Shortcut Trick

Required percentage = [r ÷ (100 + r)] × 100

Given r = 16%

Required % = [16 ÷ (100 + 16)] × 100

Required % = [16 ÷ 116] × 100 = 400 ÷ 29 ≈ 13.79%

∴ The correct answer is 13.79%.

Alternate Method

Given:

A's salary is 16% more than B's salary.

⇒ Let B's salary = 100

⇒ A's salary = 100 + 16% of 100 = 116

Calculations:

⇒ Difference between A and B = 116 − 100 = 16

⇒ B's salary is less than A's salary by = (Difference ÷ A's salary) × 100

⇒ Percentage less = (16 ÷ 116) × 100

⇒ Percentage less = (4 ÷ 29) × 100

⇒ Percentage less = 400 ÷ 29 = 13.7931...

⇒ Correct to two decimal places = 13.79%

∴ The correct answer is 13.79%.

Additional Information

Percentage Comparison Rule

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.

Base Value Concept

The percentage change depends on the denominator (base value). When comparing B to A, A's value becomes the base.

The Correct answer is Luni River.

Key Points

• The Luni River is the largest river in the Thar Desert region of northwestern India.
• It originates from the and flows through the desert.
• The Luni River is known for its saline water, especially in the lower reaches.
• It finally disappears into the marshy lands of the Rann of Kutch without forming a delta or flowing into the sea.
• Due to its salinity, the Luni River is not suitable for drinking or irrigation, except in its upper course where the water is relatively fresh.
• The river supports life in the arid region by enabling limited agricultural activities and acts as a lifeline for the settlements along its course.
• Its tributaries include Sukri, Jawai, Bandi, and Guhiya rivers, which contribute to its seasonal flow.

 Additional Information

• Yamuna River
• The Yamuna River is a major tributary of the Ganges River and flows through northern India.
• **Its source is at the **Yamunotri glacier in the Uttarakhand Himalayas.
• The Yamuna is significant for its use in agriculture, drinking water, and industrial activities in the regions it flows through.
• It does not flow through the Thar Desert region.
• Sutlej River
• The Sutlej River is one of the five rivers of Punjab and is a major river in the Indus River system.
• It originates in the Tibet region and flows through Himachal Pradesh and Punjab.
• The Sutlej River does not pass through the Thar Desert region.
• Ghaggar River
• The Ghaggar River is a seasonal river that originates in Himachal Pradesh.
• **It flows through Haryana and Punjab and is believed to be the remnant of the ancient **Saraswati River.
• It does not flow through the Thar Desert region.

The given code language is:           

Thus, 'S' is coded as '3'.

Hence, the correct answer is "Option 1".

Shortcut Trick

Subtract the two scenarios: (19 chairs + 19 tables) − (15 chairs + 19 tables) = ₹5,700 − ₹5,300.

This gives 4 chairs = ₹400 ⇒ Cost of 1 chair = ₹100.

From the second set: 19 tables = 5,700 − (19 × 100) = ₹3,800 ⇒ Cost of 1 table = ₹200.

Required cost for 12 chairs and 13 tables = (12 × 100) + (13 × 200) = 1,200 + 2,600 = ₹3,800.

∴ The correct answer is ₹3,800.

Alternate Method

Given:

Cost of 15 chairs and 19 tables = ₹5,300

Cost of 19 chairs and 19 tables = ₹5,700

Formula Used:

Total Cost = (Number of chairs × Price per chair) + (Number of tables × Price per table)

Calculations:

⇒ Let C be the cost of one chair and T be the cost of one table.

⇒ 15C + 19T = 5300 — (Eq. 1)

⇒ 19C + 19T = 5700 — (Eq. 2)

⇒ Subtracting Eq. 1 from Eq. 2:

⇒ (19C − 15C) + (19T − 19T) = 5700 − 5300

⇒ 4C = 400

⇒ C = 100

⇒ Substitute C = 100 in Eq. 2:

⇒ 19(100) + 19T = 5700

⇒ 1900 + 19T = 5700

⇒ 19T = 5700 − 1900 = 3800

⇒ T = 3800 ÷ 19 = 200

⇒ Cost of 12 chairs and 13 tables = 12C + 13T

⇒ Cost = 12(100) + 13(200) = 1200 + 2600 = 3800

∴ The correct answer is ₹3,800.

Additional Information

Simultaneous Linear Equations

Equations with two variables can be solved using the elimination method (removing one variable) or the substitution method.

Unitary Method

Finding the value of a single unit first helps in calculating the value of any required number of units easily.

Total Cost Formula

Total Cost = ∑ (Rate_i × Quantity_i). For multiple items, sum the individual costs of each item category.

Given:

1 is added to each odd digit and 1 is subtracted from each even digit in the number 8146752.

The new number formed is 7235861.

The last digit from the left is 1.

The first digit from the left is 7.

Difference between the last and first digits = 7 - 1 = 6.

Hence, the correct answer is "Option 2".

The Correct answer is 5 years.

Key Points

• The Kartarpur Corridor is a landmark initiative between India and Pakistan to facilitate Sikh pilgrims' travel to the holy site of Gurdwara Darbar Sahib, Kartarpur, in Pakistan.
• It connects Dera Baba Nanak in the Gurdaspur district of Punjab, India, to Kartarpur Sahib in Pakistan.
• In October 2024, both nations agreed to extend the agreement for an additional 5 years, showing mutual cooperation and respect for religious sentiments.
• The corridor was originally inaugurated on 9^th November 2019, marking the 550^th birth anniversary of Guru Nanak Dev Ji.
• The agreement allows Sikh pilgrims to visit the site without a visa, provided they obtain a permit.
• The extension reflects the corridor's significance in fostering peaceful relations and promoting cultural diplomacy between the two nations.
• The Gurdwara Kartarpur Sahib is where Guru Nanak Dev Ji spent the last years of his life, making it one of the most revered places for Sikhs.
• The corridor is a symbol of cross-border religious harmony, and its extension for 5 years ensures continued access for pilgrims to this holy site.

The Correct answer is Devabhuti.

Key Points

• Devabhuti was the last ruler of the Shunga Dynasty, which ruled parts of India from approximately 185 BCE to 75 BCE.
• The Shunga Dynasty was founded by Pushyamitra Shunga, who overthrew the Mauryan Empire.
• Devabhuti's reign marked the decline of the Shunga Dynasty, and he is believed to have been overthrown by his minister Vasudeva Kanva, who established the Kanva Dynasty.
• Historical accounts suggest that Devabhuti was known for his indulgent lifestyle, which led to political instability and weakened the Shunga state.
• The Shunga Dynasty is significant in Indian history for its contributions to art, culture, and education, including the development of Buddhist stupas like those at Sanchi and Bharhut.
• The dynasty primarily ruled over areas of present-day northern and central India.

 Additional Information

• Agnimitra
• Agnimitra was the second ruler of the Shunga Dynasty and the son of Pushyamitra Shunga, the dynasty's founder.
• He is a prominent character in the Sanskrit play Mālavikāgnimitram, written by Kalidasa.
• Agnimitra is credited with consolidating the Shunga rule and expanding its influence.
• Bhagabhadra
• Bhagabhadra was a ruler of the Shunga Dynasty who is known from historical inscriptions, including a Heliodorus pillar inscription at Vidisha.
• The Heliodorus pillar, erected by a Greek ambassador, suggests cultural exchange during his reign.
• Vasumitra
• Vasumitra is believed to have been a successor of Agnimitra in the Shunga Dynasty.
• He is mentioned in some historical texts as having participated in military campaigns to secure the Shunga Empire's frontiers.

The Correct answer is more than 1.

Key Points

• In the context of taxation, tax buoyancy measures the responsiveness of tax revenue growth to the growth of Gross Domestic Product (GDP).
• A tax buoyancy of more than 1 indicates that tax revenue grows faster than the GDP, showcasing an efficient tax system.
• As per the Ministry of Finance, the SGST (State Goods and Services Tax) revenue, including compensation to states, has consistently demonstrated a buoyancy above 1 in the post-GST period from 2017-18 to 2022-23.
• This reflects the success of the GST implementation in ensuring sustainable revenue growth for state governments while streamlining the taxation system.
• The introduction of GST replaced a complex tax structure with a unified system, enhancing compliance and improving revenue collection efficiency.
• The government has also provided compensation to states for any revenue shortfall due to the transition to GST, ensuring financial stability for states.
• This buoyancy signifies that GST has been instrumental in improving the tax-to-GDP ratio, contributing to fiscal consolidation goals.

 Additional Information

• Zero
• A tax buoyancy of zero implies no growth in tax revenue despite GDP growth, indicating inefficiency in the tax system.
• This scenario is unlikely for GST, as the tax reform has been designed to improve compliance and revenue collection.
• Less than 1
• A tax buoyancy of less than 1 would mean that tax revenue grows at a slower rate than GDP, which is not reflective of the GST revenue trends observed in India.
• GST implementation has significantly improved tax collection efficiency, ensuring buoyancy above 1.
• Greater than 10
• A tax buoyancy of greater than 10 is an extreme and unrealistic figure, as it would imply an exceptionally high growth in tax revenue compared to GDP growth.
• Such a scenario is unlikely and not observed in the context of India's GST revenue.

Shortcut Trick

Given a, b, c are in continued proportion (a : b :: b : c).

The relation is b² = ac.

Substitute b = 6 and c = 9a (so a = c/9).

6² = (c/9) × c ⇒ 36 = c² / 9 ⇒ c² = 324 ⇒ c = 18.

∴ The correct answer is 18.

Alternate Method

Given:

Proportion: a : b :: b : c

Values: b = 6, c = 9a

Formula Used:

For continued proportion, Product of extremes = Product of means ⇒ ac = b²

Calculations:

⇒ a/b = b/c

⇒ b² = a × c

⇒ Substitute c = 9a → a = c/9

⇒ 6² = (c/9) × c

⇒ 36 = c² / 9

⇒ c² = 36 × 9

⇒ c² = 324

⇒ c = √324

⇒ c = 18

∴ The correct answer is 18.

