RRB Group D Official Paper (Held On: 27 Nov, 2025 Shift 2) (Previous Year Paper)

The correct answer is Water conduction through xylem was blocked.

Key Points

• Water conduction in plants occurs through the xylem tissue, which is responsible for transporting water and minerals from the roots to other parts of the plant, including leaves.
• When a plant is uprooted, its roots are damaged or severed. This disrupts the xylem vessels, blocking the flow of water and essential nutrients to the plant's aerial parts such as stems and leaves.
• Wilting occurs when the plant loses more water through transpiration than it can absorb from the soil. In the absence of water conduction, the leaves and stems lose turgidity, making the plant appear weak and droopy.
• Exposure to sunlight exacerbates the water loss through transpiration, accelerating the wilting process. Since the plant can no longer replenish lost water, the symptoms become more pronounced.
• This highlights the critical role of the xylem in maintaining the plant's water balance and structural integrity.
• Hence, the statement "Water conduction through xylem was blocked" explains the observed wilting of the plant.

Additional Information

• Xylem Tissue:
• The xylem is a vascular tissue in plants composed of tracheids, vessels, xylem parenchyma, and xylem fibers.
• It transports water and dissolved minerals from the roots upward to the rest of the plant.
• Xylem vessels and tracheids provide structural support while facilitating water conduction.
• Wilting:
• Wilting is the loss of rigidity in non-woody parts of a plant due to excessive water loss or insufficient water uptake.
• It typically occurs due to drought stress, root damage, or issues such as blocked xylem vessels.
• Prolonged wilting can lead to plant death if the water deficit is not addressed.
• Transpiration:
• Transpiration is the process by which water evaporates from the aerial parts of the plant, mainly through stomata in the leaves.
• It plays a critical role in pulling water upward through the xylem via the cohesion-tension mechanism.
• Increased transpiration under sunlight speeds up water loss, making the plant more vulnerable to wilting if water absorption is impaired.
• Phloem vs. Xylem:
• While xylem conducts water and minerals, phloem is responsible for the translocation of food (sugars) produced during photosynthesis.
• In this scenario, the role of the phloem is unrelated to the wilting observed in the plant.

Important Points

• Mechanism of Water Transport:
• The movement of water through the xylem occurs via the cohesion-tension theory, which involves the cohesive force between water molecules and the tension created by transpiration.
• Root pressure also contributes to water movement but plays a minor role compared to transpiration pull.
• Environmental Factors Affecting Transpiration:
• Light: Increases stomatal opening, leading to higher rates of transpiration.
• Temperature: Higher temperatures increase evaporation and transpiration rates.
• Humidity: Lower humidity levels accelerate water loss through transpiration.
• Wind: Windy conditions remove the moisture layer near the leaf surface, increasing transpiration.
• Plant Adaptations:
• Plants in arid environments often have adaptations like sunken stomata, thick cuticles, or reduced leaf surface area to minimize water loss.
• Some plants store water in specialized tissues (succulents) to survive periods of drought.

The correct answer is Decrease in inter-particle space.

Key Points

• When pressure is applied to a gas, the particles are forced closer together, reducing the inter-particle space. This leads to a significant decrease in the volume of the gas as per Boyle’s Law, which states that the pressure of a gas is inversely proportional to its volume, provided temperature remains constant. Hence, this statement is correct.
• The inter-particle space decreases because gases are compressible due to the large gaps between their particles under normal conditions. Applying pressure reduces these gaps. Hence, the correct answer is related to the decrease in inter-particle space.
• The statement "Increase in inter-particle space" is incorrect because pressure compacts the particles rather than expanding the space between them.
• "Evaporation of the gas" is not a result of increasing pressure but rather a result of increasing temperature, as evaporation occurs when particles gain sufficient energy to overcome intermolecular forces. Hence, this statement is incorrect.
• Increase in kinetic energy occurs due to an increase in temperature, not pressure. When pressure increases at constant temperature, kinetic energy remains unchanged. Hence, this statement is incorrect.

Additional Information

• Boyle's Law:
• Boyle's Law, formulated by Robert Boyle in 1662, states that the pressure of a gas is inversely proportional to its volume at constant temperature.
• This relationship can be expressed mathematically as P × V = constant (where P is pressure and V is volume).
• For example, doubling the pressure on a gas will halve its volume, provided temperature remains constant.
• Compressibility of Gases:
• Gases are highly compressible due to the large spaces between their particles compared to solids and liquids.
• When pressure is applied, these particles come closer together, reducing the volume of the gas.
• Inter-particle Space:
• The inter-particle space in gases is much larger than in solids and liquids, which is why gases can be compressed more easily.
• Applying pressure decreases this space, making the particles more densely packed.
• Practical Applications:
• Compressed natural gas (CNG): CNG is stored under high pressure to reduce its volume for easier transportation and storage.
• Gas cylinders: Liquefied petroleum gas (LPG) cylinders use high pressure to compress the gas into a liquid state, maximizing storage efficiency.

Important Points

• Boyle’s Law in Real Life:
• The functioning of syringes, where the volume of air inside decreases as pressure is applied to the plunger, demonstrates Boyle’s Law.
• Submarines use the compressibility of gases for buoyancy control by adjusting air pressure in ballast tanks.
• Isothermal Process:
• An isothermal process occurs when a gas undergoes compression or expansion at constant temperature, keeping its internal energy unchanged.
• Boyle's Law is applicable in such scenarios, as temperature remains constant.

The correct answer is Burning of wood.

Key Points

• Burning of wood is the best example of the conversion of chemical energy into heat energy. Wood contains stored chemical energy in the form of cellulose and lignin, which, when burned, release heat energy.
• During combustion, the chemical bonds in wood are broken, and new bonds are formed, releasing a significant amount of thermal energy in the process.
• This process is a common natural phenomenon and is utilized in activities like cooking, heating, and industrial processes.
• Unlike other examples, such as solar cookers or electric motors, the burning of wood does not involve any intermediate conversion (e.g., electricity or sunlight) but directly transforms chemical energy into heat energy.
• Hence, burning of wood is a classic example of a direct and efficient energy conversion process.

