RRB Group D Official Paper (Held On: 04 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is To increase genetic variation for better adaptation.

Key Points

• Sexual reproduction in plants involves the fusion of male and female gametes to form a zygote, which later develops into a new organism. This process results in genetic variation.
• Genetic variation produced during sexual reproduction allows plants to adapt better to changing environmental conditions such as climate, diseases, or pests. This is critical for their survival and evolution.
• Unlike asexual reproduction, which produces identical clones, sexual reproduction ensures that offspring are genetically diverse. This diversity increases the chances of survival for the species in adverse conditions. Hence, option 1 is incorrect.
• Sexual reproduction is essential for plants to produce seeds, which are often dispersed far and wide, ensuring the propagation of the species over large areas.
• Flowers are a crucial part of sexual reproduction in plants as they facilitate pollination, leading to fertilization. Therefore, sexual reproduction does not avoid growing flowers; rather, it necessitates them. Hence, option 3 is incorrect.
• Pollination is a fundamental step in sexual reproduction for transferring pollen from male to female parts of the plant. Sexual reproduction, therefore, does not eliminate the need for pollination. Hence, option 4 is incorrect.

Additional Information

• Methods of Pollination:
• Self-pollination: Occurs when pollen from the same flower or another flower on the same plant fertilizes the ovule.
• Cross-pollination: Pollen is transferred from the flower of one plant to another plant, often facilitated by wind, water, or animals like bees, butterflies, and birds.
• Advantages of Sexual Reproduction:
• Promotes genetic diversity, which enhances adaptability and survival in changing environments.
• Helps in the elimination of harmful mutations over generations.
• Plays a significant role in evolution by introducing new traits into a population.
• Examples of Plants with Sexual Reproduction:
• Plants like sunflowers, roses, mangoes, and apple trees reproduce sexually through flowers.
• Some plants, like corn and grasses, rely on wind for pollination, while others, like orchids, depend on animals.
• Key Features of Genetic Variation:
• Genetic variation arises due to processes like meiosis, recombination, and fertilization.
• It improves a species' ability to resist diseases and adapt to environmental changes.
• Examples include the development of disease-resistant crop varieties through selective breeding.

Important Points

• Reproduction in Plants:
• Asexual reproduction occurs without the fusion of gametes, producing genetically identical offspring, such as in potatoes and onions.
• Sexual reproduction requires gamete fusion, leading to genetic diversity, as seen in flowering plants.
• Pollinators:
• Bees, butterflies, birds, and bats are common pollinators crucial for the process of sexual reproduction in plants.
• The decline in pollinator populations due to environmental changes can negatively impact plant reproduction and biodiversity.
• Role of Seeds:
• Seeds formed through sexual reproduction ensure the continuity of plant species.
• They allow plants to spread to new areas, promoting growth in diverse habitats.

The correct answer is Sound waves in air are longitudinal because particles move along the direction of propagation..

Key Points

• Longitudinal Waves: Sound waves in air are classified as longitudinal waves because the particles of the medium vibrate parallel to the direction of wave propagation. This means that the compressions and rarefactions in the wave occur along the same direction as the wave travels.
• Particle Motion: The air particles oscillate back and forth in the direction of propagation, creating regions of high pressure (compressions) and low pressure (rarefactions). Hence, the statement "particles move along the direction of propagation" is correct.
• Incorrect Options: The statement "particles move perpendicular to the direction of propagation" describes transverse waves, which are not applicable to sound waves in air. Similarly, "air molecules do not vibrate" and "particles move randomly" are incorrect because molecular vibration and organized particle motion are essential for sound propagation.
• Propagation Mechanism: Sound waves require a medium (like air, water, or solids) to travel. The longitudinal nature of sound waves is due to the way the medium's particles transfer energy, through successive compressions and rarefactions.
• Physical Characteristics: The speed, frequency, and wavelength of sound waves are influenced by the medium's properties, such as its density and elasticity. Air is the most common medium for sound waves, and its molecules vibrate parallel to the wave's propagation direction.

Additional Information

• Longitudinal vs. Transverse Waves:
• Longitudinal waves involve particle motion parallel to the direction of wave propagation. Examples include sound waves in air and seismic P-waves.
• Transverse waves involve particle motion perpendicular to the direction of wave propagation. Examples include light waves and water waves.
• Compressions and Rarefactions:
• Compressions are regions where particles are closer together, resulting in higher pressure.
• Rarefactions are regions where particles are farther apart, resulting in lower pressure.
• This alternating pattern of compressions and rarefactions allows sound waves to travel through a medium.
• Applications of Sound Waves:
• Sound waves are widely used in communication systems (e.g., telephones, radios).
• Medical applications include ultrasound imaging, which utilizes high-frequency sound waves.
• Industrial applications include material testing and acoustic sensors.
• Speed of Sound:
• The speed of sound in air is approximately 343 m/s at room temperature (20°C).
• The speed varies with the medium; it is faster in solids and liquids compared to gases due to higher density and elasticity.

Important Points

• Energy Transfer: Sound waves transfer energy from one particle to another without transporting matter over long distances. This is a key characteristic of mechanical waves.
• Human Hearing Range: Humans can hear sound frequencies between 20 Hz and 20,000 Hz, known as the audible range.
• Sound Intensity: Measured in decibels (dB), sound intensity depends on the amplitude of the wave. Higher amplitude results in louder sounds.
• Reflection and Refraction: Sound waves can reflect off surfaces (echo) and refract when passing through mediums with varying densities.

The correct answer is Kinetic energy.

Key Points

• When a body falls freely from a height under the influence of gravity, its potential energy is gradually converted into kinetic energy. This is due to the principle of energy conservation in a closed system.
• Potential energy (PE) is the energy possessed by a body due to its position or configuration. It is given by the formula PE = m × g × h, where m is mass, g is acceleration due to gravity, and h is height.
• As the body falls, the height (h) decreases, causing the potential energy to decrease proportionally. Simultaneously, the velocity of the body increases due to gravitational acceleration, causing its kinetic energy (KE) to increase.
• Kinetic energy (KE) is the energy possessed by a body due to its motion. It is given by the formula KE = 1/2 × m × v², where m is mass and v is velocity.
• At the highest point, the body has maximum potential energy and zero kinetic energy. As it falls, potential energy decreases and is entirely converted into kinetic energy just before impact, assuming no energy losses due to air resistance.
• This process demonstrates the law of conservation of energy, which states that energy in a closed system remains constant and only transforms from one form to another.

Additional Information

• Sound energy:
• Sound energy is produced when a vibrating object creates waves that propagate through a medium such as air, water, or solids.
• In the case of a falling object, sound energy may be generated upon impact with the ground, but it is not a direct conversion of potential energy during free fall.
• Thermal energy:
• Thermal energy refers to the energy possessed by a substance due to the kinetic energy of its molecules or atoms.
• In real-world scenarios, a small amount of potential energy could be converted into thermal energy due to air resistance, but this is negligible in an ideal free-fall scenario.
• Chemical energy:
• Chemical energy is stored in the bonds of chemical compounds and is released during chemical reactions.
• Free-fall motion does not involve any chemical reactions, so potential energy is not converted into chemical energy.
• Law of Conservation of Energy:
• This fundamental law states that energy cannot be created or destroyed; it can only be transformed from one form to another.
• In the case of free fall, the total mechanical energy (sum of potential and kinetic energy) remains constant, neglecting air resistance.
• Practical Applications:
• The conversion of potential energy to kinetic energy is utilized in systems like hydroelectric power plants, where water stored at height (potential energy) is used to generate electricity through turbines (kinetic energy).
• It is also observed in roller coasters, where the initial height determines the energy available for motion.

Important Points

• Gravitational force: The force of gravity acts as the driving force for the conversion of potential energy to kinetic energy during free fall.
• Air resistance: In real-world conditions, air resistance reduces the efficiency of energy conversion, leading to a small loss of energy as heat.
• Velocity at impact: The final velocity of a falling object can be calculated using the formula v = √(2 × g × h), where g is the acceleration due to gravity and h is the height.
• Understanding energy transformations is crucial in engineering, physics, and various practical applications like designing efficient machines and energy systems.

The correct answer is Temperature.

Key Points

• Temperature-dependent sex determination (TSD) is a phenomenon observed in certain reptiles, including turtles, crocodiles, and some lizards, where the sex of the offspring is determined by the temperature during the incubation period of the eggs.
• In species with TSD, higher incubation temperatures typically produce one sex, while lower temperatures yield the other sex. For example, in many turtles, warmer temperatures produce females, while cooler temperatures produce males.
• This mechanism is distinct from genetic sex determination found in mammals and birds, where sex chromosomes dictate the sex of an individual. (Hence, this statement is correct.)
• The pivotal temperature is the temperature at which an equal ratio of males and females are produced. Variations around this temperature lead to skewed sex ratios. (This is an essential concept related to TSD.)
• TSD is believed to have evolved as an adaptive mechanism to maximize reproductive success under varying environmental conditions. For example, producing more females during favorable conditions can ensure population growth. (This highlights the evolutionary advantage of TSD.)

Additional Information

• Temperature and Reptile Reproduction:
• Reptiles, such as turtles, crocodiles, and some lizards, exhibit TSD rather than chromosomal sex determination.
• Species like the Red-eared slider turtle and American alligator are classic examples of organisms with temperature-dependent sex determination.
• In crocodiles, intermediate temperatures produce males, while extreme temperatures (both high and low) tend to produce females.
• Key Characteristics of Other Options:
• Wind velocity: Wind does not influence the sex determination of reptile eggs. While it plays a role in environmental factors like plant pollination and seed dispersal, it is irrelevant to sex determination. (This statement is incorrect.)
• Touch: Touch or physical interaction does not affect the sex determination of reptiles. It is more relevant to behavioral aspects in other species. (This statement is incorrect.)
• Water: Water availability may impact the survival and development of embryos but does not influence sex determination. (This statement is incorrect.)
• Importance of TSD in Ecology:
• TSD allows reptiles to adapt to environmental changes, ensuring population stability by balancing sex ratios.
• Climate change poses a threat to species with TSD, as rising temperatures may lead to skewed sex ratios, potentially endangering populations.
• Examples of TSD:
• Leatherback turtles: Warmer temperatures produce females, while cooler temperatures produce males.
• American alligators: Intermediate temperatures produce males, while extreme temperatures lead to females.

Important Points

• Impact of Climate Change: Rising global temperatures could severely disrupt TSD mechanisms, leading to imbalanced sex ratios and threatening species survival.
• Pivotal Temperature: This is a critical concept in TSD and varies among species. Research on pivotal temperatures helps in understanding and conserving reptiles.
• Conservation Efforts: Efforts to protect habitats and regulate temperatures during incubation are vital for species with TSD. Conservationists often monitor and manage nesting sites to ensure balanced sex ratios.

The correct answer is increases.

Key Points

• Pressure is defined as the force exerted per unit area. Mathematically, it is expressed as:
  

Pressure = Force/Area.

• When the thrust (force acting perpendicular to a surface) is kept constant, the pressure is inversely proportional to the area of contact.
• If the area of contact decreases while the thrust remains constant, the pressure increases because the same amount of force is concentrated over a smaller area.
• This principle is commonly observed in daily life. For example, sharp knives cut better because they apply the same force over a smaller surface area, increasing pressure.
• Hence, in this scenario, when the area of contact decreases while maintaining constant thrust, the pressure increases.

Additional Information

• Pressure:
• It is a scalar quantity and is measured in Pascals (Pa) in the SI system. 1 Pascal is equal to 1 Newton per square meter (N/m²).
• Pressure is crucial in many applications such as hydraulics, pneumatics, and fluid mechanics.
• Thrust:
• Thrust is a force that acts perpendicular to a surface.
• In physics, thrust is an essential factor in understanding pressure and is applied in various fields like aerodynamics and engineering.
• Examples of Pressure Variation:
• High Heels: They exert more pressure on the ground because the contact area is smaller compared to flat shoes.
• Snowshoes: These increase the surface area, reducing pressure and preventing a person from sinking into the snow.
• Applications of Pressure:
• Hydraulics: Pressure is used to lift heavy objects using hydraulic machines.
• Aerodynamics: Understanding pressure helps in designing aircraft wings for efficient flight.

