RRB NTPC Graduate Level CBT-II Official Paper (Held On: 10 Jul, 2026) (Previous Year Paper)

Shortcut Trick

When two parallel lines are intersected by a transversal, the corresponding angles are always equal.

In the figure, X and the given angle 55° are corresponding angles.

∴ X = 55.

∴ The correct answer is 55.

Alternate Method

Given: AB ∥ CD and RS is a transversal.

Formula Used: Property of Corresponding Angles in parallel lines.

Calculations:

⇒ Let the point of intersection of RS with AB be P and with CD be Q.

⇒ Since AB ∥ CD, the angles at the same relative position at each intersection are corresponding angles.

⇒ ∠BPS and ∠DQS are corresponding angles.

⇒ Property: ∠BPS = ∠DQS

⇒ X = 55°

∴ The correct answer is 55.

Additional Information

Alternate Interior Angles

When a transversal cuts parallel lines, the pairs of angles on opposite sides of the transversal and between the parallel lines are equal.

Consecutive Interior Angles

Also known as co-interior angles, these are on the same side of the transversal and their sum is always 180° for parallel lines.

Vertically Opposite Angles

Angles opposite each other at a vertex where two lines intersect are always equal, regardless of parallel lines.

The correct answer is

Power.

Key Points

• Power is defined as the rate at which work is done or the rate at which energy is transferred or transformed.
• It describes how fast a specific amount of work is completed over a given time interval.
• The mathematical expression for power is Power = Work ÷ Time.
• The SI unit of power is the Watt (W), which is equivalent to one Joule per second (J/s).
• Power is a scalar quantity, meaning it has only magnitude and no specific direction.
• The dimensional formula for power is [ML²T⁻³].

Additional Information

• Energy is the total capacity to do work. While energy tells us how much work can be done, power tells us how quickly it is done.
• Force is a vector quantity that represents a push or pull upon an object, calculated as the product of mass and acceleration (F = m × a).
• Displacement refers to the change in position of an object in a specific direction; it is the shortest straight-line distance between the start and end points.
• Work is done when a force acts upon an object to cause a displacement (Work = Force × Displacement × cosθ).
• Common non-SI units of power include Horsepower (hp), where 1 hp ≈ 746 Watts.

Shortcut Trick

Height of kite from ground (using Amit's data) = 100 × sin 30° = 100 × (1/2) = 50 m.

Vertical height of kite relative to Nitin's roof = 50 m − 10 m = 40 m.

Nitin's string length = Relative Height ÷ sin 45° = 40 ÷ (1/√2) = 40√2 m.

∴ The correct answer is 40√2 m.

Alternate Method

Given:

Length of Amit's string = 100 m

Angle of elevation for Amit = 30°

Height of building (Nitin's position) = 10 m

Angle of elevation for Nitin = 45°

Calculations:

Let the kite be at point P and its vertical height from ground be H.

In the right-angled triangle for Amit:

⇒ sin 30° = H ÷ 100

⇒ 1/2 = H ÷ 100

⇒ H = 50 m

Now, Nitin is on a building of height 10 m. Let his string length be L.

Vertical distance from Nitin's level to the kite = 50 − 10 = 40 m

In the right-angled triangle for Nitin:

⇒ sin 45° = 40 ÷ L

⇒ 1/√2 = 40 ÷ L

⇒ L = 40 × √2

⇒ L = 40√2 m

∴ The correct answer is 40√2 m.

Additional Information

Trigonometric Ratios

For a right-angled triangle: sin θ = Perpendicular ÷ Hypotenuse; cos θ = Base ÷ Hypotenuse; tan θ = Perpendicular ÷ Base.

Standard Values

sin 30° = 1/2; sin 45° = 1/√2; sin 60° = √3/2.

Angle of Elevation

The angle formed by the line of sight with the horizontal when looking up at an object.

Shortcut Trick

Ratio of Height (h) to Base (b) is 1 : 3.

Let height = h, then base = 3h.

Area = Base × Height ⇒ 216 = 3h × h ⇒ 216 = 3h².

h² = 72 ⇒ h = √72 = √(36 × 2) = 6√2 cm.

∴ The correct answer is 6√2 cm.

Alternate Method

Given:

Area of the parallelogram = 216 cm²

Relation: Base (b) = 3 × Height (h)

Formula Used:

Area of a parallelogram = Base × Height

Calculations:

⇒ Area = b × h

⇒ 216 = (3h) × h

⇒ 216 = 3h²

⇒ h² = 216 ÷ 3

⇒ h² = 72

⇒ h = √72

⇒ h = √(36 × 2)

⇒ h = 6√2 cm

∴ The correct answer is 6√2 cm.

Additional Information

Area and Perimeter of Parallelogram

The Area is calculated as Base × Perpendicular Height. The Perimeter is 2 × (Sum of adjacent sides).

Properties of a Parallelogram

Opposite sides are equal and parallel. Diagonals of a parallelogram bisect each other, but they are not necessarily equal in length unless it is a rectangle.

Relationship with Triangles

A diagonal divides a parallelogram into two congruent triangles. Thus, Area of parallelogram = 2 × Area of triangle formed by the diagonal.

The correct answer is - Bird populations may decline due to reduced insect prey, altering energy flow

Key Points

• Disruption of Trophic Levels
• In a typical agricultural food web, insects often occupy the role of primary consumers (eating plants) or secondary consumers (predatory insects).
• Many bird species are insectivores, meaning they rely almost exclusively on insects as their primary food source.
• When a pesticide significantly reduces the insect population, it removes a critical energy source for these birds, leading to starvation, reduced reproductive success, and an eventual decline in bird populations.
• Energy Flow Alteration
• Energy flows from producers (plants) to primary consumers (insects) and then to secondary consumers (birds).
• A sharp decrease in the biomass of one trophic level (insects) creates a bottleneck, preventing energy from efficiently reaching the higher trophic levels (birds).
• This alteration can lead to a top-down or bottom-up effect that destabilizes the entire ecosystem structure.

Additional Information

• Food Web Dynamics
• A food web consists of all the food chains in a single ecosystem. Each living thing in an ecosystem is part of multiple food chains.
• The removal of a single species or group (like insects) can have a cascading effect on organisms that are not directly targeted by the pesticide.
• Pesticide Impact
• While pesticides are intended to protect crops, they can lead to bioaccumulation and biomagnification, where toxic substances become more concentrated as they move up the food chain.
• Indirect effects, such as habitat loss or prey depletion, are often more damaging to biodiversity than the direct toxicity of the chemical itself.
• Role of Other Organisms
• Decomposers: While they break down dead matter, a sudden drop in living insect populations doesn't necessarily mean a long-term increase in decomposers, as the total organic matter input might eventually decrease.
• Plants: Although some plants rely on insects for pollination, many agricultural crops are wind-pollinated. Therefore, a decline in birds due to food shortage is a more direct and universal consequence of insect loss in this context.

The correct answer is Elephanta Island.

Key Points

• Elephanta Island, locally known as Gharapuri, is located in the Mumbai Harbour approximately 10 kilometers east of Mumbai city.
• It is renowned for the Elephanta Caves, a collection of seven rock-cut caves carved between the 5^th and 8^th centuries, showcasing high-quality sculptural art.
• The site was designated as a UNESCO World Heritage Site in 1987 due to its cultural significance and the preservation of Shiva-centric iconography.
• The most iconic feature is the 7-meter-high Trimurti sculpture, which depicts Lord Shiva as the Creator (Vamadeva), Preserver (Tatpurusha), and Destroyer (Aghora).
• The architectural style reflects the Kalachuri dynasty influence, featuring massive pillars, spacious halls, and intricate panels depicting Hindu mythological narratives.

 Additional Information

• Butcher Island
• Officially known as Jawahar Dweep, it serves as an essential oil terminal for the Mumbai Port Trust.
• It lacks historical monuments and is primarily used for industrial and logistical purposes related to crude oil storage.
• Khanderi Island
• Located south of Mumbai off the coast of Thal, it is famous for the Khanderi Fort built by Chhatrapati Shivaji Maharaj in 1679.
• The island, along with its twin Underi, played a strategic role in Maratha naval defense against the Siddhis and the British.
• Salsette Island
• It is the 14^th most populous island in the world and contains the entire Mumbai Metropolitan Region, including Thane.
• While it houses the Kanheri Caves within Sanjay Gandhi National Park, it is generally referred to by its geographic name rather than as a specific heritage island site like Elephanta.

The correct answer is Ganga.

Key Points

• The Ganga River basin is the largest drainage basin in India, covering a vast catchment area of approximately 8,61,452 square kilometers.
• It accounts for nearly 26.3% of the total geographical area of the country, stretching across the states of Uttarakhand, Uttar Pradesh, Bihar, West Bengal, Haryana, Himachal Pradesh, Rajasthan, Madhya Pradesh, Chhattisgarh, and the Union Territory of Delhi.
• The basin is a part of the larger Ganges-Brahmaputra-Meghna (GBM) river system and is fed by both Himalayan snow-melt and Peninsular rainfall, ensuring it remains a perennial water source.
• The Ganga originates from the Gangotri Glacier near Gaumukh in the Uttarkashi district of Uttarakhand, where it is initially known as the Bhagirathi.
• The fertile Indo-Gangetic Plain formed by this basin is one of the most densely populated and agriculturally productive regions in the world due to the deposition of alluvial soil.

 Additional Information

• Godavari:
• It is the second-largest river basin in India, often referred to as the Dakshina Ganga.
• The catchment area spans approximately 3,12,812 square kilometers, covering parts of Maharashtra, Telangana, Andhra Pradesh, Chhattisgarh, and Odisha.
• Krishna:
• This is the third-largest basin in India, with a total catchment area of about 2,58,948 square kilometers.
• It drains large parts of Maharashtra, Karnataka, Telangana, and Andhra Pradesh before entering the Bay of Bengal.
• Narmada:
• It is the largest west-flowing river of Peninsular India, originating from the Amarkantak plateau.
• Its basin is relatively smaller, covering about 98,796 square kilometers, primarily across Madhya Pradesh and Gujarat.