Additional Information

Continued Proportion

Three numbers a, b, and c are in continued proportion if a/b = b/c. Here, b is called the mean proportional between a and c.

Mean Proportional Formula

The mean proportional b between two numbers a and c is calculated as b = √(ac).

Third Proportional

In the proportion a : b :: b : c, the term c is known as the third proportional to a and b, where c = b²/a.

Shortcut Trick

Interest for 1 year = 10% of ₹2,500 = ₹250

Total interest to be fetched = ₹1,000

Time = Total Interest ÷ Interest per year

⇒ Time = 1000 ÷ 250 = 4 years

∴ The correct answer is 4 years.

Alternate Method

Given:

Principal (P) = ₹2,500

Simple Interest (SI) = ₹1,000

Rate (R) = 10% per annum

Formula Used:

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

Calculations:

⇒ 1000 = (2500 × 10 × T) ÷ 100

⇒ 1000 = 250 × T

⇒ T = 1000 ÷ 250

⇒ T = 4 years

∴ The correct answer is 4 years.

Additional Information

Total Amount in Simple Interest

Amount (A) is the sum of Principal (P) and Interest (I). Formula: A = P + (P × R × T) ÷ 100 or A = P(1 + RT/100).

Compound Interest (CI) vs Simple Interest (SI)

Simple interest is calculated only on the principal, whereas CI is calculated on the principal plus the accumulated interest of previous periods.

Rule of 72

A quick way to estimate how long it takes to double your money: Time ≈ 72 ÷ Rate (usually applied to Compound Interest).

The Correct answer is Laws for reorganisation of states.

Key Points

• Article 4 of the Indian Constitution deals specifically with the reorganisation of states and the related laws.
• It states that any laws enacted under Article 2 (admission or establishment of new states) or Article 3 (formation of new states and alteration of areas, boundaries, or names of existing states) are not considered amendments to the Constitution.
• This provision under Article 4 ensures that such laws can be passed by a simple majority in the Parliament rather than requiring the more stringent process of a constitutional amendment.
• The reorganisation of states may involve altering the boundaries, names, or territories of states based on administrative, linguistic, cultural, or other considerations.
• Examples of state reorganisation include the creation of Telangana in 2014 and the bifurcation of Andhra Pradesh.

 Additional Information

• Emergency Provisions
• Emergency Provisions are covered under Articles 352 to 360 of the Indian Constitution.
• They allow the President to declare an emergency in the event of war, external aggression, armed rebellion, financial instability, or failure of constitutional machinery in a state.
• There are three types of emergencies: National Emergency, State Emergency, and Financial Emergency.
• Fundamental Rights
• Fundamental Rights are enshrined in Part III of the Indian Constitution and cover Articles 12 to 35.
• These rights include the Right to Equality, Right to Freedom, Right against Exploitation, Right to Freedom of Religion, Cultural and Educational Rights, and the Right to Constitutional Remedies.
• They protect the citizens against arbitrary actions of the state and ensure individual liberty.
• Financial Powers
• Financial Powers are primarily covered under Articles 264 to 293 of the Indian Constitution.
• These articles deal with the distribution of financial resources between the Union and the States, taxation powers, and the management of consolidated funds.
• It also includes the role of institutions like the Finance Commission in ensuring equitable distribution of revenues.

The correct answer is Bihar.

Key Points

• Bihar recorded the highest Multidimensional Poverty Index (MPI) as per the National Multidimensional Poverty Index 2023.
• The National Multidimensional Poverty Index is based on multiple deprivations experienced by households in areas such as health, education, and standard of living.
• Bihar, being one of India's most populous states, has faced persistent challenges in reducing poverty levels due to limited industrialization and dependency on agriculture.
• The state has shown significant efforts in improving various sectors, but it still struggles with key areas like malnutrition, lack of access to quality education, and inadequate infrastructure.
• In Bihar, a large percentage of the population experiences deprivation in aspects such as sanitation, electricity, and housing, leading to its poor MPI ranking.
• The MPI serves as a tool for assessing poverty holistically and identifying areas that require targeted intervention.

 Additional Information

• Uttar Pradesh
• Uttar Pradesh is the most populous state in India and has made strides in reducing poverty, but it still faces challenges in areas like education, healthcare, and employment opportunities.
• The state has implemented several welfare schemes to improve living standards, but the sheer size of the population poses difficulties.
• Rajasthan
• Rajasthan has a significant rural population, and poverty reduction efforts have been focused on improving water resources, agriculture, and rural development.
• The state has vast deserts, which create unique challenges in terms of infrastructure development.
• Madhya Pradesh
• Madhya Pradesh has a large tribal population, and poverty alleviation in these areas often requires targeted interventions addressing livelihood opportunities and education.
• The state has shown improvement in poverty reduction through government programs, but challenges remain in remote areas.

The Correct answer is Rajnigandha Achievers.

Key Points

• Rajnigandha Achievers emerged victorious in the Aditya Birla Memorial Polo Cup 2025, showcasing exceptional skill and teamwork.
• The team is known for its strategic gameplay and consistency in high-stakes polo tournaments.
• The Aditya Birla Memorial Polo Cup is a prestigious event that draws some of the best polo teams from across the country.
• The tournament is held annually and is recognized as one of the significant events in the Indian polo circuit.
• Polo is a fast-paced equestrian sport that requires a combination of athleticism, strategy, and coordination between players and their horses.
• Winning such a tournament reflects the dedication and commitment of the team toward excellence in the sport.

 Additional Information

• Jindal Panther
• Jindal Panther is a polo team associated with the Jindal Group, a prominent industrial conglomerate in India.
• The team is known for its strong performance in various polo tournaments across the country.
• Aditya Birla Polo Team
• This team represents the Aditya Birla Group, a leading multinational conglomerate.
• The team actively participates in polo tournaments and promotes the sport of polo in India.
• New Delhi Polo Team
• New Delhi Polo Team is a well-established polo team based in the capital city of India.
• They often compete in regional and national-level polo tournaments.

Given : Each of P, Q, R, S, T, U and V has an exam on a different day of a week starting from Monday and ending on Sunday of the same week.

1. Q has the exam on Sunday.

2. Only four people have exams between P and U.

3. T has the exam immediately after P.

4. R has the exam before S but after V.

Final arrangement:

 

There is 1 person who has an exam between V and P.

Hence, the correct answer is "Option 1".

Shortcut Trick

Upstream speed (U) = 37.2 km / (27/60) h = 82.67 kmph

Ratio Boat : Stream = 8 : 5 → Upstream ratio = 8 − 5 = 3 units; Downstream ratio = 8 + 5 = 13 units.

3 units = 82.66 kmph ⇒ 1 unit = 27.55 kmph ⇒ Downstream (13 units) = 358.22 kmph.

Total Time = (79.4 / 82.66) + (85.8 / 358.22) ≈ 0.96 + 0.24 = 1.2 hours.

∴ The correct answer is 1.2 hours.

Alternate Method

Given: Upstream Distance = 37.2 km, Time = 27 min, Ratio of Boat speed (v) to Stream speed (u) = 8 : 5.

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

⇒ U = 37.2 ÷ (27/60) = (37.2 × 60) ÷ 27 = 2232 ÷ 27 = 82.666... kmph.

⇒ Let v = 8x and u = 5x. Then U = 8x − 5x = 3x.

⇒ 3x = 82.666... ⇒ x = 27.555... kmph.

⇒ Downstream Speed (D) = 8x + 5x = 13x = 13 × 27.555... = 358.222... kmph.

⇒ Time for 79.4 km upstream = 79.4 ÷ 82.666... = 0.9605 hours.

⇒ Time for 85.8 km downstream = 85.8 ÷ 358.222... = 0.2395 hours.

⇒ Total Time = 0.9605 + 0.2395 = 1.2 hours.

∴ The correct answer is 1.2 hours.

Additional Information

Relative Speed in Water

Upstream speed is the difference (v − u) and Downstream speed is the sum (v + u) of the boat and stream speeds.

Conversion of Time

To convert minutes to hours, divide by 60. To calculate speed in kmph, distance in km must be divided by time in hours.

Boat Speed Calculation

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

Shortcut Trick

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

Ratio of sides = 3 : 10

Ratio of areas = 3² : 10²

⇒ Ratio of areas = (3 × 3) : (10 × 10) = 9 : 100

∴ The correct answer is 9 : 100.

Alternate Method

Given:

Ratio of corresponding sides of two similar triangles = 3 : 10

Formula Used:

Area₁ / Area₂ = (Side₁ / Side₂)²

Calculations:

Let the areas of the two triangles be A₁ and A₂.

Let the corresponding sides be s₁ and s₂.

The Correct answer is A computer where both data and instructions are stored in the same primary memory for sequential execution.

Key Points

• The Von Neumann architecture introduced the concept of a stored-program computer.
• In this architecture, both data and instructions are stored together in the same memory unit, enabling the processor to access them sequentially.
• This design uses a single memory space for storing both programs (instructions) and data, eliminating the need for separate storage units.
• The stored-program concept allows flexibility, enabling computers to execute different programs without hardware modifications.
• The architecture consists of key components such as a Central Processing Unit (CPU), memory, and I/O devices.
• The CPU executes instructions sequentially, fetching them from memory, decoding them, and performing operations as specified.
• This model laid the foundation for modern computing systems and is used in almost all general-purpose computers today.
• The concept revolutionized computing by introducing programmability and reducing dependency on external hardware for processing instructions.

 Additional Information

• A computer that can only execute a fixed set of instructions hardwired into its circuitry
• This describes an early computing model, such as those used in the pre-Von Neumann era.
• These systems, often referred to as hardwired computers, lacked flexibility and required physical rewiring to change instructions.
• A computer that relies primarily on external storage devices for program execution
• This describes systems that depend heavily on external media such as tapes or disks for program storage and execution.
• Such systems are less efficient compared to the stored-program model introduced by Von Neumann.
• A computer that utilises separate memory units for data and instructions to optimise processing speed
• This describes the Harvard architecture, which is distinct from the Von Neumann model.
• In the Harvard architecture, separate memory units for data and instructions enable parallel access, improving processing speed.