Important Points

• Factors Affecting Pressure:
• Force: Greater force leads to higher pressure if the area remains constant.
• Area: Smaller area leads to higher pressure when force is constant.
• Practical Applications:
• Syringes: Applying pressure to the plunger forces liquid out through a narrow nozzle.
• Nails and Screws: Their pointed ends reduce the area of contact, making it easier to penetrate materials.
• Real-Life Demonstrations:
• Walking on sand with bare feet sinks deeper due to greater pressure compared to wearing flat wooden planks.
• Using a hydraulic press, where a small force applied over a small area can lift heavy loads over a larger area.

The correct answer is Pomfret.

Key Points

• Pomfret is one of the most popular and widely consumed marine fish varieties in India. It is commonly found in the coastal regions of the country.
• Pomfret belongs to the family Stromateidae and is known for its delicate flavor, soft texture, and high demand in Indian cuisine.
• This fish is predominantly found in the Arabian Sea, Bay of Bengal, and Indian Ocean, making it a vital resource for the fishing industry in India.
• Pomfret is rich in protein, omega-3 fatty acids, and essential nutrients, making it a healthy choice for seafood lovers.
• The fish is often used in various traditional Indian recipes such as Pomfret fry, curry, and grilled dishes, making it a staple in coastal culinary traditions.

Additional Information

• Catla:
• Catla is a freshwater fish native to the rivers and lakes of India. It belongs to the carp family.
• It is an important species in aquaculture and is farmed in various parts of the country.
• Catla is known for its large size and is widely consumed in states like West Bengal and Assam.
• Mrigal:
• Mrigal is another freshwater fish commonly found in Indian rivers and reservoirs.
• It is a popular species in Indian aquaculture, especially in composite fish farming systems.
• Mrigal is appreciated for its mild taste and nutritional value, including being a good source of protein.
• Rohu:
• Rohu is a widely cultivated freshwater fish in India, belonging to the carp family.
• It is highly prized for its flavor and texture and is a significant part of the cuisine in eastern and northern India.
• Rohu is rich in omega-3 fatty acids, vitamins, and minerals, making it a healthy dietary choice.
• Marine vs. Freshwater Fish:
• Marine fish like Pomfret are found in the ocean, while freshwater species such as Catla, Mrigal, and Rohu are native to rivers and lakes.
• Marine fish are often higher in iodine and omega-3 fatty acids, whereas freshwater fish are a good source of protein and minerals.

Important Points

• Significance of Marine Fish: Marine fish like Pomfret play a crucial role in India's coastal economy, supporting livelihoods through fishing and export activities.
• Marine ecosystems: Pomfret and other marine fish are integral to maintaining the balance of marine biodiversity.
• Health benefits: Marine fish are an excellent source of essential nutrients like omega-3 fatty acids, which are beneficial for heart and brain health.
• Fishing industry: Pomfret is a valuable catch for Indian fishermen, contributing significantly to the seafood market.

The correct answer is Mitochondria.

Key Points

• Mitochondria, often referred to as the “powerhouse of the cell,” are responsible for the production of Adenosine Tri Phosphate (ATP), the primary energy currency of cells.
• ATP is produced through a process called cellular respiration, which occurs in the mitochondria. This process involves the breakdown of glucose and oxygen to release energy.
• Mitochondria have a unique double-membrane structure. The inner membrane is folded into structures called cristae, which increase the surface area for ATP production.
• The mitochondria are semi-autonomous organelles, meaning they have their own DNA and ribosomes, enabling them to produce some of their own proteins and enzymes required for ATP synthesis.
• ATP molecules produced in the mitochondria are used in various cellular processes, such as muscle contraction, active transport, and biochemical reactions essential for life activities.

Additional Information

• Golgi Apparatus:
• The Golgi apparatus is primarily involved in the modification, sorting, and packaging of proteins and lipids for secretion or use within the cell.
• It does not play a role in the production of ATP.
• It is also responsible for the formation of lysosomes and the transport of lipids.
• Lysosomes:
• Lysosomes are membrane-bound organelles that contain digestive enzymes to break down waste materials and cellular debris.
• They are often referred to as the “suicidal bags” of the cell due to their role in autophagy and apoptosis.
• Lysosomes do not produce ATP.
• Vacuoles:
• Vacuoles are storage organelles found in both plant and animal cells, though they are larger in plant cells.
• They store nutrients, waste products, and other substances but are not involved in ATP synthesis.
• In plants, they help maintain turgor pressure, which is essential for structural support.
• ATP (Adenosine Triphosphate):
• ATP is a nucleotide consisting of three components: adenine (a nitrogenous base), ribose (a sugar), and three phosphate groups.
• The energy stored in the high-energy phosphate bonds is released when ATP is hydrolyzed to ADP (Adenosine Diphosphate) and inorganic phosphate.
• This energy is used in various cellular processes, such as active transport, metabolism, and mechanical work.
• Important Points:
• The mitochondria are unique among organelles because they are inherited maternally (from the mother).
• Examples of ATP-utilizing processes include muscle contraction, nerve impulse transmission, and biosynthesis of macromolecules.
• The human body synthesizes and uses its own weight in ATP molecules daily.
• Damaged or dysfunctional mitochondria can lead to energy deficits and are implicated in several disorders, such as mitochondrial myopathies and neurodegenerative diseases.

The correct answer is mass can neither be created nor destroyed in a chemical reaction.

Key Points

• The Law of Conservation of Mass states that mass can neither be created nor destroyed in a chemical reaction. This principle is fundamental in the study of chemistry and physics.
• It was proposed by Antoine Lavoisier in 1789, who is considered the "Father of Modern Chemistry."
• According to this law, the total mass of reactants in a chemical reaction is equal to the total mass of the products. For example, in a reaction where carbon reacts with oxygen to form carbon dioxide, the mass of carbon plus oxygen equals the mass of carbon dioxide.
• This law is valid for both closed systems (where no matter enters or leaves the system) and in isolated systems, ensuring mass balance in chemical equations.
• This principle forms the basis for the stoichiometric calculations used in chemical equations, ensuring that the quantities of reactants and products are balanced.

Additional Information

• Antoine Lavoisier's Contribution:
• Lavoisier demonstrated the law through experiments involving combustion and concluded that the total mass of substances involved in a chemical reaction remained constant.
• He debunked the phlogiston theory and laid the groundwork for modern chemical nomenclature and analytical techniques.
• Applications of the Law:
• It is crucial in industries such as pharmaceuticals, where precise quantities of reactants are needed to produce the desired products.
• Used in environmental science to balance chemical reactions in pollution control and waste treatment processes.
• Mathematical Representation:
• The law can be expressed mathematically as: Σm_reactants = Σm_products, where m represents mass.
• Important Limitations:
• While the law is valid in classical physics and chemistry, it does not account for mass-energy equivalence as explained by Einstein's theory of relativity (E = mc²), where mass can be converted into energy in nuclear reactions.
• It is not applicable in systems where nuclear transformations occur, such as in nuclear fission and fusion.
• Comparison with Other Conservation Laws:
• While the Law of Conservation of Mass focuses on mass, other laws like the Law of Conservation of Energy and Law of Conservation of Momentum address energy and momentum in physical systems.

Important Points

• Option 1: Volume conservation is not universal in chemical reactions. For example, gases may change volume depending on conditions like pressure and temperature. Incorrect.
• Option 2: Temperature is a variable, not a conserved quantity. It can increase or decrease during reactions due to energy exchange. Incorrect.
• Option 3: Pressure is also not conserved and depends on factors like the number of gas molecules and volume. Incorrect.
• Option 4: The mass of reactants equals the mass of products in a chemical reaction, making this the correct answer.

The correct answer is 24 cm.

Key Points

• A concave lens always forms a virtual, erect, and diminished image.
• The magnification of a lens is given by: m = v/u
• Since the image is one-third the size of the object, m = +1/3.
• Therefore, v = u/3.
• Using the lens formula: 1/f = 1/v − 1/u
• For a concave lens:
• f = –12 cm
• v = u/3
• Substituting,
  

1/(-12) = 1/(u/3) − 1/u

−1/12 = 3/u − 1/u

−1/12 = 2/u

u = −24 cm.

• The negative sign indicates that the object is placed in front of the lens. Hence, the object distance is 24 cm.
• Hence, the correct answer is 24 cm.

Additional Information

• Lens Formula:
• 1/f = 1/v − 1/u
• Applicable to both convex and concave lenses using the Cartesian sign convention.
• Concave Lens:
• Has a negative focal length.
• Always produces a virtual, erect, and diminished image for a real object.
• Magnification:
• m = v/u = h_i/h_o
• For a concave lens, the magnification is positive but less than 1.

The correct answer is Exposure to moisture.

Key Points

• Corrosion is a natural process where metals deteriorate due to environmental factors such as moisture, oxygen, and pollutants. Preventing corrosion involves techniques that create a barrier or reduce the reactivity of the metal surface.
• Painting: Applying paint on metal surfaces forms a protective layer that prevents direct contact with moisture and oxygen, which are primary causes of corrosion. Hence, painting is an effective method to prevent corrosion.
• Galvanisation: This process involves coating the metal with a layer of zinc. Zinc acts as a protective barrier, and even if the coating gets scratched, zinc acts as a sacrificial metal, preventing corrosion of the underlying metal. Hence, galvanisation is an effective anti-corrosion method.
• Applying sacrificial anodes: In this method, a more reactive metal (e.g., zinc or magnesium) is attached to the metal that needs protection. The sacrificial anode corrodes instead of the protected metal. Hence, this is another effective corrosion prevention method.
• Exposure to moisture: Moisture accelerates the electrochemical reaction that causes corrosion. Allowing metals to come into direct contact with moisture will increase their rate of corrosion rather than prevent it. Hence, exposure to moisture is NOT an effective method to prevent corrosion.

Additional Information

• Corrosion Process:
• Corrosion is an electrochemical process where a metal reacts with its environment, leading to the formation of oxides, hydroxides, or other compounds.
• For example, iron reacts with oxygen and water to form iron oxide (rust), which weakens the metal over time.
• Common Corrosion Prevention Methods:
• Electroplating: Coating the surface of a metal with a layer of another metal using electrolysis (e.g., chrome or nickel plating).
• Anodising: A process used for aluminum where the surface is oxidized to create a protective oxide layer.
• Greasing or Oil Coating: Lubricants provide a physical barrier that protects metals from moisture and oxygen.
• Factors Influencing Corrosion:
• Environmental factors: Presence of moisture, oxygen, and pollutants like sulfur dioxide or salt.
• Metal properties: Reactivity, purity, and surface area of the metal significantly influence corrosion rate.
• Temperature: Higher temperatures generally accelerate the corrosion process.
• Examples of Corrosion:
• Rusting of Iron: Formation of reddish-brown iron oxide in the presence of moisture and oxygen.
• Green Patina on Copper: Copper reacts with carbon dioxide and moisture to form copper carbonate, giving it a green appearance.

Important Points

• Economic Impact of Corrosion: Corrosion leads to significant economic losses, including maintenance, repair, and replacement costs. According to studies, the global cost of corrosion is estimated to be 3-4% of the global GDP.
• Corrosion-resistant Materials:
• Stainless Steel: Contains chromium, which forms a protective oxide layer and resists rusting.
• Aluminum: Forms a natural oxide layer that protects against further corrosion.
• Alloys: Materials like brass and bronze are more resistant to corrosion compared to pure metals.
• Environmental Protection: Preventing corrosion not only saves costs but also minimizes environmental damage caused by the disposal of corroded materials and the extraction of new metals.

The correct answer is 18.52 ms−1ms−1

.

Key Points

• Total Distance: The car travels 100 km in one direction and returns another 100 km, making the total distance covered 200 km.
• Time Taken: For the onward journey, the car travels at an average speed of 50 km/h, which takes 2 hours. The return trip covers the same 100 km in 1 hour, making the total time for the journey 3 hours.
• Average Speed Formula: Average speed is calculated using the formula:

Average Speed=Total DistanceTotal Time

Average Speed=Total DistanceTotal Time

Plugging in the values:

Average Speed=200 km3 hours=66.67 km/h

Average Speed=200 km3 hours=66.67 km/h

.

• **Conversion to ms−1ms−1

  :** To convert the speed from km/h to ms−1ms−1

  , multiply the value by

 

:

66.67×10003600=18.52ms−1

66.67×10003600=18.52ms−1

.

• Correct Answer: Hence, the average speed of the car for the entire journey is 18.52 ms−1ms−1

  .

Additional Information

• Average Speed Concept: Average speed is a measure of the total distance traveled divided by the total time taken, irrespective of variations in speed during the trip. It is calculated using the formula:

Average Speed=Total DistanceTotal Time

Average Speed=Total DistanceTotal Time

.

