÷ 3 = 2209. We know 2209 = 47².

∴ 6627 = 3 × 47². The unpaired factor is 3.

∴ The correct answer is 3.

Alternate Method

Given: Number = 6627.

Step-by-step Calculation:

⇒ Sum of digits of 6627 = 6 + 6 + 2 + 7 = 21 (Divisible by 3).

⇒ 6627 ÷ 3 = 2209.

⇒ 2209 is a perfect square since √2209 = 47.

⇒ Prime factorisation of 6627 = 3¹ × 47².

⇒ For a number to be a perfect square, the power of every prime factor must be even.

⇒ Here, 3 has a power of 1. To make it even, we must multiply by 3.

∴ The correct answer is 3.

Additional Information

Perfect Square Condition

A number is a perfect square if all its prime factors appear in pairs (even exponents).

Smallest Multiplier/Divisor

To find the smallest number to multiply/divide to make a perfect square, identify the unpaired factors in the prime factorisation.

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

The Correct answer is Indian Famine Commission.

Key Points

• The Indian Famine Commission was established in 1880 to study recurring famines in India and recommend measures to prevent them.
• The commission highlighted the importance of establishing rural credit institutions to help farmers during times of distress.
• The recommendations of the Indian Famine Commission influenced the enactment of the Cooperative Credit Societies Act of 1904, which aimed to provide affordable credit to farmers and prevent exploitation by moneylenders.
• The act was a landmark step in the development of cooperative movements in India, especially in rural areas.
• It laid the foundation for the establishment of cooperative societies, which were designed to provide credit at reasonable rates and improve the financial stability of farmers.
• The initiative was crucial for addressing the agricultural distress caused by recurring famines and boosting rural development.

 Additional Information

• Royal Commission on Agriculture
• The Royal Commission on Agriculture was set up in 1926 to study agricultural issues in India and recommend solutions for improving the agrarian economy.
• Its recommendations were more focused on land tenure systems, irrigation, marketing, and agricultural practices, rather than rural credit.
• Hunter Commission
• The Hunter Commission, established in 1882, dealt with education reforms in India and had no direct connection to agricultural credit or the Cooperative Credit Societies Act.
• It primarily focused on expanding and improving primary and secondary education in India.
• Simon Commission
• The Simon Commission, formed in 1927, focused on political reforms in India and assessing the working of the Government of India Act, 1919.
• It did not address agricultural credit or cooperative movements.

The Correct answer is President of India

Key Points

• Under Article 75(4) of the Constitution of India, the oath of office and secrecy is administered to the Union Ministers by the President of India.
• The oath is taken in accordance with the provisions of the Third Schedule of the Constitution.
• The President ensures that the Union Ministers pledge to uphold the Constitution of India, maintain secrecy, and perform their duties faithfully.
• The President is the head of the state and serves as the constitutional authority responsible for appointing Union Ministers as per the advice of the Prime Minister.
• This process reflects the principle of maintaining the integrity and solemnity of the parliamentary democracy in India.

 Additional Information

• Chief Justice of India
• The Chief Justice of India (CJI) primarily heads the judiciary and does not administer the oath of office and secrecy to Union Ministers.
• The Chief Justice is responsible for interpreting the Constitution and overseeing the functioning of the Supreme Court of India.
• However, the CJI administers the oath to the President of India and Governors in specific circumstances.
• Prime Minister
• The Prime Minister acts as the head of the government and advises the President on the appointment of Union Ministers.
• While the Prime Minister plays a significant role in governance, he/she does not administer the oath to Union Ministers.
• Vice President
• The Vice President of India is primarily responsible for officiating in the absence of the President and serves as the Chairman of the Rajya Sabha.
• The Vice President does not have the constitutional authority to administer the oath to Union Ministers.

The Correct answer is ₹5 lakh.

Key Points

• As of April 2025, the Reserve Bank of India (RBI) has introduced flexibility in transaction limits for UPI payments made at in-person merchant locations.
• The revised limits range from ₹2 lakh to ₹5 lakh, enhancing the usability of UPI for higher-value transactions.
• Unified Payments Interface (UPI) is a real-time payment system developed by the National Payments Corporation of India (NPCI), enabling instant money transfer between bank accounts.
• This move aims to encourage the adoption of digital payments, particularly for high-value purchases, fostering a cashless economy in India.
• The decision aligns with the RBI’s focus on promoting financial inclusion and supporting the growing digital infrastructure in merchant transactions.
• Such flexibility is expected to benefit sectors like retail, e-commerce, and luxury goods, where higher transaction limits are essential.
• The introduction of higher limits shows the RBI’s confidence in the security and reliability of UPI systems, ensuring safe transactions for users.

Shortcut Trick

Effective price percentage = (100 − d₁)% × (100 − d₂)% × (100 − d₃)%

Effective price = 100 × (92/100) × (84/100) × (97/100)

Effective price = 0.92 × 0.84 × 97% = 0.7728 × 97%

Effective price = 74.9616% ≈ 74.96%

∴ The correct answer is 74.96%.

Alternate Method

Given: Three consecutive discounts of 8%, 16%, and 3%.

Formulas: Selling Price (SP) = Marked Price (MP) × (100 − d₁)/100 × (100 − d₂)/100 × (100 − d₃)/100

⇒ Let the Marked Price be 100 units.

⇒ Price after 8% discount = 100 − 8 = 92 units.

⇒ Price after 16% discount = 92 − (16% of 92) = 92 − 14.72 = 77.28 units.

⇒ Price after 3% discount = 77.28 − (3% of 77.28) = 77.28 − 2.3184 = 74.9616 units.

⇒ Effective Price Percentage = (74.9616/100) × 100 ≈ 74.96%.

∴ The correct answer is 74.96%.

Additional Information

Effective Discount Rate

To find the total discount, use the formula for successive discounts: a + b − (ab/100). Subtracting this from 100 gives the final price.

Relationship between MP and SP

Selling Price (SP) = Marked Price (MP) × (1 − Discount%/100). The effective price percentage is essentially the ratio of SP to MP expressed as a percentage.

Shortcut Trick

Given secθ + tanθ = 2. Then secθ − tanθ = 1/2.

Adding both: 2secθ = 2 + 1/2 = 5/2 ⇒ secθ = 5/4.

Find: 3secθ + 4 = 3(5/4) + 4 = 15/4 + 16/4 = 31/4.

∴ The correct answer is 31/4.

Alternate Method

Given: secθ + tanθ = 2

Formula Used: sec²θ − tan²θ = 1 ⇒ (secθ + tanθ)(secθ − tanθ) = 1

⇒ 2 × (secθ − tanθ) = 1

⇒ secθ − tanθ = 1/2

⇒ (secθ + tanθ) + (secθ − tanθ) = 2 + 1/2

⇒ 2secθ = 5/2

⇒ secθ = 5/4

⇒ 3secθ + 4 = 3 × (5/4) + 4

⇒ 15/4 + 4 = (15 + 16)/4 = 31/4

∴ The correct answer is 31/4.

Additional Information

Trigonometric Identity

The identity 1 + tan²θ = sec²θ leads to the reciprocal relationship (secθ + tanθ) = 1 / (secθ − tanθ).

Cosecant Identity

Similarly, 1 + cot²θ = cosec²θ, meaning (cosecθ + cotθ) = 1 / (cosecθ − cotθ).

Pythagorean Triplets

Since secθ = 5/4 (Hypotenuse/Base), the Perpendicular is √(5² − 4²) = 3, forming a 3-4-5 triplet.

Shortcut Trick

Given Surface Area = 4πr² = 25π ⇒ r² = 25/4 ⇒ r = 5/2 cm.

Volume = (4/3)πr³ = (4/3)π(5/2)³ = (4/3)π(125/8).

On simplifying, Volume = (125π)/6 cm³.

∴ The correct answer is 125π/6 cm³.

Alternate Method

Given:

Surface Area of Sphere = 25π cm²

Formula Used:

⇒ Surface Area (S) = 4πr²

⇒ Volume (V) = (4/3)πr³

Calculations:

⇒ 4πr² = 25π

⇒ r² = 25/4

⇒ r = √(25/4) = 5/2 cm

⇒ Volume = (4/3) × π × (5/2)³

⇒ Volume = (4/3) × π × (125/8)

⇒ Volume = (1 × π × 125) / (3 × 2)

⇒ Volume = 125π/6 cm³

∴ The correct answer is 125π/6 cm³.

Additional Information

Properties of Sphere

A sphere is a perfectly round geometrical object in three-dimensional space. Surface Area = 4πr² and Volume = (4/3)πr³.

Hemisphere Formulas

For a solid hemisphere, Curved Surface Area = 2πr², Total Surface Area = 3πr², and Volume = (2/3)πr³.

Ratio relations

If the ratio of radii of two spheres is a:b, then the ratio of their surface areas is a²:b² and the ratio of their volumes is a³:b³.

The Correct answer is Chanakya.