The logic follows here,

• 39 + 13 = 52
• 52 + 13 = 65
• 65 + 13 = 78
• 78 + 13 = 91
• 91 + 13 = 104

Thus, the missing term in the series is 104.

Hence, the correct answer is 'Option 4';

Shortcut Trick

Use the Successive Multiplier method to find the final price.

Final SP = Initial CP × (1 + Profit₁/100) × (1 − Loss₂/100)

Final SP = 40,000 × 1.20 × 0.90

Final SP = 40,000 × 1.08 = 43,200

∴ The correct answer is ₹43,200.

Alternate Method

Given: Initial Cost Price (CP) = ₹40,000, First Profit = 20%, Second Loss = 10%

Step 1: Calculate the first selling price (SP₁) after 20% profit.

⇒ SP₁ = CP + (Profit % of CP)

⇒ SP₁ = 40,000 + (20/100 × 40,000)

⇒ SP₁ = 40,000 + 8,000 = ₹48,000

Step 2: This SP₁ becomes the Cost Price for the second transaction.

⇒ CP₂ = ₹48,000

Step 3: Calculate the final selling price (SP₂) after 10% loss.

⇒ Final SP = CP₂ − (Loss % of CP₂)

⇒ Final SP = 48,000 − (10/100 × 48,000)

⇒ Final SP = 48,000 − 4,800 = ₹43,200

∴ The correct answer is ₹43,200.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the effective net change is given by [a + b + (a × b)/100] %.

Profit and Loss Multiplier

For a profit of P%, the selling price multiplier is (1 + P/100). For a loss of L%, the multiplier is (1 − L/100).

Effective Profit Percentage

In this case, net change = [20 − 10 + (20 × −10)/100] = [10 − 2] = 8% profit. Final SP = 40,000 × 1.08 = ₹43,200.

Preparing the family tree diagram by using the following symbols:

Checking : V % Q # K + Z − T × L

 

Based on the tree for expression I, Q, Z, and T are siblings. Q and T are females. T is the wife of L. Therefore, Q is the sister of L's wife.

Hence, the correct answer is 'Option 3'.

The correct answer is National Biodiversity Authority.

Key Points

• The National Biodiversity Authority (NBA) is a statutory body established by the Government of India in 2003 to implement the provisions of the Biological Diversity Act, 2002.
• In March 2026, the NBA constituted a specialized committee dedicated to the management and mitigation of Invasive Alien Species (IAS), which pose a significant threat to India’s endemic flora and fauna.
• The primary objective of this committee is to develop a comprehensive national strategy for the early detection, rapid response, and systematic eradication of species that disrupt local ecosystems.
• This move is part of India’s commitment to the Kunming-Montreal Global Biodiversity Framework, which aims to reduce the introduction and establishment of known invasive alien species by at least 50% by 2030.
• The NBA, headquartered in Chennai, functions as a regulatory and advisory body to the Central Government on issues of conservation and the equitable sharing of benefits arising from biological resources.

 Additional Information

• Central Pollution Control Board (CPCB):
• Established in September 1974 under the Water (Prevention and Control of Pollution) Act, 1974.
• It serves as a technical wing of the Ministry of Environment, Forest and Climate Change, focusing on monitoring air quality and water pollution levels across the country.
• Indian Council of Forestry Research and Education (ICFRE):
• An autonomous body headquartered in Dehradun, it oversees forestry research, education, and extension.
• It operates several regional institutes and centers to address location-specific forestry and silviculture challenges.
• Invasive Alien Species (IAS):
• These are non-native species, such as Lantana camara, Prosopis juliflora, and Water Hyacinth, that cause ecological or economic harm.
• In India, these species often outcompete native vegetation, leading to a loss of biodiversity and affecting the livelihoods of forest-dependent communities.

The correct answer is Brahmi.

Key Points

• Emperor Ashoka, who ruled the Mauryan Empire from approximately 268 to 232 BCE, utilized inscriptions on rocks and pillars to disseminate his message of Dhamma (righteousness).
• While these edicts were written in regional languages like Prakrit, Greek, and Aramaic, the Brahmi script was the most widely used script for the majority of the inscriptions found across the Indian subcontinent.
• The Brahmi script is an ancient writing system written from left to right and is considered the mother of almost all modern Indian scripts, including Devanagari, Bengali, and Tamil.
• In 1837, James Prinsep, a British antiquarian and colonial administrator, successfully deciphered the Brahmi script, which provided the key to understanding Ashokan history and his royal titles like Devanampiya (Beloved of the Gods).
• The use of Brahmi was uniform across the central and southern parts of the empire, making it the predominant script of the Mauryan administrative and religious communications.

 Additional Information

• Kharosthi:
• This script was used primarily in the northwestern part of the Mauryan Empire (modern-day Pakistan and Afghanistan), such as in the Shahbazgarhi and Mansehra edicts.
• Unlike Brahmi, Kharosthi was written from right to left and was influenced by the Aramaic script.
• Devanagari:
• This is a much later script that evolved around the 7^th century CE from the Gupta script.
• It is currently used for writing Sanskrit, Hindi, and Marathi, but it did not exist during Emperor Ashoka's reign.
• Gupta Script:
• It was used to write Sanskrit during the Gupta Empire (4^th to 6^th century CE).
• It represents a cursive evolution of Brahmi and served as a link between ancient and medieval Indian scripts.

Shortcut Trick

Total sum of interior angles of a 10-sided figure = (10 − 2) × 180 = 1440 degrees.

Sum of the five given angles = 5 × 185 = 925 degrees.

Sum of the remaining five equal angles = 1440 − 925 = 515 degrees.

Each remaining angle = 515 ÷ 5 = 103 degrees.

∴ The correct answer is 103 degrees.

Alternate Method

Given: Number of sides (n) = 10; 5 angles = 185 degrees each; other 5 angles are equal.

Formula: Sum of interior angles of a polygon = (n − 2) × 180 degrees.

For a 10-sided figure (decagon):

⇒ Total sum of interior angles = (10 − 2) × 180°

⇒ Total sum = 8 × 180° = 1440°

Let each of the five equal remaining angles be x.

⇒ 5 × 185° + 5 × x = 1440°

⇒ 925° + 5x = 1440°

⇒ 5x = 1440° − 925°

⇒ 5x = 515°

⇒ x = 515° ÷ 5 = 103°

∴ The correct answer is 103 degrees.

Additional Information

Sum of Interior Angles

For any n-sided simple polygon, the sum of all its interior angles is given by the formula (n − 2) × 180 degrees.

Sum of Exterior Angles

The sum of the exterior angles of any convex or concave polygon, taking one at each vertex, is always 360 degrees.

Concave Polygon

A polygon is concave if at least one of its interior angles is a reflex angle (greater than 180 degrees).

Regular Polygon Angle

Each interior angle of a regular polygon with n sides is calculated as [(n − 2) × 180] ÷ n.

Shortcut Trick

Interest for 3 years = ₹4,500

Interest for 1 year = 4500 ÷ 3 = ₹1,500

Rate (x) = (Interest for 1 year ÷ Principal) × 100

x = (1500 ÷ 30000) × 100 = 5

∴ The correct answer is 5.

Alternate Method

Given:

Principal (P) = ₹30,000

Time (T) = 3 years

Simple Interest (SI) = ₹4,500

Rate (R) = x% per annum

Formula Used:

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

Calculations:

⇒ 4500 = (30000 × x × 3) ÷ 100

⇒ 4500 = 300 × 3 × x

⇒ 4500 = 900 × x

⇒ x = 4500 ÷ 900

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Simple Interest Formula

Simple Interest is calculated only on the initial principal amount: SI = (P × R × T) ÷ 100.

Calculating Rate

The rate of interest per annum can be found using R = (SI × 100) ÷ (P × T).

Total Amount

The total amount (A) payable after the time period is the sum of Principal and Interest: A = P + SI.

Given: 

• A certain number of people are sitting in a row, facing north.

• E sits at the extreme left end of the row.

• Exactly seven people sit between E and F.

• A sits third to right of F.

• Exactly four people sit between C and B.

• C sits exactly in between E and F.

• Exactly four people sit between B and D.

• D sits at one of the extreme ends

The Final Arrangement is, 

The total number of people seated in the row corresponds to the position of the last person, D.

The total number of people is 15.

Hence, the correct answer is "Option 2"

The correct answer is Ousmane Dembélé.

Key Points

• The 2025 Ballon d’Or ceremony took place at the Théâtre du Châtelet in Paris on 22 September 2025, marking the 69^th edition of the annual awards presented by France Football.
• Ousmane Dembélé, a star winger for Paris Saint-Germain (PSG) and the France national team, was honored as the world's best player for the 2024-25 season.
• The French international was recognized for his extraordinary playmaking abilities, explosive pace, and his pivotal role in guiding PSG to multiple domestic honors and a deep run in the UEFA Champions League.
• The voting was conducted by a select jury of international journalists from the top 100 FIFA-ranked nations, who assessed players based on individual talent, team trophies, and exemplary conduct.
• Winning this accolade cements Dembélé’s legacy as one of the few French players to receive the Golden Ball, following in the footsteps of legends like Michel Platini and Zinedine Zidane.

 Additional Information

• Kylian Mbappé:
• A world-renowned forward and France national team captain who joined Real Madrid in 2024; he remains a dominant force in La Liga and continues to be a perennial contender for the Ballon d'Or.
• Jude Bellingham:
• The English midfielder playing for Real Madrid who previously won the Kopa Trophy and has become a vital component of the England national team midfield.
• Erling Haaland:
• A Norwegian striker representing Manchester City, widely regarded as one of the best finishers in the world and a frequent recipient of the Gerd Müller Trophy for most goals.
• Ballon d'Or Evolution:
• Founded in 1956 by the magazine France Football, it was originally restricted to European players before expanding its eligibility globally in 2007.
• The award is distinct from FIFA’s The Best awards, although the two were merged into the FIFA Ballon d'Or between 2010 and 2015.