• Unit Conversion: Speed in km/h can be converted to ms−1ms−1

  by multiplying it by

 

or approximately . This is because 1 km = 1000 meters and 1 hour = 3600 seconds.

• Practical Applications: Calculating average speed is essential in transportation, logistics, and physics to determine efficiency, travel times, and performance metrics.
• Difference Between Average and Instantaneous Speed: Average speed considers the entire journey, while instantaneous speed measures the rate at which an object is moving at a specific moment.
• Real-World Examples: Average speed calculations are used in traffic analysis, determining fuel efficiency, and optimizing travel routes.

Important Points

• Speed and Velocity: While speed is a scalar quantity (magnitude only), velocity is a vector quantity (magnitude and direction).

• Uniform vs Non-Uniform Motion: In uniform motion, an object covers equal distances in equal time intervals, whereas in non-uniform motion, distances covered vary.

• SI Unit: The SI unit for speed is ms−1ms−1

  , which stands for meters per second.

• Applications in Physics: Understanding speed and its variations is fundamental in mechanics, kinematics, and dynamics.

The correct answer is Propene.

Key Points

• Unsaturated compounds are defined as organic compounds containing one or more double bonds (C=C) or triple bonds (C≡C) between carbon atoms in their molecular structure.
• Propene (C₃H₆) is an alkene that contains a carbon-carbon double bond, making it an unsaturated compound. Hence, it satisfies the criteria for being unsaturated.
• Butane (C₄H₁₀) and Propane (C₃H₈) are alkanes with only single bonds between carbon atoms, classifying them as saturated hydrocarbons.
• Methane (CH₄) is the simplest alkane and also a saturated hydrocarbon, as it contains no double or triple bonds.
• Since Propene contains a double bond, it is the only option among the given compounds that qualifies as an unsaturated hydrocarbon.

Additional Information

• Saturated hydrocarbons:
• These are hydrocarbons that contain only single bonds between carbon atoms.
• They are also called alkanes and follow the general formula C_nH_2n+2.
• Examples include methane (CH₄), ethane (C₂H₆), and propane (C₃H₈).
• Unsaturated hydrocarbons:
• These are hydrocarbons containing one or more double bonds (alkenes) or triple bonds (alkynes).
• They are more reactive than saturated hydrocarbons due to the presence of π-bonds.
• Examples include ethene (C₂H₄), propene (C₃H₆), and ethyne (C₂H₂).
• Characteristics of alkenes:
• Alkenes contain at least one C=C double bond.
• They follow the general formula C_nH_2n.
• Alkenes undergo addition reactions, such as hydrogenation, halogenation, and hydration, due to the presence of the double bond.
• Applications of unsaturated hydrocarbons:
• They are used as raw materials in the petrochemical industry for the production of plastics, synthetic rubber, and fibers.
• Ethene is used in the manufacture of ethanol, polyethylene, and other chemicals.
• Alkynes like ethyne (acetylene) are used in welding and cutting metals due to their high flame temperature.

Important Points

• Hydrocarbons classification:
• Alkanes: Saturated hydrocarbons with single bonds (e.g., methane, propane).
• Alkenes: Unsaturated hydrocarbons with double bonds (e.g., ethene, propene).
• Alkynes: Unsaturated hydrocarbons with triple bonds (e.g., ethyne).
• Testing for unsaturation: Unsaturated hydrocarbons decolorize bromine water or Baeyer's reagent (KMnO₄), indicating the presence of double or triple bonds.
• Environmental impact: Unsaturated hydrocarbons, when incompletely combusted, can release pollutants like carbon monoxide and soot, contributing to air pollution.

The correct answer is Apiary.

Key Points

• An apiary is a location where bees are kept and raised for the purpose of commercial honey production and other bee-related products like beeswax and royal jelly.
• Beekeeping in an apiary is also called apiculture, which involves the maintenance of bee colonies, typically in man-made hives.
• The main species of bees used for honey production include Apis mellifera (European honey bee) and Apis cerana (Asian honey bee).
• Apiaries are carefully managed environments that provide bees with adequate shelter, food, and water to ensure their health and productivity.
• Commercial apiaries can vary in size from a few hives to hundreds of hives, depending on the scale of honey production and other bee-related activities.

Additional Information

• Aquarium:
• An aquarium is a container or facility for keeping aquatic animals and plants, such as fish, in a controlled environment.
• It is not related to bees or honey production, making it an incorrect option for this question.
• Aviary:
• An aviary is an enclosed space where birds are kept for conservation, breeding, or display purposes.
• Since it deals with birds and not bees, it is also an incorrect option.
• Sanctuary:
• A sanctuary is a protected area established for the conservation of wildlife, including animals and plants.
• It does not specifically relate to commercial honey production or bees, making it an incorrect choice.
• Importance of Apiaries:
• Apiaries play a critical role in pollination, as bees are essential pollinators for many crops and plants.
• Beekeeping also provides economic benefits by producing honey, beeswax, and other products used in food, cosmetics, and medicine.
• Apiaries contribute to the environment by supporting biodiversity and the reproduction of flowering plants.
• Notable Practices in Beekeeping:
• Modern beekeepers use Langstroth hives, which consist of removable frames for easier management of bee colonies.
• Beekeeping requires regular monitoring of hives to prevent diseases and pests like Varroa mites.
• Proper location selection for an apiary is essential to ensure access to nectar-rich flowers and clean water sources.

The correct answer is The optical centre of the lens.

Key Points

• In the sign convention for lenses, all distances are measured from the optical centre of the lens. This ensures uniformity and clarity in calculations.
• The optical centre is the point on the axis of the lens where the principal axis passes through the lens without bending. It acts as a reference point for all measurements.
• Distances measured towards the object (on the left side of the lens) are taken as negative, while those measured towards the image (on the right side of the lens) are taken as positive.
• The sign convention follows the Cartesian coordinate system, where distances above the principal axis are positive, and distances below are negative.
• This convention is crucial for deriving lens formulas, such as the lens maker's formula and the magnification formula, and for accurately predicting the nature and position of images formed by lenses.

Additional Information

• Principal Axis:
• The principal axis is the straight line passing through the optical centre and perpendicular to the lens's surface.
• It serves as a baseline for measuring the focal length, object distance, and image distance in lenses.
• Focal Point:
• The focal point is the point on the principal axis where parallel rays of light converge (convex lens) or appear to diverge (concave lens) after passing through the lens.
• The distance between the optical centre and the focal point is known as the focal length.
• Sign Conventions:
• Distances measured along the direction of the incident light are positive, while those measured against it are negative.
• Heights of objects and images are positive if measured upwards from the principal axis and negative if measured downwards.
• Lens Formula:
• The lens formula is given by (1/f = 1/v - 1/u), where:
• f = Focal length
• v = Image distance
• u = Object distance
• This formula is derived considering the optical centre as the origin for all measurements.
• Applications of Lenses:
• Convex Lenses: Used in magnifying glasses, cameras, projectors, and human eye correction for hypermetropia.
• Concave Lenses: Used in correcting myopia, peepholes in doors, and certain optical instruments.

Important Points

• Option 1: The pole of the lens is not used as a reference for measuring distances in the sign convention. Hence, this option is incorrect.
• Option 2: The focus of the lens is an important point but is not used as the origin for measuring distances. Hence, this option is incorrect.
• Option 3: The optical centre is the correct reference point for measuring distances in the Cartesian coordinate system for lenses. This is the correct answer.
• Option 4: The principal axis is a reference line, not a reference point for measuring distances. Hence, this option is incorrect.

The correct answer is Leishmania.

Key Points

• Binary fission is a mode of asexual reproduction in which a single organism divides into two identical daughter organisms. This method is commonly observed in unicellular organisms.
• Leishmania, a protozoan parasite, reproduces through binary fission. It divides longitudinally, ensuring each daughter cell receives a copy of the nucleus and cytoplasmic components. Hence, Leishmania reproduces via binary fission.
• Other organisms mentioned in the options such as Butterfly, Rose plant, and Neem plant, do not reproduce by binary fission. They reproduce sexually or through other methods like vegetative propagation.
• Butterflies reproduce sexually through the process of egg fertilization, while Rose and Neem plants reproduce through seeds (sexual reproduction) or vegetative propagation (e.g., cuttings in roses). Hence, these options are incorrect.
• Binary fission is a characteristic feature of simpler organisms like bacteria, amoeba, and certain protozoans. It is not observed in complex multicellular organisms like plants and animals.

Additional Information

• Binary Fission Process:
• During binary fission, the DNA of the organism is replicated first.
• The cytoplasm then divides, and the organism splits into two equal-sized daughter cells.
• Each daughter cell is genetically identical to the parent cell.
• Leishmania Details:
• Leishmania is a protozoan parasite responsible for diseases like Leishmaniasis, which affects humans and animals.
• It is transmitted through the bite of infected female sandflies (Phlebotomus species).
• Leishmania exhibits a specific mode of binary fission, where the cell divides longitudinally, unlike bacteria, which divide by transverse binary fission.
• Other Examples of Binary Fission:
• Bacteria: They reproduce by transverse binary fission, dividing into two identical cells.
• Amoeba: A unicellular organism that divides into two daughter cells after replicating its nucleus.
• Paramecium: A ciliate protozoan that undergoes binary fission along its longitudinal axis.
• Reproduction in Other Organisms:
• Butterfly: Reproduces sexually; eggs are laid, which hatch into larvae (caterpillars) and undergo metamorphosis to become adult butterflies.
• Rose Plant: Can reproduce sexually through seeds and asexually through vegetative propagation (e.g., stem cuttings).
• Neem Plant: Reproduces sexually through seeds formed after pollination.

Important Points

• Binary Fission vs. Mitosis:
• Binary fission is simpler and occurs in prokaryotic organisms like bacteria and protozoans.
• Mitosis occurs in eukaryotic cells, involving the division of the nucleus followed by cytokinesis.
• Types of Binary Fission:
• Simple Binary Fission: Observed in Amoeba; cell divides in any plane.
• Longitudinal Binary Fission: Seen in Leishmania; division occurs along the longitudinal axis.
• Transverse Binary Fission: Seen in Paramecium; division occurs along the transverse axis.
• Oblique Binary Fission: Observed in certain protozoans; division occurs at an oblique angle.
• Significance of Binary Fission:
• Allows rapid population growth under favorable conditions.
• Ensures genetic consistency in offspring as they are clones of the parent.
• Helps organisms survive and adapt in specific ecological niches.

The correct answer is 13 amu.

Key Points

• Atomic mass is the total mass of protons, neutrons, and electrons in an atom. However, since the mass of electrons is negligible, the atomic mass is primarily the sum of protons and neutrons in the nucleus.
• Carbon-13 is an isotope of carbon that contains 6 protons and 7 neutrons. The sum of these nucleons gives its atomic mass: 6 (protons) + 7 (neutrons) = 13 atomic mass units (amu).
• Isotopes are atoms of the same element with the same number of protons but a different number of neutrons. Hence, while the atomic number of all carbon isotopes is 6, their atomic masses differ due to the variation in the number of neutrons.
• The most common isotope of carbon is Carbon-12, which has an atomic mass of 12 amu. However, Carbon-13 occurs naturally in smaller amounts, approximately 1.1% of all carbon atoms.
• In scientific terms, the atomic mass of Carbon-13 is specifically measured as 13.00335 amu, but for simplicity, it is rounded to 13 amu.

Additional Information

• Isotopes of Carbon:
• Carbon-12: The most abundant isotope, comprising approximately 98.9% of carbon atoms. It contains 6 protons and 6 neutrons, giving it an atomic mass of 12 amu.
• Carbon-13: A stable isotope of carbon, which makes up about 1.1% of carbon atoms. It is used in various scientific fields, including carbon dating and nuclear magnetic resonance (NMR) spectroscopy.
• Carbon-14: A radioactive isotope with 6 protons and 8 neutrons. It has an atomic mass of 14 amu and is widely used in radiocarbon dating to determine the age of fossils and archaeological samples.
• Applications of Carbon-13:
• Carbon-13 is extensively used in NMR spectroscopy to study the molecular structure of organic compounds. Its nuclear spin properties make it useful for analyzing chemical environments.
• It is utilized in metabolic studies to trace biochemical pathways by incorporating Carbon-13-labeled compounds into biological systems.
• Carbon-13 is also used in climatology to analyze carbon cycles and understand historical changes in atmospheric carbon levels.
• Atomic Mass Unit (amu):
• An atomic mass unit is defined as one-twelfth of the mass of a Carbon-12 atom. It is the standard unit for expressing atomic and molecular masses.
• The exact value of 1 amu is 1.660539066 × 10⁻²⁷ kilograms.
• Using amu simplifies calculations in chemistry and physics by providing a convenient scale for comparing the masses of atoms and molecules.