Key Points

• Chanakya, also known as Kautilya or Vishnugupta, was a renowned ancient Indian philosopher, economist, and statesman.
• He played a pivotal role in assisting Chandragupta Maurya in overthrowing the Nanda dynasty and establishing the Maurya Empire.
• Chanakya is credited with authoring the Arthashastra, an ancient Indian treatise on statecraft, economic policy, and military strategy.
• He was a master strategist who used his knowledge and wit to mentor Chandragupta Maurya and unite the fragmented regions of India under a single empire.
• Chanakya’s guidance was instrumental in planning and executing the strategies to defeat the Nandas, who were powerful rulers at the time.
• Chanakya’s contributions to Indian political thought and governance are considered pioneering and are still studied widely.
• He is often referred to as the “Machiavelli of India” for his diplomatic and political acumen.

 Additional Information

• Megasthenes
• Megasthenes was a Greek ambassador to the Mauryan court during the reign of Chandragupta Maurya.
• He authored the book Indica, which provides insights into the Mauryan Empire and ancient Indian society.
• However, he did not play any role in the overthrow of the Nanda dynasty.
• Bindusara
• Bindusara was the son of Chandragupta Maurya and the second ruler of the Maurya Empire.
• He succeeded Chandragupta and expanded the empire further but was not involved in the overthrow of the Nandas.
• Bimbisara
• Bimbisara was a ruler of the Haryanka dynasty in ancient India and a contemporary of the Buddha.
• He ruled the kingdom of Magadha but lived long before Chandragupta and Chanakya.

The Correct answer is Part IX-A.

Key Points

• The 74th Amendment Act of the Indian Constitution, enacted in 1992, specifically introduced Part IX-A, which deals with the governance of urban local bodies.
• This amendment provides a constitutional status to urban local self-governments such as Municipal Corporations, Municipalities, and Nagar Panchayats.
• It aims to ensure decentralization in urban governance and encourage grassroots democracy in urban areas.
• Article 243-P to Article 243-ZG were added to the Constitution under Part IX-A, detailing the composition, powers, and responsibilities of urban local bodies.
• It mandates the establishment of Ward Committees in municipalities with a population of over 3 lakh, ensuring greater citizen participation.
• The amendment also outlines provisions for the reservation of seats for Scheduled Castes, Scheduled Tribes, and women in urban local bodies.
• A State Finance Commission is to be constituted to recommend measures for the financial health of these bodies.
• The 74th Amendment is significant for promoting good governance and addressing the challenges of urbanization in India.

 Additional Information

• Part X
• Part X of the Constitution deals with the Scheduled and Tribal Areas.
• It includes provisions for the administration and governance of these areas to protect the interests of Scheduled Tribes.
• It covers Articles 244 and 244A.
• Part IX
• Part IX of the Constitution deals with Panchayati Raj Institutions and was added through the 73rd Amendment Act, 1992.
• It focuses on rural local governance and includes Articles 243 to 243-O.
• Part XI
• Part XI of the Constitution deals with the relations between the Union and the States.
• It includes provisions related to legislative and administrative relations, covered under Articles 245 to 263.

Shortcut Trick Filling time (A) = 6 hours Net filling time (A − Leak) = 7(1/5) = 36/5 hours Leak empty time = (Product of times) ÷ (Difference of times) Leak empty time = (6 × 36/5) ÷ (36/5 − 6) = (216/5) ÷ (6/5) = 36 hours ∴ The correct answer is 36 hr. Alternate Method Person / Pipe Time (hr) Total Work (LCM) Efficiency Pump (A) 6 36 units 6 Pump + Leak (A + L) 36/5 5 Let the total capacity of the tank be 36 units (LCM of 6 and 36/5). ⇒ Efficiency of Pump (A) = 36 ÷ 6 = 6 units/hr ⇒ Combined Efficiency (A + Leak) = 36 ÷ (36/5) = 5 units/hr ⇒ Efficiency of Leak (L) = (A + L) − A = 5 − 6 = −1 unit/hr ⇒ Time taken by leak to empty = Total Work ÷ Efficiency of Leak ⇒ Time = 36 ÷ 1 = 36 hours ∴ The correct answer is 36 hr. Additional Information Pipes & Cisterns Basics Efficiency of an inlet pipe is taken as positive, while that of an outlet pipe or leak is taken as negative. Time Calculation Formula If pipe A fills in 'x' hours and pipe B empties in 'y' hours (y > x), then net filling time = (xy) ÷ (y − x). Total Work Assumption Always take the LCM of individual time durations to represent the total capacity of the tank for easier calculations.

The Correct answer is Bihar.

Key Points

• The Khelo India Youth Games is a flagship initiative by the Government of India aimed at promoting sports culture in the country.
• In 2025, the state of Bihar was chosen to host the prestigious event, showcasing its growing commitment to sports infrastructure and development.
• Bihar is focusing on creating a conducive environment for young athletes by improving sports facilities and encouraging participation at grassroots levels.
• The selection of Bihar as the host state highlights its efforts to rejuvenate its image as an emerging hub for sports events in India.
• The event is expected to bring together thousands of young athletes from across India, competing in various disciplines, thus fostering unity and sportsmanship.
• Khelo India Youth Games serves as a platform for identifying and nurturing talent, ensuring that promising athletes receive the support and training they need to excel at national and international levels.
• The initiative aligns with the government's vision of making India a global sporting powerhouse by promoting fitness and competitive spirit among its youth.

 Additional Information

• Uttar Pradesh
• Uttar Pradesh is one of India's largest states and has hosted several important events, but it was not selected to host the Khelo India Youth Games in 2025.
• The state is known for its efforts in promoting sports, with initiatives like the Major Dhyan Chand Sports University in Meerut.
• West Bengal
• West Bengal has a rich history of sports, particularly in football and cricket, but it was not the host for the 2025 Khelo India Youth Games.
• The state is known for iconic venues like the Eden Gardens, one of the largest cricket stadiums in the world.
• Madhya Pradesh
• Madhya Pradesh has been actively promoting sports and was the host for the Khelo India Youth Games in 2023, but it was not selected for 2025.
• The state has invested in building sports infrastructure, including the international-level stadiums and training centers.

Shortcut Trick

Let initial Cost Price (CP₁) be 100 units.

Initial Selling Price (SP₁) at 22% loss = 100 − 22 = 78 units.

New Cost Price (CP₂) at 5% less = 95 units.

New Selling Price (SP₂) at 15% gain = 95 × 115 ÷ 100 = 109.25 units.

Difference in SP = 109.25 − 78 = 31.25 units ⇒ 935.

1 unit = 935 ÷ 31.25 = 29.92 ∴ 100 units = ₹2,992.

∴ The correct answer is ₹2,992.

Alternate Method

Formulas: SP = CP × (100 ± P/L)% | CP = SP × 100 ÷ (100 ± P/L)%

⇒ Let the initial cost price be x. ⇒ Initial selling price SP₁ = x × (100 − 22) ÷ 100 = 0.78x. ⇒ New cost price CP₂ = x × (100 − 5) ÷ 100 = 0.95x. ⇒ New selling price SP₂ = 0.95x × 115 ÷ 100 = 1.0925x. ⇒ According to the question: SP₂ − SP₁ = 935. ⇒ 1.0925x − 0.78x = 935 ⇒ 0.3125x = 935. ⇒ x = 935 ÷ 0.3125 = 2,992.

∴ The correct answer is ₹2,992.

Additional Information

Percentage Change in CP and SP

When both CP and SP are varied, calculate the new values relative to the original CP (usually 100) to find the absolute difference in units.

Profit and Loss Basics

Profit % is always calculated on CP unless stated otherwise. Profit % = (Profit ÷ CP) × 100.

The Correct answer is Decreased from 6% to 3.2%.

Key Points

• The unemployment rate refers to the percentage of individuals in the labor force who are actively seeking work but are unable to find employment.
• The trend from 2017-18 to 2023-24 indicates a significant decrease in the unemployment rate for individuals aged 15 years and above, dropping from 6% to 3.2%.
• This decline demonstrates an improvement in the employment scenario, possibly due to various government initiatives, schemes, and an increase in economic activities.
• Factors such as skill development programs, enhanced focus on manufacturing, and promotion of entrepreneurship may have contributed to this positive trend.
• The data is often sourced from reports such as the Periodic Labour Force Survey (PLFS), which provides critical insights into employment and unemployment patterns.
• This reduction reflects efforts to enhance job creation and improve the economic stability of the country.

 Additional Information

• Government efforts and schemes
• Programs such as Skill India Mission, Make in India, and promotion of self-employment have likely played a role in reducing unemployment.
• Enhanced focus on vocational training and creation of job opportunities in both urban and rural areas have contributed to this trend.

Given:

$\frac{?}{48} = \frac{243}{?}$

Formula used:

If $\frac{a}{b} = \frac{c}{a}$, then $a^2 = b \times c$

Calculation:

⇒ $?^2 = 48 \times 243$

⇒ $?^2 = 11664$

⇒ $? = 108$

∴ The positive number is 108.

Given:

Samrath's rank from the top = 31st.

Samrath's rank from the bottom = 13th.

Total persons = (Rank from Top) + (Rank from Bottom) - 1

Total students = 31 + 13 - 1

= 44 - 1 = 43

Thus, there are 43 students in the class.