The Correct answer is Bitra.

Key Points

• Bitra is the smallest inhabited island in the Union Territory of Lakshadweep.
• It has an area of approximately 0.105 square kilometers, making it the tiniest inhabited landmass in the region.
• The island is known for its pristine natural beauty and is part of the archipelago of 36 islands, out of which only 10 are inhabited.
• Bitra is one of the most sparsely populated islands in Lakshadweep.
• The island is located in the Arabian Sea, and its inhabitants primarily rely on fishing and coconut cultivation for their livelihood.
• Bitra is surrounded by a rich ecosystem, including coral reefs and marine biodiversity, making it significant for ecological studies.
• It is part of the Lakshadweep Islands, which are governed under the administration of India.

 Additional Information

• Agatti
• Agatti is another inhabited island in the Lakshadweep archipelago.
• It is known for its beautiful beaches and coral reefs, making it a popular destination for tourism.
• The island has an area of approximately 3.84 square kilometers.
• Agatti is famous for its marine biodiversity, including species of fish and turtles.
• Kavaratti
• Kavaratti is the capital of the Union Territory of Lakshadweep.
• It is one of the most developed islands and serves as the administrative headquarters.
• The island covers an area of 3.93 square kilometers.
• Kavaratti is known for its water sports activities and its marine aquarium.
• Minicoy
• Minicoy is the second-largest island in Lakshadweep.
• It has an area of 4.8 square kilometers.
• Minicoy is culturally distinct, as it has influences from the Maldive Islands.
• The island is known for its lighthouse, which is one of the oldest landmarks in the region.

Shortcut Trick

Ratio method: Let the Cost Price (CP) be 100%.

First selling price (SP₁) = 100% − 40% = 60%.

Desired selling price (SP₂) = 100% + 55% = 155%.

Given: 60% → 1,20,000. So, 1% → 1,20,000 ÷ 60 = 2,000.

Required SP₂ = 155 × 2,000 = 3,10,000.

∴ The correct answer is ₹3,10,000.

Alternate Method

Given: Selling Price (SP₁) = ₹1,20,000, Loss% = 40%, Required Profit% = 55%.

Formulas: CP = SP × 100 ÷ (100 − Loss%); SP = CP × (100 + Profit%) ÷ 100.

⇒ Cost Price (CP) = 1,20,000 × 100 ÷ (100 − 40) = 1,20,000 × 100 ÷ 60.

⇒ Cost Price (CP) = 2,00,000.

⇒ New Selling Price (SP₂) = 2,00,000 × (100 + 55) ÷ 100.

⇒ New Selling Price (SP₂) = 2,00,000 × 155 ÷ 100 = 2,000 × 155.

⇒ New Selling Price (SP₂) = 3,10,000.

∴ The correct answer is ₹3,10,000.

Additional Information

Cost Price & Selling Price Relationship

When an object is sold at a loss, the Cost Price is always greater than the Selling Price. CP = SP + Loss.

Direct SP Calculation

To find the new SP directly: SP₂ = SP₁ × (100 + Profit%) ÷ (100 − Loss%).

The logic followed here is.

Option (1) 'PN - JD':

Option (2) 'LJ - EZ':

Option (3) 'QO - JE':

Option (4) 'GE - ZU':

Thus, option 1 is odd one.

Hence, the correct answer is "Option 1".

Shortcut Trick

Put θ = 30° in the given expression.

cot 30° = √3 and cosec 30° = 2.

Expression = [√3 / (2 + 1)] + [(2 + 1) / √3] = [√3 / 3] + [3 / √3] = [1 / √3] + √3 = (1 + 3) / √3 = 4 / √3.

Check options for θ = 30°: 2sec 30° = 2 × (2 / √3) = 4 / √3.

∴ The correct answer is 2secθ.

Alternate Method

Given: [cot θ / (cosec θ + 1)] + [(cosec θ + 1) / cot θ]

Formula Used: cosec²θ − cot²θ = 1 ⇒ cot²θ = cosec²θ − 1 = (cosec θ + 1)(cosec θ − 1)

⇒ Taking LCM: [cot²θ + (cosec θ + 1)²] / [cot θ(cosec θ + 1)]

⇒ [cot²θ + cosec²θ + 1 + 2cosec θ] / [cot θ(cosec θ + 1)]

⇒ [(cosec²θ − 1) + cosec²θ + 1 + 2cosec θ] / [cot θ(cosec θ + 1)]

⇒ [2cosec²θ + 2cosec θ] / [cot θ(cosec θ + 1)]

⇒ [2cosec θ (cosec θ + 1)] / [cot θ (cosec θ + 1)]

⇒ 2cosec θ / cot θ = 2(1 / sin θ) / (cos θ / sin θ)

⇒ 2 / cos θ = 2sec θ

∴ The correct answer is 2secθ.

Additional Information

Trigonometric Identities

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

Reciprocal Relations

sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = cos θ/sin θ.

Shortcut Trick

Directly apply r² = Volume ÷ (π × h).

r² = 8470 ÷ ((22/7) × 2.2)

r² = (8470 × 7) ÷ (22 × 2.2) = (8470 × 7) ÷ 48.4 = 1225.

r = √1225 = 35 cm.

∴ The correct answer is 35 cm.

Alternate Method

Given:

Volume (V) = 8470 cm³

Height (h) = 2.2 cm

π = 22/7

Formula Used:

Volume of Cylinder = πr²h

Calculations:

⇒ 8470 = (22/7) × r² × 2.2

⇒ 8470 = (22 × 2.2 / 7) × r²

⇒ 8470 = (48.4 / 7) × r²

⇒ r² = (8470 × 7) ÷ 48.4

⇒ r² = 59290 ÷ 48.4

⇒ r² = 1225

⇒ r = √1225

⇒ r = 35 cm

∴ The correct answer is 35 cm.

Additional Information

Curved Surface Area (CSA) of Cylinder

The lateral area excluding the top and bottom bases: CSA = 2πrh.

Total Surface Area (TSA) of Cylinder

The sum of the CSA and the area of the two circular bases: TSA = 2πr(h + r).

Volume of Hollow Cylinder

The volume of the material between two concentric cylinders: V = π(R² − r²)h.

The Correct answer is Pulicat.

Key Points

• Pulicat was the site where the Dutch East India Company established a factory in 1610.
• Pulicat later became the headquarters of the Dutch directorate for the Coromandel region.
• The Coromandel region is located along the southeastern coast of India, known for its strategic importance in trade during the colonial era.
• The Dutch East India Company had a significant presence in India, operating through trading posts and factories, particularly in coastal areas.
• Pulicat was a notable port for the Dutch, facilitating trade in spices, textiles, and other commodities.
• The Dutch also constructed a fort in Pulicat, known as Fort Geldria, to protect their trading interests.
• Pulicat served as a major trading center and played a key role in establishing Dutch influence in the region during the 17^th century.

 Additional Information

• Surat
• Surat, located in Gujarat, was a significant trading port during the Mughal era.
• The Dutch, along with the Portuguese, British, and French, operated factories in Surat, but it was not the headquarters for the Coromandel region.
• Bombay
• Bombay, now known as Mumbai, became an important trading hub under British rule.
• It was not associated with the Dutch directorate for the Coromandel region.
• Cochin
• Cochin, located in Kerala, was a major port for the Portuguese and later the Dutch.
• While the Dutch had a presence here, it did not serve as the headquarters for their Coromandel operations.

The correct answer is mv oldname.txt newname.txt.

Key Points

• The mv command in Linux is used for both moving and renaming files or directories.
• When you use mv with the syntax mv oldname.txt newname.txt, it effectively renames the file named oldname.txt to newname.txt.
• This operation does not create a new file; instead, it modifies the metadata of the existing file, changing its name.
• The mv command is part of the standard set of Linux utilities and is available on almost all Linux distributions.
• Renaming with mv is a lightweight operation because it simply updates the file name in the filesystem without altering the file's contents.
• The mv command can also be used to move files between directories. For example: mv file.txt /path/to/destination/.
• To rename a file, you must have the write permissions for the directory containing the file.
• The mv command is versatile and widely used for file and directory management in Linux environments.

 Additional Information

• touch oldname.txt newname.txt
• The touch command is used to create empty files or update the timestamps of existing files.
• It does not rename files or move them. Using touch oldname.txt newname.txt will create two separate empty files named oldname.txt and newname.txt, but it will not rename an existing file.
• cp oldname.txt newname.txt
• The cp command is used to copy files. It creates a duplicate file named newname.txt with the same content as oldname.txt.
• It does not rename the original file; instead, it creates a new file.
• Create oldname.txt newname.txt
• This is not a valid Linux command. Linux does not have a Create command for managing files directly.
• File creation is typically done using commands like touch or text editors like nano, vim, or cat.

Shortcut Trick Let the longer side be L. Then the shorter side is L − 21. Area = 7 × Perimeter ⇒ L × (L − 21) = 7 × 2 × [L + (L − 21)] L2 − 21L = 14(2L − 21) ⇒ L2 − 21L = 28L − 294 L2 − 49L + 294 = 0. Solving this: (L − 42)(L − 7) = 0. Since L > 21, the only possible value for the longer side is 42 cm. ∴ The correct answer is 42 cm.

Alternate Method

Given: ⇒ Let Longer side = x cm ⇒ Shorter side = (x − 21) cm Formula Used: ⇒ Area of Rectangle = Length × Breadth ⇒ Perimeter of Rectangle = 2 × (Length + Breadth) Calculations: ⇒ Area = 7 × Perimeter ⇒ x(x − 21) = 7 × 2[x + (x − 21)] ⇒ x2 − 21x = 14(2x − 21) ⇒ x2 − 21x = 28x − 294 ⇒ x2 − 49x + 294 = 0 ⇒ x2 − 42x − 7x + 294 = 0 ⇒ x(x − 42) − 7(x − 42) = 0 ⇒ (x − 42)(x − 7) = 0 ⇒ x = 42 or x = 7 ⇒ Since the side must be greater than 21 (as x − 21 must be positive), x = 42 cm. ∴ The correct answer is 42 cm.