Important Points

• Mass Number: The mass number is the total number of protons and neutrons in the nucleus of an atom. It is always a whole number.
• Atomic Number: The atomic number represents the number of protons in an atom and determines the chemical identity of an element. For carbon, the atomic number is 6.
• Natural Abundance: Isotopes of an element have different natural abundances. Carbon-12 is the most abundant isotope of carbon, while Carbon-13 and Carbon-14 occur in much smaller amounts.
• Stable vs. Radioactive Isotopes: Carbon-12 and Carbon-13 are stable isotopes, whereas Carbon-14 is radioactive, decaying over time with a half-life of approximately 5730 years.

The correct answer is Loosely packed.

Key Points

• Connective tissue is characterized by the presence of an abundant intercellular (extracellular) matrix.
• According to NCERT, the cells of connective tissue are loosely packed and are embedded in the matrix.
• The matrix may be fluid, jelly-like, or solid depending on the type of connective tissue and provides support, binding, protection, and transport.
• Unlike epithelial tissue, connective tissue has more intercellular space, which keeps its cells loosely arranged.
• Hence, the correct answer is Option 2 – Loosely packed.

Additional Information

• Connective tissue connects, supports, binds, and protects different tissues and organs of the body.
• Common types of connective tissue include areolar tissue, adipose tissue, tendon, ligament, cartilage, bone, and blood.
• Epithelial tissue has tightly packed cells with very little intercellular space, whereas connective tissue has a relatively larger amount of extracellular matrix.

Important Points

• NCERT wording: "The cells of connective tissue are loosely packed and embedded in an intercellular matrix."
• Exam Tip: If the options include both "loosely packed" and "far away from each other", choose "loosely packed" as it follows the standard NCERT terminology.
• Features of Connective Tissue:
• Cells are loosely arranged.
• Abundant extracellular matrix is present.
• Provides support, binding, protection, storage, and transport.

The correct answer is only protons.

Key Points

• The atomic number of an element is defined as the total number of protons present in the nucleus of its atom. This is a fundamental property of an element and remains constant for all atoms of that element.
• The atomic number is symbolized by the letter Z. For example, for Hydrogen, Z = 1, as it has one proton in its nucleus.
• Neutrons do not determine the atomic number. Although neutrons are present in the nucleus, their count can vary in isotopes of an element without affecting the atomic number.
• The atomic number is used to uniquely identify an element in the periodic table. For example, Carbon has an atomic number of 6, which means it has 6 protons in its nucleus.
• Electrons in a neutral atom are equal in number to the protons, but they are not considered in defining the atomic number.

Additional Information

• Protons:
• Protons are positively charged particles found in the nucleus of an atom.
• The number of protons determines the chemical identity of an element. For instance, Oxygen always has 8 protons.
• Each proton has a charge of +1e (positive elementary charge) and a mass of approximately 1 atomic mass unit (amu).
• Neutrons:
• Neutrons are neutral particles (no charge) also located in the nucleus of an atom.
• The number of neutrons can vary among atoms of the same element, leading to the formation of isotopes.
• For example, Carbon has three common isotopes: Carbon-12 (6 neutrons), Carbon-13 (7 neutrons), and Carbon-14 (8 neutrons).
• Electrons:
• Electrons are negatively charged particles that orbit the nucleus in different energy levels or shells.
• In a neutral atom, the number of electrons is equal to the number of protons.
• Electrons play a crucial role in chemical bonding and reactions but do not affect the atomic number.
• Isotopes:
• Isotopes are atoms of the same element with the same number of protons but different numbers of neutrons.
• For example, Hydrogen has three isotopes: Protium (1 proton, 0 neutrons), Deuterium (1 proton, 1 neutron), and Tritium (1 proton, 2 neutrons).
• Despite having different masses, isotopes of the same element have identical chemical properties.
• Periodic Table:
• The periodic table is arranged in order of increasing atomic number (number of protons).
• Elements in the same group have similar chemical properties because they have the same number of valence electrons.
• For instance, Group 1 elements like Sodium and Potassium have a single valence electron and exhibit similar reactivity.

Important Points

• Atomic Mass Number (A): It is the sum of the number of protons and neutrons in the nucleus of an atom. For example, for Carbon-12, A = 6 (protons) + 6 (neutrons) = 12.
• Neutral Atom: In a neutral atom, the number of protons equals the number of electrons, maintaining electrical neutrality.
• Ions: When an atom loses or gains electrons, it becomes an ion. For example, Sodium (Na) loses one electron to form Na⁺, and Chlorine (Cl) gains one electron to form Cl^-.
• Discovery of Protons: Protons were discovered by Ernest Rutherford in 1917 during his gold foil experiment.

The correct answer is decreasing reactivity.

Key Points

• The reactivity series is a list of metals arranged in the order of their decreasing reactivity. This means the most reactive metal is placed at the top, and the least reactive metal is at the bottom.
• Metals higher in the series can displace metals lower in the series from their compounds during chemical reactions, such as displacement reactions.
• The reactivity is determined based on how easily a metal can lose electrons to form positive ions (cations). Metals that lose electrons easily are more reactive.
• The series is critical in predicting the outcomes of chemical reactions, such as the reaction of metals with water, acids, and other chemicals.
• For example, in the reactivity series, potassium and sodium are at the top because they are highly reactive, while gold and platinum are at the bottom because they are least reactive.

Additional Information

• Reactivity Series:
• The reactivity series is also called the activity series, and it includes metals such as potassium, sodium, calcium, magnesium, aluminum, zinc, iron, lead, copper, silver, and gold, arranged in decreasing order of reactivity.
• Metals at the top (e.g., potassium and sodium) react vigorously with water and air, while metals at the bottom (e.g., gold and platinum) do not react easily.
• This series is determined experimentally through reactions of metals with water, acids, and other salts.
• Examples of Reactivity:
• Potassium and sodium react violently with water, forming hydroxides and releasing hydrogen gas.
• Iron reacts slowly with dilute acids, producing hydrogen gas, but does not react with water at room temperature.
• Gold and platinum are chemically inert and do not react with most acids or water.
• Applications of the Reactivity Series:
• It is used to predict the feasibility of a displacement reaction. For example, zinc can displace copper from copper sulfate solution, but copper cannot displace zinc.
• The series helps in the extraction of metals. Highly reactive metals like sodium and potassium are extracted using electrolysis, while less reactive metals like iron can be extracted using reduction with carbon.
• It is applied in designing corrosion-resistant materials and galvanization processes.
• Displacement Reaction:
• In a displacement reaction, a more reactive metal displaces a less reactive metal from its compound. For example: Zn + CuSO₄ → ZnSO₄ + Cu
• This reaction occurs because zinc is higher in the reactivity series than copper.

Important Points

• The reactivity series also includes hydrogen, which is a reference point. Metals above hydrogen can displace it from acids, while metals below cannot.
• Metals like aluminum and zinc are used in sacrificial protection to prevent rusting of iron structures.
• Metals at the bottom of the series, such as silver and gold, are used in jewelry due to their low reactivity, making them resistant to tarnishing.
• The position of a metal in the reactivity series is also critical in determining its method of extraction. For example, electrolysis is used for highly reactive metals, while reduction is used for moderately reactive metals.

The correct answer is Carbon dioxide.

Key Points

• When dilute hydrochloric acid (HCl) reacts with sodium hydrogencarbonate (NaHCO₃), the reaction produces carbon dioxide gas (CO₂), along with water (H₂O) and sodium chloride (NaCl).
• The chemical reaction for this process is: NaHCO₃ + HCl → NaCl + H₂O + CO₂. This is a classic acid-carbonate reaction where carbon dioxide gas is released.
• The evolution of carbon dioxide can be observed as effervescence or bubbling during the reaction.
• This reaction is commonly used in experiments to demonstrate the presence of carbon dioxide, as the gas can turn limewater (Ca(OH)₂) milky when bubbled through it, due to the formation of calcium carbonate (CaCO₃).
• None of the other options (Nitrogen, Oxygen, or Hydrogen) is evolved in this reaction. The reaction specifically releases carbon dioxide due to the decomposition of the bicarbonate ion (HCO₃^-).

Additional Information

• Sodium Hydrogencarbonate (NaHCO₃):
• Also known as baking soda, it is a weak base commonly used in cooking, cleaning, and as an antacid.
• When it reacts with an acid like HCl, it releases carbon dioxide gas, which is responsible for the characteristic fizzing or bubbling.
• It is also used in baking to help dough rise, as the released carbon dioxide creates air pockets in the mixture.
• Hydrochloric Acid (HCl):
• HCl is a strong acid commonly used in laboratories and industries for various purposes, including cleaning, pH regulation, and in reactions with bases or carbonates.
• In the human body, HCl is a component of gastric acid, aiding in digestion by breaking down food and killing harmful microorganisms.
• Carbon Dioxide (CO₂):
• CO₂ is a colorless, odorless gas that is naturally present in Earth's atmosphere.
• It is a byproduct of respiration in animals and combustion of organic matter, and it plays a crucial role in the carbon cycle and photosynthesis in plants.
• In practical applications, it is used in the production of carbonated beverages, fire extinguishers, and as a refrigerant.
• Characteristics of Other Gases:
• Nitrogen (N₂): A colorless, odorless, and inert gas, it makes up about 78% of Earth's atmosphere. It does not react with HCl or NaHCO₃.
• Oxygen (O₂): Essential for respiration, it is a highly reactive gas but is not produced in this reaction.
• Hydrogen (H₂): A flammable gas, it is evolved in reactions between acids and metals, not in the reaction between HCl and NaHCO₃.

Important Points

• Acid-Carbonate Reactions: These reactions are commonly used in laboratories to generate carbon dioxide gas for experiments.
• Testing for CO₂: Carbon dioxide can be confirmed by passing it through limewater. If the solution turns milky, it indicates the presence of CO₂.
• Applications of the Reaction: This reaction is utilized in household cleaning, baking, and in fire extinguishers where the rapid release of CO₂ helps to smother flames.
• Environmental Role of CO₂: While it is essential for photosynthesis, excessive CO₂ in the atmosphere contributes to global warming and climate change.

The correct answer is Both statements are correct, and Statement II correctly explains Statement I..

Key Points

• Statement I: Zinc displaces copper from copper sulphate solution, but copper cannot displace zinc from zinc sulphate solution.
• This statement is correct.
• When zinc is added to copper sulphate solution, it reacts to form zinc sulphate and copper metal. This occurs because zinc is more reactive than copper.
• However, when copper is added to zinc sulphate solution, no reaction occurs as copper cannot displace zinc due to its lower reactivity.
• Statement II: Zinc is more reactive than copper in the reactivity series.
• This statement is also correct.
• The reactivity series is a list of metals arranged in order of their reactivity. Zinc is positioned above copper, indicating that it is more reactive.
• The higher reactivity of zinc allows it to displace copper from its compounds in aqueous solutions.
• Explanation:
• Statement II correctly explains Statement I because the ability of zinc to displace copper is directly due to its higher position in the reactivity series, which reflects its greater reactivity.
• Thus, both statements are correct, and Statement II provides the reason for the observation in Statement I.

Additional Information

• Reactivity Series:
• The reactivity series is a chart that ranks metals based on their reactivity with water, acids, and other substances.
• Zinc is placed above copper in the reactivity series, meaning it has a stronger tendency to lose electrons and form positive ions.
• Some highly reactive metals in the series include potassium, sodium, and calcium, while less reactive ones like gold and silver are found at the bottom.
• Displacement Reactions:
• A displacement reaction occurs when a more reactive metal displaces a less reactive metal from its compound in an aqueous solution.
• For example, zinc displaces copper from copper sulphate solution: Zn + CuSO₄ → ZnSO₄ + Cu.
• However, the reverse (Cu + ZnSO₄) does not occur because copper is less reactive than zinc.
• Applications of Reactivity Series:
• The reactivity series is used in industrial processes like metal extraction and refining.
• It helps predict the feasibility of chemical reactions, such as displacement and redox reactions.
• It is also used to prevent corrosion by selecting metals for galvanization or sacrificial protection (e.g., using zinc to protect iron from rusting).
• Examples of Displacement Reactions:
• Iron displacing copper: Fe + CuSO₄ → FeSO₄ + Cu.
• Magnesium displacing zinc: Mg + ZnSO₄ → MgSO₄ + Zn.