Hence, the correct answer is "Option 1".

The Correct answer is 2015.

Key Points

• The Pradhan Mantri Awas Yojana (PMAY) was launched on 25th June 2015 by the Government of India.
• It is a flagship mission aimed at providing housing for all by the year 2022, with a focus on the economically weaker sections (EWS), low-income groups (LIG), and middle-income groups (MIG).
• PMAY consists of two components: PMAY-Urban (PMAY-U) and PMAY-Gramin (PMAY-G), addressing housing needs in urban and rural areas respectively.
• The mission is implemented under four verticals: In-situ Slum Redevelopment (ISSR), Credit Linked Subsidy Scheme (CLSS), Affordable Housing in Partnership (AHP), and Beneficiary-led Construction (BLC).
• Under the Credit Linked Subsidy Scheme (CLSS), interest subsidies are provided on housing loans for eligible beneficiaries.
• The scheme also promotes the use of sustainable and eco-friendly construction technologies.
• PMAY supports the vision of the Housing for All initiative, aligning with the goals of inclusive growth and urban development.
• It has helped millions of families across India to achieve their dream of owning a home, especially those belonging to vulnerable sections of society.

 Additional Information

• 2018
• 2018 was not the year of the launch of PMAY, but during this year, several milestones were achieved under the scheme, including increased housing allocations and implementation advancements.
• 2014
• While 2014 marked the beginning of several new initiatives under the leadership of the Narendra Modi government, PMAY was officially launched in 2015.
• 2022
• 2022 was the target year for achieving the Housing for All goal under PMAY, but it was not the year of the scheme's launch.

Given number: 8731542

Performing the operations:

The new number formed is: 7842631

Highest digit in the new number: 8

Lowest digit in the new number: 1

Difference between the highest and lowest digits: 8 - 1 = 7.

Hence, the correct answer is "Option 3".

Shortcut Trick

Upstream Speed (U) = 59.4 km ÷ (18/60) h = 198 kmph.

Ratio of Still Water (u) to Stream (v) = 3 : 2 ⇒ Upstream ratio (u−v) = 3 − 2 = 1 unit.

1 unit = 198 kmph ⇒ u = 3 × 198 = 594 kmph, v = 2 × 198 = 396 kmph.

Downstream Speed (D) = u + v = 5 unit = 5 × 198 = 990 kmph.

Total Time = (29.3 ÷ 198) + (51.5 ÷ 990) ≈ 0.148 + 0.052 = 0.2 hours.

∴ The correct answer is 0.2 hours.

Alternate Method

Given: Upstream Distance = 59.4 km, Time = 18 min = 0.3 hours. Ratio u : v = 3 : 2.

Formula Used: Speed = Distance ÷ Time; Upstream Speed (U) = u − v; Downstream Speed (D) = u + v.

⇒ Upstream Speed (U) = 59.4 ÷ 0.3 = 198 kmph.

⇒ Let u = 3x and v = 2x. Then U = 3x − 2x = x.

⇒ x = 198 kmph.

⇒ Downstream Speed (D) = 3x + 2x = 5x = 5 × 198 = 990 kmph.

⇒ Time for 29.3 km Upstream = 29.3 ÷ 198 ≈ 0.14798 hours.

⇒ Time for 51.5 km Downstream = 51.5 ÷ 990 ≈ 0.05202 hours.

⇒ Total Time = 0.14798 + 0.05202 = 0.2 hours.

∴ The correct answer is 0.2 hours.

Additional Information

Relative Speed in Water

Upstream Speed = Speed of boat (u) − Speed of stream (v). Downstream Speed = Speed of boat (u) + Speed of stream (v).

Still Water and Stream Speed Formulas

Speed of boat in still water (u) = (Downstream + Upstream) ÷ 2. Speed of stream (v) = (Downstream − Upstream) ÷ 2.

Time, Speed, Distance Relation

Time = Distance ÷ Speed. Ensure all units are consistent (e.g., km and hours or m and seconds) before calculation.

Given series:

(Left) @ * # % * 8 1 4 2 2 & 4 8 9 8 8 # 6 $$ * 3 (Right)

Condition to be checked:

Number → Number → Symbol

@ * # % * 8 1 4 2 2 & 4 8 9 8 8 # 6 $$ * 3

There are two such numbers in the above arrangement (2 2 &, 8 8 #).

Hence, the correct answer is "Option 3".

The pattern followed here is.

Thus, the missing term in the series is INQ 21.

Hence, the correct answer is Option 1.

The Correct answer is Drafting the Indian Penal Code and promoting English education.

Key Points

• Lord Thomas Babington Macaulay was a British historian, politician, and lawmaker who played a pivotal role in shaping the administrative and educational policies of British India.
• He is best known for drafting the Indian Penal Code (IPC), which became the backbone of the Indian legal system and is still in use today with modifications.
• The IPC was a comprehensive code of criminal law for India, and it was officially enacted in 1860.
• In addition to his work on the IPC, Macaulay strongly advocated for the promotion of English education in India.
• His famous "Minute on Education" (1835) laid the foundation for the use of English as the medium of instruction in Indian schools and colleges.
• Macaulay believed that English education would create a class of Indians who were "Indian in blood and color, but English in taste, in opinions, in morals, and in intellect."
• His efforts significantly influenced the spread of Western education, science, and literature in India.

 Additional Information

• Framing the Government of India Act, 1858
• The Government of India Act, 1858 marked the formal transfer of power from the East India Company to the British Crown.
• This Act was introduced after the Indian Rebellion of 1857 to improve governance in India.
• However, Macaulay was not directly involved in drafting this Act.
• Introducing the Vernacular Press Act
• The Vernacular Press Act of 1878 was introduced by Lord Lytton, not Macaulay.
• This Act aimed to curb the freedom of the Indian-language press and suppress criticism of British policies.
• Leading the revolt of 1857
• The Indian Rebellion of 1857 was a large-scale uprising against British rule, but Macaulay was not involved in leading it.
• By the time of the revolt, Macaulay had already returned to England and was not in India.

The logic followed here is:

Thus, the missing term in the series is 368.

Hence, the correct answer is "Option 4";

Shortcut Trick

Subtract the remainders from the given numbers: 104 − 8 = 96 and 133 − 5 = 128.

Find the HCF of 96 and 128.

HCF(96, 128) = 32 because 32 × 3 = 96 and 32 × 4 = 128.

∴ The correct answer is 32.

Alternate Method

Given: Numbers are 104 and 133, remainders are 8 and 5 respectively.

Formula Used: Required number = HCF(Number₁ − Remainder₁, Number₂ − Remainder₂)

⇒ First number = 104 − 8 = 96

⇒ Second number = 133 − 5 = 128

⇒ Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

⇒ Factors of 128: 1, 2, 4, 8, 16, 32, 64, 128

⇒ Highest Common Factor (HCF) = 32

∴ The correct answer is 32.

Additional Information

HCF Definition

The Highest Common Factor (HCF) is the largest positive integer that divides each of the integers without a remainder.

Remainder Rule for HCF

To find the greatest number that leaves remainders a and b when dividing x and y, always find HCF(x−a, y−b).

Euclid Division Lemma

For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.

Shortcut Trick

Remaining percentage after rent = 100% − 76% = 24%

Total of other expenses and savings = 627 + 3947 + 712 = ₹5286

24% → ₹5286

100% → (5286 ÷ 24) × 100 = 220.25 × 100 = ₹22,025

∴ The correct answer is ₹22,025.

Alternate Method

Given:

Rent Expenditure = 76% of Salary

Conveyance Expenditure = ₹627

Grocery Expenditure = ₹3,947

Monthly Savings = ₹712

Calculations:

⇒ Total non-rent amount = Conveyance + Grocery + Savings

⇒ Total non-rent amount = 627 + 3947 + 712 = ₹5,286

⇒ Since Rent = 76%, Remaining amount percentage = 100% − 76% = 24%

⇒ 24% of Monthly Salary = ₹5,286

⇒ Monthly Salary = (5286 ÷ 24) × 100

⇒ Monthly Salary = 220.25 × 100 = ₹22,025

∴ The correct answer is ₹22,025.

Additional Information

Income Expenditure Formula

Income = Total Expenditure + Savings. Any component can be calculated if others are given.

Percentage Base Change

If a value is distributed into segments, the sum of all segment percentages must equal 100% of the base value.

Successive Percentage

If the problem states 'percentage of remaining', we use the formula: Final = Initial × (1 − r₁/100) × (1 − r₂/100).

The logic followed in the given number pairs is:

Logic: The first number is five times the second number.

Now,

For, 10 : 2

⇒ 10 = 2 × 5

For, 45 : 9

⇒ 45 = 9 × 5

Applying the same logic for the third pair:

For, 50 : ?

⇒ 50 = ? × 5

⇒ ? = 50 / 5

⇒ ? = 10

Hence, the correct answer is "Option 4".

Shortcut Trick

For any cubic equation of the form ax³ + bx² + cx + d = 0, the sum of roots is always given by −b/a.

Here, the coefficient of x² (b) is 7 and the coefficient of x³ (a) is 2.

Applying the trick: Sum = − (7 / 2).