Additional Information

• Batteries lighting a bulb:
• This involves the conversion of chemical energy into electrical energy, which is then converted into light and heat energy in the bulb.
• While heat energy is a by-product, the primary conversion here is from chemical to electrical energy.
• A motor running on electricity:
• Here, electrical energy is converted into mechanical energy.
• This is not an example of chemical to heat energy conversion, as the motor's primary function is to generate motion.
• Solar cooker heating food:
• The solar cooker uses solar energy (light energy) and converts it directly into heat energy for cooking.
• It does not involve any chemical energy, making it unrelated to the question's context.
• Combustion Process in Detail:
• During combustion, wood reacts with oxygen in the air to produce carbon dioxide, water vapor, and heat.
• This exothermic reaction releases energy stored within the chemical bonds of the fuel (wood).
• The heat produced is utilized for various purposes like cooking, heating, and energy production.

Important Points

• Combustion Reaction:
• A combustion reaction is a type of chemical reaction where a substance combines with oxygen to release energy in the form of heat and light.
• Examples include burning wood, coal, or fossil fuels.
• Applications of Combustion:
• Used for generating heat in households (e.g., wood stoves).
• Provides energy for industrial processes like smelting and manufacturing.
• Utilized in power generation through steam turbines.
• Environmental Impact:
• Burning wood releases carbon dioxide (CO2), a greenhouse gas contributing to global warming.
• Incomplete combustion can result in the release of harmful pollutants like carbon monoxide (CO).

The correct answer is It combines the DNA of two parents in the offspring..

Key Points

• Sexual reproduction involves the combination of DNA from two parents: This process is central to sexual reproduction, where the genetic material from both parents (male and female gametes) combines during fertilization to produce genetically unique offspring. Hence, this statement is correct.
• It does not produce a large number of offspring: Unlike asexual reproduction, sexual reproduction typically results in fewer offspring, as the process involves mating and a more complex developmental process. Hence, this statement is incorrect.
• It does not pass DNA from only one parent: Sexual reproduction ensures that offspring inherit genetic material from both parents, creating genetic diversity. This is a fundamental feature of sexual reproduction. Hence, this statement is incorrect.
• Offspring are not exactly similar to parents: Due to the genetic recombination and variation occurring during meiosis and fertilization, the offspring are genetically unique and not identical to either parent. Hence, this statement is incorrect.

Additional Information

• Advantages of sexual reproduction:
• Genetic diversity: Sexual reproduction results in genetic variation within a population, which is beneficial for adaptation and survival in changing environments.
• Evolution: It drives evolution by introducing new combinations of alleles, leading to the emergence of new traits that may provide a survival advantage.
• Key processes in sexual reproduction:
• Meiosis: A specialized type of cell division that reduces the chromosome number by half, producing haploid gametes (sperm and egg cells).
• Fertilization: The fusion of male and female gametes to form a diploid zygote, which develops into the offspring.
• Examples of organisms reproducing sexually:
• Humans and animals: Most mammals, birds, reptiles, and amphibians reproduce sexually.
• Plants: Flowering plants undergo sexual reproduction through the process of pollination and fertilization.
• Comparison with asexual reproduction:
• Asexual reproduction: Involves a single parent, produces genetically identical offspring, and is faster but lacks genetic diversity.
• Sexual reproduction: Requires two parents, produces genetically diverse offspring, and is slower but promotes adaptability.

Important Points

• Fertilization types:
• External fertilization: Occurs outside the body, as seen in amphibians and fish.
• Internal fertilization: Occurs inside the body, as seen in mammals, birds, and reptiles.
• Chromosomal contribution: Each parent contributes half of the chromosomes to the offspring. For instance, in humans, both the egg and sperm contribute 23 chromosomes each, resulting in a total of 46 chromosomes in the zygote.
• Role of genetic recombination: During meiosis, crossing over and independent assortment create new genetic combinations, contributing to variation in the offspring.
• Significance of genetic variation: Genetic diversity is crucial for a species' ability to adapt to environmental changes, resist diseases, and avoid the risks associated with inbreeding.

The correct answer is Rainwater harvesting and watershed management.

Key Points

• Rainwater Harvesting:
• Rainwater harvesting involves collecting and storing rainwater for agricultural, domestic, or industrial use. It helps in improving water availability for crops, especially in regions with irregular rainfall patterns.
• This method reduces dependency on groundwater and promotes sustainable water use in agriculture.
• Rainwater harvesting structures like check dams, percolation tanks, and rooftop harvesting systems are commonly used.
• Watershed Management:
• Watershed management is an integrated approach to managing water resources within a defined geographical area.
• It includes soil conservation, afforestation, and water conservation practices to enhance the availability of water for irrigation and other purposes.
• Watershed projects aim to improve the productivity of land and ensure sustainable water use for agricultural needs.
• Both Rainwater Harvesting and Watershed Management are eco-friendly and cost-effective techniques for enhancing water availability for crops.
• These methods align with sustainable agricultural practices and environmental conservation goals.
• Hence, the correct answer is Rainwater harvesting and watershed management.

Additional Information

• Drip Irrigation:
• Drip irrigation is a micro-irrigation system that delivers water directly to the roots of plants through a network of pipes, emitters, and valves.
• It is highly efficient in water usage and helps in reducing wastage, but it is not directly related to rainwater harvesting or watershed management.
• Drip irrigation is ideal for areas with water scarcity, but it involves higher installation costs.
• Crop Rotation:
• Crop rotation refers to the practice of growing different crops in a sequence on the same land to improve soil fertility and reduce pest issues.
• While beneficial for soil health, it does not directly address water availability for crops.
• Sprinkler Irrigation:
• Sprinkler irrigation mimics natural rainfall by spraying water over crops. It is another method to irrigate crops efficiently.
• Although useful, it is not part of rainwater harvesting or watershed management.
• Manuring and Chemical Spraying:
• Manuring involves adding organic or chemical fertilizers to the soil to enhance its fertility.
• Chemical spraying is used for pest control but does not contribute to improving water availability.
• Benefits of Rainwater Harvesting and Watershed Management:
• They promote sustainable agricultural practices by conserving water resources.
• These methods help in reducing soil erosion and improving groundwater recharge.
• Rainwater harvesting and watershed management are particularly beneficial in drought-prone and semi-arid regions.
• They contribute to long-term water security and environmental sustainability.

Important Points

• Rainwater Harvesting Techniques:
• Rooftop Rainwater Harvesting: Collecting rainwater from rooftops and storing it in tanks.
• Check Dams: Structures built across streams to hold rainwater.
• Percolation Tanks: Tanks designed to allow water to percolate into the ground.
• Contour Trenches: Dug along contours to trap rainwater and reduce soil erosion.
• Watershed Management Components:
• Soil Conservation: Preventing soil erosion through vegetation and contour farming.
• Water Conservation: Creating reservoirs, ponds, and check dams to store water.
• Afforestation: Planting trees to maintain the ecological balance of watersheds.
• Community Participation: Involvement of local communities in planning and executing watershed projects.
• Government Initiatives:
• Programs like the Mahatma Gandhi National Rural Employment Guarantee Act (MGNREGA) often include watershed development components.
• Integrated Watershed Management Programme (IWMP) is a government initiative focused on sustainable watershed management.
• Global Perspective:
• Countries like Israel and Australia have implemented advanced water management techniques for agriculture.
• Rainwater harvesting is widely promoted in South Asia and Africa to address water scarcity.