Important Points

• Key Factors in Reactivity:
• The reactivity of a metal depends on its ability to lose electrons and form positive ions.
• Metals higher in the reactivity series are more electropositive and readily participate in chemical reactions.
• Galvanization:
• Galvanization is the process of coating iron or steel with a layer of zinc to prevent rusting.
• This process works because zinc, being more reactive, corrodes in place of the iron, acting as a sacrificial metal.
• Extraction of Metals:
• Less reactive metals like gold are found in nature in their elemental form, while more reactive ones like zinc are extracted from their ores through reduction or electrolysis.

The correct answer is Lactic Acid.

Key Points

• Anaerobic respiration occurs in the absence of oxygen and is a process by which cells generate energy. In animals, the end product of anaerobic respiration is lactic acid. This occurs primarily during intense physical activities when oxygen availability is limited.
• Lactic acid formation: During anaerobic respiration, glucose is broken down into pyruvate through glycolysis. In the absence of oxygen, pyruvate is converted into lactic acid by the enzyme lactate dehydrogenase.
• Lactic acid accumulation: The buildup of lactic acid in muscle tissues leads to muscle fatigue and soreness. This is commonly experienced during strenuous exercises such as sprinting or heavy lifting.
• Energy yield: Anaerobic respiration yields only 2 ATP molecules per glucose molecule, which is significantly lower compared to aerobic respiration (which yields 36-38 ATP molecules).
• This process is temporary and reversible; once oxygen becomes available, lactic acid is transported to the liver where it is converted back into glucose through the Cori cycle.

Additional Information

• Anaerobic respiration in animals:
• This process is vital during high-energy demands when oxygen supply is insufficient, such as during short bursts of physical exertion.
• It allows cells to continue producing energy temporarily despite the lack of oxygen.
• It is common in muscle cells, especially during activities like sprinting, weightlifting, or intense sports.
• Comparison of aerobic and anaerobic respiration:
• Aerobic respiration: Requires oxygen, produces carbon dioxide and water as end products, and yields high energy (36-38 ATP molecules).
• Anaerobic respiration: Does not require oxygen, produces lactic acid in animals (or ethanol and carbon dioxide in yeast), and yields lower energy (2 ATP molecules).
• Lactic acid and exercise:
• During intense exercise, lactic acid buildup causes a burning sensation in muscles and fatigue.
• Post-exercise, oxygen-rich blood removes lactic acid, aiding recovery and reducing soreness.
• Proper hydration and stretching can help reduce the effects of lactic acid accumulation.
• Other end products of anaerobic respiration:
• In plants and yeast: The end products of anaerobic respiration are ethanol and carbon dioxide. This process is known as fermentation.
• In bacteria: Different bacteria produce various end products such as methane, hydrogen, or acetic acid, depending on their metabolism.

Important Points

• Lactic acid in clinical context: Elevated levels of lactic acid in the blood, termed lactic acidosis, can indicate conditions like hypoxia, sepsis, or metabolic disorders.
• Cori cycle: This metabolic pathway involves the conversion of lactic acid back into glucose in the liver, demonstrating the body's ability to recycle metabolic byproducts.
• Role of enzymes: The enzyme lactate dehydrogenase plays a crucial role in the conversion of pyruvate to lactic acid during anaerobic respiration.
• Adaptation in animals: Some animals, such as diving mammals, have adapted mechanisms to tolerate higher levels of lactic acid during prolonged periods of oxygen deprivation.

The correct answer is Reaction force of the bullet on the gun.

Key Points

• When a gun is fired, the bullet is propelled forward with a high velocity due to the explosive force generated by the combustion of gunpowder within the barrel. This action creates an equal and opposite reaction force on the gun, causing it to recoil backward. This phenomenon is explained by Newton's Third Law of Motion, which states: "For every action, there is an equal and opposite reaction."
• The backward motion of the gun, known as recoil, is directly proportional to the forward momentum of the bullet. This relationship is governed by the principle of conservation of momentum, which states that the total momentum of an isolated system remains constant if no external forces act on it.
• The mass of the gun is significantly larger than the bullet, so the gun's recoil velocity is much smaller in magnitude than the bullet's forward velocity. This inverse relationship ensures that the total momentum before and after firing remains balanced.
• Neither air resistance, gravity, nor friction between the gun and the person holding it plays a role in the gun's recoil. These factors may affect the gun's trajectory or stability but are not responsible for the recoil motion.
• This concept is critical in firearm design, as the recoil force must be managed to ensure the shooter’s safety and maintain accuracy during subsequent shots.

Additional Information

• Newton's Third Law of Motion:
• Newton's Third Law explains the interaction between two objects. If object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A.
• This law governs not only firearm recoil but also other phenomena such as rocket propulsion, swimming, and walking.
• Conservation of Momentum:
• The total momentum of an isolated system remains constant if no external forces act on it.
• In the context of firearms, the momentum of the bullet moving forward is equal in magnitude but opposite in direction to the momentum of the recoiling gun.
• Firearm Recoil Management:
• Recoil can be reduced by using techniques such as adding weight to the firearm, using recoil pads, or employing gas-operated mechanisms in semi-automatic weapons.
• Suppressors or muzzle brakes are also commonly used to mitigate recoil by redirecting gases expelled during firing.
• Applications of Newton's Laws:
• Newton’s Third Law is instrumental in the design and operation of rockets, jet engines, and even vehicles, where propulsion relies on the expulsion of gases or forces in one direction to generate movement in the opposite direction.
• Misconceptions about Recoil:
• Some people mistakenly believe that air resistance or friction is responsible for the recoil of a gun. In reality, these factors are negligible compared to the reaction force caused by the bullet's forward motion.

Important Points

• Firearm Safety:
• Recoil management is crucial for accurate shooting and preventing injuries. Proper stance, grip, and understanding of the firearm’s behavior are essential for safe handling.
• Physics in Everyday Life:
• The principles of Newton's laws and momentum conservation are not limited to firearms but are integral to understanding a wide range of mechanical systems and daily activities.
• Historical Context:
• The study of firearm recoil and ballistics has been a significant area of research since the development of gunpowder-based weapons. Modern advancements in materials and design have enhanced both safety and performance.

The correct answer is 20 V.

Key Points

• The relationship between work done (W), charge (Q), and potential difference (V) is given by the formula: V = W/Q.
• Here, the work done (W) is given as 30 J, and the charge (Q) is 1.5 C.
• Using the formula: V = W/Q = 30 J / 1.5 C = 20 V.
• Therefore, the potential difference between the two points is 20 V. The calculation is correct based on the given values.
• Option 4: 20 V is the correct answer.

Additional Information

• Electric Potential Difference:
• The potential difference is the work done in moving a unit charge from one point to another in an electric field.
• It is measured in volts (V), where 1 volt = 1 joule/coulomb.
• Potential difference is a scalar quantity and is represented mathematically as V = W/Q.
• Work Done in Electric Fields:
• Work done refers to the energy required to move a charge against an electric field.
• It is given by W = Q × V, where Q is the charge and V is the potential difference.
• The SI unit of work is the joule (J).
• Charge:
• Charge is a fundamental property of matter that causes it to experience a force in an electric field.
• It is measured in coulombs (C), where 1 coulomb is the charge transported by a current of 1 ampere in 1 second.
• Applications of Potential Difference:
• Electrical Circuits: Potential difference drives current through circuits, powering devices like lights, motors, and appliances.
• Electrochemical Processes: It is crucial in batteries and electroplating, where chemical reactions are driven by electric potential differences.
• Measurement: Devices like voltmeters measure the potential difference across components in electrical circuits.

Important Points

• Formula Recap: Always remember the formula V = W/Q for calculating potential difference, where W is in joules and Q is in coulombs.
• Units:
• The unit of potential difference is volt (V), work is joules (J), and charge is coulombs (C).
• 1 volt = 1 joule/coulomb.
• Practical Example: The potential difference of a standard AA battery is approximately 1.5 V, while a car battery typically has a potential difference of 12 V.
• When solving problems, ensure consistent units (e.g., joules, coulombs) to avoid calculation errors.

The correct answer is the current flowing through the loop.

Key Points

• The magnetic field at the center of a current-carrying circular loop is determined using Ampere’s Law or Biot-Savart Law. It is given by the formula: B = (μ₀ * I) / (2 * R), where:
• B: Magnetic field at the center (measured in Tesla).
• μ₀: Permeability of free space (4π × 10⁻⁷ T·m/A).
• I: Current flowing through the loop.
• R: Radius of the loop.
• From the formula, it is evident that the magnetic field B is directly proportional to the current (I). Hence, an increase in current will lead to a proportional increase in the magnetic field.
• The radius of the loop affects the magnetic field inversely. However, the question specifically asks about direct proportionality, which is only true for the current flowing through the loop.
• The other options, such as the square of the radius or the square of the current, are not correct because the magnetic field does not depend on these quantities in that manner.
• Hence, the correct answer is the current flowing through the loop.

Additional Information

• Biot-Savart Law:
• The Biot-Savart law describes the magnetic field produced by a current element. It states that the magnetic field (dB) due to a small segment of current (IdL) at a point is directly proportional to the current and inversely proportional to the square of the distance between the segment and the point.
• Mathematically: dB = (μ₀/4π) * (I * dL × r̂) / r².
• Ampere’s Law:
• Ampere’s Law relates the integrated magnetic field around a closed loop to the total electric current passing through the loop.
• It is expressed as: ∮B·dl = μ₀I, where I is the current enclosed by the loop.
• Impact of Radius:
• The magnetic field at the center is inversely proportional to the radius of the loop. If the radius increases, the field decreases and vice versa.
• This relationship is evident from the formula: B = (μ₀ * I) / (2 * R).
• Applications of Magnetic Fields in Circular Loops:
• Used in electromagnets for generating strong magnetic fields.
• Widely applied in electric motors, MRI machines, and particle accelerators.
• Units and Constants:
• Magnetic field (B) is measured in Tesla (T).
• Permeability of free space (μ₀) has a value of 4π × 10⁻⁷ T·m/A.

Important Points

• Option 1: The square of the radius of the circular loop is not involved in the formula for the magnetic field. Thus, this option is incorrect.
• Option 3: The inverse of the current is incorrect because the magnetic field is directly proportional to the current.
• Option 4: The square of the current does not influence the magnetic field; hence, this is also incorrect.

Shortcut Trick

Initial Income = 100%, Savings = 25.5%, Expenditure = 74.5%.

New Income = 100 × 1.2 = 120%.

New Expenditure = 74.5 × 1.4 = 104.3%.

New Savings = 120% − 104.3% = 15.7%.

Change in Savings = 15.7% − 25.5% = −9.8% (Decrease).

Value decrease = 9.8% of ₹25,000 = ₹2,450.

∴ The correct answer is decrease by ₹2,450.

Alternate Method

Given:

Initial Income = ₹25,000

Initial Savings % = 25.5%

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Initial Savings = 25.5% of 25,000 = ₹6,375

⇒ Initial Expenditure = 25,000 − 6,375 = ₹18,625

⇒ New Income = 25,000 × (120/100) = ₹30,000

⇒ New Expenditure = 18,625 × (140/100) = ₹26,075

⇒ New Savings = 30,000 − 26,075 = ₹3,925

⇒ Change in Savings = 3,925 − 6,375 = −₹2,450

∴ The correct answer is decrease by ₹2,450.

Additional Information

Percentage Change in Savings

If income increases by x% and expenditure increases by y%, the new savings is calculated by subtracting the new expenditure from the new income.

Base Value Stability

When calculating absolute changes, always refer back to the original values unless successive changes are mentioned.

Expenditure Component

Expenditure = Income × (100 − Savings%). Any percentage increase in expenditure applies to this calculated component.

Shortcut Trick

Given tan θ = 3/4, which represents Perpendicular/Base.

Using Pythagorean Triplets (3, 4, 5), the Hypotenuse is 5.

sin θ = Perpendicular/Hypotenuse = 3/5.

∴ The correct answer is 3/5.