∴ The correct answer is -7/2.

Alternate Method

Given:

The polynomial equation is 2x³ + 7x² − 9x + 12 = 0.

Formula Used:

According to Vieta's Formulas, for a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, and γ:

Sum of roots (α + β + γ) = −b / a

Calculations:

Comparing the given equation with the standard form:

⇒ a = 2

⇒ b = 7

⇒ c = −9

⇒ d = 12

Substituting these values into the formula for the sum of roots:

⇒ Sum of roots = −(7) / 2

⇒ Sum of roots = −7/2

∴ The correct answer is -7/2.

Additional Information

Product of Roots (Taken Two at a Time)

For the cubic equation ax³ + bx² + cx + d = 0, the sum of the product of roots taken two at a time is given by αβ + βγ + γα = c/a.

Product of All Roots

The product of all three roots for a cubic equation is given by αβγ = −d/a.

General Relation for n-degree Polynomial

For any polynomial a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₀ = 0, the sum of roots is always −a_n−1 / a_n.

Given:

Eshwar starts from Point Y and drives 13 km towards the south.

He then takes a left turn, drives 27 km, turns right and drives 51 km.

He then takes a left turn and drives 18 km. He takes a left turn, drives 35 km.

He then turns left, drives 45 km, turns left and drives 18 km to stop at Point Z.

Movement of Eshwar is,

Since Eshwar's final position (Point Z) is 47 km South and 0 km East/West of his starting point (Point Y), he needs to travel a distance of 47 km in the North direction to get back to Point Y.

Hence, the correct answer is "Option 2".

The Correct answer is Swarts reaction.

Key Points

• Swarts reaction is a method for the preparation of haloalkanes, particularly fluorinated compounds.
• In this reaction, metal fluorides like AgF, SbF3, or Hg2F2 are used to replace halogens (chlorine or bromine) in alkyl halides.
• Freon-12 (Dichlorodifluoromethane) is commonly prepared using the Swarts reaction as it involves the replacement of chlorine atoms in chlorinated hydrocarbons with fluorine atoms.
• The reaction is widely applied in the industrial production of Freon compounds, which are used as refrigerants.
• Freon-12 has been extensively utilized in air conditioning and refrigeration systems due to its physical properties, such as low boiling point and non-flammability.
• However, its use has been phased out in many countries due to its ozone-depleting potential under the Montreal Protocol.

 Additional Information

• Substitution reaction
• In a substitution reaction, one atom or group in a molecule is replaced by another atom or group.
• It is a common reaction in organic chemistry and involves mechanisms like SN1 and SN2 reactions.
• While substitution reactions are used for synthesizing various compounds, they are not the primary method for producing fluorinated hydrocarbons like Freon-12.
• Wurtz reaction
• The Wurtz reaction is a method used to prepare higher alkanes by reacting alkyl halides with sodium in dry ether.
• This reaction is not suitable for the production of fluorinated hydrocarbons like Freon-12.
• Elimination reaction
• Elimination reactions involve the removal of atoms or groups from a molecule, resulting in the formation of double or triple bonds.
• These reactions do not involve the direct substitution of halogens with fluorine and are unrelated to the synthesis of Freon compounds.

The given numbers are:

107, 788, 634, 346, 488

The highest number is 788.

The first digit of the highest number (788) is 7.

The lowest number is 107.

The third digit of the lowest number (107) is 7.

Sum = 7 + 7 = 14.

The resultant is 14.

Hence, the correct answer is "Option 1".

Given:

\(\left[ \left(17 \times 3\right) \times \left{ 7 \div 7 \times \left(18 - \frac{14}{2}\right) \right} \right].\)

Formula used:

Simplify using the BODMAS rule.

Calculations:

[ (17 × 3) × { 7 ÷ 7 × (18 - (14 ÷ 2)) } ]

⇒ [ (17 × 3) × { 7 ÷ 7 × (18 - 7) } ]

⇒ [ (17 × 3) × { 7 ÷ 7 × 11 } ]

⇒ [ (17 × 3) × { 1 × 11 } ]

⇒ [ (17 × 3) × 11 ]

⇒ (51 × 11)

⇒ 561

∴ The correct answer is option (3).

Given:

• Harry's rank from the top = 34th
• Harry's rank from the bottom = 25th

The formula for the total number of persons in a row (or class) when one person's rank from both ends is known is:

Total persons = (Rank from one end) + (Rank from other end) - 1

Using this formula for Harry's ranks:

Total students = (Harry's rank from top) + (Harry's rank from bottom) - 1

= 34 + 25 - 1

= 59 - 1 = 58 students.

Hence, the correct answer is "Option 2".

The Correct answer is 51%.

Key Points

• According to the World Bank's State of Social Protection Report 2025, social protection coverage in low- and middle-income countries increased significantly between 2010 and 2022.
• The coverage rose from 41% to 51%, showing a notable improvement in efforts to ensure economic security and welfare for vulnerable populations.
• Social protection programs often include cash transfers, unemployment benefits, health insurance, and pensions, which are crucial for reducing poverty and inequality.
• The increase in social protection coverage reflects global initiatives aimed at achieving Sustainable Development Goals (SDGs), particularly Goal 1 (No Poverty) and Goal 10 (Reduced Inequalities).
• The rise to 51% underscores the importance of government policies, international aid, and community-based programs in addressing social and economic challenges.
• This progress is particularly significant for countries in regions like Sub-Saharan Africa, South Asia, and Latin America, which often face higher poverty rates.
• The report highlights the role of organizations like the World Bank, UNICEF, and other global entities in supporting social protection frameworks.

 Additional Information

• Social Protection Coverage
• Social protection refers to programs designed to reduce poverty and vulnerability by promoting efficient labor markets and diminishing people's exposure to risks.
• Examples include conditional cash transfers, food assistance programs, old-age pensions, and disaster relief funds.
• It is critical for achieving inclusive economic growth and protecting against economic shocks.
• World Bank Reports
• The World Bank publishes periodic reports to assess progress in areas such as economic development, poverty alleviation, and social protection.
• The State of Social Protection Report provides insights into global trends and challenges in expanding coverage.
• This report serves as a benchmark for policymakers to evaluate their strategies and improve their frameworks.

Given: Dipti, Priti, Neeti, Reeti, Swati, Kirti and Jyoti are sitting around a circular table, facing the centre.

Dipti sits second to the right of Priti.

Jyoti sits to the immediate left of Priti.

Swati is an immediate neighbour of both Jyoti and Neeti.

Reeti is an immediate neighbour of Priti.

And Kirti sits at the remaining place.

Dipti sits second to the left of Neeti.

Hence, the correct answer is "Option 4".

The pattern followed here is.

Thus, the missing term in the series is PIN65.

Hence, the correct answer is Option 2.

The Correct answer is Maikala Range

Key Points

• The Shillong Plateau is a significant geographical feature located in the northeastern part of India.
• It is primarily divided into three distinct regions: the Khasi Hills, Jaintia Hills, and Garo Hills.
• The Maikala Range, however, is not a part of the Shillong Plateau. It is located in central India, primarily in the states of Chhattisgarh and Madhya Pradesh.
• The Maikala Range is an extension of the Satpura Range and is geographically distinct from the Shillong Plateau.
• The Shillong Plateau is known for its unique geological features, being composed of Precambrian rocks and forming part of the Indian Shield.
• In contrast, the Maikala Range is known for its forests and biodiversity, and is home to the Amarkantak Plateau, which is the origin of rivers like the Narmada and Son.
• Thus, the Maikala Range is geographically unrelated to the Shillong Plateau, making it the correct answer.

 Additional Information

• Khasi Hills
• The Khasi Hills form a significant part of the Shillong Plateau.
• It is known for its scenic beauty, heavy rainfall, and biodiversity.
• Cherrapunji and Mawsynram, located in the Khasi Hills, are among the wettest places on Earth.
• Garo Hills
• The Garo Hills are located in the western part of the Shillong Plateau.
• They are known for their tribal culture, forests, and the presence of unique wildlife.
• It is one of the regions inhabited predominantly by the Garo tribe.
• Mikir Hills
• While geographically close to the Shillong Plateau, the Mikir Hills are technically part of the larger region around the Kaziranga National Park in Assam.
• They are known for their rolling hills and proximity to areas of ecological importance.

Shortcut Trick

Mahesh age = (Sum of ages) − (Ramesh age)

Mahesh age = (2x² − 3x²y + y² + 3xy²) − (x² + 3x²y + y² + 3xy²)

Canceling common terms (y² and 3xy²): (2x² − 3x²y) − (x² + 3x²y)

Subtracting like terms: x² − 6x²y

∴ The correct answer is (x² − 6x²y) years.

Alternate Method

Given:

Sum of ages of Ramesh and Mahesh = 2x² − 3x²y + y² + 3xy²

Age of Ramesh = x² + 3x²y + y² + 3xy²

Formula Used:

Age of Mahesh = Total Sum − Age of Ramesh

Calculations:

⇒ Age of Mahesh = (2x² − 3x²y + y² + 3xy²) − (x² + 3x²y + y² + 3xy²)

⇒ Age of Mahesh = 2x² − 3x²y + y² + 3xy² − x² − 3x²y − y² − 3xy²

⇒ Grouping like terms: (2x² − x²) + (−3x²y − 3x²y) + (y² − y²) + (3xy² − 3xy²)

⇒ Age of Mahesh = x² − 6x²y + 0 + 0

⇒ Age of Mahesh = x² − 6x²y

∴ The correct answer is (x² − 6x²y) years.