Given:   715 438 961 127 821 

• The third digit of the highest number is: 1 (961)
• The second digit of the lowest number is: 2 (127)

So, the addition of the third digit of the highest number and the second digit of the lowest number is: 1 + 2 = 3.

Hence, the correct answer is 'Option 3'.

The correct answer is Kuchipudi.

Key Points

• Kuchipudi is a prominent Indian classical dance form that originated in the Krishna district of Andhra Pradesh, specifically in the village of Kuchelapuram.
• Historically, the dance-drama traditions of the Andhra region were generically called Yakshagaana. Over time, the specific style developed by the Brahmin troupes of this village became distinctively known as Kuchipudi.
• The tradition was significantly refined by Siddhendra Yogi during the 17^th century, who is credited with formalizing the dance and composing the famous Bhama Kalapam.
• Originally, it was performed exclusively by men (known as Bhagavatulu), who enacted various roles from the Bhagavata Purana and other mythological texts in a theater-style presentation.
• Characteristic features of this dance include Tarangam (dancing on the rim of a brass plate) and a blend of Nritta (pure dance), Nritya (expressive dance), and Natya (drama).

 Additional Information

• Mohiniyattam:
• A classical dance form from Kerala, performed by women as a solo recital.
• The term refers to the Dance of the Enchantress, based on the Mohini avatar of Lord Vishnu.
• Bharatanatyam:
• Originating in Tamil Nadu, it was anciently known as Sadir Attam or Dasi Attam.
• It was traditionally performed by Devadasis in South Indian temples as a form of worship.
• Kathak:
• Hailing from North India, its name is derived from the word Katha, meaning story.
• It was traditionally performed by nomadic storytellers known as Kathakars in temples and later in Mughal courts.

The correct answer is Spreadsheet.

Key Points

• A Spreadsheet is an application software used for organizing, analyzing, and storing data in a tabular format consisting of rows and columns.
• The most distinctive feature of a spreadsheet is Automatic Recalculation, which ensures that when a user changes a value in a cell, the software instantly updates all results derived from that data through formulas.
• This capability allows for What-if Analysis, where users can modify variables (such as prices or interest rates) and observe the immediate impact on totals or financial projections without manual re-calculation.
• Spreadsheets use Cell Referencing (e.g., A1, B5) to link different data points, creating a dynamic environment where data flows logically throughout the Worksheet.

 Additional Information

• Database software:
• Focuses on managing large, structured datasets and ensuring Data Integrity.
• Common examples include MySQL, Oracle, and Microsoft Access.
• While it stores data, viewing changes often requires running specific Queries rather than providing an instant grid-based update.
• Presentation tool:
• Used to create Slide Shows for visual communication (e.g., Microsoft PowerPoint, Google Slides).
• It is designed for displaying information and lacks the mathematical engine required for real-time data modeling.
• Word processor:
• Specialized for Text Editing and document formatting (e.g., Microsoft Word).
• Although it can include tables, it does not possess the computational logic to automatically update mathematical results when a table value is changed.

Shortcut Trick

Apply the Mean Proportional formula: b² = a × c.

Here, a = (x + 3), c = (x − 3), and b = 4.

4² = (x + 3)(x − 3) ⇒ 16 = x² − 9.

x² = 25 ⇒ x = 5 (since x > 0).

∴ The correct answer is 5.

Alternate Method

Given:

First term (a) = (x + 3)

Second term (c) = (x − 3)

Mean proportional (b) = 4

Formula Used:

If 'b' is the mean proportional between 'a' and 'c', then b = √(a × c) or b² = a × c.

Calculations:

⇒ 4² = (x + 3) × (x − 3)

⇒ 16 = x² − 3² [Using identity (a + b)(a − b) = a² − b²]

⇒ 16 = x² − 9

⇒ x² = 16 + 9

⇒ x² = 25

⇒ x = √25

⇒ x = 5 (since it is given that x > 0)

Thus, the value of x is 5, which corresponds to Option (4).

∴ The correct answer is 5.

Additional Information

Mean Proportional

The mean proportional between two numbers 'a' and 'c' is the square root of their product, given by b = √(ac).

Third Proportional

If a : b = b : c, then 'c' is called the third proportional to 'a' and 'b', calculated as c = b²/a.

Algebraic Identity

The identity (a + b)(a − b) = a² − b² is a fundamental tool for solving quadratic relations in proportions.

Shortcut Trick

Identify the cube roots of the cubic terms: √³(27x³) = 3x and √³(343y³) = 7y.

Check if all terms in the expression are positive, which indicates the (a + b)³ form.

Verify that the middle terms match 3a²b and 3ab² for a = 3x and b = 7y.

The resulting expression is simply (3x + 7y)³.

∴ The correct answer is (3x + 7y)³.

Alternate Method

Given: 27x³ + 343y³ + 189x²y + 441xy²

Formula Used: (a + b)³ = a³ + b³ + 3a²b + 3ab²

⇒ Let us rewrite the terms in the standard identity form:

⇒ 27x³ = (3x)³

⇒ 343y³ = (7y)³

⇒ 189x²y = 3 × (3x)² × (7y) = 3 × 9x² × 7y = 189x²y

⇒ 441xy² = 3 × (3x) × (7y)² = 3 × 3x × 49y² = 441xy²

⇒ Expression = (3x)³ + (7y)³ + 3(3x)²(7y) + 3(3x)(7y)²

⇒ Expression = (3x + 7y)³

∴ The correct answer is (3x + 7y)³.

Additional Information

Cubic Difference Identity

The expression for (a − b)³ is expanded as a³ − b³ − 3a²b + 3ab².

Sum of Two Cubes

The factorization of a³ + b³ is (a + b)(a² − ab + b²).

Difference of Two Cubes

The factorization of a³ − b³ is (a − b)(a² + ab + b²).

Preparing the family tree diagram by using the following symbols:

 

From the above breakdown, M is the mother of H, and H is the mother of F. This makes M the maternal grandmother of F.

Therefore, M is the Mother’s mother of F.

Hence, the correct answer is 'Option 3'.

Shortcut Trick

Ratio of distances X : Y = 750 : 740 = 75 : 74

Ratio of distances Y : Z = 444 : 438 = 74 : 73

Combining the ratios, X : Y : Z = 75 : 74 : 73

In a 150 m race, X covers 75 × 2 = 150 m, so Z covers 73 × 2 = 146 m.

Winning distance = 150 − 146 = 4 m.

∴ The correct answer is 4 m.

Alternate Method

Given:

Distance of Race 1 (X and Y) = 750 m | X beats Y by 10 m.

Distance of Race 2 (Y and Z) = 444 m | Y beats Z by 6 m.

Distance of Race 3 (X and Z) = 150 m.

Calculations:

When X covers 750 m, Y covers (750 − 10) = 740 m.

⇒ Ratio of speeds V_x/V_y = 750/740 = 75/74

When Y covers 444 m, Z covers (444 − 6) = 438 m.

⇒ Ratio of speeds V_y/V_z = 444/438 = (444 ÷ 6) / (438 ÷ 6) = 74/73

To find the relation between X and Z:

⇒ V_x/V_z = (V_x/V_y) × (V_y/V_z)

⇒ V_x/V_z = (75/74) × (74/73) = 75/73

In a race of 150 m, distance covered by X = 150 m.

⇒ If 75 units = 150 m, then 1 unit = 2 m.

⇒ Distance covered by Z = 73 × 2 = 146 m.

⇒ X beats Z by = 150 − 146 = 4 m.

∴ The correct answer is 4 m.

Additional Information

Concepts of Races

In a race where A beats B by d meters, A covers the full distance L while B covers L − d in the same time. The ratio of their speeds is V_a/V_b = L/(L − d).

Successive Ratios

If A:B = x:y and B:C = y:z, then the combined ratio A:B:C = x:y:z. If the common term B is not the same, equate it by multiplying the ratios with suitable constants.

Beating by Time

If A beats B by t seconds, it means A reaches the finish line t seconds before B. Distance by which A beats B = Speed of B × Time t.

The correct answer is Black soil rich in clay and moisture.

Key Points

• Black soil, also popularly known as Regur soil, is renowned for its high clay content, which allows it to retain moisture for an extended period, making it ideal for rain-fed cotton cultivation.
• This soil type is predominantly found in the Deccan Trap region, covering states like Maharashtra, Gujarat, and parts of Madhya Pradesh, formed from the weathering of basaltic lava rocks.
• The self-ploughing characteristic of this soil, where it develops deep cracks during the dry season, facilitates oxygenation and deep aeration of the soil layers.
• Chemically, black soil is rich in essential nutrients like calcium carbonate, magnesium, potash, and lime, which support the robust growth of the cotton plant.

 Additional Information

• Alluvial Soil
• Deposited by the Indus, Ganga, and Brahmaputra river systems.
• Covers nearly 40 percent of India's total land area and is highly fertile for wheat, paddy, and sugarcane.
• Laterite Soil
• Formed due to intense leaching caused by heavy tropical rains in areas like the Western Ghats and Odisha.
• It is typically acidic in nature and is preferred for plantation crops like cashew nuts, tea, and coffee.
• Desert Soil
• Found in arid and semi-arid regions such as Western Rajasthan.
• Exhibits high salinity and low humus content, requiring intensive irrigation for agricultural productivity.

Shortcut Trick

Check the validity of each statement based on quadrilateral properties:

A. Isosceles trapezium: Non-parallel sides are equal, not parallel sides. (False)

B. Kite: Pairs of adjacent sides are equal; Parallelogram: Opposite sides are equal. (False)

C. Square: Has 4 equal sides. Rhombus: Has 4 equal sides. Thus, all squares are rhombuses. (True)

D. Varignon’s Theorem: Joining mid-points of any quadrilateral always forms a parallelogram. (False)

∴ The correct answer is Only C.

Alternate Method

Analyze each statement by geometric definitions:

Statement A: In an isosceles trapezium, the non-parallel sides (legs) are equal in length. If the parallel sides were equal, the shape would be a parallelogram, not a trapezium.

Statement B: A kite is a quadrilateral with two distinct pairs of equal adjacent sides. A parallelogram must have two pairs of parallel and equal opposite sides. Thus, a kite is not a type of parallelogram.