Additional Information Properties of a Rectangle A rectangle is a quadrilateral with four right angles. Opposite sides are equal and parallel. Diagonals are equal and bisect each other. Diagonal Formula The length of the diagonal (d) of a rectangle is given by d = √(L2 + B2). Quadratic Root Selection In geometry word problems, always check if the calculated side length makes all other dimensions (like L − 21) positive real numbers.

Given:

Mr. D starts from Point P and drives 11 km towards the South.

He then takes aleft turn, drives 6 km, turns right and drives 12 km.

He then takes a left turn anddrives 1 km. He takes a final left turn derives 23 km and stops at Point Q. 

Thus, D has to drive 7km west to reach point P.

Hence, the correct answer is "Option 3".

Shortcut Trick

Total frequency N = 148, so N/2 = 74.

Cumulative frequencies: 12, 28, 58, 78, 95, 133, 148.

The Median Class is 50−60 (since 78 ≥ 74).

Median = 50 + [(74 − 58) ÷ 20] × 10 = 50 + 8 = 58.

∴ The correct answer is 58 years.

Alternate Method

Given: Frequency distribution of patients by age.

Formula Used: Median = L + [((N/2) − cf_prev) ÷ f] × h

Calculations:

⇒ Total patients (N) = 12 + 16 + 30 + 20 + 17 + 38 + 15 = 148

⇒ N/2 = 148 ÷ 2 = 74

⇒ Cumulative Frequencies (cf):

20−30: 12 | 30−40: 28 | 40−50: 58 | 50−60: 78 | 60−70: 95 | 70−80: 133 | 80−90: 148

⇒ Median Class = 50−60 (since cf 78 is just greater than 74)

⇒ L (Lower limit) = 50,

Shortcut Trick

Arrange the data in ascending order: 13, 18, 25, 27, 31, 39, 56, 80, 87, 92, 98.

Total number of observations (n) = 11 (which is Odd).

Median = Middle term = ((11 + 1) / 2)^th term = 6^th term.

The 6^th term in the sorted list is 39.

∴ The correct answer is 39.

Alternate Method

Given: Observations = 87, 56, 27, 31, 13, 39, 18, 80, 98, 92, 25.

Formula Used: Median = ((n + 1) / 2)^th term (when n is odd).

⇒ Arrange observations in ascending order: 13, 18, 25, 27, 31, 39, 56, 80, 87, 92, 98.

⇒ Count of observations (n) = 11.

⇒ Median = ((11 + 1) / 2)^th term.

⇒ Median = 6^th term.

⇒ 6^th term = 39.

∴ The correct answer is 39.

Additional Information

Median for Even n

If the number of observations (n) is even, the median is the average of the (n/2)^th and (n/2 + 1)^th terms.

Mean (Average)

Mean = (Sum of all observations) ÷ (Total number of observations).

Mode

The mode is the value that appears most frequently in a data set. A set can have one mode, more than one mode, or no mode.

Given series:

(Left) & # © @ 7 2 5 1 4 © 4 2 £ & 9 1 % # 4 4 & 1 (Right)

Condition to be checked:

Number → Symbol → Number

& # © @ 7 2 5 1 4 © 4 2 £ & 9 1 % # 4 4 & 1

There is one such symbol in the above arrangement (4 © 4, 4 & 1).

Hence, the correct answer is "Option 1".

Given:

Mr. STU starts from Point A and drives 33 km towards the east.

He then takes a left turn,drives 21 km, turns left and drives 41 km. He then takes a left turn and drives 41 km.

He takes a final left turn, drives 8 km and stops at Point P.

Thus, STU has to drive 20km north to reach point A.

Hence, the correct answer is "Option 4".

Shortcut Trick

Time to empty = (Filling time × Emptying time) ÷ (Emptying time − Filling time)

Required time = (10 × 8) ÷ (10 − 8)

Required time = 80 ÷ 2 = 40 hours.

∴ The correct answer is 40 hr.

Alternate Method

Given:

Filling pipe (A) = 10 hours

Emptying pipe (B) = 8 hours

Formula Used:

Total Work = LCM of individual times

Efficiency = Total Work ÷ Time

⇒ Net efficiency of (A + B) = +4 − 5 = −1 unit/hr

⇒ Negative sign indicates the cistern is being emptied.

⇒ Time to empty full cistern = Total Capacity ÷ Net Efficiency

⇒ Time = 40 ÷ 1 = 40 hours.

∴ The correct answer is 40 hr.

Additional Information

Inlet and Outlet Pipes

An Inlet pipe fills the tank (positive efficiency), while an Outlet pipe empties it (negative efficiency).

Net Work Done

If efficiency of the outlet pipe is greater than the inlet pipe, a full tank will always be emptied.

Total Work Assumption

Total capacity is usually taken as the LCM of time taken by individual pipes for easier calculations.

Shortcut Trick

Required value = √[(Sum of squares + 2 × Product) ÷ (Sum of squares − 2 × Product)]

Ratio = √[(90 + 2 × 27) ÷ (90 − 2 × 27)]

Ratio = √[144 ÷ 36] = 12 ÷ 6 = 2

∴ The correct answer is 2.

Alternate Method

Given:

⇒ a² + b² = 90

⇒ ab = 27

Formula Used:

⇒ (a + b)² = a² + b² + 2ab

⇒ (a − b)² = a² + b² − 2ab

Calculations:

⇒ (a + b)² = 90 + 2(27) = 144

⇒ a + b = √144 = 12

⇒ (a − b)² = 90 − 2(27) = 36

⇒ a − b = √36 = 6

⇒ Required fraction = (a + b) ÷ (a − b) = 12 ÷ 6 = 2

∴ The correct answer is 2.

Additional Information

Algebraic Identity for 4ab

The relationship between the square of the sum and the square of the difference is given by (a + b)² − (a − b)² = 4ab.

Sum of Squares

The sum of squares a² + b² can be expressed as ((a + b)² + (a − b)²) ÷ 2.

Difference of Squares

The product of the sum and difference of two numbers results in the difference of their squares: (a + b)(a − b) = a² − b².

Shortcut Trick

Net discount for 29% and 18% = 29 + 18 − (29 × 18) ÷ 100 = 47 − 5.22 = 41.78%.

Now, combine 41.78% and 9% = 41.78 + 9 − (41.78 × 9) ÷ 100.

= 50.78 − 3.7602 = 47.0198%.

Rounding to two decimal places, we get 47.02%.

∴ The correct answer is 47.02%.

Alternate Method

Given → Three successive discounts d₁ = 29%, d₂ = 18%, and d₃ = 9%. Formulas: Net Discount = 100 − [100 × (100−d₁)/100 × (100−d₂)/100 × (100−d₃)/100].

⇒ Final Price = 100 × (71/100) × (82/100) × (91/100) ⇒ Final Price = 100 × 0.71 × 0.82 × 0.91 ⇒ Final Price = 52.9802 ⇒ Single Equivalent Discount = 100 − 52.9802 = 47.0198% ⇒ Equivalent Discount ≈ 47.02%

∴ The correct answer is 47.02%.

Additional Information

Successive Discounts

When a series of discounts are applied one after another on the reducing balance, they are called successive discounts. The order in which successive discounts are applied does not change the final result.

Single Equivalent Discount Formula

For two discounts a% and b%, the equivalent discount is (a + b − ab/100)%. For three, apply the formula to the first two, then to the result and the third.

The Correct answer is 14%.

Key Points

• The Global Risks Perception Survey (GRPS) is conducted annually as part of the World Economic Forum's Global Risks Report.
• It gathers insights from a diverse group of stakeholders, including business leaders, policymakers, academics, and risk experts.
• In the 2025 survey, 14% of respondents identified 'risk associated with extreme weather events' as a key concern.
• Extreme weather events are becoming more frequent due to the impact of climate change.
• These events include floods, hurricanes, heatwaves, and droughts, which have significant social, economic, and environmental consequences.
• The survey highlights the growing awareness of climate-related risks and emphasizes the need for global collaboration in addressing these challenges.
• The percentage (0.14 or 14%) reflects the proportion of respondents who prioritize this risk in their assessment for the upcoming year.

 Additional Information

• Global Risks Perception Survey (GRPS)
• The GRPS is an integral part of the World Economic Forum's annual risk assessment initiative.
• It aims to identify and rank the major global risks that could impact societies, economies, and industries.
• The survey results influence the development of strategies to mitigate these risks at the global and regional levels.
• Extreme Weather Events
• Examples of extreme weather include hurricanes, typhoons, heatwaves, floods, and droughts.
• These events are often linked to climate change and can lead to severe disruptions in livelihoods, infrastructure, and ecosystems.
• Global efforts such as the Paris Agreement aim to combat the underlying causes of these events.

Shortcut Trick

For the equation 5x² − 12x − 17 = 0, find two numbers that multiply to (5 × −17) = −85 and add to −12.

The numbers are −17 and +5.

The roots are obtained by changing the signs of these numbers and dividing by the coefficient of x² (a = 5).

Roots = −(−17)/5 and −(5)/5 ⇒ 17/5 and −1.

∴ The correct answer is -1, 17/5.

Alternate Method

Given Equation: 5x² − 12x − 17 = 0

Method: Factorization by splitting the middle term

⇒ 5x² − 17x + 5x − 17 = 0

⇒ x(5x − 17) + 1(5x − 17) = 0

⇒ (5x − 17)(x + 1) = 0

Now, set each factor equal to zero:

⇒ 5x − 17 = 0 ⇒ 5x = 17 ⇒ x = 17/5

⇒ x + 1 = 0 ⇒ x = −1

The roots of the equation are −1 and 17/5.

∴ The correct answer is -1, 17/5.