Additional Information

Algebraic Subtraction

When subtracting algebraic expressions, change the sign of every term in the expression being subtracted and then combine like terms.

Like Terms

Like terms are terms that have the same variables raised to the same powers. Only like terms can be added or subtracted (e.g., x²y and x²y).

The correct answer is Dutch East India Company.

Key Points

• The Battle of Colachel was fought in 1741 between the forces of the Dutch East India Company and the Kingdom of Travancore, led by its ruler Marthanda Varma.
• This battle was significant as it marked the first-ever defeat of a European colonial power by an Indian kingdom in a direct military confrontation.
• The Dutch forces aimed to secure a monopoly over the lucrative pepper trade in the Malabar region, which was a key economic resource during that time.
• However, the Travancore army under Marthanda Varma's leadership not only repelled the Dutch forces but also captured several of their officers, including their commander, Eustachius De Lannoy.
• The victory at Colachel was a strategic turning point, significantly weakening the influence of the Dutch East India Company in the region.
• Eustachius De Lannoy later became a trusted military advisor to Marthanda Varma, helping modernize the Travancore army with European military techniques.
• This battle is a symbol of resistance against colonial domination and showcases the military prowess of Indian rulers during the pre-colonial period.

 Additional Information

• Austrian East India Company
• The Austrian East India Company was a trading company established in the late 18th century but did not have significant influence or operations in the Malabar region.
• Its focus was primarily on limited trade routes and commodities, and it lacked the military strength to engage in battles like Colachel.
• English East India Company
• The English East India Company was a major colonial power in India but was not involved in the Battle of Colachel.
• It primarily focused on establishing control over Bengal, Madras, and Bombay during the 18th century.
• Spanish East India Company
• The Spanish East India Company was primarily involved in trade routes connecting Spain, the Americas, and the Philippines.
• It had no notable presence or military engagement in the Indian subcontinent.

Given: Seven boxes, A, B, C, D, E, F and G, are kept one above the other. The lowermost box is numbered 1 and the topmost is numbered 7.

1. D is kept second from the bottom.

2. Only three boxes are kept between D and E.

3. F is kept in one of the positions below D.

4. Only A is kept between E and G.

5. B is kept in one of the positions above E.

6. The only remaining box is C, and the only remaining position is 3. So, C is on Position 3.

Therefore, One box is kept between A and B.

Hence, the correct answer is "Option 4".

Given: Seven friends, K, L, M, N, O, P and S, are sitting in a row facing north.

Only two people sit to the left of S.

Only three people sit between S and O.

K sits second to the right of S.

Only two people sit between K and M.

L sits at some place to the right of N but at some place to the left of P.

Three people sit to the right of L.

Hence, the correct answer is "Option 2".

Given series:

(Left) 3 £ & 1 2 © 2 5 $ 9 3 © 9 4 3 % 9 3 # © 6 2 (Right)

Condition to be checked:

Number → Symbol → Symbol

3 £ & 1 2 © 2 5 $ 9 3 © 9 4 3 % 9 3 # © 6 2

There are two such symbols in the above arrangement (3 £ &, 3 # ©).

Hence, the correct answer is "Option 4".

BODMAS Rule:

According to the question, the signs are interchanged as follows:

Now, given the equation: 96 × 8 − 11 ÷ 83 + 22 = ?

After interchanging the specified signs, the new equation is:

96 ÷ 8 × 11 + 83 − 22 = ?

Perform operations according to BODMAS:

⇒ 96 ÷ 8 × 11 + 83 − 22

⇒ 12 × 11 + 83 − 22

⇒ 132 + 83 − 22

⇒ 215 − 22 = 193

Hence, the correct answer is "Option 2".

Shortcut Trick

Since the given Mode is exactly 125, which lies on the boundary of the classes 120−125 and 125−130, the frequencies of these two adjacent classes must be equal to maintain symmetry.

⇒ Frequency of class (120−125) = Frequency of class (125−130)

⇒ 39 = x

⇒ x = 39

∴ The correct answer is 39.

Alternate Method

Given:

Mode of the data = 125

Formula Used:

Mode = L + [ (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) ] × h

Where:

L = Lower limit of the modal class

f₁ = Frequency of the modal class

f₀ = Frequency of the class preceding the modal class

f₂ = Frequency of the class succeeding the modal class

h = Class width (size of class interval)

Calculations:

Since the mode is 125, it lies on the boundary of the classes 120−125 and 125−130. We can analyze both cases:

Case 1: Assuming 120−125 is the modal class

⇒ L = 120, f₁ = 39, f₀ = 31, f₂ = x, h = 5

⇒ 125 = 120 + [ (39 − 31) ÷ (2 × 39 − 31 − x) ] × 5

⇒ 5 = [ 8 ÷ (78 − 31 − x) ] × 5

⇒ 1 = 8 ÷ (47 − x)

⇒ 47 − x = 8

⇒ x = 39

Case 2: Assuming 125−130 is the modal class

⇒ L = 125, f₁ = x, f₀ = 39, f₂ = 15, h = 5

⇒ 125 = 125 + [ (x − 39) ÷ (2x − 39 − 15) ] × 5

⇒ 0 = [ (x − 39) ÷ (2x − 54) ] × 5

⇒ x − 39 = 0

⇒ x = 39

Both cases consistently yield x = 39.

Why other options are incorrect:

Option (1) 48: If x = 48, the modal class is 125−130. Mode = 125 + [ (48 − 39) ÷ (2 × 48 − 39 − 15) ] × 5 = 125 + [ 9 ÷ 42 ] × 5 ≈ 126.07 ≠ 125.

Option (2) 40: If x = 40, the modal class is 125−130. Mode = 125 + [ (40 − 39) ÷ (2 × 40 − 39 − 15) ] × 5 = 125 + [ 1 ÷ 26 ] × 5 ≈ 125.19 ≠ 125.

Option (3) 31: If x = 31, the modal class is 120−125. Mode = 120 + [ (39 − 31) ÷ (2 × 39 − 31 − 31) ] × 5 = 120 + [ 8 ÷ 16 ] × 5 = 122.5 ≠ 125.

∴ The correct answer is 39.

Additional Information

Modal Class

The class interval in a grouped frequency distribution that has the highest frequency.

Empirical Relationship

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

Symmetric Distribution

In a perfectly symmetric distribution, Mean = Median = Mode.

The Correct answer is Tibet (China).

Key Points

• The Lipulekh Pass is a strategic mountain pass located in the Himalayas.
• It lies at the tri-junction of India, Nepal, and China (Tibet).
• The pass has historically been used for pilgrimage routes, including the Kailash Mansarovar Yatra.
• It serves as an important point for border trade between India and Tibet (China).
• In recent years, the area has been a subject of geopolitical discussions due to its strategic significance.
• The Lipulekh Pass is located in the Pithoragarh district of Uttarakhand, India.
• It is situated at an elevation of approximately 5,200 meters above sea level.
• The pass has a historical significance as it was mentioned in treaties like the 1815 Treaty of Sugauli.

 Additional Information

• Bangladesh
• Bangladesh shares a border with India but does not have mountain passes like Lipulekh as its terrain is mostly plains and delta regions.
• It is located to the east of India and is separated by the Radcliffe Line.
• The country is known for rivers like the Ganges, Brahmaputra, and Meghna.
• Bhutan
• Bhutan is a landlocked country sharing a border with India but does not have the Lipulekh Pass.
• It lies to the northeast of India and is known for the Himalayan mountain range.
• The country has close ties with India under the Indo-Bhutan Friendship Treaty.
• Afghanistan
• Afghanistan shares no common border with India and is separated by Pakistan.
• The terrain of Afghanistan primarily includes mountains and arid regions.
• India engages with Afghanistan through development projects and historical ties, but it does not involve any direct mountain passes like Lipulekh.

The given code language is:

Thus, 'F' is coded as '4'.

Hence, the correct answer is "Option 4".

Shortcut Trick

For supplementary angles, the sum is always 180°.

Let the angles be L (larger) and S (smaller).

L + S = 180° and L − S = 60°

Smaller Angle (S) = (Sum − Difference) ÷ 2

S = (180° − 60°) ÷ 2 = 120° ÷ 2 = 60°

∴ The correct answer is 60°.

Alternate Method

Given:

Sum of supplementary angles = 180°

Difference between the two angles = 60°

Formula Used:

Sum of angles on a straight line = 180°

Calculations:

Let the smaller angle be x.

According to the question, the larger angle = x + 60°

⇒ x + (x + 60°) = 180°

⇒ 2x + 60° = 180°

⇒ 2x = 180° − 60°

⇒ 2x = 120°

⇒ x = 120° ÷ 2

⇒ x = 60°

∴ The correct answer is 60°.