Statement C: A square is defined as a quadrilateral with four equal sides and four right angles. A rhombus is defined as a quadrilateral with four equal sides. Since every square meets the criteria of having four equal sides, it is always a rhombus.

Statement D: According to Varignon’s Theorem, the mid-points of the sides of any arbitrary quadrilateral, when joined in order, form a parallelogram, which is a more specific property than being a trapezium.

⇒ Statement A is incorrect.

⇒ Statement B is incorrect.

⇒ Statement C is correct (Square ⊆ Rhombus).

⇒ Statement D is incorrect.

∴ The correct answer is Only C.

Additional Information

Varignon’s Theorem

The quadrilateral formed by joining the mid-points of the sides of any quadrilateral is always a parallelogram, and its area is exactly half the area of the original quadrilateral.

Hierarchy of Quadrilaterals

Every square is a rhombus and a rectangle. Every rhombus and rectangle is a parallelogram. Every parallelogram is a trapezium (based on the inclusive definition).

Isosceles Trapezium Properties

Key features include equal non-parallel sides, equal base angles, and equal diagonals. It is also a cyclic quadrilateral.

Shortcut Trick

Ratio of Rahul to Son = 7 : 2

Given Rahul's age (7 units) = 49 years

Value of 1 unit = 49 ÷ 7 = 7 years

Son's age (2 units) = 2 × 7 = 14 years

∴ The correct answer is 14 years.

Alternate Method

Given:

Rahul's age = 49 years

Ratio of ages (Rahul : Son) = 7 : 2

Formula Used:

Total Value = Sum of parts × Unit value | Individual Value = Part × Common Factor (x)

Calculations:

⇒ Let the common ratio factor be x.

⇒ Rahul's age = 7x

⇒ Son's age = 2x

⇒ According to the question: 7x = 49

⇒ x = 49 ÷ 7 = 7

⇒ Son's age = 2 × 7

⇒ Son's age = 14 years

∴ The correct answer is 14 years.

Additional Information

Ratio Definition

A ratio is a comparison of two quantities of the same unit, expressed as a : b or a/b.

Proportion Rule

If a : b = c : d, then the product of extremes equals the product of means (a × d = b × c).

Age Difference Consistency

The difference between the ages of two persons remains constant regardless of the number of years passed.

Shortcut Trick

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

Here, the sum triples (n = 3) in T = 7 years.

We want the sum to become 27 times.

⇒ 27 = 3³, so k = 3.

⇒ Required time = k × T = 3 × 7 = 21 years.

∴ The correct answer is 21 years.

Alternate Method

Given:

Initial principal = P

Time (T₁) = 7 years

Amount (A₁) = 3P

Target Amount (A₂) = 27P

Formula Used:

A = P(1 + r/100)ⁿ

Calculations:

⇒ 3P = P(1 + r/100)⁷

⇒ 3 = (1 + r/100)⁷ ... (i)

Let the required time be 't' years for the sum to become 27 times.

⇒ 27P = P(1 + r/100)^t

⇒ 27 = (1 + r/100)^t

⇒ 3³ = [(1 + r/100)⁷]^t/7

Substituting the value from equation (i):

⇒ 3³ = 3^t/7

Equating the powers:

⇒ 3 = t/7

⇒ t = 3 × 7 = 21 years.

∴ The correct answer is 21 years.

Additional Information

Multiplicative Growth in CI

In compound interest, the amount grows by the same factor over equal intervals of time. If it doubles in 'x' years, it will quadruple in '2x' years.

Annual vs Half-Yearly Compounding

For half-yearly compounding, the rate is halved (r/2) and the time periods are doubled (2n) in the formula A = P(1 + r/200)²ⁿ.

Rule of 72

A quick way to estimate the time required to double an investment: Time ≈ 72 ÷ Interest Rate.

CONCEPT:

• Thermal Decomposition is a chemical process where a single compound breaks down into two or more simpler substances upon the application of heat.
• The observation of brown fumes during the heating of an inorganic salt is a classic diagnostic test for the presence of the nitrate (NO₃⁻) ion, specifically indicating the evolution of nitrogen dioxide (NO₂) gas.

EXPLANATION:

• When Lead(II) nitrate crystals are heated in a dry test tube, they undergo a decomposition reaction.
• The balanced chemical equation for this thermal decomposition is:

$2Pb(NO_3)_2(s) \xrightarrow{\text{Heat}} 2PbO(s) + 4NO_2(g) + O_2(g)$

• In this reaction:
• Lead(II) oxide (PbO) is formed as a solid residue which is yellow when cold and reddish-brown when hot.
• Nitrogen dioxide (NO₂) is evolved as a gas, which is visible as characteristic reddish-brown fumes.
• Oxygen (O₂) is also evolved, which is a colorless and odorless gas.
• Analysis of other substances:
• Calcium carbonate (CaCO₃) decomposes into calcium oxide and carbon dioxide (a colorless gas).
• Silver chloride (AgCl) typically decomposes into silver and chlorine gas (pale yellow-green) under the influence of light (photolytic decomposition).
• Sodium chloride (NaCl) is highly stable and does not decompose upon simple heating in a laboratory burner.

Therefore, the compound that emits brown fumes upon heating is Lead(II) nitrate, Pb(NO₃)₂.

Given:  A, B, C, D, E, F, G and H are sitting around a square table facing the centre of the table. Some of them are sitting at the corners while some are sitting at the exact centre of the sides.

1. B is seated second to the right of H. B is not seated at the centre of any side.
2. F is the immediate neighbour of E and B.
3. D is seated third to the left of H.
4. G is seated in a corner.
5. C is seated second to the right of A

The Final Arrangement is, 

There are five people, E, F, B, C, and H seated between D and A.

Hence, the correct answer is "Option 1"

Shortcut Trick

Ratio of speeds of Sushant and Varun = (1/100) : (1/80) = 80 : 100 = 4 : 5.

In a circular track, if two people run in opposite directions with speeds in the ratio a : b, they meet at (a + b) distinct points.

Total meeting points = 4 + 5 = 9 points (including the starting point).

The question asks for the number of meetings before they return to the start together.

Number of meetings = 9 − 1 = 8 times.

∴ The correct answer is 8.

Alternate Method

Given:

Length of the circular track = 400 m

Time taken by Sushant = 100 sec

Time taken by Varun = 80 sec

Formula Used:

1. Speed = Distance ÷ Time
2. Time to meet at starting point = LCM of individual times
3. Time for each meeting = Track length ÷ Relative Speed

Calculations:

⇒ Speed of Sushant (V_s) = 400 ÷ 100 = 4 m/s

⇒ Speed of Varun (V_v) = 400 ÷ 80 = 5 m/s

⇒ Total time after which they meet at the starting point = LCM(100, 80) = 400 sec

⇒ Since they run in opposite directions, Relative Speed = V_s + V_v = 4 + 5 = 9 m/s

⇒ Time for one meeting = Track Length ÷ Relative Speed = 400 / 9 sec

⇒ Total number of meetings in 400 seconds = 400 ÷ (400 / 9) = 9

⇒ The 9^th meeting occurs exactly at 400 seconds at the starting point.

⇒ Meetings before returning to the start = 9 − 1 = 8

∴ The correct answer is 8.

Additional Information

Meeting Points in Opposite Direction

If two people run in opposite directions with speeds in the ratio a : b (simplified), they will meet at (a + b) distinct points on the track.

Meeting Points in Same Direction

If two people run in the same direction with speeds in the ratio a : b (simplified), they will meet at |a − b| distinct points on the track.

Time for First Meeting

The time for the first meeting anywhere on the track is Track Length ÷ Relative Speed.

LCM of Times

The LCM of the times taken by individuals to complete one round gives the time when they will all meet again at the starting point.

The logic followed here is based on the positional values of the letters in the English alphabetical series.

Logic: (1^st letter + 5 = 2^nd letter) and (2^nd letter + 3 = 3^rd letter).

For OSV,

For JOR,

For HMP,

For  DIL,

Therefore, the letter-cluster that is different from the others is OSV.

Hence, the correct answer is 'Option 1'.

Shortcut Trick

Subtract the mixed recurring decimals by observing that the recurring parts (0.0666...) are identical.

3x = 5.4666... − 2.3666...

3x = 3.1

3x = 31/10

x = 31 / (10 × 3) = 31/30

∴ The correct answer is 31/30.

Alternate Method

Given:

Linear equation: 3x + 2.3666... = 5.4666...

Formula Used:

To convert a mixed recurring decimal to a vulgar fraction: (Whole number without decimal − Non-recurring part) / (9 for each recurring digit followed by 0 for each non-recurring decimal digit).

Calculations:

⇒ Convert 2.3666... to fraction: (236 − 23) / 90 = 213/90

⇒ Convert 5.4666... to fraction: (546 − 54) / 90 = 492/90

⇒ Substitute in equation: 3x + 213/90 = 492/90

⇒ 3x = 492/90 − 213/90

⇒ 3x = (492 − 213) / 90

⇒ 3x = 279/90

⇒ 3x = 31/10 (simplifying by 9)

⇒ x = 31 / (10 × 3)

⇒ x = 31/30

∴ The correct answer is 31/30.

Additional Information

Mixed Recurring Decimals

Decimals in which at least one digit after the decimal point is not recurring and some digits are recurring (e.g., 2.3666...).

Vulgar Fraction

A fraction where both the numerator and denominator are integers, expressed in the form a/b.

Conversion Rule

For a mixed recurring decimal like 0.ab c(bar), the fraction is (abc − ab) / 900. The number of 9s equals the number of recurring digits, and the number of 0s equals the number of non-recurring decimal digits.

The solution for the given number/symbol coding problem is as follows.

According to the given condition:

Given: 3 % @ 1 4

• 3 →  coded as ©
• %  → coded as H
• @→coded as  O
• 1:  → coded as M
• 4 → coded as ©

Thus, the code for the group 3 % @ 1 4 coded as © H O M ©.

Hence, the correct answer is 'Option 4'$.

The correct answer is WCL.