The correct answer is Tendrils.

Key Points

• Tendrils are specialized structures found in climbing plants, including the pea plant (Pisum sativum), that help them attach to and climb up other plants or objects.
• They are modified organs, such as leaves, leaflets, or stems, which coil around a support structure to provide stability and support to the plant.
• In pea plants, the tendrils are modified leaflets of the compound leaves, allowing the plant to climb efficiently.
• When tendrils come into contact with a support, they exhibit a growth response called thigmotropism, coiling around the object for attachment.
• This adaptation is crucial for pea plants, as it allows them to reach sunlight for photosynthesis while conserving energy that would otherwise be spent on producing a thick stem for support.

Additional Information

• Thigmotropism:
• Thigmotropism is a directional growth response of a plant organ to touch or physical contact with a solid object.
• In tendrils, thigmotropism helps them detect support structures and coil around them for stability.
• Other Structures in Plants:
• Shoot: The shoot is the above-ground part of the plant, which includes the stem, leaves, and flowers. It provides structural support and is responsible for photosynthesis and reproduction. However, it is not specifically adapted for climbing.
• Runner: Runners are horizontal stems that grow along the soil surface, giving rise to new plants. They are used for vegetative propagation, not climbing.
• Sucker: Suckers are shoots that arise from the base of a plant or its underground parts. They help in vegetative propagation but are not involved in climbing.
• Climbing Mechanisms in Plants:
• Plants use various mechanisms to climb, such as twining stems, adhesive roots, hooks, and tendrils, depending on their species and habitat.
• Tendrils are common in climbers like pea plants, grapevines, and cucumbers, while twining stems are seen in plants like morning glory.
• Examples of Plants with Tendrils:
• Pea Plant (Pisum sativum): Has leaf tendrils to climb supports.
• Grapevine (Vitis vinifera): Uses tendrils to attach to trellises or other structures.
• Cucumber (Cucumis sativus): Utilizes tendrils for climbing and support.

The correct answer is Density – Kilogram per cubic metre.

Key Points

• Density – Kilogram per cubic metre:
• Density is defined as the mass per unit volume of a substance and is expressed in the SI unit of kilograms per cubic metre (kg/m³).
• The formula for density is Density = Mass / Volume, where mass is measured in kilograms (kg) and volume in cubic metres (m³).
• Correct statement: The unit of density in the SI system is indeed kilogram per cubic metre.
• Temperature – Calorie:
• This is an incorrect statement. The SI unit of temperature is the Kelvin (K), not the calorie.
• Calorie is a unit of energy and is used to measure heat, not temperature.
• In scientific contexts, the calorie is often replaced by the joule (J), the SI unit of energy.
• Volume – Cubic centimetre:
• This is an incorrect statement. The SI unit of volume is the cubic metre (m³).
• While cubic centimetre (cm³) is commonly used in smaller volume measurements, it is not the standard SI unit.
• 1 cubic metre is equal to 1,000,000 cubic centimetres (cm³).
• Pressure – Newton:
• This is also an incorrect statement. The SI unit of pressure is the Pascal (Pa), not the Newton.
• Pressure is defined as force per unit area, and 1 Pascal is equal to 1 Newton per square metre (N/m²).
• While Newton is a unit of force, it is not used to measure pressure directly.

Additional Information

• SI Unit System:
• The International System of Units (SI) is the modern form of the metric system and is the most widely used system of measurement worldwide.
• The SI system is built on seven base units: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, Kelvin (K) for temperature, mole (mol) for the amount of substance, and candela (cd) for luminous intensity.
• Density:
• Density is a fundamental property of matter, essential in fields like physics, chemistry, and engineering.
• Common examples include water, which has a density of approximately 1000 kg/m³ at standard temperature and pressure (STP).
• The concept of density is applied in numerous industries, such as material science, construction, and manufacturing.
• Temperature:
• Kelvin (K) is the SI unit of temperature. It is an absolute temperature scale that starts at absolute zero (0 K), the theoretical point where molecular motion ceases.
• Other units of temperature include Celsius (°C) and Fahrenheit (°F), but Kelvin is used in scientific measurements.
• Volume:
• While the SI unit of volume is the cubic metre (m³), other units like litres (L) and millilitres (mL) are also commonly used in daily life.
• 1 cubic metre equals 1000 litres.
• Pressure:
• The SI unit of pressure, the Pascal (Pa), is named after the French mathematician and physicist Blaise Pascal.
• Pressure is an important parameter in physics, meteorology, and engineering, playing a key role in understanding and designing systems like hydraulics and barometers.

The correct answer is Glucose.

Key Points

• Concentrated sulphuric acid acts as a strong dehydrating agent due to its high affinity for water molecules. It removes water from compounds by absorbing moisture or chemically reacting to eliminate water.
• Glucose, being a carbohydrate (C₆H₁₂O₆), contains hydrogen and oxygen atoms in a ratio of 2:1, similar to water (H₂O). When treated with concentrated sulphuric acid, glucose undergoes dehydration, forming a black carbon residue.
• This reaction is a chemical demonstration of sulphuric acid's ability to remove water. The dehydration process breaks down glucose into elemental carbon and water, which gets absorbed by the acid.
• Propane, a hydrocarbon (C₃H₈), does not contain water molecules within its structure. Hence, concentrated sulphuric acid does not act as a dehydrating agent for propane.
• Sodium hydroxide (NaOH) and ammonia gas (NH₃) do not contain water molecules in their composition in a form that sulphuric acid can dehydrate. Thus, sulphuric acid does not act as a dehydrating agent for these substances.