Alternate Method

Given:

tan θ = 3/4

Formula Used:

tan θ = Perpendicular (P) / Base (B)

sin θ = Perpendicular (P) / Hypotenuse (H)

H² = P² + B²

Calculations:

⇒ P = 3, B = 4

⇒ H² = 3² + 4²

⇒ H² = 9 + 16 = 25

⇒ H = √25 = 5

⇒ sin θ = P/H = 3/5

∴ The correct answer is 3/5.

Additional Information

Pythagorean Identity

For any angle θ, sin²θ + cos²θ = 1. This can be used to find sine if cosine is known.

Trigonometric Ratios

cos θ = Base/Hypotenuse and cot θ = 1/tan θ. In this case, cos θ = 4/5.

Angle of Elevation

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

Shortcut Trick

The area of a quadrilateral with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄) is calculated using the Shoelace Formula:

Area = 1/2 |(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) − (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁)|

Area = 1/2 |(2×7 + 5×6 + 10×2 + 7×3) − (3×5 + 7×10 + 6×7 + 2×2)|

Area = 1/2 |(14 + 30 + 20 + 21) − (15 + 70 + 42 + 4)|

Area = 1/2 |85 − 131| = 1/2 |−46| = 23 sq. units.

∴ The correct answer is 23 sq. units.

Alternate Method

Given: Vertices of the quadrilateral are A(2,3), B(5,7), C(10,6), and D(7,2).

Formula Used: Area of Quadrilateral ABCD = Area of ΔABC + Area of ΔADC

⇒ Area of ΔABC = 1/2 |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

⇒ Area of ΔABC = 1/2 |2(7−6) + 5(6−3) + 10(3−7)|

⇒ Area of ΔABC = 1/2 |2(1) + 5(3) + 10(−4)| = 1/2 |2 + 15 − 40| = 11.5 sq. units.

⇒ Area of ΔADC = 1/2 |x₁(y₄−y₃) + x₄(y₃−y₁) + x₃(y₁−y₄)|

⇒ Area of ΔADC = 1/2 |2(2−6) + 7(6−3) + 10(3−2)|

⇒ Area of ΔADC = 1/2 |2(−4) + 7(3) + 10(1)| = 1/2 |−8 + 21 + 10| = 11.5 sq. units.

⇒ Total Area = 11.5 + 11.5 = 23 sq. units.

∴ The correct answer is 23 sq. units.

Additional Information

Shoelace Formula

A general algorithm to find the area of any polygon by using the coordinates of its vertices in order.

Area of a Triangle

For vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the area is 1/2 |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.

Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is d = √((x₂−x₁)² + (y₂−y₁)²).

Shortcut Trick

Expenditure ratio is calculated by multiplying the Price ratio and Quantity ratio.

Expenditure (Lemon : Strawberry) = (28 × 15) : (26 × 14)

Simplify: (2 × 15) : (26 × 1) ⇒ 30 : 26 ⇒ 15 : 13

Required Ratio (Strawberry : Lemon) = 13 : 15

∴ The correct answer is 13 : 15.

Alternate Method

Given:

Ratio of Unit Prices (Lemon : Strawberry) = 28 : 26

Ratio of Quantities (Lemon : Strawberry) = 15 : 14

Formula Used:

Expenditure = Price × Quantity

Calculations:

⇒ Expenditure on Lemon Chocolate = 28 × 15 = 420

⇒ Expenditure on Strawberry Chocolate = 26 × 14 = 364

⇒ Ratio (Strawberry : Lemon) = 364 : 420

⇒ Dividing both sides by 28:

⇒ 364 ÷ 28 = 13

⇒ 420 ÷ 28 = 15

⇒ Required Ratio = 13 : 15

∴ The correct answer is 13 : 15.

Additional Information

Compound Ratio

When two or more ratios are multiplied term by term, the result is called a compound ratio. If a:b and c:d are two ratios, their compound ratio is (a×c) : (b×d).

Expenditure Relationship

Total Expenditure is directly proportional to both the price per unit and the total quantity consumed. E ∝ P and E ∝ Q.

Inverse Variation

If the total expenditure remains constant, the price is inversely proportional to the quantity (P ∝ 1/Q).

Shortcut Trick

Calculate the Total Effective Interest Rate by summing the products of individual rates and their respective time periods.

Effective Interest Rate = (6 years × 8%) + (3 years × 3%) = 48% + 9% = 57%

Total Simple Interest = 57% of ₹3,500

Total Simple Interest = 0.57 × 3500 = ₹1,995

∴ The correct answer is ₹1,995.

Alternate Method

Given:

Principal (P) = ₹3,500

Case 1: Time (T₁) = 6 years, Rate (R₁) = 8% p.a.

Case 2: Time (T₂) = 3 years, Rate (R₂) = 3% p.a.

Formula Used:

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

Calculations:

⇒ SI₁ (for first 6 years) = (3500 × 8 × 6) ÷ 100

⇒ SI₁ = 35 × 48 = ₹1,680

⇒ SI₂ (for last 3 years) = (3500 × 3 × 3) ÷ 100

⇒ SI₂ = 35 × 9 = ₹315

⇒ Total SI = SI₁ + SI₂

⇒ Total SI = 1680 + 315 = ₹1,995

∴ The correct answer is ₹1,995.

Additional Information

Total Amount Formula

The total Amount (A) to be paid at the end of the term is the sum of the Principal and the Total Simple Interest: A = P + SI.

Variable Interest Rates

When interest rates change over time, the total interest is calculated as SI = P × (R₁T₁ + R₂T₂ + ... + R_nT_n) ÷ 100.

Simple vs Compound Interest

In Simple Interest, the interest is calculated only on the initial principal, whereas in Compound Interest, interest is calculated on the principal plus any accumulated interest from previous periods.

Shortcut Trick

Identify the Pythagorean Triplet in the right-angled triangle formed by the altitude.

Half of the base = 60 ÷ 2 = 30 cm; Equal side (hypotenuse) = 78 cm.

Simplify ratio: 30 : 78 → 5 : 13 (divided by 6). The triplet is (5, 12, 13).

Actual altitude = 12 × 6 = 72 cm.

Area = 1/2 × base × height = 1/2 × 60 × 72 = 2160 cm².

∴ The correct answer is 2160 cm².

Alternate Method

Given: Equal sides (a) = 78 cm, Base (b) = 60 cm.

Formula Used: Area of isosceles triangle = 1/2 × base × height, where height (h) = √(a² − (b/2)²).

⇒ Semi-base = 60 ÷ 2 = 30 cm

⇒ Height (h) = √(78² − 30²)

⇒ h = √((78 − 30) × (78 + 30))

⇒ h = √(48 × 108) = √(16 × 3 × 36 × 3)

⇒ h = √(16 × 36 × 9) = 4 × 6 × 3 = 72 cm

⇒ Area = 1/2 × 60 × 72

⇒ Area = 30 × 72 = 2160 cm²

∴ The correct answer is 2160 cm².

Additional Information

Heron's Formula

Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a + b + c) / 2. Useful for any triangle where all sides are known.

Properties of Isosceles Triangle

The altitude drawn to the base of an isosceles triangle also acts as the median and the angle bisector.

Direct Area Formula

Area = (b/4) × √(4a² − b²), where 'a' represents equal sides and 'b' represents the base.

Shortcut Trick

Check odd numbers between 440 and 452: 441, 443, 445, 447, 449, 451.

441 → 4+4+1=9 (divisible by 9); 445 → ends in 5; 447 → 4+4+7=15 (divisible by 3).

451 → 11 × 41. Remaining are 443 and 449.

∴ The correct answer is 2.

Alternate Method

Given: Range of numbers is (440, 452).

Formula Used: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

List of numbers between 440 and 452:

⇒ 441: 4+4+1 = 9, so divisible by 3 and 9. (Composite)

⇒ 442: Even number. (Composite)

⇒ 443: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19. (Prime)

⇒ 444: Even number. (Composite)

⇒ 445: Divisible by 5. (Composite)

⇒ 446: Even number. (Composite)

⇒ 447: 4+4+7 = 15, so divisible by 3. (Composite)

⇒ 448: Even number. (Composite)

⇒ 449: Not divisible by any prime up to √449 ≈ 21. (Prime)

⇒ 450: Even number. (Composite)

⇒ 451: 11 × 41 = 451. (Composite)

⇒ Total prime numbers found = 2 (443 and 449).

∴ The correct answer is 2.

Additional Information

Prime Number Property

To check if N is prime, test divisibility by primes up to √N.

Divisibility Rule of 11

Difference between sum of digits at odd places and even places is 0 or multiple of 11. (e.g., 451 → (4+1)−5 = 0).

Co-prime Numbers

Two numbers are co-prime if their HCF is 1, regardless of whether the numbers themselves are prime.

Given:

Value 1 = 1 ÷ 20

Value 2 = 2 ÷ 50

Formula Used:

Percentage = (Value 1 ÷ Value 2) × 100

Calculation:

⇒ (1 ÷ 20) ÷ (2 ÷ 50)

⇒ = (1 ÷ 20) × (50 ÷ 2)

⇒ = 50 ÷ 40

⇒ = 5 ÷ 4

⇒ Percentage = (5 ÷ 4) × 100

⇒ = 125

∴ The correct answer is 125%.

Shortcut Trick

Cost Price (CP) : Selling Price (SP) ratio = 7500 : 8750

Divide by 1250 → 6 : 7

Profit = 7 − 6 = 1 unit on base of 6 units

Profit % = (1 ÷ 6) × 100 = 16.666...

∴ The correct answer is 16.67%.

Alternate Method

Given:

Cost Price (CP) = ₹7,500

Selling Price (SP) = ₹8,750

Formula Used:

Profit = Selling Price − Cost Price

Profit % = (Profit ÷ Cost Price) × 100

Calculations:

⇒ Profit = 8750 − 7500

⇒ Profit = ₹1,250

⇒ Profit % = (1250 ÷ 7500) × 100

⇒ Profit % = (1 ÷ 6) × 100

⇒ Profit % = 16.666...% ≈ 16.67%

∴ The correct answer is 16.67%.

Additional Information

Profit and Loss Basics

Profit occurs when SP > CP. Loss occurs when CP > SP. Both are always calculated on the Cost Price unless stated otherwise.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Relationship Formula

CP / MP = (100 − Discount %) / (100 + Profit %).

Shortcut Trick

Total amount = ₹20 using ₹2 and ₹5 coins.

Condition: At least one of each denomination must be used.

If ₹5 coins = 1 → ₹2 coins = (20 − 5)/2 = 7.5 (Not possible)

If ₹5 coins = 2 → ₹2 coins = (20 − 10)/2 = 5 coins.

If ₹5 coins = 3 → ₹2 coins = (20 − 15)/2 = 2.5 (Not possible)

∴ The correct answer is 5 coins.

Alternate Method

Given: Total amount = ₹20, Denominations = ₹2 and ₹5.

Formula: 2x + 5y = 20 (where x and y are the number of coins, x, y ≥ 1).

⇒ 2x = 20 − 5y

⇒ x = (20 − 5y) / 2

⇒ Since x must be an integer, (20 − 5y) must be even.

⇒ This implies 5y must be even, so y must be an even number.

⇒ Let y = 2: x = (20 − 5 × 2) / 2 = 10 / 2 = 5.

⇒ Let y = 4: x = (20 − 5 × 4) / 2 = 0 / 2 = 0 (Violates 'at least one' condition).

∴ The correct answer is 5 coins.

Additional Information

Linear Diophantine Equations

Equations of the form ax + by = c where solutions must be integers. The number of positive integer solutions depends on the GCD(a, b).

Integer Constraints

In coin problems, the number of coins must always be a natural number (1, 2, 3...) if at least one is required.

Parity Rule

If the sum of two terms is even (20) and one term is even (2x), the other term (5y) must also be even.

Shortcut Trick

Apply the direct algebraic identity: X² + Y² = (X − Y)² + 2XY

Substitute the given values: 20² + 2(12)

Calculate: 400 + 24 = 424

∴ The correct answer is 424.

Alternate Method

Given:

Difference of two numbers (X − Y) = 20

Product of two numbers (XY) = 12

Formula Used:

We use the square of a binomial identity: (X − Y)² = X² + Y² − 2XY

Calculations:

⇒ (X − Y)² = X² + Y² − 2XY

⇒ (20)² = X² + Y² − 2 × 12

⇒ 400 = X² + Y² − 24

⇒ X² + Y² = 400 + 24

⇒ X² + Y² = 424

∴ The correct answer is 424.