Key Points

• The Coal India Inter-Company Powerlifting, Weightlifting, and Bodybuilding Championship 2025-26 was organized under the auspices of Coal India Limited (CIL) to foster sportsmanship among its employees.
• The tournament was held in October 2025 at the Western Coalfields Limited (WCL) Headquarters and associated sports facilities in Nagpur, Maharashtra.
• Western Coalfields Limited (WCL) emerged as the Overall Champion by securing the maximum number of points across the three contested disciplines: Powerlifting, Weightlifting, and Bodybuilding.
• Athletes from various subsidiaries of Coal India, including SECL, MCL, NCL, ECL, BCCL, and CCL, participated in the high-intensity competition.
• The event highlighted the athletic prowess within the coal sector, with WCL maintaining its stronghold as the host team to clinch the top honors.

 Additional Information

• Coal India Limited (CIL)
• It is a Maharatna Public Sector Undertaking under the Ministry of Coal, headquartered in Kolkata, West Bengal.
• It operates through several subsidiaries spread across different states to manage India's vast coal resources.
• Western Coalfields Limited (WCL)
• Headquartered in Nagpur, it manages coal mining operations in Maharashtra and parts of Madhya Pradesh.
• The company frequently hosts inter-company sporting events to promote the Fit India Movement within the industrial workforce.
• Other Participating Subsidiaries
• SECL (South Eastern Coalfields Limited): Headquartered in Bilaspur, Chhattisgarh; it is one of the largest coal-producing subsidiaries.
• MCL (Mahanadi Coalfields Limited): Based in Sambalpur, Odisha; it contributes significantly to India's total coal output.
• SCCL (Singareni Collieries Company Limited): While often involved in coal sector meets, it is a joint venture between the Government of Telangana and the Government of India.

Based on the given information, we can determine the rank of each person (A, B, C, M, N, and O) from highest to lowest marks:

• Only two people have scored less than B.No one has scored less than O

 

• C has scored immediately more than M.
• Only two people have scored between M and O.
• A has scored more than N

The Final Arrangement is,

To find how many people scored more than N, we look at those ranked above Position 5. These people are A, C, M, and B.

Total number of people who scored more than N = 4.

Hence, the correct answer is 'Option 4'.

The correct answer is Ensures universal viewing without requiring PowerPoint software.

Key Points

• The Portable Document Format (PDF) is a universal file type developed by Adobe that preserves the formatting, fonts, and layout of a document exactly as it was created, regardless of the software or hardware used to view it.
• The primary advantage of converting a PowerPoint (.pptx) presentation to PDF is that it allows anyone to open and read the content using a simple Web Browser or a free PDF Reader, eliminating the dependency on Microsoft Office licenses.
• By using PDF, users avoid the common issue of Font Substitution or Layout Shifting, which frequently occurs when a PowerPoint file is opened on a computer that lacks the specific fonts used by the original creator.
• This format is highly Cross-Platform compatible, meaning it functions identically across Windows, macOS, Linux, and mobile operating systems like Android and iOS.

 Additional Information

• File Size Reduction:
• While PDF conversion can involve Compression techniques to reduce file size for easier Email Sharing, this is not always guaranteed if the presentation contains high-resolution graphics.
• Security and Encryption:
• PDFs offer advanced security features like Digital Signatures and Password Encryption to restrict printing or editing, but these are secondary features compared to the core benefit of Portability.
• Editing Constraints:
• Unlike PowerPoint, which is an Authoring Tool, PDF is intended as a Final Distribution format. It is generally difficult to perform Cross-platform editing on a PDF without specialized software like Adobe Acrobat Pro.
• Loss of Dynamic Content:
• A notable disadvantage of the PDF format is that it does not support PowerPoint Animations, Slide Transitions, or Live Video Embeds, converting them into Static Images.

Concept:

• Magnification (m) produced by a lens is defined as the ratio of the height of the image to the height of the object. It is also given by the ratio of the image distance to the object distance.
• Since it is a ratio of two similar quantities, magnification is a dimensionless quantity and has no SI unit.
• Dimensional Formula of magnification: [M⁰L⁰T⁰]
• Formula: m = v / u
• Object Distance (u) is the distance from the optical center of the lens to the object. According to the New Cartesian Sign Convention, the object distance is always taken as negative since the object is placed in front of the lens.
• SI Unit of distance: Meter (m)
• Dimensional Formula of distance: [M⁰L¹T⁰]
• Image Distance (v) is the distance from the optical center of the lens to the formed image. If the image is formed on the same side as the object, it is a virtual image and its distance is taken as negative according to the sign convention.

Calculation:

Given Data:

• Object distance, u = −10 cm
• Image distance, v = −30 cm

The formula for the magnification of a lens is:

m = v / u

Substituting the given values into the formula:

m = −30 / −10

⇒ m = 3

Hence, the magnification produced by the lens is 3.

The correct answer is Record a macro for each commonly used sort order..

Key Points

• Macros are automated sequences of instructions in Microsoft Excel that use Visual Basic for Applications (VBA) to perform repetitive tasks instantly.
• By recording a macro, a user can capture the exact steps of a complex sorting operation, including multi-level sorts (sorting by multiple columns simultaneously) and specific custom lists.
• Once the macro is saved, it can be assigned to a Keyboard Shortcut or a Form Control Button, allowing the user to trigger the specific sort order with a single action without ever opening the Sort dialogue box.
• This method is the most efficient for large datasets because it ensures consistency, eliminates the time spent re-selecting columns, and prevents manual entry errors during frequent data analysis.
• To use macros effectively, the workbook must be saved in a macro-enabled format such as .xlsm or .xlsb to retain the automation scripts.

 Additional Information

• Slicers:
• Originally introduced in Excel 2010, these are visual filtering tools mainly designed for PivotTables and Excel Tables. They are optimized for filtering (hiding/showing data) rather than reordering or sorting records.
• Quick Access Toolbar (QAT):
• The QAT is a customizable toolbar that sits above the Ribbon. While it can hold the Sort command icon to save a few clicks, it does not store the specific criteria (like which column to sort) and still requires manual configuration for each use.
• Quick Sort Buttons:
• These are the Sort A to Z and Sort Z to A commands found in the Data tab. They are useful for single-column sorting but cannot handle multi-criteria sorting efficiently without repetitive manual selection.

CONCEPT:

• Chemical Weathering is the process by which rocks are broken down by chemical reactions that change their mineral composition.
• Carbonation is a specific type of chemical weathering involving the reaction between carbon dioxide, water, and carbonate minerals.

EXPLANATION:

• When rain falls through the atmosphere, it absorbs carbon dioxide (CO₂) from the air. Similarly, as water moves through the soil, it picks up additional CO₂ produced by decaying organic matter and root respiration.
• The dissolved carbon dioxide reacts with water (H₂O) to form a weak acid known as carbonic acid (H₂CO₃):

H₂O + CO₂ → H₂CO₃

• This carbonic acid is highly effective at reacting with minerals like calcium carbonate (CaCO₃), which is the primary constituent of limestone and marble.
• The acid converts the insoluble calcium carbonate into calcium bicarbonate [Ca(HCO₃)₂], which is soluble in water and can be easily washed away.
• The entire sequence of CO₂ combining with water to act as a weathering agent on rocks is technically termed Carbonation.
• This process is responsible for creating unique karst landscapes, including features such as sinkholes, underground caves, and pillars.

Therefore, the process is known as Carbonation.

Given:

Expression = $\frac{2^4 \times 3^4 \times 5^4}{2^2 \times 3^2 \times 5^2}$

Formula used:

According to the Law of Indices:

$\frac{a^m}{a^n} = a^{m-n}$

Calculations:

To simplify the given expression, we apply the formula for exponents:

⇒ $(\frac{2^4}{2^2}) \times (\frac{3^4}{3^2}) \times (\frac{5^4}{5^2})$

⇒ 2⁴⁻² × 3⁴⁻² × 5⁴⁻²

⇒ 2² × 3² × 5²

⇒ 4 × 9 × 25

⇒ 36 × 25

⇒ 900

∴ The correct answer is option (4).

The correct answer is Atomic Energy Regulatory Board.

Key Points

• The Sustainable Harnessing and Advancement of Nuclear Energy for Transforming India (SHANTI) Act, 2025, is a pivotal legislative framework aimed at modernizing India’s nuclear energy sector to achieve Net Zero emissions by 2070.
• Under the SHANTI Act, 2025, the Atomic Energy Regulatory Board (AERB) is identified as the central authority responsible for the safety and regulatory oversight of all nuclear power initiatives, including the integration of private sector players.
• The Act seeks to provide the Atomic Energy Regulatory Board with a more robust statutory foundation, ensuring its functional independence from the Department of Atomic Energy (DAE) to enhance public and international confidence.
• A primary function of the Board under this new Act is to develop safety protocols and licensing procedures for Small Modular Reactors (SMRs) and Bharat Small Reactors (BSRs), which are key to decentralized energy production.
• The legislation empowers the AERB to enforce rigorous radiation protection standards and oversee the decommissioning of older plants, aligning India with the guidelines of the International Atomic Energy Agency (IAEA).
• The SHANTI Act also mandates the Board to manage regulatory approvals for dual-use nuclear technology in medical and industrial applications, ensuring security and non-proliferation.

 Additional Information

• Atomic Energy Regulatory Board (AERB):
• Established on November 15, 1983, by the President of India under the Atomic Energy Act, 1962.
• The Board is headquartered in Mumbai and consists of a full-time Chairman, an ex-officio Member, and three part-time Members.
• Its mission includes the development of safety codes and standards for nuclear installations such as those at Tarapur, Rawatbhata, and Kalpakkam.
• Department of Atomic Energy (DAE):
• Established in 1954, it operates directly under the Prime Minister of India.
• It is responsible for the research and development of nuclear power technology through institutions like the Bhabha Atomic Research Centre (BARC) and Indira Gandhi Centre for Atomic Research (IGCAR).
• National Nuclear Safety Board:
• This concept is often associated with the Nuclear Safety Regulatory Authority (NSRA) Bill, which was previously proposed to replace the AERB with a fully independent body.