Additional Information

Discriminant and Nature of Roots

The discriminant D is given by b² − 4ac. Since D = (−12)² − 4(5)(−17) = 144 + 340 = 484 > 0, the roots are real and distinct.

Sum and Product of Roots

For ax² + bx + c = 0, sum of roots = −b/a and product of roots = c/a. Here, sum = 12/5 and product = −17/5.

Quadratic Formula

Roots can also be found using x = [−b ± √(b² − 4ac)] / 2a.

The Correct answer is Jakarta, Indonesia.

Key Points

• The 67^th Governing Body Meeting of the Asian Productivity Organization (APO) was held in Jakarta, Indonesia in May 2025.
• The Asian Productivity Organization (APO) is an intergovernmental organization committed to enhancing the productivity of its member countries.
• Indonesia, the host country, is one of the founding members of the APO, which was established in 1961.
• Jakarta, the capital of Indonesia, is a major hub for international conferences and events, reflecting its role as a prominent economic and political center in the Southeast Asian region.
• The APO Governing Body Meeting is an annual event where delegates from all member countries gather to discuss and plan strategies for productivity enhancement, innovation, and sustainable development.
• Indonesia has been actively involved in APO programs, focusing on productivity improvement in sectors like agriculture, manufacturing, and services.
• The event in Jakarta underscored Indonesia's commitment to fostering regional cooperation and addressing productivity challenges in the Asia-Pacific region.

 Additional Information

• Bangkok, Thailand
• Bangkok is the capital and largest city of Thailand.
• It is known for its vibrant street life, cultural landmarks, and bustling markets.
• Thailand is also an APO member and has hosted several APO events in the past.
• Tokyo, Japan
• Tokyo is the capital of Japan and serves as the headquarters of the Asian Productivity Organization (APO).
• Japan is a key member of the APO and often plays a leading role in its initiatives and programs.
• New Delhi, India
• New Delhi is the capital of India, another founding member of the APO.
• India has been an active participant in APO initiatives, particularly in areas like agriculture and industrial productivity.

Given:

Seven people, A, B, L, M, N, S, and T are sitting in a row, facing north.

Only two people sit to the right of T. Only two people sit between T and N.

Only two people sit between M and A. A sits to the immediate left of T.

B sits to the immediate right of S.

Thus, L sits at third position from the left end of the line.

Hence, the correct answer is "Option 2".

Shortcut Trick

Assume Cost Price (CP) = 100. Then Selling Price (SP) = 179.

When CP and SP are interchanged, New CP = 179 and New SP = 100.

Loss = 179 − 100 = 79.

Loss% = (79 ÷ 179) × 100.

Calculation: 7900 ÷ 179 ≈ 44.134%.

∴ The correct answer is 44.13%.

Alternate Method

Given: Profit = 79%, New CP = Old SP, New SP = Old CP.

Formulas: SP = CP × (100 + P%) ÷ 100 | Loss% = (Loss ÷ CP) × 100

⇒ Let Initial Cost Price (CP₁) = 100 units.

⇒ Initial Profit = 79%, so Initial Selling Price (SP₁) = 179 units.

⇒ After interchanging, New Cost Price (CP₂) = 179 units and New Selling Price (SP₂) = 100 units.

⇒ New Loss = CP₂ − SP₂ = 179 − 100 = 79 units.

⇒ Loss Percentage = (Loss ÷ CP₂) × 100 = (79 ÷ 179) × 100.

⇒ Loss% = 44.13407...% ≈ 44.13%.

∴ The correct answer is 44.13%.

Additional Information

Profit and Loss Percentage

Profit or Loss is always calculated on the Cost Price unless specified otherwise. Formula: Profit% = (P/CP)×100 and Loss% = (L/CP)×100.

Interchanging CP and SP

If a transaction results in a profit, interchanging CP and SP will always result in a loss. The absolute value of the profit/loss remains the same, but the percentage changes because the base (CP) changes.

Given:

2 is added to each even digit and 1 is subtracted from each odd digit in the number 74135296.

The new number formed is 66024488.

The digits 6, 4, and 8 appear more than once.

Thus, three digits appear more than once.

Hence, the correct answer is "Option 3".

The Correct answer is Geetika Lidder.

Key Points

• Geetika Lidder is the author of the book titled 'I am a Soldier's Wife'.
• The book is a tribute to the life of soldiers and their families, focusing on the challenges faced by a soldier’s spouse.
• Geetika Lidder is known for her emotional and inspirational writing, bringing to light the sacrifices made by military families.
• The book is an example of literary works that highlight the human side of military life, emphasizing resilience and courage.
• 'I am a Soldier's Wife' provides insights into the emotional aspects and the unspoken realities of being married to a soldier.
• The book has gained recognition for its poignant and heartfelt storytelling, making it relatable to readers who value stories of dedication and perseverance.

 Additional Information

• Soumitra Chatterjee
• Soumitra Chatterjee was a renowned actor from India, primarily known for his work in Bengali cinema.
• He collaborated extensively with Satyajit Ray, starring in iconic films like 'The World of Apu'.
• Soumitra Chatterjee was awarded the Dadasaheb Phalke Award, India's highest award in cinema, for his contributions to Indian cinema.
• Dr. Aishwarya Pandit
• Dr. Aishwarya Pandit is a historian and academic known for her work in the field of Indian history and culture.
• She has contributed to research and publications on Indian socio-political history, but is not associated with the book 'I am a Soldier's Wife.'
• Namita Gokhale
• Namita Gokhale is an Indian writer and co-founder of the Jaipur Literature Festival.
• She is known for her novels like 'Paro: Dreams of Passion' and 'The Himalayan Love Story.'
• Her works often explore themes of contemporary Indian society and mythology.

Given Expression: E + F - G ÷ H x K

E + F: E is the son of F.

F - G: F is the sister of G.

G ÷ H: G is the father of H.

H x K: H is the wife of K.

According to the given information Family Tree Representation:

 

Thus, E is the Wife's father's sister's son of K.

Hence, the correct answer is "Option 1"

The Correct answer is Republic of the Congo.

Key Points

• The Republic of the Congo had its FIFA suspension lifted in May 2025 after resolving issues related to third-party interference in the governance of its football federation.
• FIFA suspensions are typically imposed when there is a violation of FIFA statutes, particularly concerning political or third-party interference.
• The Republic of the Congo’s football federation worked closely with FIFA to address these issues and implement reforms to ensure compliance with FIFA’s rules.
• The lifting of the suspension allowed the country to resume participation in international football events, including qualification matches for FIFA tournaments.
• This decision reflects the importance of autonomy and proper governance in sports organizations, as emphasized by FIFA.
• The resolution of this issue is expected to foster better development opportunities for football in the Republic of the Congo.

 Additional Information

• Democratic Republic of the Congo
• The Democratic Republic of the Congo (DRC) is a separate country in Central Africa, distinct from the Republic of the Congo.
• It has its own football federation and has had a relatively stronger presence in African football, including winning the Africa Cup of Nations twice.
• The DRC was not involved in any FIFA suspension during the specified time period.
• Cameroon
• Cameroon is known for its successful football history, including multiple Africa Cup of Nations titles and appearances in the FIFA World Cup.
• While governance challenges have arisen in the past, Cameroon was not under FIFA suspension in May 2025.
• Nigeria
• Nigeria is one of Africa’s football powerhouses, with notable achievements such as winning the Africa Cup of Nations and strong performances in the FIFA World Cup.
• Although it has faced governance issues in the past, Nigeria was not suspended by FIFA in May 2025.

The logic followed here is:

YAGK : PRXB

OQWA : FHNR

 

Applying the logic to NPVZ:

Thus, NPVZ is related to EGMQ.

Hence, the correct answer is "Option 4".

The Correct answer is RNA has multiple roles including catalytic functions.

Key Points

• RNA is a versatile molecule that performs various functions essential for life.
• In addition to being a genetic material in certain organisms (like viruses), RNA also plays a major role in protein synthesis as messenger RNA (mRNA), transfer RNA (tRNA), and ribosomal RNA (rRNA).
• Certain types of RNA, such as ribozymes, exhibit catalytic functions. Ribozymes catalyze biochemical reactions, such as cleaving RNA molecules or forming peptide bonds during protein synthesis.
• The discovery of catalytic RNA challenged the traditional belief that only proteins could act as enzymes.
• RNA is involved in processes such as gene regulation, splicing, and editing, highlighting its multifunctional nature.
• The ability of RNA to fold into complex three-dimensional structures is crucial for its catalytic and regulatory roles.
• Examples of RNA molecules with catalytic functions include self-splicing introns and the ribosome, where rRNA catalyzes peptide bond formation.

 Additional Information

• DNA is not present in viruses
• Some viruses, such as retroviruses (e.g., HIV), use RNA as their genetic material instead of DNA.
• However, many viruses, including adenoviruses and herpesviruses, contain DNA as their genetic material. Hence, the statement is incorrect.
• DNA acts only as a messenger molecule
• DNA is primarily the repository of genetic information in most organisms. It does not function as a messenger molecule; this role is performed by RNA (mRNA).
• RNA acts solely as genetic material in all living organisms
• In most living organisms, DNA serves as the primary genetic material, while RNA plays supporting roles. Only certain viruses use RNA as their genetic material.

The Correct answer is Uparika.

Key Points

• During the reign of the Gupta Empire, judicial and administrative systems were highly organized and decentralized.
• The term Uparika referred to provincial governors who held authority in the provinces.
• Uparikas were responsible for overseeing judicial matters and ensuring the enforcement of laws within their respective provinces.
• They functioned as the representative of the central government in the provinces and played a key role in governance.
• The Gupta administration was known for its efficiency and division of responsibilities, which contributed to the prosperity and stability of the empire.
• The judicial system under the Guptas was influenced by Hindu legal texts like the Manusmriti and was deeply rooted in Dharma (moral and religious law).
• Uparikas ensured that disputes were resolved fairly and that the legal and moral order was maintained across the provinces.