Additional Information

Sum Square Identity

The identity for the square of a sum is (X + Y)² = X² + Y² + 2XY.

Difference of Squares

The factors of the difference of two squares are given by X² − Y² = (X − Y)(X + Y).

Relationship between Sum and Difference

The sum and difference of two numbers are related by (X + Y)² = (X − Y)² + 4XY.

Shortcut Trick

Discount % = (Marked Price − Selling Price) ÷ Marked Price × 100

Discount = 758 − 256 = 502

Discount % = (502 ÷ 758) × 100 ≈ 66.2269%

Rounding to two decimal places, we get 66.23%.

∴ The correct answer is 66.23%.

Alternate Method

Given:

Marked Price (MP) = ₹758

Selling Price (SP) = ₹256

Formula Used:

Discount = Marked Price − Selling Price

Discount Percentage = (Discount ÷ Marked Price) × 100

Calculations:

⇒ Discount = 758 − 256 = ₹502

⇒ Discount % = (502 ÷ 758) × 100

⇒ Discount % = 50200 ÷ 758

⇒ Discount % = 66.22691...

⇒ Discount % ≈ 66.23%

∴ The correct answer is 66.23%.

Additional Information

Successive Discounts

When two discounts d₁ and d₂ are given, the equivalent discount is [d₁ + d₂ − (d₁ × d₂)/100]%.

Marked Price and Cost Price Relation

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

Selling Price Calculation

SP = Marked Price × (100 − Discount%)/100.

Shortcut Trick

Identify the Modal Class (highest frequency): 16 − 20 where f₁ = 34.

Calculate frequency differences: d₁ (f₁ − f₀) = 34 − 18 = 16 and d₂ (f₁ − f₂) = 34 − 26 = 8.

Apply direct ratio: Mode = Lower Limit + [d₁ / (d₁ + d₂)] × Class Width.

Mode = 16 + [16 / (16 + 8)] × 4 = 16 + (16 / 24) × 4 = 16 + 2.67 = 18.67.

∴ The correct answer is 18.67.

Alternate Method

Given:

Modal Class (Class with max frequency 34) = 16 − 20

Lower limit (l) = 16 | Frequency of modal class (f₁) = 34

Frequency of preceding class (f₀) = 18 | Frequency of succeeding class (f₂) = 26

Class interval height (h) = 4

Formula Used:

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Calculations:

⇒ Mode = 16 + [(34 − 18) / (2 × 34 − 18 − 26)] × 4

⇒ Mode = 16 + [16 / (68 − 44)] × 4

⇒ Mode = 16 + [16 / 24] × 4

⇒ Mode = 16 + [2 / 3] × 4

⇒ Mode = 16 + 2.666...

⇒ Mode ≈ 18.67

∴ The correct answer is 18.67.

Additional Information

Empirical Relationship

For a moderately skewed distribution: Mode = 3 Median − 2 Mean.

Median for Grouped Data

Median = l + [(n/2 − cf) / f] × h, where cf is the cumulative frequency of the preceding class.

Mean for Grouped Data

Mean (χ) = Σf_ix_i / Σf_i, where x_i is the class mark (midpoint) of each interval.

Shortcut Trick

Net efficiency = (1/6) − (1/12) = 1/12 per minute.

Time to fill full tank = 12 minutes.

Time to fill 1/4th tank = 12 × (1/4) = 3 minutes.

∴ The correct answer is 3 minutes.

Alternate Method

Given: Filling pipe (A) time = 6 min, Emptying pipe (B) time = 12 min.

Formula Used: Efficiency = Total Work ÷ Time; Required Time = Required Work ÷ Total Efficiency.

⇒ Combined Efficiency of (A + B) = 2 − 1 = 1 unit/min

⇒ Total Work (Capacity) = 12 units

⇒ Target Work = 1/4 × 12 = 3 units

⇒ Required Time = 3 units ÷ 1 unit/min = 3 minutes

∴ The correct answer is 3 minutes.

Additional Information

Pipes & Cisterns Sign Convention

Always take the efficiency of a filling pipe as positive (+) and the efficiency of an emptying pipe (outlet) as negative (−).

Net Time Formula

If pipe A fills in 'x' and pipe B empties in 'y', net time T = (x × y) ÷ (y − x). For a fraction 'k' of the tank, time = k × T.

Work & Wages

Wages are always distributed in the ratio of the work done or the ratio of efficiencies when time is constant.

Shortcut Trick

Calculate the semi-perimeter (s) of the triangle: s = (102 + 154 + 80) ÷ 2 = 168 cm.

Use Heron's Formula for Area: Area = √(168 × 66 × 14 × 88) = 3696 cm².

Altitude (h) to the base 154 cm: h = (2 × Area) ÷ Base = (2 × 3696) ÷ 154.

Simplifying the calculation: h = 3696 ÷ 77 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given: Sides of the triangle are a = 102 cm, b = 154 cm, and c = 80 cm.

Formula Used:

1. Semi-perimeter (s) = (a + b + c) ÷ 2
2. Area of triangle (A) = √[s(s − a)(s − b)(s − c)]
3. Area of triangle (A) = (1 ÷ 2) × base × altitude

⇒ Semi-perimeter (s) = (102 + 154 + 80) ÷ 2 = 336 ÷ 2 = 168 cm

⇒ Area = √[168 × (168 − 102) × (168 − 154) × (168 − 80)]

⇒ Area = √(168 × 66 × 14 × 88)

⇒ Area = √[(14 × 12) × (6 × 11) × 14 × (8 × 11)]

⇒ Area = √[14² × 11² × (12 × 6 × 8)]

⇒ Area = √(14² × 11² × 576) = 14 × 11 × 24 = 3696 cm²

⇒ 3696 = (1 ÷ 2) × 154 × Altitude

⇒ 3696 = 77 × Altitude

⇒ Altitude = 3696 ÷ 77 = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Heron's Formula

Used to find the area of a scalene triangle when all three sides are known: Area = √[s(s−a)(s−b)(s−c)].

Altitude Property

The altitude is the perpendicular distance from a vertex to the opposite side. The area is always (1 ÷ 2) × base × corresponding altitude.

Right-Angled Triplets

In this triangle, the altitude divides the base 154 into segments of 90 cm and 64 cm, forming two right triangles with sides (48, 90, 102) and (48, 64, 80).

Given:

$30\%$ of $40\%$ of $\frac{5}{9}$ of number = 704

Formula Used:

$x\% = \frac{x}{100}$

Calculation:

⇒ $\frac{30}{100} \times \frac{40}{100} \times \frac{5}{9} \times N = 704$

⇒ $\frac{1200}{10000} \times \frac{5}{9} \times N = 704$

⇒ $\frac{6}{50} \times \frac{5}{9} \times N = 704$

⇒ $\frac{30}{450} \times N = 704$

⇒ $\frac{1}{15} \times N = 704$

⇒ $N = 704 \times 15 = 10560$

∴ The correct answer is 10560.

Shortcut Trick

Apply the Successive Percentage Formula for Net Change: x + y + (x × y)/100

Here, Markup (x) = +51% and Discount (y) = −8%.

Net Profit % = 51 − 8 + (51 × (−8))/100

Net Profit % = 43 − 4.08 = 38.92%

∴ The correct answer is 38.9%.

Alternate Method

Given: Marked Price (MP) is 51% above Cost Price (CP). Discount is 8% on MP.

Let Cost Price (CP) = 100

⇒ Marked Price (MP) = 100 + 51% of 100 = 151

⇒ Selling Price (SP) = MP − Discount

⇒ SP = 151 − (8% of 151)

⇒ SP = 151 − 12.08 = 138.92

⇒ Profit = SP − CP = 138.92 − 100 = 38.92

⇒ Profit Percentage = (38.92 / 100) × 100 = 38.92%

⇒ Rounded to 1 decimal place = 38.9%

∴ The correct answer is 38.9%.

Additional Information

Marked Price and Cost Price Relation

The relationship between CP and MP can be expressed as: MP/CP = (100 + Profit%)/(100 − Discount%).

Effective Discount

Two successive discounts of a% and b% result in a single equivalent discount of (a + b − ab/100)%.

Markup Percentage

Markup % = [(MP − CP) / CP] × 100. It is always calculated on the Cost Price.

Given:

(2x³y⁵z) × (5xy⁶z²) ÷ (15x²yz²)

Formula Used:

a^m × aⁿ = a^m+n

a^m ÷ aⁿ = a^m-n

Calculation:

(2x³y⁵z × 5xy⁶z²) ÷ (15x²yz²)

⇒ (10x⁴y¹¹z³) ÷ (15x²yz²)

⇒ (10 ÷ 15) × x^4-2 × y^11-1 × z^3-2

⇒ (2 ÷ 3) × x² × y¹⁰ × z

∴ The correct answer is $\frac{2x^2 y^{10} z}{3}$.

Shortcut Trick

Since the train covers its own length in both cases, the distance is constant.

Ratio of times taken = 30 : 44 = 15 : 22

Therefore, the ratio of relative speeds = 22 : 15

Let the speed of the train be S km/hr. Since they walk in the same direction, relative speeds are (S − 5.4) and (S − 18).

⇒ (S − 5.4) ÷ (S − 18) = 22 ÷ 15 ⇒ 15S − 81 = 22S − 396 ⇒ 7S = 315 ⇒ S = 45 km/hr

∴ The correct answer is 45 km/hr.

Alternate Method

Given:

Speed of first person (V₁) = 5.4 km/hr = 5.4 × (5/18) = 1.5 m/s

Speed of second person (V₂) = 18 km/hr = 18 × (5/18) = 5 m/s

Time taken to overtake first person (t₁) = 30 seconds

Time taken to overtake second person (t₂) = 44 seconds

Formula Used:

Distance = Relative Speed × Time

When moving in the same direction, Relative Speed = Speed of Train − Speed of Person

Calculations:

Let the speed of the train be S m/s and length of the train be L meters.

⇒ L = (S − 1.5) × 30

⇒ L = (S − 5) × 44

Equating the length of the train from both equations:

⇒ (S − 1.5) × 30 = (S − 5) × 44

⇒ 30S − 45 = 44S − 220

⇒ 14S = 175

⇒ S = 12.5 m/s

Converting speed of train into km/hr:

⇒ S = 12.5 × (18/5) = 45 km/hr

∴ The correct answer is 45 km/hr.

Additional Information

Relative Speed

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

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Shortcut Trick

Combine the ratios: First : Second = 16 : 9 and Second : Third = 15 : 21 (or 5 : 7).

Common ratio (1st : 2nd : 3rd) = (16×5) : (9×5) : (9×7) = 80 : 45 : 63.

Total units = 80 + 45 + 63 = 188 units → 4512.

Difference (1st − 3rd) = 80 − 63 = 17 units.

Value = (4512 ÷ 188) × 17 = 24 × 17 = 408.

∴ The correct answer is 408.

Alternate Method

Given:

Sum of three numbers (A + B + C) = 4512

Ratio A : B = 16 : 9

Ratio B : C = 15 : 21 = 5 : 7

Calculations:

⇒ To equalize B, multiply (A : B) by 5 and (B : C) by 9.

⇒ A : B : C = (16×5) : (9×5) : (7×9) = 80 : 45 : 63

⇒ Let the numbers be 80x, 45x, and 63x.

⇒ 80x + 45x + 63x = 4512

⇒ 188x = 4512

⇒ x = 4512 ÷ 188 = 24

⇒ Difference between first and third number = 80x − 63x = 17x

⇒ Difference = 17 × 24 = 408

∴ The correct answer is 408.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is the compound ratio. If A:B and B:C are given, the combined ratio A:B:C is found by equalizing the common term B.

Ratio Proportion Rule

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

Difference Calculation

To find the difference between two parts in a ratio total, multiply the total value by (difference in ratio units ÷ sum of all ratio units).

Shortcut Trick

Efficiency ratio of Man (M) to Woman (W): 2M = 9W → M/W = 9/2.

Total Work = 2 × 9 × 9 = 162 units.

Work done by 2 men in 5 days = 2 × 9 × 5 = 90 units.

Remaining Work = 162 − 90 = 72 units.

Required women (N) to complete 72 units in 9 days with efficiency 2: N × 2 × 9 = 72 ⇒ N = 4.

∴ The correct answer is 4.

Alternate Method

Given: 2 men can complete the job in 9 days, or 9 women can complete the job in 9 days.

Formula Used: Total Work = Number of workers × Efficiency × Time

Let efficiency of 1 man = m and 1 woman = w.