Shortcut Trick

Divide the given equation by cosA to directly obtain an equation in tanA.

10 tanA + 1 = √6 tanA

Rearranging gives tanA(√6 − 10) = 1, so tanA = 1 / (√6 − 10).

On rationalizing the denominator: tanA = (√6 + 10) / (6 − 100) = (−10 − √6) / 94.

∴ The correct answer is (−10 − √6) / 94.

Alternate Method

Given: 10sinA + cosA = √6 sinA

Formula Used: tanA = sinA / cosA; Rationalization: 1 / (a − b) = (a + b) / (a² − b²)

⇒ 10sinA + cosA = √6 sinA

⇒ cosA = √6 sinA − 10sinA

⇒ cosA = sinA(√6 − 10)

⇒ 1 = (sinA / cosA)(√6 − 10)

⇒ 1 = tanA(√6 − 10)

⇒ tanA = 1 / (√6 − 10)

⇒ Multiplying by conjugate (√6 + 10): tanA = (√6 + 10) / [(√6 − 10)(√6 + 10)]

⇒ tanA = (√6 + 10) / (6 − 100)

⇒ tanA = (√6 + 10) / (−94)

⇒ tanA = (−10 − √6) / 94

∴ The correct answer is (−10 − √6) / 94.

Additional Information

Trigonometric Ratio tanA

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, expressed as tanA = sinA / cosA.

Rationalization Process

To rationalize a denominator containing surds like (a − √b), multiply both numerator and denominator by its conjugate (a + √b).

Identity of Reciprocals

The cotangent ratio is the reciprocal of the tangent ratio: cotA = 1 / tanA = cosA / sinA.

Logic followed here is the comparison of heights based on the conditions provided in the statements to identify the tallest person among H, I, J, K, and L.

From Statement I:

• J is taller than H and I but shorter than K.
• K > J > H and K > J > I

 

From Statement II:

• L is shorter than K but taller than J.
• K > L > J

After Combining Statement I and Statement II:

• From Statement I: K > J > (H, I)
• From Statement II: K > L > J

Combining these into a single sequence: K > L > J > H/I > I/H

• Therefore, K is definitively the tallest person among the five.
• Both statements I and II together (and not statement I alone or statement II alone) are sufficient to answer the question.

​Hence. the correct answer is "Option 3"

Shortcut Trick

When a value is first increased by x% and then decreased by the same x%, the net result is always a decrease.

Shortcut Formula: Net Percentage Change = (x² / 100)% decrease.

Given x = 20, the net change = (20² / 100)% = 400 / 100 = 4%.

Since this formula always represents a loss, the change is a 4% decrease.

∴ The correct answer is 4% decrease.

Alternate Method

Given: Percentage increase = 20%, Percentage decrease = 20%.

Formula Used: Net Change % = x + y + (x × y / 100), where x and y are the percentage changes.

⇒ Let the Initial Price be 100.

⇒ Price after 20% increase = 100 + (20% of 100) = 100 + 20 = 120.

⇒ Price after 20% decrease = 120 − (20% of 120) = 120 − 24 = 96.

⇒ Net change = Initial Price − Final Price = 100 − 96 = 4.

⇒ Percentage Change = (Net Change / Initial Price) × 100 = (4 / 100) × 100 = 4%.

∴ The correct answer is 4% decrease.

Additional Information

Successive Percentage Change

If a value is changed by x% and then by y%, the net percentage change is given by (x + y + xy/100)%. Use positive values for increase and negative for decrease.

Multiplying Factor Method

For a 20% increase, the multiplying factor is 1.20 (1 + 20/100). For a 20% decrease, it is 0.80 (1 − 20/100). Net factor = 1.20 × 0.80 = 0.96, which indicates a 4% decrease.

Percentage Base Rule

A percentage increase followed by the same percentage decrease never results in the original value because the second change is calculated on a larger base.

Given:

The logic followed here is:

First Letter: +5 position change in the English alphabetical series

Second Letter: +5 position change in the English alphabetical series

Number: +2 addition

For CM24 is related to HR26 ,

For AT15 is related to FY17,

Applying to the same logic CR12,

Hence, the correct answer is "Option 2"

The correct answer is - Storing starch produced through photosynthesis

Key Points

• Leucoplasts
• The colorless plastids observed in the potato tuber are a type of Leucoplast.
• Leucoplasts are non-pigmented organelles found in non-photosynthetic parts of plants, such as roots, bulbs, and tubers.
• Their primary function is the storage of nutrients like starch, lipids, and proteins.
• Amyloplasts
• In a potato tuber, the specific type of leucoplast present is the Amyloplast.
• Amyloplasts are responsible for the synthesis and storage of starch grains.
• The glucose produced during photosynthesis in the leaves is converted into sucrose, transported to the tuber, and then converted into starch for long-term energy storage within these plastids.

Additional Information

• Types of Plastids
• Chloroplasts: Contain green pigment (chlorophyll) and are responsible for photosynthesis.
• Chromoplasts: Contain carotenoid pigments (red, orange, yellow) and provide color to flowers and fruits to attract pollinators.
• Leucoplasts: Colorless plastids categorized based on what they store:
• Amyloplasts: Store starch.
• Elaioplasts: Store oils and fats.
• Aleuroplasts: Also known as proteinoplasts, they store proteins.
• Potato Tuber
• A potato is a modified underground stem used by the plant to survive the winter and provide energy for regrowth.
• The high concentration of amyloplasts in its tissue makes it a significant source of carbohydrates (starch) in the human diet.

Shortcut Trick

In triangle XYZ, since XT is the angle bisector of angle X, the Angle Bisector Theorem applies.

The ratio of sides is equal to the ratio of segments of the opposite side: XY ÷ XZ = YT ÷ TZ.

Given YZ = 6 cm and YT = 3.2 cm, we find TZ = 6 − 3.2 = 2.8 cm.

Applying the ratio: 5.6 ÷ XZ = 3.2 ÷ 2.8.

⇒ XZ = (5.6 × 2.8) ÷ 3.2 = 4.9 cm.

∴ The correct answer is 4.9 cm.

Alternate Method

Given: In ΔXYZ, XY = 5.6 cm, YZ = 6 cm, and YT = 3.2 cm. XT is the bisector of ∠X.

Formula Used: According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

⇒ XY ÷ XZ = YT ÷ TZ

Calculations:

⇒ Total length of side YZ = 6 cm.

⇒ Segment YT = 3.2 cm.

⇒ Segment TZ = YZ − YT

⇒ TZ = 6 − 3.2 = 2.8 cm.

⇒ From the theorem, XY ÷ XZ = YT ÷ TZ

⇒ 5.6 ÷ XZ = 3.2 ÷ 2.8

⇒ XZ = (5.6 × 2.8) ÷ 3.2

⇒ XZ = 15.68 ÷ 3.2

⇒ XZ = 4.9 cm.

∴ The correct answer is 4.9 cm.

Additional Information

Internal Angle Bisector Theorem

The angle bisector of an interior angle of a triangle divides the opposite side into segments proportional to the sides adjacent to the angle.

Length of Angle Bisector

The length of the internal bisector XT can be calculated using the formula: XT² = (XY × XZ) − (YT × TZ).

External Angle Bisector Theorem

The bisector of an exterior angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle.

**Given:**Seven people A, B, C, D, S, T and U were born in different months: January, February, April, May, June, September, and December.starting from January and ending in December

1. D was born in June.

2. U was born in February.

3. Only S is younger than B.

4. C is elder than T but younger than A.

The Final Arrangement is, 

• Total number of people born between B and A is 4.
• Thus, there are Four people born between B and A.

Hence, the correct answer is 'Option 4'.

The correct answer is Article 243B.

Key Points

• Article 243B of the Constitution of India specifically provides for the mandatory constitution of Panchayats at three levels: the village, intermediate, and district levels.
• This provision was incorporated through the 73^rd Constitutional Amendment Act, 1992, which came into effect on April 24, 1993, granting constitutional status to Panchayati Raj Institutions (PRIs).
• The Article establishes a uniform three-tier system across the country to ensure decentralized governance and direct participation of rural populations in development.
• According to Clause (2) of this Article, the intermediate level (often known as the Block or Mandal level) may not be constituted in a State that has a population not exceeding twenty lakhs.
• The primary objective of Article 243B is to strengthen grassroots democracy by ensuring that local bodies are structured consistently across different administrative layers.

 Additional Information

• Other relevant Articles under Part IX of the Constitution include:
• Article 243A: This Article defines the Gram Sabha, which consists of persons registered in the electoral rolls relating to a village comprised within the area of Panchayat at the village level.
• Article 243D: Focuses on the reservation of seats for Scheduled Castes and Scheduled Tribes in proportion to their population, and mandates that at least one-third of the total seats must be reserved for women.
• Article 243H: Empowers the State Legislature to authorize Panchayats to levy, collect, and appropriate specific taxes, duties, tolls, and fees, and provides for the creation of Funds for the Panchayats.
• Article 243G: Outlines the powers, authority, and responsibilities of Panchayats, enabling them to function as institutions of self-government regarding economic development and social justice.

Given: 

• Omaira starts from point F and walks 45 m towards south.
• She takes a left turn, walks 30 m and then takes a left turn, walks 20 m, reaches point C and stops.
• Anjan starts from point Q which is 15 m towards the north of point C.
• He walks 30 m towards east, takes a left turn, walks 10 m and stops at point U.

The Final Arrangement is, 

Shortest distance between F and U = 60 m

Hence, the correct answer is "option 2"

The logic follows here, 

The cluster that completes the series is PXT69.

Hence, the correct answer is "option 1".

The correct answer is South and East Asia.