 Additional Information

• King
• The King was the supreme authority in the Gupta Empire, responsible for making laws and overseeing the overall administration.
• While the King held the ultimate judicial power, day-to-day judicial matters were delegated to officials like the Uparikas in the provinces.
• Gramika
• Gramika refers to the headman of a village.
• They were responsible for local administration, resolving minor disputes, and maintaining order at the village level.
• They did not handle judicial matters at the provincial level.
• Ayuktaka
• The term Ayuktaka referred to officers responsible for revenue collection and administrative duties.
• They primarily focused on economic and administrative activities rather than judicial matters.

Shortcut Trick

For 10-monthly compounding, rate per period = 12% × (10/12) = 10%.

Total periods for 2.5 years (30 months) = 30 ÷ 10 = 3 periods.

Using Ratio Method: 10% = 1/10 ⇒ Amount/Principal = (11/10)³ = 1331/1000.

1331 units = ₹83,853 ⇒ 1 unit = ₹83,853 ÷ 1331 = ₹63.

Principal (x) = 1000 units = 1000 × 63 = ₹63,000.

∴ The correct answer is ₹63,000.

Alternate Method

Given:

Amount (A) = ₹83,853 | Annual Rate (R) = 12% | Time (T) = 2.5 years = 30 months.

Compounding = 10-monthly.

Formula Used:

A = P(1 + r/100)ⁿ

Calculations:

⇒ Effective rate (r) = 12% × (10/12) = 10% per period

⇒ Number of periods (n) = 30 ÷ 10 = 3

⇒ 83,853 = x(1 + 10/100)³

⇒ 83,853 = x(1.1)³

⇒ 83,853 = x × 1.331

⇒ x = 83,853 ÷ 1.331

⇒ x = 83,853,000 ÷ 1331

⇒ x = 63,000

∴ The correct answer is ₹63,000.

Additional Information

Effective Rate for Compounding

When interest is compounded k-monthly, the rate used is (R ÷ 12) × k and the number of periods is (Total months ÷ k).

Compound Interest Formula

Amount A = P(1 + r/100)ⁿ, where P is principal, r is rate per period, and n is the number of periods.

Successive Percentage Method

For 3 periods of 10% each, the effective percentage increase is 10 + 10 + (10×10)/100 = 21%, then 21 + 10 + (21×10)/100 = 33.1%.

The logic followed here is:

Thus, the missing letter in the series is LOQ.

Hence, the correct answer is "Option 2".

The Correct answer is It opens the file in a read-only mode, disabling most editing features.

Key Points

• Protected View is a security feature in Microsoft PowerPoint and other Office applications.
• When a file is downloaded or received from an external source, it opens in Protected View to safeguard the system from potential risks.
• In this mode, the file is opened in a read-only format, preventing accidental modifications and limiting editing capabilities.
• This feature is particularly useful for files from unknown or untrusted sources, as it prevents the execution of malicious macros or scripts.
• Users can enable editing manually if they trust the source, by clicking the "Enable Editing" button in the notification bar.
• The goal of Protected View is to enhance security by reducing the risk of malware or viruses affecting the system through potentially unsafe files.
• This feature is available in Microsoft Office 2010 and later versions.

 Additional Information

• Macros are enabled automatically
• Macros are not enabled in Protected View. Instead, this mode blocks the execution of macros to safeguard the system from potentially harmful code.
• Enabling macros must be done manually by the user if they trust the file source.
• The file is corrupted
• A file being in Protected View does not necessarily mean it is corrupted.
• Corrupted files might not open at all or could display error messages, while Protected View is a proactive security measure.
• The presentation is encrypted and cannot be viewed
• Encrypted files require a password or decryption key to be accessed, which is different from the functionality of Protected View.
• In Protected View, the file is accessible but with limited functionality.

Shortcut Trick

The sum of eight numbers = Average × Count = 80 × 8 = 640.

The least number is 1/16 of the total sum.

Value = 640 ÷ 16 = 40.

∴ The correct answer is 40.

Alternate Method

Given:

Total count of numbers = 8

Average of numbers = 80

Condition: Least number = 1/16 × Sum

Formula:

Total Sum = Average × Number of observations

Calculations:

⇒ Total Sum of 8 numbers = 80 × 8 = 640

⇒ Least number = (1/16) × 640

⇒ Least number = 640 ÷ 16

⇒ Least number = 40

∴ The correct answer is 40.

Additional Information

Concept of Average

The average is the central value of a set of data, calculated by Sum ÷ Count.

Summation Property

If the average of 'n' numbers is 'x', then their total sum is always n × x.

Fractional Parts

To find a specific fractional part of a total, multiply the total by the numerator and divide by the denominator.

The logic followed here is:

Logic: Second Number = First Number × Third Number.

For, (23, 253, 11)

⇒ 23 × 11 = 253.

For, (46, 1104, 24)

⇒ 46 × 24 = 1104.

Option 1) (1, 12, 9)

⇒ 1 × 9 = 9. (Given Second number is 12, 12 ≠ 9)

Here, logic not followed.

Option 2) (4, 11, 2)

⇒ 4 × 2 = 8. (Given Second number is 11, 11 ≠ 8)

Here, logic not followed.

Option 3) (25, 525, 21)

⇒ 25 × 21 = 525.

Here, logic followed.

Option 4) (25, 253, 10)

⇒ 25 × 10 = 250. (Given Second number is 253, 253 ≠ 250)

Here, logic not followed.

Hence, the correct answer is "Option 3".

The Correct answer is Article 49.

Key Points

• Article 49 of the Indian Constitution falls under the Directive Principles of State Policy (DPSP).
• It directs the State to ensure the protection of every monument or place or object of artistic or historic interest that is of national importance.
• This Article emphasizes that such monuments or objects should not be spoliated, disfigured, destroyed, removed, disposed of, or exported.
• The objective of this Article is to preserve and maintain the rich cultural heritage of India for future generations.
• This provision is part of the Directive Principles of State Policy, meaning it is not enforceable by the courts but provides important guidelines for the government.
• It reinforces the government's responsibility to promote and protect India's artistic, cultural, and historical legacy.
• The protection of monuments is also supplemented by laws such as the Ancient Monuments and Archaeological Sites and Remains Act, 1958.

 Additional Information

• Article 48
• It directs the State to organize agriculture and animal husbandry on modern and scientific lines.
• It also promotes the preservation and improvement of breeds and prohibits the slaughter of cows, calves, and other milch and draught cattle.
• Article 50
• This Article mandates the separation of the judiciary from the executive in the public services of the State.
• It aims to ensure judicial independence and the rule of law.
• Article 48A
• Added by the 42^nd Amendment Act, 1976, it directs the State to protect and improve the environment and safeguard forests and wildlife.
• It highlights the importance of environmental conservation as part of the Directive Principles.

The Correct answer is maximum point.

Key Points

• The crest of a wave is the point where the disturbance reaches its maximum displacement in an oscillatory motion.
• It is the highest point the wave reaches above its mean position or equilibrium line.
• This concept is fundamental in understanding wave mechanics, including water waves, sound waves, and electromagnetic waves.
• The crest is followed by the trough, which is the lowest point or the point of maximum negative displacement.
• In mathematical terms, for a sinusoidal wave, the crest corresponds to the point where the amplitude of the wave is at its positive maximum value.
• The concept of the crest is essential in various fields, including physics, engineering, and oceanography, as it helps in analyzing wave behavior, energy, and motion.
• Understanding the crest is crucial in determining the wavelength (distance between two consecutive crests) and the frequency of the wave.

 Additional Information

• Minimum point
• The minimum point in a wave corresponds to the trough, which is the lowest point in the wave cycle or the point of maximum negative displacement.
• Troughs are essential for calculating the amplitude and determining the wave's energy.
• Zero point
• The zero point refers to the mean position or equilibrium line of the wave, where there is no displacement.
• It is the point of zero disturbance in the wave's cycle.
• Constant point
• This term is not typically used in wave mechanics as waves are inherently oscillatory and do not remain constant.
• The motion of waves is characterized by continuous variations in displacement, making "constant point" an incorrect description.

The Correct answer is 4.0%.

Key Points

• The Monetary Policy Statement is issued periodically by the Reserve Bank of India (RBI) to outline the economic and financial outlook for the country.
• The Consumer Price Index (CPI) measures the average change in the prices of goods and services consumed by households over time.
• For the financial year 2025-26, the CPI inflation rate has been projected at 4% based on current economic conditions and policy measures.
• The projection indicates the central bank’s focus on maintaining price stability while supporting economic growth.
• A 4% inflation rate is considered within the target range set by the RBI under its flexible inflation targeting framework (2%-6%).
• This projection helps policymakers, investors, and businesses to plan their activities by providing a benchmark for expected price levels.
• Maintaining inflation near the target ensures the purchasing power of the currency remains stable, thereby fostering sustainable economic development.

 Additional Information

• Consumer Price Index (CPI) is an important economic indicator widely used for purposes such as:
• Measuring inflation: It reflects the rate at which the general level of prices is rising.
• Policy formulation: The RBI uses CPI data to design its monetary policies, including interest rate decisions.
• Wage adjustments: CPI is often used to adjust salaries and pensions to offset the impact of inflation.
• The Monetary Policy Committee (MPC), a statutory body within the RBI, is responsible for setting the inflation target and adjusting key policy rates like the repo rate to achieve it.
• Other projections or data points in the Monetary Policy Statement typically include:
• GDP growth rates,
• fiscal deficit trends,
• and global economic outlooks.

Shortcut Trick

Target Average = 80. Calculate Net Deviation of given marks from 80.

Deviations: (78−80) + (85−80) + (69−80) + (90−80) ⇒ −2 + 5 − 11 + 10 = +2.

To keep average at 80, Net Deviation must be 0. So, 5^th subject deviation must be −2.

5^th subject score = Target Average + Required Deviation ⇒ 80 + (−2) = 78.

∴ The correct answer is 78 marks.

Alternate Method

Given: Marks in 4 subjects = 78, 85, 69, 90; Required Average for 5 subjects = 80.