Additional Information

• Dehydrating Agent:
• A dehydrating agent is a substance that removes water or moisture from another compound or substance. Concentrated sulphuric acid is a classic example, widely used in laboratory reactions to dehydrate organic and inorganic compounds.
• It is often employed in chemical processes like the preparation of esters, dehydration of alcohols, and in demonstrating carbonization reactions with carbohydrates.
• Properties of Concentrated Sulphuric Acid:
• It is a highly viscous and dense liquid with strong hygroscopic properties, meaning it readily absorbs water from the atmosphere or other substances.
• The chemical formula for sulphuric acid is H₂SO₄, and it is a diprotic acid with two hydrogen ions available for reaction.
• Its ability to absorb water makes it a powerful reagent in drying gases and dehydrating compounds.
• Reaction with Glucose:
• When glucose (C₆H₁₂O₆) is treated with concentrated H₂SO₄, it gets dehydrated, leaving behind black carbon residue and water.
• This reaction can be written as: C₆H₁₂O₆ → 6C + 6H₂O.
• The water produced in the reaction is absorbed by sulphuric acid, demonstrating its strong dehydrating nature.
• Applications of Sulphuric Acid:
• Used in the manufacturing of fertilizers, particularly phosphates like superphosphate.
• Employed in petroleum refining, chemical synthesis, and production of detergents and paints.
• Acts as a drying agent in laboratories for gases like hydrogen chloride (HCl) and carbon dioxide (CO₂).

Important Points

• Comparison of Options:
• Glucose: Contains water molecules chemically bonded, making it prone to dehydration by sulphuric acid.
• Propane: A hydrocarbon lacking water in its structure; sulphuric acid does not act as a dehydrating agent for it.
• Sodium Hydroxide: Inorganic base with no water molecules available for dehydration.
• Ammonia Gas: Composed of nitrogen and hydrogen, without water molecules for dehydration.
• Safety Considerations:
• Concentrated sulphuric acid is highly corrosive and can cause severe burns upon contact with skin.
• Always handle it with proper protective gear, including gloves and goggles, in a well-ventilated area.
• In case of spills, neutralize it with a base like sodium bicarbonate before cleaning.

The correct answer is Four times the original.

Key Points

• Resistance of a wire is determined by the formula: R = ρ × (L/A), where:
• R: Resistance of the wire
• ρ: Resistivity (material property, constant for a given material)
• L: Length of the wire
• A: Cross-sectional area of the wire
• When the length of the wire is doubled, the new length becomes 2L.
• When the cross-sectional area is halved, the new area becomes A/2.
• Substituting these changes into the formula for resistance:
• New Resistance (R') = ρ × ((2L) / (A/2))
• R' = ρ × (2L × 2 / A)
• R' = 4 × (ρ × L/A)

Hence, the new resistance becomes 4 times the original resistance.

• Thus, the resistance of the wire increases to four times its original value.

Additional Information

• Factors affecting resistance:
• Length of the wire (L): Resistance is directly proportional to the length. Doubling the length doubles the resistance.
• Cross-sectional area (A): Resistance is inversely proportional to the cross-sectional area. Halving the area doubles the resistance.
• Material of the wire: Different materials have different resistivities (ρ), affecting their resistance.
• Temperature: Resistance changes with temperature, usually increasing for conductors as temperature rises.
• Ohm's Law:
• Ohm's law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points.
• Mathematically: V = IR, where R is the resistance.
• Applications of resistance:
• Resistance is utilized in devices like resistors, heaters, and filaments in light bulbs to control current flow.
• Higher resistance is used in devices to convert electrical energy into heat (e.g., electric irons).
• Resistivity:
• It is a material-specific property that quantifies how strongly a given material opposes the flow of electric current.
• Unit: Ohm meter (Ω·m).
• Materials with low resistivity, like copper and silver, are good conductors, while materials with high resistivity, like rubber, are insulators.
• Practical Example:
• When designing electrical wiring in buildings, larger cross-sectional areas are chosen to reduce resistance and prevent overheating.
• Long-distance power transmission lines use materials with low resistivity and large cross-sectional areas to minimize energy loss.

The correct answer is Each trait is controlled by two factors..

Key Points

• Mendel’s Principle of Inheritance: Gregor Mendel postulated that each trait in organisms is controlled by two factors, which are now known as alleles. These alleles could be either dominant or recessive.
• Mendel’s experiments demonstrated that one factor (allele) in a pair could mask the expression of the other, leading to the concept of dominant and recessive traits.
• Law of Segregation: Mendel proposed that during the formation of gametes, the two factors (alleles) for a trait segregate and only one allele is passed on to the offspring. This ensures genetic variation.
• Law of Independent Assortment: Traits are inherited independently of one another if they are controlled by different genes. This observation was derived from Mendel’s dihybrid cross experiments.
• Each parent contributes one factor for each trait, which combines during fertilization to form a pair. These pairs of factors are responsible for the expression of specific traits in offspring.

Additional Information

• Mendel’s Experiments:
• Mendel conducted experiments on pea plants (Pisum sativum) to study inheritance patterns.
• He chose pea plants due to their distinct traits, ease of cultivation, and ability to self-pollinate.
• Seven contrasting traits, such as flower color, seed shape, and pod color, were studied to understand inheritance.
• Factors and Genes:
• Mendel referred to the hereditary units controlling traits as “factors.” These were later identified as genes with the discovery of DNA.
• Each gene exists in two forms called alleles, which can be dominant (expressed) or recessive (masked).
• Dominant and Recessive Traits:
• If both alleles are the same (homozygous), the trait is expressed according to the type of allele (dominant or recessive).
• If the alleles are different (heterozygous), the dominant allele determines the trait.
• Examples of Mendelian Traits:
• Dominant Traits: Purple flower color, round seeds.
• Recessive Traits: White flower color, wrinkled seeds.
• Significance of Mendel’s Work:
• Mendel’s principles formed the foundation of genetics, a field that studies inheritance and variation.
• His work laid the groundwork for understanding genetic disorders, selective breeding, and biotechnology.

Important Points

• Statement 1: Each trait is controlled by one factor. This is incorrect because Mendel showed that traits are controlled by two factors (alleles).
• Statement 2: Each trait is controlled by three genes. This is incorrect as Mendel identified only two factors controlling each trait.
• Statement 3: Each trait is controlled by two factors. This is correct and aligns with Mendel’s findings.
• Statement 4: Each trait is controlled by one chromosome. This is incorrect because chromosomes contain multiple genes, and traits are determined by specific genes.

The correct answer is collagen and elastin fibres.

Key Points

• Fat cells, also known as adipocytes, are primarily found in connective tissues, which serve as the structural framework for tissues and organs in animals.
• In connective tissue, fat cells are organized into lobules, or clusters, which are separated by partitions made of collagen and elastin fibres.
• Collagen fibres provide strength and structural support to the tissue, while elastin fibres contribute elasticity, allowing the tissue to stretch and return to its original shape.
• The combination of collagen and elastin ensures the connective tissue is both strong and flexible, enabling it to withstand mechanical forces and maintain the integrity of the fat lobules.
• This specific arrangement of connective tissue ensures proper storage and insulation of fat, as well as cushioning against physical trauma. Hence, the correct answer is collagen and elastin fibres.