Key Points

• South and East Asia represent the most densely populated regions globally, housing over half of the world’s total population due to favorable geographical and economic factors.
• The presence of extensive fertile plains, such as the Indo-Gangetic Plain in India and the North China Plain, provides the foundation for intensive alluvial agriculture and paddy cultivation, which can support high caloric demands.
• Perennial river systems including the Ganges, Brahmaputra, Yangtze, and Yellow River ensure a consistent water supply for both irrigation and large-scale industrial processes.
• Rapid industrialization in countries like China, India, and Vietnam has transformed these regions into 'the world’s factory,' creating massive labor markets that drive high population concentration.
• Unprecedented urban development has led to the rise of megacities such as Tokyo, Delhi, Mumbai, and Shanghai, where population densities often exceed several thousand people per square kilometer.
• The monsoon climate facilitates multiple cropping seasons per year, unlike the seasonal limitations found in higher latitudes or arid zones.

 Additional Information

• Other regions mentioned have low population densities due to environmental constraints:
• Sahara Desert: Characterized by extreme aridity and hyper-arid conditions, making it unsuitable for large-scale permanent settlements or agriculture.
• Greenland: Dominated by a massive ice sheet and a polar climate; it has the lowest population density of any territory in the world.
• Amazon Basin: While possessing vast freshwater resources, the dense tropical rainforest and leached soils limit industrial expansion and high-density urban clustering compared to Asian river valleys.

The correct answer is Allan Octavian Hume.

Key Points

• Allan Octavian Hume, commonly known as A.O. Hume, was a retired British officer of the Indian Civil Service who is credited as the primary founder of the Indian National Congress (INC).
• The first official session of the Indian National Congress was held from December 28 to December 31, 1885, at Gokuldas Tejpal Sanskrit College in Bombay (now Mumbai).
• Hume served as the General Secretary of the organization for its first twenty-two years, helping to stabilize and expand its reach across the Indian subcontinent.
• The assembly was attended by 72 delegates representing various provinces of India, where Womesh Chandra Bonnerjee was elected as the first President of the Congress.
• Hume's initiative is often associated with the Safety Valve Theory, which suggests that the British aimed to create a platform for educated Indians to voice their grievances peacefully to prevent a violent uprising like the Revolt of 1857.

 Additional Information

• Lord Curzon:
• Served as the Viceroy of India from 1899 to 1905.
• He is most notably remembered for the Partition of Bengal in 1905, which triggered the Swadeshi Movement.
• Lord Dufferin:
• He was the Viceroy of India from 1884 to 1888, during the actual foundation of the INC in 1885.
• He famously dismissed the Congress as representing only a microscopic minority of the Indian population.
• Lord Ripon:
• Viceroy from 1880 to 1884, often hailed as the Father of Local Self-Government in India.
• He is known for the Ilbert Bill controversy and for repealing the Vernacular Press Act in 1882.

The correct answer is Right-click selection > Table Properties > Cell tab > Vertical alignment: Center.

Key Points

• In MS Word 2021, table cell alignment is distinct from standard paragraph alignment as it involves two dimensions: Horizontal and Vertical.
• The Table Properties dialog box is the primary administrative interface for managing table-specific dimensions, borders, and cell behaviors. By right-clicking a selection, users access this context-sensitive menu.
• The Cell tab within Table Properties specifically controls how content interacts with the cell boundaries. The Vertical alignment setting offers three options: Top, Center, and Bottom, which determine the text position relative to the upper and lower cell borders.
• Unlike horizontal alignment, which shifts text between the left and right margins, Vertical alignment: Center ensures that the white space above and below the text inside the cell is equal.

 Additional Information

• Home tab > Paragraph group:
• This group contains Align Left, Align Right, Center, and Justify. These commands only modify the Horizontal alignment of the text.
• Using these options inside a table will move the text toward the side walls of the cell but will not change its vertical position relative to the top or bottom borders.
• Table Layout Ribbon Tab:
• When a table is selected, a contextual Layout tab appears in the ribbon. It contains an Alignment group with nine grid-based icons.
• These icons allow for one-click alignment combinations, such as Align Top Left, Align Center (which centers both horizontally and vertically), and Align Bottom Right.
• Cell Margins:
• Inside Table Properties > Options, users can also adjust Cell Margins to fine-tune the internal padding, which further affects the perceived centering of text.

The correct answer is File Transfer Protocol (FTP).

Key Points

• File Transfer Protocol (FTP) is a standard communication protocol used specifically for the transfer of computer files from a server to a client on a computer network.
• It is built on a client-server model architecture and uses separate control and data connections between the client and the server.
• FTP is the most suitable service for large files because it is designed to handle bulk data transfers efficiently, offering better reliability and speed than standard web-based downloads for massive datasets.
• The protocol allows for bi-directional transfer, meaning users can both upload files to a server and download files from it using dedicated software known as FTP clients (e.g., FileZilla).
• A key feature of FTP is its ability to resume interrupted transfers, ensuring that large files do not need to be restarted from the beginning if the connection drops.
• It typically functions using Port 21 for sending commands and Port 20 for the actual movement of data.

 Additional Information

• World Wide Web (WWW):
• An information system where documents and other web resources are identified by Uniform Resource Locators (URLs).
• While it supports file downloads via HTTP/HTTPS, it is primarily designed for viewing hypertext documents rather than specialized high-volume file management.
• E-mail (Electronic Mail):
• A method of exchanging messages between people using electronic devices via protocols like SMTP, POP3, and IMAP.
• Most E-mail providers impose strict attachment size limits (usually between 10 MB and 25 MB), making it unsuitable for very large files.
• Instant Messaging (IM):
• A type of online chat that offers real-time text transmission over the internet.
• While modern IM apps support file sharing, they are often optimized for short-form communication and small media files rather than large-scale data archiving or transfer.

Shortcut Trick

Find the LCM of 8, 7, and 11: 8 × 7 × 11 = 616.

Largest 4-digit number is 9999. Dividing 9999 by 616 gives a remainder of 143.

Largest multiple of 616 below 9999 = 9999 − 143 = 9856.

Required number = Multiple + Remainder = 9856 + 4 = 9860.

∴ The correct answer is 9860.

Alternate Method

Given: Divisors = 8, 7, 11 and common remainder = 4.

Formula Used: Required Number = [LCM(divisors) × k] + Remainder, where k is an integer.

⇒ LCM(8, 7, 11) = 8 × 7 × 11 = 616

⇒ The general form of the number is 616k + 4.

⇒ To find the largest 4-digit number, we divide 9999 by 616.

⇒ 9999 ÷ 616 ≈ 16.23

⇒ Taking the integer value, k = 16.

⇒ Largest multiple = 616 × 16 = 9856.

⇒ Required number = 9856 + 4 = 9860.

∴ The correct answer is 9860.

Additional Information

LCM of Co-prime Numbers

When numbers have no common factors other than 1, their LCM is simply the product of those numbers (e.g., 7, 8, and 11).

Common Remainder Rule

If a number N divided by a, b, and c leaves the same remainder r, then N = LCM(a, b, c) × k + r.

Division Algorithm

Any number can be expressed as Dividend = Divisor × Quotient + Remainder. Here, 9860 = 616 × 16 + 4.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and even positions is 0 or a multiple of 11.

The logic followed here ,​

 

For CF - GI - KL:

 

For MP - QS - UV:

 

Applying for the same logic, for the triad OR - SU - WX:

Hence, the correct answer is "Option 3"

The correct answer is Kurichiya Revolt.

Key Points

• The Kurichiya Revolt was a significant tribal uprising that took place in 1812 in the Wayanad region of Kerala against the British East India Company.
• The Kurichiyas are traditionally known as the first agricultural tribe of Kerala, and they were the primary force behind this movement, along with the Kurumbas.
• The rebellion was fundamentally triggered by the British agrarian exploitation, specifically the insistence on collecting land revenue in the form of cash rather than the traditional system of paying in kind (agricultural produce).
• The economic distress caused by high taxation and the failure of the British to acknowledge the tribal social structure led the Kurichiyas to declare a total war against the colonial administration.
• The movement was led by prominent tribal leaders such as Rama Nambi, who organized the tribesmen into a formidable force that used guerilla warfare and traditional weapons like bows and arrows.
• The British authorities, led by Thomas Harvey Baber, eventually suppressed the revolt by June 1812 through brutal military intervention and the execution of key rebel leaders.

 Additional Information

• Kadar Tribal Revolt:
• The Kadars are a small tribal community inhabiting the Anamalai and Parambikulam ranges in the Western Ghats.
• They were primarily hunter-gatherers and did not lead a major organized agricultural revolt in 1812 like the Kurichiyas.
• Koya Viplavam:
• This refers to the Koya tribal uprisings, most notably the Rampa Rebellion, which occurred in the Godavari region of Andhra Pradesh.
• These revolts were typically directed against forest laws and the exploitation by mansabdars during the late 19^th century.
• Kurumba Uprising:
• While the Kurumba tribe participated alongside the Kurichiyas in the Wayanad struggle of 1812, historical records predominantly identify the event as the Kurichiya Revolt due to the leadership and majority involvement of the Kurichiya tribe.

The correct answer is Fund Resilience, Not Disasters.

Key Points

• The International Day for Disaster Risk Reduction (IDDRR) is observed globally every year on October 13 to promote a culture of risk awareness and disaster reduction.
• The 2025 theme, Fund Resilience, Not Disasters, highlights the urgent need to shift financial resources from reactive post-disaster spending toward proactive investments in disaster mitigation and preparedness.
• This theme is deeply rooted in the objectives of the Sendai Framework for Disaster Risk Reduction (2015-2030), specifically focusing on Target F, which advocates for enhanced international cooperation to developing countries.
• The focus on funding resilience aims to reduce the economic losses caused by disasters, which disproportionately affect low and middle-income nations, by building climate-resilient infrastructure and robust community safety nets.
• The United Nations Office for Disaster Risk Reduction (UNDRR) leads the coordination of this day, encouraging governments and stakeholders to integrate disaster risk into all financial decision-making processes.

 Additional Information

• Historical Context:
• The day was established by the United Nations General Assembly in 1989, initially as part of the International Decade for Natural Disaster Reduction.
• In 2009, the General Assembly officially designated October 13 as the fixed date for the annual observance.
• Key Related Concepts:
• Sendai Framework: A 15-year agreement adopted in Japan that sets seven global targets to reduce disaster risk and losses in lives, livelihoods, and health.
• Early Warning Systems: While not the 2025 theme, Strengthening Early Warning Systems was a primary focus of the 2022 observance under the 'Early Warning for All' initiative.
• Resilience Investment: Every 1 dollar invested in risk reduction and prevention can save up to 15 dollars in post-disaster recovery costs, according to UNDP statistics.