Formula Used: Total Sum = Average × Number of Observations.

Calculations:

⇒ Total marks required for 5 subjects = 80 × 5 = 400.

⇒ Sum of marks in 4 subjects = 78 + 85 + 69 + 90.

⇒ Sum of 4 subjects = 322.

⇒ Marks required in 5^th subject = Total required marks − Current sum.

⇒ Marks in 5^th subject = 400 − 322 = 78.

∴ The correct answer is 78 marks.

Additional Information

Arithmetic Mean

It is the sum of all observations divided by the total number of observations. Formula: Mean (μ) = Σx / n.

Net Deviation Rule

The algebraic sum of deviations of a set of observations from their arithmetic mean is always zero. This is a powerful tool for solving missing value problems in averages.

New Observation Entry

New Observation = (New Average × New Count) − (Old Average × Old Count).

Shortcut Trick

Combined efficiency of A + B = 1/2 work/day

Efficiency of A = 1/38 work/day

Efficiency of B = 1/2 − 1/38 = (19 − 1)/38 = 18/38 = 9/19 work/day

Time for B to do 9-times work = (Total work × 9) ÷ Efficiency of B = 9 ÷ (9/19) = 19 days

∴ The correct answer is 19 days.

Alternate Method

Given:

Time taken by A + B = 2 days

Time taken by A alone = 38 days

Formula Used:

Total Work = LCM of time taken

Efficiency = Total Work ÷ Time Taken

Calculations:

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

⇒ Efficiency of B = 19 − 1 = 18

⇒ New work to be done = 9 × Total Work = 9 × 38

⇒ Time taken by B = (9 × 38) ÷ 18

⇒ Time taken by B = 342 ÷ 18 = 19 days

∴ The correct answer is 19 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete a work. If a person completes work in 'n' days, their 1-day work is 1/n.

Combined Work Formula

If A takes 'x' days and B takes 'y' days, together they take (x × y) ÷ (x + y) days to complete the same work.

Given:

1 is added to each even digit and 2 is subtracted from each odd digit in the number 2756493.

The new number formed is 3537571.

The second digit from the left in the new number is 5.

Hence, the correct answer is "Option 1".

Shortcut Trick

Total time = 50 minutes. Breakdown: 13 mins (x), 25 mins (2x), and remaining (50 − 13 − 25) = 12 mins (x).

Total Equivalent Time at speed x = 13 + (25 × 2) + 12 = 13 + 50 + 12 = 75 minutes.

Distance = Speed × Time ⇒ 30 km = x × (75/60) hours.

x = (30 × 60) ÷ 75 = 1800 ÷ 75 = 24 km/hr.

∴ The correct answer is 24 km/hr.

Alternate Method

Given: Total Distance = 30 km, Total Time = 50 min.

Formula Used: Distance = Speed × Time

⇒ Third part time = 50 − (13 + 25) = 12 minutes.

⇒ Distance₁ = x × (13/60) km

⇒ Distance₂ = 2x × (25/60) km

⇒ Distance₃ = x × (12/60) km

⇒ Total Distance = (13x/60) + (50x/60) + (12x/60) = 30

⇒ 75x/60 = 30

⇒ 5x/4 = 30

⇒ 5x = 120

⇒ x = 24 km/hr

∴ The correct answer is 24 km/hr.

Additional Information

Average Speed Formula

Average Speed = Total Distance ÷ Total Time. In this case, 30 km ÷ (50/60) hours = 36 km/hr.

Unit Conversion

To convert minutes to hours, divide by 60. To convert km/hr to m/s, multiply by 5/18.

Distance with Variable Speeds

Total Distance = (S₁ × T₁) + (S₂ × T₂) + ... + (S_n × T_n), ensuring all units are consistent.

The Correct answer is Kerala.

Key Points

• Vidarbha defeated Kerala in the final of the 2024-25 Ranji Trophy, marking a significant achievement in Indian domestic cricket.
• The Ranji Trophy is India's premier domestic cricket tournament, organized by the Board of Control for Cricket in India (BCCI).
• Kerala has emerged as a strong competitor in recent years, showcasing improvement in its cricketing infrastructure and player development.
• The victory for Vidarbha highlights their consistent performance in domestic cricket, as they have become a formidable team in the tournament.
• Vidarbha's success can be attributed to their balanced team composition, including experienced players and promising young talent.
• The 2024-25 Ranji Trophy final was held under challenging conditions, testing the resilience and skills of both teams.
• Vidarbha's triumph against Kerala adds to their legacy in domestic cricket and boosts their confidence for future tournaments.

 Additional Information

• Tamil Nadu
• Tamil Nadu is a historically strong team in Indian domestic cricket, having won the Ranji Trophy multiple times.
• The state is known for its cricketing infrastructure and has produced renowned Indian cricketers like Ravichandran Ashwin and Dinesh Karthik.
• Bengal
• Bengal is another strong competitor in domestic cricket, known for its rich cricketing legacy.
• The team has won the Ranji Trophy multiple times and produced players like Sourav Ganguly, a former Indian captain.
• Mumbai
• Mumbai holds the record for winning the Ranji Trophy the most number of times.
• The team is considered a powerhouse in Indian domestic cricket and has produced legendary players like Sachin Tendulkar and Sunil Gavaskar.

Shortcut Trick

Prime Factorization of given numbers:

48 = 2⁴ × 3¹

88 = 2³ × 11¹

LCM = 4752 = 2⁴ × 3³ × 11¹

Since the LCM has 3³, and neither 48 nor 88 contains 3³, x must be a multiple of 3³ (27). Only 216 is divisible by 27.

∴ The correct answer is 216.

Alternate Method

Given: Numbers are 48, 88, and x. LCM = 4752.

Step 1: Find the prime factors of the given numbers.

⇒ 48 = 16 × 3 = 2⁴ × 3¹

⇒ 88 = 8 × 11 = 2³ × 11¹

Step 2: Find the prime factors of the LCM.

⇒ 4752 = 2 × 2376 = 2² × 1188 = 2³ × 594 = 2⁴ × 297

⇒ 297 = 27 × 11 = 3³ × 11¹

⇒ LCM (48, 88, x) = 2⁴ × 3³ × 11¹

Step 3: Compare factors. LCM is the product of highest powers of all prime factors present in the numbers.

⇒ Power of 2 in LCM is 2⁴ (comes from 48).

⇒ Power of 11 in LCM is 11¹ (comes from 88).

⇒ Power of 3 in LCM is 3³. However, 48 has only 3¹ and 88 has 3⁰.

⇒ Therefore, x must contain 3³ = 27 as a factor.

Step 4: Check options for multiples of 27.

⇒ 216 = 27 × 8 (Satisfies the condition).

∴ The correct answer is 216.

Additional Information

LCM Definition

Least Common Multiple is the product of the highest powers of all prime factors involved in the numbers.

HCF vs LCM Relation

For two numbers a and b: a × b = HCF(a, b) × LCM(a, b). This does not directly apply to three numbers.

Divisibility Rule of 27

A number is divisible by 27 if the sum of its digits in blocks of three from right to left is divisible by 27.

The Correct answer is Mohiniattam.

Key Points

• Swati Tirunal was a celebrated patron of the arts and a ruler of the princely state of Travancore (modern-day Kerala).
• He was instrumental in the revival and promotion of Mohiniattam, a classical dance form of Kerala.
• Mohiniattam is characterized by its graceful and lyrical movements.
• This dance form derives its name from "Mohini," a mythological enchantress avatar of Lord Vishnu.
• It is traditionally performed by women and showcases themes of love and devotion.
• Swati Tirunal composed numerous music compositions, including padams and varnams, specifically for Mohiniattam performances.
• His works contributed to the enrichment and preservation of Kerala’s cultural heritage.
• Swati Tirunal’s patronage of Mohiniattam ensured its survival and development during a time when many traditional art forms were declining.

 Additional Information

• Kathakali
• Kathakali is another classical dance-drama of Kerala, known for its elaborate costumes, heavy makeup, and expressive gestures.
• It primarily depicts stories from the Mahabharata and Ramayana.
• Unlike Mohiniattam, Kathakali is traditionally performed by male artists.
• Bharatanatyam
• Bharatanatyam is a classical dance form from Tamil Nadu.
• It is known for its intricate footwork, dramatic expressions, and spiritual themes.
• It is one of the oldest dance forms in India and is often performed as a form of devotional expression.
• Odissi
• Odissi is a classical dance form from Odisha.
• It is characterized by its fluid movements, sculpturesque poses, and themes derived from Hindu mythology.
• Odissi performances are often accompanied by Odissi music, which has its own distinct style.

The Correct answer is 5 years.

Key Points

• The tenure of elected members serving on the board of a co-operative society is generally prescribed under the respective State Co-operative Societies Acts or the Multi-State Co-operative Societies Act (for multi-state societies).
• As per the Indian Co-operative Societies Act, the elected members of the board typically serve for a term of 5 years.
• This ensures a balance between continuity of governance and the opportunity for members to bring in new leadership or policy direction.
• The 5-year term allows boards to implement long-term plans and strategies effectively for the betterment of the society and its members.
• The election process for board members is conducted in a democratic manner, ensuring representation and fairness for all members of the co-operative society.
• In certain cases, the tenure can vary depending on the specific state laws or bylaws of the co-operative society, but 5 years remains the standard term.
• The governance system in co-operative societies emphasizes transparency, accountability, and member participation, aligning with their democratic principles.

 Additional Information

• 3 years
• The 3-year term is not standard for co-operative societies in India. However, some organizations or committees with shorter project timelines may adopt this tenure.
• In certain countries, the tenure for elected members in similar entities might be 3 years, but this is not typically applicable to Indian co-operative societies.
• 4 years
• A 4-year term is more commonly seen in corporate governance or public office elections in some countries, but it is not a standard tenure for co-operative society boards in India.
• 6 years
• A 6-year term is generally considered too long for elected positions in co-operative societies, as it might hinder the democratic principle of allowing members to elect new representatives periodically.
• It is uncommon for co-operative societies in India to have a tenure of 6 years for board members.