Additional Information

• Collagen:
• Collagen is the most abundant protein in the animal body, particularly in connective tissues such as skin, tendons, cartilage, and bones.
• It provides mechanical strength and structural integrity to tissues, preventing them from tearing under tension.
• Collagen fibres are made up of triple-helical protein structures, which allow them to be both strong and flexible.
• Elastin:
• Elastin is a protein that provides elasticity and resilience to connective tissues like skin, lungs, and blood vessels.
• It allows tissues to stretch and then return to their original shape, making it crucial in areas that experience frequent deformation.
• Elastin fibres are thinner than collagen fibres and are often found interwoven with them to provide a balance of strength and flexibility.
• Fat Cells (Adipocytes):
• Adipocytes are specialized cells that store energy in the form of fat and play an important role in insulation and cushioning.
• They are classified into two types: white adipose tissue (WAT), which stores energy, and brown adipose tissue (BAT), which generates heat.
• Adipocytes are embedded within connective tissue, which supports their structure and function.
• Other Options Explained:
• Skeletal fibres and tendons: Tendons are composed of dense connective tissue that connects muscles to bones. They do not form partitions in fat tissue.
• Yellow fibres and lymph: Yellow fibres refer to elastin, but lymph is a fluid, not a structural component, and does not form partitions in fat lobules.
• Reticular fibres and plasma: Reticular fibres are found in lymphoid organs like the spleen and lymph nodes, while plasma is a component of blood, not connective tissue.

Important Points

• Connective tissues are classified into loose connective tissue (e.g., adipose tissue), dense connective tissue (e.g., tendons), cartilage, bone, and blood.
• Loose connective tissue is characterized by a loose arrangement of fibres and serves as a packing material in the body, while dense connective tissue has tightly packed fibres.
• The extracellular matrix (ECM) of connective tissues contains proteins like collagen and elastin, as well as glycoproteins and proteoglycans, which provide structural and functional support.

The correct answer is To increase the time of impact so that the force on the hands is reduced.

Key Points

• When a cricket player lowers his hands while catching a fast-moving ball, he increases the time duration of the impact between the ball and his hands.
• According to the impulse-momentum theorem, impulse is equal to the change in momentum of an object, and impulse is also equal to the product of force and the time during which the force is applied: Impulse = Force × Time.
• By increasing the time of impact, the force exerted on the hands decreases because the product of force and time remains constant. This makes the catch less painful and reduces the risk of injury.
• The momentum of the ball must decrease to zero when caught, but by lowering the hands, the player ensures that the deceleration occurs gradually over a longer period, thereby reducing the intensity of the force.
• This principle is a practical application of Newton’s second law of motion, which states that force is the rate of change of momentum: F = Δp/Δt. Here, a larger time interval (Δt) results in a smaller force (F).

Additional Information

• Impulse-Momentum Theorem:
• The theorem states that the impulse experienced by an object is equal to the change in its momentum. Mathematically: Impulse = Δp = m(v - u), where m is the mass, v is the final velocity, and u is the initial velocity.
• In sports like cricket, this theorem is used to minimize the impact on the player’s body by increasing the time of contact.
• Newton’s Second Law of Motion:
• This law explains that the force acting on an object is equal to the rate of change of its momentum: F = Δp/Δt.
• By increasing the time (Δt), the force (F) required to stop the ball decreases significantly.
• Applications of Increasing Time of Impact:
• This concept is widely applied in other areas like airbags in cars, shock absorbers in vehicles, and cushioned surfaces in sports fields to reduce impact forces.
• For example, airbags in vehicles extend the time of impact during a collision, thereby reducing the force on passengers and minimizing injuries.
• Role of Momentum in Sports:
• Momentum is a measure of the motion of an object and is given by the formula p = mv, where p is momentum, m is mass, and v is velocity.
• In cricket, the fast-moving ball has significant momentum due to its mass and velocity, and players use techniques like lowering their hands to manage this momentum safely.

Important Points

• Newton’s Laws in Sports:
• Newton’s laws of motion are fundamental in understanding and analyzing movements in sports activities like catching, throwing, and hitting the ball.
• For example, in catching, the player applies the first law (inertia), the second law (force and momentum), and the third law (action-reaction).
• Practical Examples:
• Goalkeepers in football also use this technique by moving their hands backward when catching a fast-moving ball to reduce the impact force.
• Similarly, gymnasts land on cushioned mats to increase the time of impact and reduce the force exerted on their bodies.
• Everyday Applications:
• Packaging materials like bubble wrap and foam are designed to increase the time of impact when fragile items are dropped, minimizing damage.
• Helmets and protective gear in sports use the same principle to dissipate impact forces and reduce injuries.

The correct answer is Just outside the plasma membrane in a plant cell.

Key Points

• The cell wall is a rigid, protective layer located just outside the plasma membrane in plant cells, fungi, bacteria, and some protists. Hence, option 3 is correct.
• It provides structural support to the cell, maintaining its shape and preventing it from bursting in a hypotonic environment.
• The cell wall is absent in animal cells, which rely on the plasma membrane and cytoskeleton for structure and support.
• In plant cells, the cell wall is primarily made up of cellulose, a polysaccharide, while in fungi, it consists of chitin, and in bacteria, it is composed of peptidoglycan.
• Its functions include protection against mechanical stress, regulation of cell growth, and acting as a barrier to pathogens.

Additional Information

• Plasma Membrane:
• The plasma membrane, also known as the cell membrane, is a semipermeable lipid bilayer that encloses the cytoplasm of the cell and regulates the movement of substances in and out of the cell.
• It is present in both plant and animal cells and plays a crucial role in maintaining cellular homeostasis.
• Differences between Cell Wall and Plasma Membrane:
• The cell wall is rigid and provides structural support, while the plasma membrane is flexible and selectively permeable.
• The cell wall is found in plants, fungi, and bacteria, whereas the plasma membrane is present in all cell types.
• Animal Cells:
• Animal cells lack a cell wall, which allows for a wider variety of cell shapes and the ability to form specialized tissues.
• Instead, they have an extracellular matrix composed of proteins like collagen for structural support.
• Composition of Plant Cell Wall:
• The primary cell wall is made of cellulose, hemicellulose, and pectin.
• Some plant cells also have a secondary cell wall that contains lignin, which provides additional strength and rigidity.
• Examples of Organisms with Cell Walls:
• Plants: Cellulose-based cell walls (e.g., trees, flowers).
• Bacteria: Peptidoglycan-based cell walls (e.g., E. coli).
• Fungi: Chitin-based cell walls (e.g., mushrooms).