The logic followed here is ,

• Code for happy = fv
• Code for everyone = cv
• So, the code for 'happy everyone' is fv cv.

Hence, the correct answer is "Option 3"

Given Series: S R E S 2 U ^ $ + X B 7 N M 7 4 C Z 5 @ U 4 D / 

Required Conditions : Letter − Number − Symbol

(Left) S R E S 2 U ^ $ + X B 7 N M 7 4 C Z 5 @ U 4 D / (Right)

There is only one such numbers : Z 5 @.

Hence, the correct answer is "option 1"

CONCEPT:

• Atomic Notation (also known as nuclide notation) is a standard method used to represent the composition of an atom or its nucleus.
• An element is typically represented using the general formula ${^A_Z X}$, where each symbol represents a specific physical quantity related to the atom.

EXPLANATION:

• In the standard notation ${^A_Z X}$:
• X is the Chemical Symbol of the element (e.g., H for Hydrogen, C for Carbon).
• A is the Mass Number, which represents the total number of protons and neutrons in the nucleus.
• Z is the Atomic Number.
• The Atomic Number (Z) is fundamental to chemistry because it defines the identity of the element. It indicates the total number of protons found in the nucleus of every atom of that element.
• In a neutral atom, the number of protons (Z) is equal to the number of electrons orbiting the nucleus.
• The number of neutrons (N) can be calculated using the formula: N = A − Z.
• Since Z uniquely identifies the number of protons and determines the element's position in the Periodic Table, it is always referred to as the Atomic number.

Therefore, the symbol Z in atomic notation refers to the Atomic number.

Shortcut Trick

Internal dimensions of the box: Length = 56 − 20 = 36 cm, Width = 40 − 20 = 20 cm, Height = 10 cm.

Volume of the closed box = 36 × 20 × 10 = 7200 cm³.

Volume of one matchbox = 4.0 × 2.0 × 1.5 = 12 cm³.

Number of boxes = Total Volume ÷ Volume of one matchbox = 7200 ÷ 12 = 600.

∴ The correct answer is 600 boxes.

Alternate Method

Given:

Length of metallic sheet = 56 cm

Width of metallic sheet = 40 cm

Side of the square cut from each corner = 10 cm

Dimensions of a matchbox = 4.0 cm × 2.0 cm × 1.5 cm

Calculations:

When squares of side 10 cm are cut from each corner and flaps are folded:

⇒ Length of the resulting box (L) = 56 − (10 + 10) = 36 cm

⇒ Width of the resulting box (W) = 40 − (10 + 10) = 20 cm

⇒ Height of the resulting box (H) = Side of square cut = 10 cm

⇒ Volume of the box = L × W × H

⇒ Volume of the box = 36 × 20 × 10 = 7200 cm³

⇒ Volume of one matchbox = 4.0 × 2.0 × 1.5 = 12 cm³

⇒ Number of matchboxes = (Volume of the box) ÷ (Volume of one matchbox)

⇒ Number of matchboxes = 7200 ÷ 12 = 600

∴ The correct answer is 600 boxes.

Additional Information

Volume of a Cuboid

The volume of a rectangular box (cuboid) is calculated as Length × Breadth × Height.

Surface Area of an Open Box

For an open box created by cutting corners, Surface Area = Area of Base + 2(Area of Side 1 + Area of Side 2) = LB + 2(BH + LH).

Number of Small Cuboids in a Large Cuboid

Number of items = (L × W × H) of container ÷ (l × w × h) of item, provided the dimensions are divisible to ensure a perfect fit.

CONCEPT:

Electrolytic Conduction

• An electrolyte is a substance that dissociates into free ions when dissolved in a solvent (like water) or in a molten state, making the resulting liquid electrically conductive.
• In metallic conductors, electricity is carried by the flow of electrons. However, in solutions, the charge is carried by the migration of ions.
• Acids are chemical substances that, when dissolved in water, release Hydrogen ions (H⁺), thereby acting as electrolytes.

EXPLANATION:

• When an acid is dissolved in water, it undergoes ionization or dissociation. For example, Hydrochloric acid (HCl) dissociates into its constituent ions:

HCl → H⁺(aq) + Cl⁻(aq)

• These ions are charged particles. The H⁺ ions (which typically exist as hydronium ions, H₃O⁺ in water) and the corresponding acid radical ions are free to move throughout the solution.
• When a potential difference (voltage) is applied to the solution using electrodes, these charged ions begin to migrate:
• The positively charged ions (cations) move toward the negative electrode (cathode).
• The negatively charged ions (anions) move toward the positive electrode (anode).
• This continuous movement of charged ions through the solution constitutes the electric current.
• Neutral molecules cannot conduct electricity because they do not carry a net electric charge. Gas bubbles are usually the product of the chemical reactions occurring at the electrodes during electrolysis, rather than the cause of the current flow.

Therefore, the flow of electric current in acidic solutions is due to the ions present in the solution.

BODMAS Table:

By substituting the operators in place of the letters and applying the BODMAS rule:

= 24 + 16 − 8 ÷ 4 × 2

= 24 + 16 − 2 × 2

= 24 + 16 − 4

= 40 − 4

= 36

The final value is 36.

Hence, the correct answer is "Option 4"

Given : TON TAR DEV TAB 

(Left) TON TAR DEV TAB (Right)

• The second letter of the second word from the right is DEV = E
• The second letter of the first word from the left is TON = O

The letters between E and O are: F, G, H, I, J, K, L, M, and N.

There are exactly 9 letters between them.

Hence, the correct answer is "Option 3"

Shortcut Trick

Total cost without discount = 150 × 20 = ₹3,000.

Since the quantity (150) is more than 100, a 2% discount applies.

Net payable percentage = 100% − 2% = 98%.

Total Amount = 98% of 3,000 = 0.98 × 3,000 = ₹2,940.

∴ The correct answer is ₹2,940.

Alternate Method

Given:

Price per unit = ₹20

Quantity purchased = 150 units

Discount rate = 2% (Applicable for purchases > 100 units)

Formula:

Total Payable = (Quantity × Price) − Discount

⇒ Total Purchase Value = 150 × 20 = ₹3,000

⇒ Discount Amount = 2% of 3,000 = (2 ÷ 100) × 3,000

⇒ Discount Amount = ₹60

⇒ Total Amount Payable = 3,000 − 60 = ₹2,940

∴ The correct answer is ₹2,940.

Additional Information

Trade Discount

A reduction in the list price offered by a wholesaler to a retailer, often based on the quantity of goods purchased.

Net Price Calculation

The actual price paid after deducting all discounts: Net Price = List Price × (1 − d), where d is the discount rate.

Cash Discount vs. Quantity Discount

Cash discount is for prompt payment, while quantity discount is for bulk buying to encourage higher volume sales.

Shortcut Trick

The Centroid is the average of the vertices.

To find the third vertex directly: 3 × Centroid − Sum of other two vertices.

x-coordinate ⇒ 3 × 4 − (1 + 4) = 12 − 5 = 7

y-coordinate ⇒ 3 × 5 − (5 + 7) = 15 − 12 = 3

The required vertex is C (7, 3).

∴ The correct answer is C (7, 3).

Alternate Method

Given: Vertices A(1, 5), B(4, 7), and Centroid D(4, 5).

Formula Used: Centroid (G_x, G_y) = ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3)

⇒ Let the coordinates of vertex C be (x, y).

⇒ For the x-coordinate: 4 = (1 + 4 + x) ÷ 3

⇒ 4 × 3 = 5 + x

⇒ 12 = 5 + x

⇒ x = 12 − 5 = 7

⇒ For the y-coordinate: 5 = (5 + 7 + y) ÷ 3

⇒ 5 × 3 = 12 + y

⇒ 15 = 12 + y

⇒ y = 15 − 12 = 3

⇒ Vertex C = (7, 3)

∴ The correct answer is C (7, 3).

Additional Information

Centroid Definition

The Centroid is the point of intersection of the three medians of a triangle and always lies inside the triangle.

Median Ratio Rule

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

Area of Triangle (Coordinates)

Area = 0.5 × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. For ΔABC, Area = 0.5 × |1(7 − 3) + 4(3 − 5) + 7(5 − 7)| = 0.5 × |4 − 8 − 14| = 9 sq. units.

Shortcut Trick

The difference between the outer perimeter and the inner perimeter of a path of uniform width (w) around a rectangle is always constant and independent of the rectangle dimensions.

Shortcut Formula: Difference = 8 × width

⇒ Difference = 8 × 2.4 m

⇒ Difference = 19.2 m

∴ The correct answer is 19.2 m.

Alternate Method

Given: Length of field = 68.5 m, Breadth of field = 45.6 m, Width of path = 2.4 m (Outside).

Formula Used: Perimeter of Rectangle = 2 × (Length + Breadth)

⇒ Original Perimeter (P₁) = 2 × (68.5 + 45.6) = 2 × 114.1 = 228.2 m

⇒ Outer Length = 68.5 + (2 × 2.4) = 68.5 + 4.8 = 73.3 m

⇒ Outer Breadth = 45.6 + (2 × 2.4) = 45.6 + 4.8 = 50.4 m

⇒ Outer Perimeter (P₂) = 2 × (73.3 + 50.4) = 2 × 123.7 = 247.4 m

⇒ Difference = P₂ − P₁ = 247.4 − 228.2 = 19.2 m

∴ The correct answer is 19.2 m.

Additional Information

Perimeter Change Rule

For any path of uniform width w added around a rectangle (inside or outside), the difference in perimeter is always 8w.

Path Area (Outside)

The area of a path constructed outside a rectangle is calculated as Area = 2w(L + B + 2w).

Path Area (Inside)

The area of a path constructed inside a rectangle is calculated as Area = 2w(L + B − 2w).