The Correct answer is United States of America.

Key Points

• The doctrine of judicial review originated in the United States of America.
• The concept was established through the landmark case of Marbury v. Madison (1803), which was presided over by Chief Justice John Marshall.
• This case established the principle that the Supreme Court of the United States has the authority to review and invalidate laws and actions of the executive and legislative branches if they conflict with the U.S. Constitution.
• Judicial review is considered a fundamental aspect of the separation of powers and ensures that no branch of government exceeds its constitutional limits.
• The doctrine serves as a mechanism to protect the rights of individuals and maintain the supremacy of the Constitution in the governance of the country.
• In the United States, judicial review plays a critical role in maintaining the principle of checks and balances among the three branches of government: the legislative, executive, and judiciary.

 Additional Information

• United Kingdom
• The United Kingdom follows the principle of parliamentary sovereignty, meaning Parliament is the supreme legal authority.
• Unlike the United States, the concept of judicial review is limited in the UK. Courts cannot overrule legislation passed by Parliament but can review whether public authorities act within their legal powers.
• India
• India adopted the concept of judicial review from the United States, and it is enshrined in the Indian Constitution.
• Articles 13, 32, and 136 of the Indian Constitution empower courts to review the constitutionality of laws and executive actions.
• France
• France follows the civil law system, and judicial review as in the U.S. does not exist.
• The Constitutional Council of France examines the constitutionality of laws but only before they are enacted, not after.

Shortcut Trick

Mode is the value that appears most frequently in a data set.

Count the frequency of each observation in the given data set:

Observation 3 appears 4 times, which is the highest frequency among all numbers.

∴ The correct answer is 3.

Alternate Method

Given observations: 3, 6, 5, 7, 8, 4, 3, 3, 2, 5, 3, 6, 4, 8, 5

Formula Used:

Mode = Value with the maximum frequency in the given data set.

Let us arrange the data in ascending order to count the frequencies easily:

2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8

Now, let us list the frequency of each observation:

⇒ Frequency of 2 = 1

⇒ Frequency of 3 = 4

⇒ Frequency of 4 = 2

⇒ Frequency of 5 = 3

⇒ Frequency of 6 = 2

⇒ Frequency of 7 = 1

⇒ Frequency of 8 = 2

Since the observation 3 has the maximum frequency of 4, the mode of the given data is 3.

Option Analysis:

⇒ Option (1) - 6: Frequency of 6 is 2, which is less than the maximum frequency of 4.

⇒ Option (2) - 7: Frequency of 7 is 1, which is less than the maximum frequency of 4.

⇒ Option (3) - 5: Frequency of 5 is 3, which is less than the maximum frequency of 4.

⇒ Option (4) - 3: Frequency of 3 is 4, which is the highest frequency. Hence, this is the correct option.

∴ The correct answer is 3.

Additional Information

Mean, Median, and Mode Relationship

For a moderately skewed distribution, the empirical relationship is: Mode = 3 × Median − 2 × Mean.

Types of Mode Distributions

A data set can have one mode (unimodal), two modes (bimodal), three modes (trimodal), or no mode if all values occur with the same frequency.

Median of Data

The median is the middlemost value when data is arranged in ascending or descending order. For N = 15 (odd), Median = ((15 + 1) ÷ 2)^th term = 8^th term = 5.

The Correct answer is Eliminating all private sector participation in defence manufacturing.

Key Points

• The updated Defence Acquisition Procedure (DAP) 2024 aims to strengthen India's self-reliance in defence manufacturing under the 'Aatmanirbhar Bharat' initiative.
• The government is actively encouraging private sector participation in defence manufacturing to boost innovation, competition, and efficiency.
• Rather than eliminating private sector involvement, the DAP 2024 promotes collaboration between Indian defence firms and global Original Equipment Manufacturers (OEMs).
• The procedure emphasizes streamlined procurement timelines, reducing bureaucratic hurdles, and improving operational efficiency.
• The focus is on higher indigenisation requirements, ensuring that defence contracts support local manufacturing and technology development.
• Encouraging private sector participation aligns with India's goal of becoming a global hub for defence production.
• Eliminating private sector participation would contradict the government's vision for fostering in defence manufacturing.

 Additional Information

• Streamlining procurement timelines and reducing bureaucratic hurdles
• One of the primary objectives of DAP 2024 is to simplify and expedite procurement processes.
• This includes reducing bureaucratic delays to ensure faster acquisition of defence equipment.
• The streamlined approach is expected to support India's armed forces in acquiring modern and indigenous equipment without unnecessary delays.
• Increasing collaboration between Indian defence firms and global OEMs
• The updated DAP emphasizes partnerships between Indian defence companies and international Original Equipment Manufacturers (OEMs).
• Such collaborations aim to transfer advanced technology to India while promoting local manufacturing capabilities.
• It also helps Indian firms gain access to global markets and improve their competitiveness.
• Promoting higher indigenisation requirements for defence contracts
• The DAP 2024 emphasizes the need for greater indigenisation in defence contracts.
• Indigenisation requirements ensure that a significant portion of components, technology, and manufacturing is sourced locally.
• This boosts India's domestic defence industry and reduces dependence on foreign imports.

Shortcut Trick

Let the longer side be L and shorter side be (L − 18).

Given: Area = 6 × Perimeter

L(L − 18) = 6 × 2(L + L − 18) ⇒ L(L − 18) = 12(2L − 18)

Test options: If L = 36, then 36(18) = 648 and 12(72 − 18) = 12(54) = 648. Both match.

∴ The correct answer is 36 cm.

Alternate Method

Given:

Let Longer side (Length) = x cm

Shorter side (Breadth) = (x − 18) cm

Formula Used:

Area of Rectangle = Length × Breadth

Perimeter of Rectangle = 2(Length + Breadth)

Calculations:

⇒ Area = 6 × Perimeter

⇒ x(x − 18) = 6 × [2(x + x − 18)]

⇒ x² − 18x = 12(2x − 18)

⇒ x² − 18x = 24x − 216

⇒ x² − 42x + 216 = 0

⇒ x² − 36x − 6x + 216 = 0

⇒ x(x − 36) − 6(x − 36) = 0

⇒ (x − 36)(x − 6) = 0

⇒ x = 36 or x = 6

⇒ Since breadth = x − 18, x cannot be 6 (as side cannot be negative).

⇒ Length = 36 cm

∴ The correct answer is 36 cm.

Additional Information

Properties of Rectangle

A rectangle has opposite sides equal and all angles are 90°. Diagonal d = √(L² + B²).

Quadratic Roots Shortcut

For ax² + bx + c = 0, if the sum of roots is −b/a and product is c/a, the factors of 216 that sum to 42 are 36 and 6.

The Correct answer is Xylem and phloem present on the same radius.

Key Points

• A conjoint vascular bundle is a type of vascular bundle where both xylem and phloem are present together on the same radius of the stem or root.
• It is a characteristic feature of most angiosperms and is commonly found in dicot stems and monocot stems.
• In this type of arrangement, the xylem is typically located towards the inner side (adaxial side), while the phloem is positioned towards the outer side (abaxial side).
• Conjoint vascular bundles are further classified into two types:
• Collateral: Xylem and phloem are present side by side in the same bundle (e.g., sunflower stem).
• Bicollateral: Xylem is sandwiched between two layers of phloem (e.g., Cucurbita stem).
• In most cases, conjoint vascular bundles are surrounded by a layer of parenchyma cells, which form the vascular cambium in dicots.
• This arrangement is efficient for the transport of water, minerals, and organic nutrients throughout the plant body.

 Additional Information

• Xylem and phloem alternate along the radii
• This arrangement describes a radial vascular bundle, where xylem and phloem are located on alternate radii.
• Radial vascular bundles are found in roots of both monocots and dicots.
• Xylem only on the outer side
• This statement is incorrect because xylem is usually located on the inner side in a conjoint vascular bundle.
• However, in open vascular bundles, the cambium allows for secondary growth, leading to the formation of additional xylem on the inside.
• Cambium is absent
• This condition is seen in closed vascular bundles, which are common in monocots.
• In such bundles, there is no cambium, so secondary growth does not occur.

The Correct answer is Alt + F.

Key Points

• The keyboard shortcut Alt + F is commonly used to open the File menu in most legacy and standard Windows applications.
• This shortcut allows users to quickly access options such as Open, Save, Save As, Print, Exit, and other file-related commands.
• It is part of the keyboard accessibility features designed to improve productivity by reducing reliance on mouse navigation.
• Legacy applications follow this convention, making Alt + F a widely recognized shortcut across many software programs.
• By pressing the Alt key, the application activates its menu bar, and the letter F corresponds to the File menu, following the underlined shortcut convention.
• This shortcut improves user efficiency, especially for tasks requiring frequent interaction with file operations.
• It is useful for accessibility purposes as it allows users with physical limitations to navigate through applications without using a mouse.