Important Points

• Statement 1: "Just outside the plasma membrane in an animal cell" is incorrect because animal cells do not have a cell wall.
• Statement 2: "Just inside the plasma membrane in a plant cell" is incorrect because the cell wall is located outside the plasma membrane.
• Statement 3: "Just outside the plasma membrane in a plant cell" is correct as this is the precise location of the cell wall in plant cells.
• Statement 4: "Just inside the plasma membrane in an animal cell" is incorrect because animal cells neither have a cell wall inside nor outside the plasma membrane.

The correct answer is According to the New Cartesian Sign Convention, the object is always placed to the left of the lens..

Key Points

• The New Cartesian Sign Convention is a widely used system to define the direction and measurement of distances in optics.
• In this convention, the object is always placed to the left of the lens or mirror, and the direction of the incident light is considered as moving from left to right. Hence, statement 2 is correct.
• The object distance (u) is typically measured from the optical center and is considered negative because it is measured opposite to the direction of the incident light. Statement 1 is incorrect.
• The light rays always travel from the object towards the optical device (lens or mirror). However, it falls on the lens from the left-hand side, not the right-hand side. Statement 3 is incorrect.
• According to the convention, the object is always assumed to be placed along the principal axis, not necessarily above it. Statement 4 is incorrect.

Additional Information

• New Cartesian Sign Convention:
• This convention simplifies calculations in ray optics by assigning specific signs to distances based on their orientation.
• Distances measured against the direction of incident light are considered negative, while those measured along the direction of light are considered positive.
• For example, focal length is positive for convex lenses and negative for concave lenses.
• Principal Axis:
• The principal axis is an imaginary straight line that passes through the optical center of the lens or mirror.
• All measurements, including object distance (u) and image distance (v), are taken along this axis.
• Object Placement:
• In the New Cartesian Sign Convention, the object is always placed to the left-hand side of the optical device.
• This placement ensures that the direction of light travel and calculations remain consistent.
• Lens Types:
• Convex lenses converge light rays and have a positive focal length.
• Concave lenses diverge light rays and have a negative focal length.
• Application of Sign Convention:
• The sign convention is crucial in solving numerical problems related to lens and mirror formulas, such as: 1/f = 1/v - 1/u.
• It helps determine the position and nature of the image formed (real, virtual, inverted, or upright).

Important Points

• Correct Interpretation: In optics, correct interpretation of distances and directions is essential to avoid calculation errors.
• Ray Diagrams: Ray diagrams are used to visually represent the path of light rays and their interaction with optical devices.
• Practical Applications: The New Cartesian Sign Convention is widely applied in designing optical devices such as cameras, telescopes, microscopes, and eyeglasses.
• Historical Context: The convention simplifies complex geometric optics problems and aligns with the principles of light propagation.

The correct answer is Regular hearing tests.

Key Points

• Ultrasound technology is widely used in various fields due to its ability to produce high-frequency sound waves that can penetrate through different materials and tissues.
• Applications of ultrasound include cleaning objects (such as delicate jewelry or medical instruments), detecting cracks in metal blocks, and medical imaging (such as sonography for examining internal organs).
• Medical imaging is one of the most common uses of ultrasound, particularly in obstetrics to visualize the fetus during pregnancy.
• Detecting cracks in metal blocks is another significant application, where ultrasound is used in industries for non-destructive testing to ensure structural integrity.
• Regular hearing tests are not an application of ultrasound. Hearing tests typically involve audiometric techniques to measure the auditory response at various frequencies and do not utilize ultrasound technology. Hence, option 3 is correct.

Additional Information

• Ultrasound Cleaning:
• Ultrasound is used to clean delicate objects such as jewelry, lenses, and surgical instruments by creating high-frequency sound waves that produce microscopic bubbles in a cleaning solution, effectively removing dirt and contaminants.
• This process is known as cavitation, where bubbles implode to dislodge particles from surfaces.
• Crack Detection in Metals:
• Ultrasound is utilized in industries for non-destructive testing of metal blocks to detect internal cracks or flaws that are not visible externally.
• The technique involves transmitting ultrasonic waves into the material and measuring the reflected waves to identify inconsistencies.
• This application is critical for ensuring the safety and reliability of structures like bridges, aircraft, and pipelines.
• Medical Imaging:
• Ultrasound imaging, also known as sonography, is commonly used for diagnosing medical conditions and monitoring pregnancies.
• It uses high-frequency sound waves to create real-time images of internal organs, blood flow, and tissues.
• One of its key advantages is that it is non-invasive and does not use ionizing radiation like X-rays.
• Examples include imaging of the abdomen, heart (echocardiography), and musculoskeletal systems.
• Hearing Tests:
• Hearing tests measure a person's ability to detect and interpret sound frequencies using techniques like pure-tone audiometry and speech audiometry.
• These tests involve transmitting audible tones at varying frequencies and intensities to evaluate hearing health.
• They do not involve ultrasound, as ultrasound frequencies are beyond the human hearing range (above 20,000 Hz).

Important Points

• Ultrasound Frequency:
• Ultrasound refers to sound waves with frequencies higher than the audible range for humans (above 20 kHz).
• It is commonly used in medical, industrial, and cleaning applications due to its ability to penetrate various materials.
• Uses in Medical Field:
• Ultrasound is extensively used for diagnostic imaging, therapeutic applications, and real-time monitoring.
• Examples include fetal monitoring during pregnancy, guided biopsies, and physiotherapy treatments.
• Limitations:
• Ultrasound has limited penetration depth and is not suitable for imaging structures behind bones or air-filled spaces like the lungs.

The correct answer is metals are converted to their oxide, sulphide or carbonate by reaction with environmental agents.

Key Points

• Corrosion is a natural process where metals react with environmental agents such as oxygen, moisture, and certain chemicals, leading to the formation of compounds like oxides, sulphides, or carbonates. Hence, the statement is correct.
• In this process, metals lose their original metallic properties and are chemically transformed into compounds, which are often less stable and less useful. For example, iron reacts with oxygen and moisture to form hydrated iron oxide (rust).
• Statement 1, which claims that metals gain weight due to reaction with acids, is incorrect as the weight gain during corrosion is due to the addition of oxygen or other elements, not acids specifically.
• Statement 3, which states that metals melt at low temperatures, is incorrect. Corrosion is a chemical reaction, not related to the melting of metals.
• Statement 4, which says that metals change their colour permanently, is partially true but not the defining feature of corrosion. The colour change is a result of the formation of new compounds during corrosion.