The logic followed here is:

Thus, the missing term in the series is 1372.

Hence, the correct answer is "Option 2";

The pattern followed here is:

JOL : FRO

Pair 2: MKH : INK

Applying the same logic to the options:

Option 4: HPD : DSG

This pair follows the same (-4, +3, +3) pattern.

Hence, the correct answer is "Option 4".

The Correct answer is Export of raw materials, import of manufactured goods.

Key Points

• During the British colonial period, India's trade structure was designed to serve the economic interests of Britain.
• India became a supplier of raw materials such as cotton, jute, tea, and spices to Britain.
• At the same time, India was made a market for the import of manufactured goods, including textiles and machinery, from Britain.
• This unequal trade structure led to the decline of India's indigenous industries, including the famous textile industry.
• British policies ensured that India remained a dependent economy incapable of self-sustained industrial growth.
• The economic exploitation drained India's wealth and contributed significantly to the poverty and backwardness observed at the time of Independence.
• The combination of and manufactured goods imports reinforced India's colonial subjugation and economic imbalance.
• This trade pattern is a classic example of colonial economic strategy focused on resource extraction and market monopolization.

 Additional Information

• Export of capital, import of labour
• This option does not accurately describe India's trade structure under British colonial rule as India's economy was focused on the export of raw materials and the import of manufactured goods.
• There was no significant export of capital or organized import of labour in this context.
• Export of industrial goods, import of food
• India's industrial sector was stifled under British rule, making it impossible to export industrial goods.
• Instead, India relied on agricultural products and raw materials for export.
• Balanced two-way trade with Europe only
• India's trade under British rule was not balanced. It was highly exploitative and one-sided, favoring Britain.
• Trade relations were not limited to Europe; India also traded with other parts of the world as per British interests.

The Correct answer is Baul.

Key Points

• Baul represents a unique form of spiritual folk music that originated in Bengal.
• The Baul tradition is deeply rooted in Bengali culture and is known for its philosophical and mystical themes.
• Baul songs emphasize love, spirituality, and human connection over material possessions and worldly matters.
• Baul music is often performed using traditional instruments like the ektara, dotara, khamak, and duggi.
• The performers, known as Bauls, are often wandering minstrels who follow a unique lifestyle based on their philosophical beliefs.
• Baul music has been recognized as part of the Intangible Cultural Heritage of Humanity by UNESCO since 2008.
• The Baul tradition is influenced by Vaishnavism, Sufism, and Tantric practices.
• Famous Baul singers include Lalon Fakir, whose songs have had a lasting impact on Bengali music and literature.

 Additional Information

• Bhakha
• The term Bhakha is not directly associated with Bengali folk music but generally refers to language or dialect in certain Indian contexts.
• It is not a recognized form of folk music or tradition in Bengal.
• Ovi
• Ovi is a form of traditional folk poetry or song from Maharashtra, not Bengal.
• Ovi is often sung by women while performing household chores or during rituals.
• Wanawan
• Wanawan refers to a form of folk singing in Kashmir and is performed during weddings and other celebrations.
• It has no connection to the spiritual folk music of Bengal.

The Correct answer is Nepal.

Key Points

• The Kalapani dispute is a territorial conflict between India and Nepal.
• It revolves around the location of the Kalapani region, a strategically important area situated in the northwestern region of Uttarakhand, India, close to the Nepal border.
• Kalapani is located near the tri-junction of India, Nepal, and China.
• The dispute arose due to differing interpretations of the origin of the Mahakali River, which serves as the boundary between the two countries as per the Sugauli Treaty of 1815.
• India claims Kalapani based on historical maps and records that show the region as part of its territory.
• Nepal, however, claims Kalapani as part of its territory, citing the river as the eastern boundary, with the region located to the east of the river.
• The disagreement escalated in 2019 when India published a new political map following the abrogation of Article 370, which showed Kalapani as part of Uttarakhand.
• The issue remains unresolved, and both nations continue to engage in diplomatic discussions to address the territorial dispute.

 Additional Information

• China
• China is located to the north of both India and Nepal.
• Although China shares borders with both nations, it is not directly involved in the Kalapani territorial dispute.
• India and China have their own border disputes, including those concerning the Line of Actual Control (LAC) and regions like Aksai Chin and Arunachal Pradesh.
• Bangladesh
• Bangladesh shares its borders with India but does not have any territorial disputes over Kalapani.
• India and Bangladesh have resolved several border issues in the past, including the Land Boundary Agreement of 2015.
• Bhutan
• Bhutan is a landlocked country located to the northeast of India.
• India and Bhutan share strong diplomatic relations, and Bhutan is not involved in the Kalapani dispute.

Given:

X varies inversely as Y and Y varies inversely as Z.

Initial values: X = 1/16 and Z = 1/32.

New value: Z = 3.

Formula Used:

X × Y = k₁ (constant for X and Y).

Y × Z = k₂ (constant for Y and Z).

Calculation:

From X × Y = k₁ and Y × Z = k₂:

Y = k₁ / X, Z = k₂ / Y.

⇒ Z = k₂ / (k₁ / X).

⇒ Z = X × k₂ / k₁.

From the initial values, substitute X = 1/16 and Z = 1/32:

k₂ / k₁ = Z / X = (1/32) / (1/16).

k₂ / k₁ = 1/32 × 16/1 = 1/2.

Now, when Z = 3:

⇒ Z = X × k₂ / k₁.

⇒ 3 = X × (1/2).

⇒ X = 3 × 2.

⇒ X = 6.

The value of X when Z = 3 is 6.

Given series:

(Left) 5 @ 6 # 8 $ % 1 9 ^ 7 ≺  2 3 ⋟ 4 (Right)

Condition to be checked:

Number → Symbol → Number

5 @ 6 # 8 $ % 1 9 ^ 7 ≺  2 3 ⋟ 4

There are more than three such symbols in the above arrangement (5 @ 6, 6 # 8, 9 ^ 7, 7 ≺  2, 3 ⋟ 4 ).

Hence, the correct answer is "Option 2".

The pattern followed here is.

Thus, the missing term in the series is DGD 125.

Hence, the correct answer is Option 1.

The least possible venn  diagram is:

Conclusion

(1) No bottles are plates → True (As, All bottles are glasses, No glasses are plates. so, No bottles are plates holds true)

 

(II) No spoons are glasses → False (As, there is no direct relation between them)

 

Thus, Only Conclusion (1) is true.

 

Hence, the correct answer is "Option 2".

Shortcut Trick

Sum of x-coordinates = 3 × Centroid_x ⇒ x + 3 − 6 = 3 × 3.5 = 10.5 ⇒ x = 13.5.

Sum of y-coordinates = 3 × Centroid_y ⇒ 5 − 4 + y = 3 × 4.5 = 13.5 ⇒ y = 12.5.

Required point (x + 3, y − 4) = (13.5 + 3, 12.5 − 4) = (16.5, 8.5).

∴ The correct answer is (16.5, 8.5).

Alternate Method

Given: Vertices of triangle are A(x, 5), B(3, −4), and C(−6, y). The centroid is G(3.5, 4.5).

Formula Used: Centroid G(X, Y) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by:

X = (x₁ + x₂ + x₃) / 3 and Y = (y₁ + y₂ + y₃) / 3

Calculations:

⇒ 3.5 = (x + 3 + (−6)) / 3

⇒ 3.5 × 3 = x − 3

⇒ 10.5 = x − 3 ⇒ x = 13.5

⇒ 4.5 = (5 + (−4) + y) / 3

⇒ 4.5 × 3 = 1 + y

⇒ 13.5 = 1 + y ⇒ y = 12.5

⇒ Required point = (x + 3, y − 4)

⇒ Point = (13.5 + 3, 12.5 − 4)

⇒ Point = (16.5, 8.5)

∴ The correct answer is (16.5, 8.5).

Additional Information

Centroid Properties

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

Area of Triangle (Coordinate Geometry)

Area = 0.5 × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If the area is zero, the points are collinear.

Section Formula

A point dividing a line segment joining (x₁, y₁) and (x₂, y₂) in ratio m:n is ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).

The Correct answer is The Centre and one or more states.

Key Points

• The original jurisdiction of the Supreme Court of India is defined under Article 131 of the Indian Constitution.
• In its original jurisdiction, the Supreme Court can adjudicate disputes between the Government of India (Centre) and one or more states.
• This jurisdiction is exclusive to the Supreme Court, meaning no other court in India can handle such disputes.
• It also includes disputes between two or more states, or between the Centre and any state(s) on one side and other states on the other side.
• This ensures the maintenance of a federal balance and resolves any legal conflicts that may arise between different levels of government.
• However, it is important to note that this jurisdiction is limited to questions of law or questions of fact on which the existence of a legal right depends.
• Some examples of cases under this jurisdiction include disputes related to taxation, water disputes, or constitutional interpretation.

 Additional Information

• Two private citizens from different states
• Disputes between private citizens are not within the original jurisdiction of the Supreme Court.
• Such matters are generally dealt with by civil courts or high courts in India.
• Two municipal corporations of different states
• Municipal corporations are governed by their respective state governments, and any disputes between them would not fall under the original jurisdiction of the Supreme Court.
• Such cases would typically be resolved within the jurisdiction of the respective state high courts.
• A state government and a private corporation
• Disputes involving private corporations are not covered under Article 131.
• Such matters are generally resolved through civil courts or other legal mechanisms, including tribunals.

Given:

Find: (60% of 40) − (2/5 of 55)

Formula used:

Percentage: (p% of q) = (p × q) ÷ 100

Fraction: (m/n of q) = (m × q) ÷ n

Calculations:

⇒ 60% of 40 = (60 × 40) ÷ 100 = 2400 ÷ 100 = 24

⇒ (2/5) of 55 = (2 × 55) ÷ 5 = 110 ÷ 5 = 22

⇒ Difference = 24 − 22 = 2

∴ 60% of 40 is greater by 2.