 Additional Information

• Alt + Tab
• The Alt + Tab combination is used to switch between open applications or windows in a Windows environment.
• This shortcut improves multitasking and workflow efficiency by allowing quick navigation between programs.
• Alt + Insert
• Alt + Insert is less commonly used but may have specific functions in certain applications, such as inserting elements or commands.
• Its functionality depends on the program in use, and it is not standardized for accessing menus.
• Alt + Delete
• Alt + Delete is not a standard shortcut in Windows for menu navigation.
• It may have specific uses in certain applications, such as editing or deleting content.

Shortcut Trick

Income = 100%

Savings = 100% − 55% = 45%

45% ratio → ₹6,408

Expenses (55%) = (6,408 ÷ 45) × 55 = 1,424 × (55 ÷ 10) = 142.4 × 55 = 7,832

∴ The correct answer is ₹7,832.

Alternate Method

Given:

Spending percentage = 55%

Monthly Savings = ₹6,408

Let the Monthly Income be x.

⇒ Savings percentage = 100% − 55% = 45%

⇒ 45% of x = 6,408

⇒ (45 ÷ 100) × x = 6,408

⇒ x = (6,408 × 100) ÷ 45

⇒ x = 14,240

⇒ Monthly Expenses = 55% of 14,240

⇒ Monthly Expenses = (55 ÷ 100) × 14,240

⇒ Monthly Expenses = 0.55 × 14,240 = 7,832

∴ The correct answer is ₹7,832.

Additional Information

Income-Expenditure Formula

Total Income = Total Expenditure + Total Savings.

Percentage Change in Income

If income increases by r%, new income = Original Income × (100 + r)/100.

Savings Ratio

Savings % = (Savings ÷ Income) × 100.

The Correct answer is Tons.

Key Points

• The Tons River is one of the largest and most important tributaries of the Yamuna River.
• It originates from the Bandarpunch Glacier in the Garhwal region of Uttarakhand.
• The Tons River merges with the Yamuna near Kalsi in the Dehradun district of Uttarakhand.
• The river is known for its rugged terrain and steep gorges, making it a significant river in the Himalayan region.
• The Tons River is also popular for adventure sports like rafting due to its high-speed water currents.
• This river contributes significantly to the water flow of the Yamuna, especially during the monsoon season.
• It flows through regions rich in biodiversity and passes through dense forests and valleys.

 Additional Information

• Son River
• The Son River is a major tributary of the Ganga, not the Yamuna.
• It originates from Amarkantak Plateau in Madhya Pradesh.
• It flows through states like Madhya Pradesh, Uttar Pradesh, Jharkhand, and Bihar.
• The river is known for its sandy riverbed and seasonal flow variations.
• Teesta River
• The Teesta River is a tributary of the Brahmaputra, not the Yamuna.
• It originates from the Tso Lhamo Lake in North Sikkim.
• The river flows through Sikkim and West Bengal, and then enters Bangladesh.
• It is a vital source of irrigation and hydroelectricity in the region.
• Gomti River
• The Gomti River is a tributary of the Ganga, not the Yamuna.
• It originates from Gomat Taal (also known as Fulhar Lake) near Madhotanda in Uttar Pradesh.
• The river flows entirely within Uttar Pradesh and passes through cities like Lucknow.
• The Gomti is considered sacred in Hindu tradition and is linked to religious practices.

The logic followed here is:

LQM : NSN

ETG : GVH

Now,

Option 1: BKL : FJL

Option 2: WHO : XZQ

Option 3: BRQ : DTR

Option 4: GHM : KMO

Thus, Option 3 follows the correct logic.

Hence, the correct answer is "Option 3".

The Correct answer is West.

Key Points

• The Malwa Plateau is located in the central-western region of India and is part of the larger Deccan Plateau.
• The Chotanagpur Plateau, on the other hand, lies in the eastern region of India, covering parts of Jharkhand, Odisha, and West Bengal.
• When comparing the relative locations of these two plateaus, the Malwa Plateau lies to the west of the Chotanagpur Plateau.
• The Malwa Plateau is primarily spread across the states of Madhya Pradesh and parts of Rajasthan.
• The Chotanagpur Plateau is rich in mineral resources such as coal, iron ore, and mica, while the Malwa Plateau is known for its black soil, which is suitable for growing crops like cotton and soybean.
• The location of these plateaus is significant for understanding the geography, resource distribution, and agricultural practices of India.
• To summarize, the Malwa Plateau lies westward when viewed relative to the position of the Chotanagpur Plateau on the map of India.

Shortcut Trick

Fractional rate of 25% = 1/4. Population increases from 4 to 5 each year.

Let population 3 years ago be P. Then 2 years ago, it was P × (5/4). Current population = P × (5/4)³.

P × (125/64) = 2,50,000 ⇒ P = 1,28,000.

Difference = (Population 2 years ago) − (Population 3 years ago) = P × (5/4 − 1) = P × 1/4.

Difference = 1,28,000 ÷ 4 = 32,000.

∴ The correct answer is 32,000.

Alternate Method

Given: Current Population = 2,50,000; Rate (R) = 25% per annum; Time (n) = 3 years.

Formula Used: Population n years ago = P / (1 + R/100)ⁿ

⇒ Population 3 years ago = 2,50,000 / (1 + 25/100)³

⇒ Population 3 years ago = 2,50,000 / (5/4)³ = 2,50,000 × (64/125) = 1,28,000

⇒ Population 2 years ago = (Population 3 years ago) × (1 + 25/100)

⇒ Population 2 years ago = 1,28,000 × 1.25 = 1,60,000

⇒ Required Difference = 1,60,000 − 1,28,000 = 32,000

∴ The correct answer is 32,000.

Additional Information

Population Growth Formula

Future Population = P(1 + R/100)ⁿ, where P is initial population, R is growth rate, and n is time.

Population Decrease Formula

If population decreases, use P(1 − R/100)ⁿ. This is similar to the Depreciation formula in accounting.

Successive Percentage Change

For two years with different rates R₁ and R₂, the final population is P(1 + R₁/100)(1 + R₂/100).

The Correct answer is Customising the top-level master ensures consistency of universal elements across all slide layouts.

Key Points

• In MS PowerPoint, the Slide Master View allows users to edit the slide master, which controls the overall design and layout of the presentation.
• Customising the top-level master slide ensures that universal design elements such as backgrounds, fonts, themes, colors, and logos are applied consistently across all individual slide layouts in the presentation.
• This approach enhances efficiency because the changes made at the top-level master are automatically inherited by all other layout slides, saving time compared to editing individual layouts.
• The top-level master is particularly helpful for maintaining brand consistency in corporate or professional presentations where uniformity is crucial.
• Universal elements added or modified at the top-level master slide create a cohesive look and feel, reducing the chances of discrepancies in formatting across slides.
• Elements like a company’s logo, header, footer, and slide numbering can be centrally managed through the top-level master slide.
• Customisation at this level does not overwrite individual layout-specific customizations unless explicitly intended, offering both flexibility and control.

Shortcut Trick

Find the net percentage difference in interest for the given time period.

Rate difference per annum = 6% − 3.5% = 2.5%

Total interest difference for 4 years = 2.5% × 4 = 10%

Required difference = 10% of ₹5,00,000 = ₹50,000

∴ The correct answer is ₹50,000.

Alternate Method

Given:

Principal (P) = ₹5,00,000 | Time (T) = 4 years

Rate Bank A (R₁) = 3.5% | Rate Bank B (R₂) = 6%

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

Calculations:

⇒ Simple Interest from Bank A = (5,00,000 × 3.5 × 4) ÷ 100

⇒ SI_A = 5,000 × 14 = ₹70,000

⇒ Simple Interest from Bank B = (5,00,000 × 6 × 4) ÷ 100

⇒ SI_B = 5,000 × 24 = ₹1,20,000

⇒ Positive Difference = SI_B − SI_A

⇒ Difference = ₹1,20,000 − ₹70,000 = ₹50,000

∴ The correct answer is ₹50,000.

Additional Information

Direct Difference Formula

When Principal and Time are constant, the difference in interest is given by: Diff = P × T × (R₂ − R₁) ÷ 100.

Effect of Simple Interest

Simple interest is calculated only on the Principal amount. It remains constant every year if the rate and principal do not change.

Annualized Rate Difference

The difference in the Total Amount (Principal + Interest) after any time T is the same as the difference in the Simple Interest paid.

The correct answer is Deepak Agarwal.

Key Points

• Deepak Agarwal was appointed as the Managing Director of National Agriculture Cooperative Marketing Federation of India Ltd (NAFED) in October 2024.
• NAFED is an apex organization under the Ministry of Agriculture and Farmers Welfare, Government of India, and was established in 1958.
• The organization plays a crucial role in promoting agricultural marketing and providing support to farmers by ensuring the procurement of their produce at minimum support prices (MSP).
• As the Managing Director, Deepak Agarwal is expected to lead initiatives to strengthen the agricultural marketing systems and further the objectives of the cooperative sector in India.
• His leadership at NAFED is pivotal in ensuring the organization’s continued commitment to improving the efficiency of marketing mechanisms and benefitting farmers across the country.
• NAFED is also involved in the implementation of various government schemes related to agriculture, such as the Price Support Scheme (PSS) and the Price Stabilization Fund (PSF).
• The appointment of a skilled professional like Deepak Agarwal reflects the organization's focus on operational excellence and innovation in the agricultural sector.