Additional Information

• Common Examples of Corrosion:
• Rusting of Iron: Iron reacts with oxygen and water to form hydrated iron oxide, commonly known as rust. This is the most familiar example of corrosion.
• Formation of Patina on Copper: Copper reacts with carbon dioxide and moisture to form copper carbonate, which gives it a green appearance.
• Corrosion of Silver: Silver reacts with sulphur compounds in the air to form silver sulphide, leading to tarnishing.
• Factors Affecting Corrosion:
• The presence of moisture and oxygen accelerates the corrosion process.
• The presence of salts (e.g., in coastal areas) increases the rate of corrosion by enhancing the conductivity of the medium.
• Acidic environments significantly increase the rate of corrosion due to the higher availability of hydrogen ions.
• Prevention of Corrosion:
• Use of coatings: Applying paint, oil, or grease to protect the metal surface from direct contact with environmental agents.
• Galvanization: Coating iron or steel with a layer of zinc to prevent rusting.
• Alloying: Mixing metals with other elements to improve their resistance to corrosion. For example, stainless steel is an alloy of iron with chromium, which prevents rusting.
• Cathodic Protection: Using a sacrificial anode (e.g., zinc or magnesium) to protect the main metal from corrosion.
• Electrochemical Nature of Corrosion: Corrosion is often an electrochemical process, involving the transfer of electrons. For example, in rusting, iron acts as the anode and loses electrons, while oxygen acts as the cathode.

Important Points

• Economic Impact: Corrosion leads to significant economic losses worldwide, as it damages infrastructure, machinery, and vehicles. Preventive maintenance is essential to mitigate these losses.
• Environmental Impact: Corrosion can lead to the release of toxic substances into the environment, especially when it occurs in industrial equipment or pipelines.
• Scientific Studies: Research into corrosion inhibitors and advanced materials (e.g., graphene coatings) is ongoing to improve the durability of metals.
• Historical Context: The study of corrosion gained importance during the Industrial Revolution when large-scale use of metals in infrastructure and machinery began to highlight the problem.

The correct answer is Isobars.

Key Points

• Isobars are atoms of different elements having the same mass number (A) but different atomic numbers (Z).
• In the given pair:
• ⁴⁰Ar: Atomic number = 18, Mass number = 40
• ⁴⁰Ca: Atomic number = 20, Mass number = 40
• Since both have the same mass number (40) but different atomic numbers (18 and 20), they are isobars.
• Hence, the correct answer is Option 1 – Isobars.

Additional Information

• Isotopes:
• Same atomic number but different mass numbers.
• Example: ³⁵Cl and ³⁷Cl.
• Isotones:
• Same number of neutrons but different atomic numbers.
• For the given pair:
• ⁴⁰Ar: Neutrons = 40 − 18 = 22
• ⁴⁰Ca: Neutrons = 40 − 20 = 20
• Since the number of neutrons is different, they are not isotones.
• Isoelectronic Species:
• Species having the same number of electrons.
• Neutral ⁴⁰Ar has 18 electrons, whereas neutral ⁴⁰Ca has 20 electrons.
• Therefore, they are not isoelectronic.

Important Points

• Isobars: Same mass number, different atomic numbers.
• Isotopes: Same atomic number, different mass numbers.
• Isotones: Same number of neutrons, different atomic numbers.
• Isoelectronic: Same number of electrons.

The correct answer is Sodium chloride is formed..

Key Points

• Sodium carbonate (Na₂CO₃) and sodium bicarbonate (NaHCO₃) react with hydrochloric acid (HCl) to produce sodium chloride (NaCl), which is common in both reactions.
• In the case of sodium carbonate, the reaction proceeds as follows: Na₂CO₃ + 2HCl → 2NaCl + CO₂ + H₂O Hence, sodium chloride is formed.
• For sodium bicarbonate, the reaction proceeds as follows: NaHCO₃ + HCl → NaCl + CO₂ + H₂O Hence, sodium chloride is formed.
• Both reactions also release carbon dioxide (CO₂) gas, which causes effervescence.
• Neither reaction produces hydrogen gas or carbon monoxide, making the production of sodium chloride the common product between both reactions.

Additional Information

• Sodium carbonate (Na₂CO₃):
• It is also known as washing soda or soda ash.
• It is widely used in the production of glass, as a cleaning agent, and in water softening processes.
• When it reacts with acids, it produces salts, water, and carbon dioxide gas.
• Sodium bicarbonate (NaHCO₃):
• Also known as baking soda, it is commonly used in baking as a leavening agent.
• It is also used in cleaning, neutralizing acids, and as an antacid for relieving heartburn.
• Reacts with acids to produce carbon dioxide gas, water, and a salt.
• Hydrochloric Acid (HCl):
• It is a strong acid commonly used in industrial processes, laboratory experiments, and digestion in the human stomach.
• Reacts with bases or salts like sodium carbonate and sodium bicarbonate to produce salts and other byproducts.
• Effervescence:
• Effervescence refers to the production of gas bubbles during a chemical reaction.
• In these reactions, effervescence is due to the release of carbon dioxide (CO₂) gas.

Important Points

• Carbon Dioxide Release: Both sodium carbonate and sodium bicarbonate release carbon dioxide gas when reacting with hydrochloric acid, leading to bubbling or fizzing during the reaction.
• Non-production of Hydrogen Gas: These reactions do not produce hydrogen gas, contrary to reactions involving metals like zinc or magnesium with acids.
• Use in Cleaning: Sodium bicarbonate and sodium carbonate are frequently used in cleaning applications due to their ability to neutralize acids and produce salts.

The correct answer is A convex lens is also called a converging lens because it brings parallel rays to a focus at a point..

Key Points

• A convex lens is also known as a converging lens because it bends incoming parallel rays of light towards a single focal point on the other side of the lens.
• This convergence occurs due to the curved shape of the lens surfaces, which refract (bend) light in such a way that the rays meet at a point.
• The focal point is where all the refracted rays converge, and it depends on the curvature and refractive index of the lens. This is a key property of convex lenses.
• Convex lenses are commonly used in devices like magnifying glasses, cameras, telescopes, and projectors, where the convergence of light is essential for forming clear images.
• Unlike concave lenses, which diverge light rays, convex lenses are specifically designed to focus light to a single point, making them ideal for applications requiring image magnification and correction of vision.