⇒ 2m × 9 = 9w × 9

⇒ 2m = 9w

⇒ 1 man = 4.5 women

⇒ Since 2 men work for 5 days, their work is equivalent to 9 women working for 5 days.

⇒ Total work in terms of woman-days = 9 women × 9 days = 81 woman-days.

⇒ Work completed by 2 men in 5 days = Work completed by 9 women in 5 days = 45 woman-days.

⇒ Remaining work = 81 − 45 = 36 woman-days.

⇒ Number of women required to complete the remaining work in 9 days = 36 ÷ 9 = 4 women.

∴ The correct answer is 4.

Additional Information

Work Equivalence Principle

When 'or' is used between different groups of workers, their total work capacities are equalized to find individual efficiency ratios (e.g., x Men = y Women).

Time and Work Basic Formula

M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂, where M is number of workers, D is days, H is hours, and W is work.

Shortcut Trick

Apply the Man-Days constant rule: M₁ × D₁ = M₂ × D₂

1200 × 12 = 1800 × D₂

D₂ = (1200 × 12) ÷ 1800 = 14400 ÷ 1800 = 8

∴ The correct answer is 8 days.

Alternate Method

Given:

Initial number of men (M₁) = 1,200

Initial provisions duration (D₁) = 12 days

Final number of men (M₂) = 1,800

Formula Used:

In problems of provisions, the total quantity remains constant. Since quantity is proportional to the number of men and number of days, we use the Inverse Proportion formula: M₁D₁ = M₂D₂.

Calculations:

⇒ Total provisions = 1200 × 12 = 14,400 man-units

⇒ New duration (D₂) = Total provisions ÷ M₂

⇒ D₂ = 14400 ÷ 1800

⇒ D₂ = 144 ÷ 18

⇒ D₂ = 8 days

∴ The correct answer is 8 days.

Additional Information

Chain Rule (Inverse Proportion)

When work is constant, men (M) and time (D) are inversely proportional. If men increase, days decrease: M₁ × D₁ = M₂ × D₂.

General Work Formula

For different work amounts and hours: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Man-Days Concept

The product of total workers and time gives the total quantum of work or food stock available.

Shortcut Trick

Total Distance = Speed × Time

Total Distance = 23 × 72 = 1656 m

Tunnel Length = Total Distance − Train Length

Tunnel Length = 1656 − 1195 = 461 m

∴ The correct answer is 461 m.

Alternate Method

Given:

Length of the train (L_t) = 1195 m

Speed of the train (S) = 23 m/s

Time taken to cross the tunnel (T) = 72 seconds

Formula Used:

Total Distance = Length of Train + Length of Tunnel

Distance = Speed × Time

Calculations:

⇒ Let the length of the tunnel be L_p.

⇒ Total Distance = L_t + L_p

⇒ 1195 + L_p = 23 × 72

⇒ 1195 + L_p = 1656

⇒ L_p = 1656 − 1195

⇒ L_p = 461 m

∴ The correct answer is 461 m.

Additional Information

Relative Speed

When two objects move in opposite directions, relative speed is S₁ + S₂. In the same direction, it is S₁ − S₂.

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Crossing Stationary Objects

To cross a pole or person, distance = length of train. To cross a platform or bridge, distance = length of train + length of object.

Shortcut Trick

Age difference remains constant (5 years).

Ratio after 8 years → John : James = 8 : 7

Ratio difference = 8 − 7 = 1 unit → 5 years

James age after 8 years = 7 units × 5 = 35 years

Present age of James = 35 − 8 = 27 years

∴ The correct answer is 27 years.

Alternate Method

Given: Age difference = 5 years; Future ratio (8 years later) = 8 : 7.

Formula Used: Ratio × constant = Age.

⇒ Let present age of James = x

⇒ Present age of John = x + 5

⇒ After 8 years, John age = x + 5 + 8 = x + 13

⇒ After 8 years, James age = x + 8

⇒ Ratio = (x + 13) / (x + 8) = 8 / 7

⇒ 7(x + 13) = 8(x + 8)

⇒ 7x + 91 = 8x + 64

⇒ x = 91 − 64 = 27

∴ The correct answer is 27 years.

Additional Information

Constant Difference Property

The difference between the ages of two persons always remains the same throughout their lives, regardless of the number of years passed.

Ratio Scaling

If the difference in age is D and the ratio is a:b, then 1 unit of the ratio = D / |a-b|.

Present Age Calculation

Always subtract the years added for future ratios (or add for past ratios) from the calculated value to find the present age.

Shortcut Trick

Assume a base average score of 55 for all 43 students.

Surplus from 10 students = 10 × (99 − 55) = 440.

Distribute this surplus across the entire class: 440 ÷ 43 ≈ 10.23.

Required Average = 55 + 10.23 = 65.23 ≈ 65.2.

∴ The correct answer is 65.2.

Alternate Method

Given:

Total students in the class = 43

Number of students with 99 average (n₁) = 10

Number of other students (n₂) = 43 − 10 = 33

Average of other students (a₂) = 55

Formula Used:

Weighted Average = (n₁ × a₁ + n₂ × a₂) ÷ (n₁ + n₂)

Calculations:

⇒ Total score of 10 students = 10 × 99 = 990

⇒ Total score of 33 students = 33 × 55 = 1815

⇒ Total score of the whole class = 990 + 1815 = 2805

⇒ Total students = 10 + 33 = 43

⇒ Average score of the class = 2805 ÷ 43

⇒ Average score ≈ 65.232

⇒ Rounded to one decimal place = 65.2

∴ The correct answer is 65.2.

Additional Information

Weighted Average

It is the average of a data set in which each data point does not contribute equally to the final average. Formula: A_w = Σ(w_i × x_i) ÷ Σw_i.

Arithmetic Mean

The simple average of a set of numbers, calculated by dividing the sum of all elements by the total number of elements.

Deviation Method

A calculation shortcut where an assumed mean is used to find the net deviation, which is then divided by the total count and added back to the assumed mean.

Given: 

1. Since all persons face the centre, each person's right = counterclockwise direction. Fix F at top (pos 1). Two people between F and G from right of F (counterclockwise).

   CodeCopy

                   F ? ? ? G ? ?
   

2. Two people between G and M from right of G (counterclockwise).

   CodeCopy

          F M ? ? G ? ?
   

3. Three people between F and N from right of F (counterclockwise).

   CodeCopy

          F M ? N G ? ?
   

 

And O sits at the remaining place.

So, I sits third to the left of O.

Hence, the correct answer is "Option 1".

The logic followed here is:

Logic: (1st number × 3rd number) = 2nd number.

Now,

For, (13, 39, 3)

⇒ (13 × 3) = 39.

For, (15, 75, 5)

⇒ (15 × 5) = 75.

Similarly, for Option 1: (12, 36, 3)

⇒ (12 × 3) = 36.

Hence, the correct answer is "Option 1".

Given series:

(Left) 7 1 8 3 3 1 5 4 8 5 1 4 4 9 5 9 6 8 7 3 5 3 6 6 9 (Right)

Condition to be checked:

Perfect Square → Digit → Perfect Square

(Left) 7 1 8 3 3 1 5 4 8 5 1 4 4 4 4 9 9 5 9 6 8 7 3 5 3 6 6 9 (Right)

Thus, there are four such digits (the 5 in 1 5 4, the 4 in 1 4 4, the 4 in 4 4 9, and the 5 in 9 5 9) that are following the above conditions.

Hence, the correct answer is "Option 2".

Mistake Points

Immediately preceded by means immediately before or to come just before.

Immediately followed by means immediately after or to come just after.

As per the question, 1 is a perfect square. The single-digit perfect squares are 1, 4, and 9.

The logic followed here is:

Position of John from the bottom = 21st

Position of John from the top = 33rd

So,

Total number of students = (Position of John from the top + Position of John from the bottom) - 1

= (33 + 21) - 1

= 54 - 1 = 53.

Hence, the correct answer is "Option 4".

The logic followed here is:

For Option 1) UX-SW:

For Option 2) RU-PT:

For Option 3) OR-MQ:

For Option 4) LO-JM:

In Options 1, 2, and 3, the vertical pattern is: the first letter shifts by -2 and the second letter shifts by -1. In

Option 4, both the first and second letters shift by -2.

Thus, the letter-cluster that DOES NOT belong to that group is LO-JM.

Hence, the correct answer is "Option 4".

The logic followed here is:

Thus, the missing letter-number cluster in the series is UQV72.

Hence, the correct answer is "Option 3".

The given interchange rules are: ‘+’ ↔ ‘−’ and ‘×’ ↔ ‘÷’.

The original equation: 48 + 36 ÷ 36 × 24 − 50 = ?

After applying the interchange rules, the new equation is:

48 − 36 × 36 ÷ 24 + 50 = ?

Now, we solve the equation following the BODMAS rule:

48 − 36 × 36 ÷ 24 + 50

⇒ 48 − 36 × 1.5 + 50

⇒ 48 − 54 + 50

⇒ 98 − 54 = 44

Hence, the correct answer is "Option 2".

The least possible Venn diagram for the given statement is:

Conclusion:

I. Some balls are vans - False (Because, No definite relation given between balls and vans, so it can be possible only).

II. All hens are vans - False (Because, All vans are hens is given, but the reverse is not necessarily true).

So, neither conclusion (I) nor (II) follows.

Hence, the correct answer is "Option 4".

Given series:

(Left) 1 # 2 3 Ω 9 % 4 7 8 & * £ 5 + @ 6 $ (Right)

Condition to be checked:

Symbol → Number → Number

Let's analyze the series for the pattern (Symbol - Number - Number):

1 # 2 3 Ω 9 % 4 7 8 & * £ 5 + @ 6 $

Thus, there are two such patterns (# 2 3 and % 4 7) where a number is preceded by a symbol and followed by a number.

Hence, the correct answer is "Option 4".

The logic followed here is:

Thus, the missing term in the series is 74.

Hence, the correct answer is "Option 3".

The logic followed here is:

By comparing the two statements, we can see that the word 'play' is the only common word in both sentences.

Similarly, in the coded side, the code 'so' is the only common term in both sequences.

Clearly, 'so' is the code for the word 'play'.

Hence, the correct answer is "Option 3".

Given:

1. One person between B and T from right of B (counterclockwise): B(1) → pos(7) → T(6).

   CodeCopy

          B ? ? ? ? T ?
   

2. One person between T and R from left of R.

   CodeCopy

          B ? ? R ? T ?
   

3. One person between B and U from right of U.

   CodeCopy

          B ? U R ? T ?
   

4. Two people between C and R from left of R.

   CodeCopy

          B ? U R ? T C
   

5. One person between A and U from right of A. And S sits at the remaining place.

   CodeCopy

          B S U R A T C
   

Therefore, T sits third to the right of S.

Hence, the correct answer is "Option 4".

The logic followed here is:

For Option 1) RO−JJ:

For Option 2) YV−RO:

For Option 3) HE−AX:

For Option 4) EB−XU:

Thus, the letter-cluster that DOES NOT belong to that group is RO−JJ.

 

Hence, the correct answer is "Option 1".

Given:

Himesh starts from Point A and drives 4 km towards the south.

He then takes a left turn, drives 7 km, turns right and drives 6 km.

He then takes a right turn and drives 7 km.

He takes a final right turn, drives 3 km and stops at Point P.

So, he should drive 7 km towards north to reach Point A again.

Hence, the correct answer is "Option 3".

The given interchange rules are: ‘+’ ↔ ‘−’ and ‘×’ ↔ ‘÷’.

The original equation: 21 ÷ 5 − 25 + 180 × 3 = ?

After applying the interchange rules, the new equation is:

21 × 5 + 25 − 180 ÷ 3 = ?

Now, we solve the equation following the BODMAS rule:

21 × 5 + 25 − 180 ÷ 3

⇒ 21 × 5 + 25 − 60

⇒ 105 + 25 − 60

⇒ 130 − 60 = 70

Hence, the correct answer is "Option 1".

The logic followed here is:

Thus, the missing term in the series is 63.

Hence, the correct answer is "Option 4".

Given:

Mr. Z starts from Point A and drives 14 km towards the North.

He then takes a left turn, drives 19 km, then turns left and drives 30 km.

He then takes a left turn and drives 21 km.

He takes a final left turn, drives 16 km, and stops at Point P.

So, he should drive 2 km to the west in order to reach Point A again.

Hence, the correct answer is "Option 2".