 Additional Information

• Praveena Rai
• Praveena Rai is the Chief Operating Officer (COO) of the National Payments Corporation of India (NPCI).
• She plays a key role in strengthening the digital payment ecosystem in India.
• Ashok Chandra
• Ashok Chandra is known for his contributions to the education sector in India.
• He has been associated with the implementation of various policy reforms in the field of education.
• Vipin Kumar
• Vipin Kumar is noted for his work in the scientific research domain.
• He has been associated with organizations such as the National Innovation Foundation, promoting grassroots innovations.

The logic followed here is.

Option (1) 'HK - GJ':

Option (2) 'JH - KL':

Option (3) 'PS - OR':

Option (4) 'LO - KN':

Thus, option 2 is odd one.

Hence, the correct answer is "Option 2".

Given,

Mr. Alex starts from Point A and drives 9 km towards the South.

He then takes a left turn, drives 3 km, turns left and drives 19 km.

He then takes a left turn and drives 7 km, turns left drives 5 km.

He then takes a right turn and drives 4 km.

He takes a final left turn, drives 5 km and stops at Point P.

Mr. Alex's final position, Point P, is 8 km West of Point A.

To return to Point A, he must drive 8 km in the East direction.

Hence, the correct answer is "Option 3".

The correct answer is Silver.

Key Points

• During the Mauryan Empire, silver was extensively used for coinage.
• The Mauryans issued coins made of punch-marked silver, which are considered one of the earliest forms of currency in ancient India.
• These coins were called Punch-marked coins, as their designs were made by punching symbols onto the metal.
• Silver was preferred due to its high intrinsic value and widespread availability during that era.
• The Mauryan Empire had access to silver mines, which facilitated the extensive use of silver for monetary purposes.
• Coins made of silver played a significant role in the economic system of the Mauryan Empire, enabling trade and commerce to flourish.
• These silver coins were used not just in local transactions but also in long-distance trade, reflecting the empire's vast reach and commercial activities.
• The use of silver coins during the Mauryan period marked an important development in the history of Indian numismatics.

 Additional Information

• Gold
• Gold coins were used in later periods of Indian history, especially during the Gupta Empire, which is known for its highly artistic gold coins.
• Gold was considered a symbol of wealth and prestige, but it was not extensively used for coinage in the Mauryan period.
• Iron
• Iron was primarily used for tools, weapons, and infrastructure during the Mauryan period.
• It was not a suitable material for coinage due to its lower intrinsic value compared to metals like silver and gold.
• Copper
• Copper coins were also issued during the Mauryan Empire, but they were less valuable and mostly used for smaller transactions.
• Copper coins were often used in conjunction with silver coins to facilitate trade and commerce.

Shortcut Trick

For similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides.

Ratio of sides (BC : QR) = 4 : 8 = 1 : 2

Perimeter of ΔABC = Perimeter of ΔPQR × (1 ÷ 2)

Perimeter of ΔABC = 32 × 0.5 = 16 cm

∴ The correct answer is 16 cm.

Alternate Method

Given:

ΔABC ∼ ΔPQR

QR = 8 cm, BC = 4 cm

Perimeter of ΔPQR = 32 cm

Formula Used:

If ΔABC ∼ ΔPQR, then:

Perimeter(ΔABC) ÷ Perimeter(ΔPQR) = AB ÷ PQ = BC ÷ QR = AC ÷ PR

P Q R A B C 8 cm 4 cm Perimeter = 32 cm Perimeter = ?

The correct answer is Density.

Key Points

• Density is defined as the mass per unit volume of a substance.
• It is mathematically expressed as Density = Mass/Volume.
• The dimensional formula for mass is [M], and for volume, it is [L³].
• Thus, the dimensional formula for density is [M L⁻³].
• This dimensional formula indicates that density has one unit of mass (M) and is inversely proportional to the cube of the length (L).
• Density is an important physical quantity used in various fields such as fluid mechanics, material science, and engineering applications.
• Examples of density include the density of water as 1000 kg/m³ and the density of air as approximately 1.225 kg/m³ at sea level.

 Additional Information

• Pressure
• Pressure is defined as the force applied per unit area.
• Its dimensional formula is [M L⁻¹ T⁻²], which is different from density.
• Pressure is measured in units such as Pascals (Pa) in the SI system.
• Applications of pressure include hydraulics, pneumatics, and weather prediction.
• Work
• Work is defined as the product of force and displacement in the direction of the force.
• Its dimensional formula is [M L² T⁻²].
• Work is measured in Joules (J) in the SI system.
• It is widely used in physics and engineering to analyze energy transfer.
• Energy
• Energy is the capacity to perform work.
• Its dimensional formula is the same as that of work, [M L² T⁻²].
• Energy is measured in Joules (J) in the SI system.
• Examples include kinetic energy, potential energy, and thermal energy.

The logic followed here is:

Thus, the missing number in the series is 418.

Hence, the correct answer is "Option 1".

The Correct answer is 5.27%.

Key Points

• The National Multidimensional Poverty Index (MPI) 2023 is based on data from the NFHS-5 (2019-21).
• The MPI measures poverty using three dimensions: health, education, and standard of living.
• In urban areas, the poverty headcount ratio for 2019-21 was calculated as 5.27%, indicating a significant reduction in multidimensional poverty in urban regions.
• The headcount ratio reflects the proportion of individuals who are deprived in at least one-third of the weighted indicators.
• The data highlights India's progress in improving living conditions and reducing poverty in urban areas.
• This reduction is attributed to various government initiatives such as housing programs, employment schemes, and improved healthcare facilities.
• The report is aligned with Sustainable Development Goals (SDGs), particularly Goal 1, which aims to eradicate poverty in all its forms.
• The findings underscore the importance of targeted policies in addressing urban poverty and ensuring equitable development.

 Additional Information

• National Multidimensional Poverty Index (MPI)
• The MPI is developed by the NITI Aayog in collaboration with Oxford Poverty and Human Development Initiative and the United Nations Development Programme (UNDP).
• It adopts a holistic approach to measuring poverty, moving beyond income-based assessments.
• The index uses 12 indicators grouped under three dimensions: health, education, and living standards.
• Indicators include access to nutrition, child mortality, school attendance, sanitation, and drinking water, among others.
• NFHS-5 (National Family Health Survey)
• The NFHS-5 was conducted during 2019-21 and provides comprehensive data on population health, nutrition, and demographics.
• It is a flagship program of the Ministry of Health and Family Welfare in India.
• The survey covers urban and rural areas, offering valuable insights into disparities and areas requiring intervention.

The given code language is:

Thus, 'T' is coded as '1'.

Hence, the correct answer is "Option 1".

Shortcut Trick

Let the present age of Sujatha be S and Vanita be V.

From the first condition: 6S = 3V + 6 → 2S − V = 2 → V = 2S − 2.

From the second condition, after 7 years: 4(V + 7) = 6(S + 7) − 6 → 4V + 28 = 6S + 36 → 4V − 6S = 8.

Substitute V = 2S − 2 into the second equation: 4(2S − 2) − 6S = 8 → 2S = 16 → S = 8 years, and V = 14 years.

The difference k = V − S = 14 − 8 = 6 years.

∴ The correct answer is 6 years.

Alternate Method

Let Sujatha's present age be S and Vanita's present age be V.

According to the first statement:

⇒ 6S = 3V + 6

⇒ 2S = V + 2

⇒ V = 2S − 2   —— (Equation 1)

According to the second statement, after 7 years:

⇒ Age of Sujatha after 7 years = S + 7

⇒ Age of Vanita after 7 years = V + 7

⇒ 4(V + 7) = 6(S + 7) − 6

⇒ 4V + 28 = 6S + 42 − 6

⇒ 4V + 28 = 6S + 36

⇒ 4V − 6S = 8

⇒ 2V − 3S = 4   —— (Equation 2)

Substitute Equation 1 into Equation 2:

⇒ 2(2S − 2) − 3S = 4

⇒ 4S − 4 − 3S = 4

⇒ S = 8

Substitute S = 8 into Equation 1:

⇒ V = 2(8) − 2 = 14

We are given that Vanita is k years older than Sujatha:

⇒ V = S + k

⇒ 14 = 8 + k

⇒ k = 6

∴ The correct answer is 6 years.

Additional Information

Linear Equations in Two Variables

Problems on ages are solved by formulating algebraic equations based on present conditions and past/future shifts, then using substitution or elimination methods.

Age Difference Invariance

The difference in age between two people remains constant throughout their lives. Thus, the value of k will always be the same regardless of the number of years passed.

The logic followed here is by observing the alphabetical position shifts from the letters of the first cluster to the corresponding letters of the second cluster.

Option (1) 'TL – BV':

Option (2) 'ZR – HB':

Option (3) 'GY – OJ':

Option (4) 'PH – XR':

In options 1, 2, and 4, the first letter of the cluster pair moves +8 positions forward and the second letter moves +10 positions forward.

In option 3, the first letter moves +8 positions forward, but the second letter moves +11 positions forward.

Hence, the correct answer is "Option 3".