Additional Information

• Properties of convex lenses:
• A convex lens is thicker at the center and thinner at the edges.
• It has a positive focal length, meaning the focal point is real and located on the side opposite to the incoming light.
• It can form both real and inverted images (when the object is beyond the focal length) or virtual and upright images (when the object is within the focal length).
• Applications of convex lenses:
• Magnifying glasses: Used to enlarge the appearance of small objects.
• Corrective lenses: Used in eyeglasses for individuals with hyperopia (farsightedness).
• Optical instruments: Found in microscopes, telescopes, and cameras to focus and magnify images.
• Option-wise explanations:
• Option 1: "It brings parallel rays to a focus at a point" is correct, as this is the defining property of a convex lens.
• Option 2: "It always forms virtual images" is incorrect, as convex lenses form both real and virtual images depending on the object's position.
• Option 3: "It does not bend light rays at all" is incorrect, as convex lenses bend (refract) light rays significantly to converge them.
• Option 4: "It diverges rays of light away from each other" is incorrect, as this property belongs to concave lenses, not convex lenses.
• Important facts about refraction:
• The bending of light when it passes from one medium to another is governed by Snell's Law.
• The refractive index of the lens material plays a crucial role in determining the focal length and convergence of light.

Important Points

• Lens Formula:
• The relationship between the focal length (f), object distance (u), and image distance (v) is given by the lens formula: 1/f = 1/v - 1/u.
• Magnification:
• Magnification (M) is defined as the ratio of the image height (h') to the object height (h): M = h'/h = v/u.

The correct answer is Chemical or physical.

Key Points

• The processes used for removing the gangue (unwanted material) from the ore are based on the differences in physical and chemical properties between the ore and the gangue.
• Physical methods like gravity separation, magnetic separation, and froth flotation are commonly used when there are distinct differences in properties such as density, magnetic susceptibility, or surface wettability.
• Chemical methods are applied when the gangue and ore have similar physical properties but different chemical reactivity. For example, leaching is used to dissolve specific components of the ore selectively.
• The choice of method depends on the nature of the ore and the gangue, as well as the desired purity of the metal extracted.
• Examples: Froth flotation is used for sulfide ores, while magnetic separation is used for ores containing magnetic minerals like magnetite. Hence, the correct answer is "Chemical or physical."

Additional Information

• Physical Properties of Ore and Gangue:
• Density: Gravity separation exploits the difference in densities between ore particles and gangue. For example, heavier metal particles settle faster in a fluid medium.
• Magnetic Properties: Magnetic separation utilizes differences in magnetic susceptibility. Magnetic ores like magnetite are separated from non-magnetic gangue.
• Surface Properties: Froth flotation is used for ores with different surface wettability. Sulfide ores are hydrophobic and attach to air bubbles, while gangue remains in water.
• Chemical Properties of Ore and Gangue:
• Reactivity: Chemical leaching utilizes differences in reactivity. For example, cyanide leaching dissolves gold selectively from its ore.
• Solubility: Specific solvents dissolve certain ores but not the gangue. For instance, alumina is dissolved in sodium hydroxide during the Bayer process.
• Examples of Common Ore Purification Processes:
• Froth Flotation: Used for sulfide ores like zinc sulfide (ZnS).
• Gravity Separation: Used for heavy ores like cassiterite (SnO₂).
• Magnetic Separation: Used for ores like magnetite (Fe₃O₄).
• Leaching: Used for aluminum extraction via the Bayer process.

Important Points

• Gangue: Refers to the unwanted rock, sand, or impurities associated with the desired mineral or ore.
• Ore: A naturally occurring solid material from which a valuable metal or mineral can be profitably extracted.
• Concentration: The process of removing gangue from the ore to increase the percentage of the desired metal.
• Examples of Metallurgical Processes:
• Calcination: Heating ore in the absence of air to remove volatile impurities.
• Roasting: Heating ore in the presence of air to oxidize impurities.
• Smelting: Extracting metal by heating ore with a reducing agent like carbon.

The correct answer is both tetravalency and catenation.

Key Points

• Tetravalency: Carbon has four valence electrons, allowing it to form four covalent bonds with other atoms, including hydrogen, oxygen, nitrogen, and other carbon atoms. This property enables the formation of complex molecules and diverse organic compounds.
• Catenation: Carbon has a unique ability to form long chains and rings of atoms by bonding with other carbon atoms. This property is known as catenation and is a key reason for the vast diversity of organic compounds.
• Combination of tetravalency and catenation: The combination of tetravalency and catenation allows carbon to form a wide range of structures, including straight chains, branched chains, and cyclic compounds. This versatility is why carbon compounds are the basis of life.
• Organic compounds: The ability to form multiple types of bonds, including single, double, and triple bonds, further enhances the diversity of organic compounds. Examples include hydrocarbons, alcohols, ketones, acids, and esters.
• Hence, the correct answer is both tetravalency and catenation.

Additional Information

• Tetravalency:
• Carbon's atomic number is 6, and its electronic configuration is 1s² 2s² 2p². The four valence electrons allow carbon to form strong covalent bonds.
• Examples include methane (CH₄), ethane (C₂H₆), and carbon dioxide (CO₂).
• The ability to form covalent bonds makes carbon compounds stable and suitable for biological processes.
• Catenation:
• Catenation is the self-linking property of carbon atoms through covalent bonds. This property is most prominent in carbon compared to other elements.
• Carbon chains can be linear (e.g., hexane), branched (e.g., isobutane), or cyclic (e.g., benzene).
• This property leads to the existence of millions of organic compounds.
• Organic Chemistry:
• Carbon forms the backbone of organic chemistry. Organic compounds consist primarily of carbon and hydrogen, along with other elements like oxygen, nitrogen, and sulfur.
• Examples include carbohydrates, proteins, lipids, and nucleic acids, which are essential for life.
• Applications of Carbon Compounds:
• Carbon compounds are widely used in industries, such as pharmaceuticals, plastics, and fuels.
• Examples include ethanol (used as a solvent), polyethylene (used in packaging), and graphite (used in pencils).

Important Points

• Hydrocarbons: Hydrocarbons are compounds made of carbon and hydrogen. They include alkanes, alkenes, and alkynes.
• Functional Groups: Organic compounds often contain functional groups like hydroxyl (-OH), carboxyl (-COOH), and amino (-NH₂), which determine their chemical properties.
• Carbon Bond Strength: Carbon-carbon and carbon-hydrogen bonds are strong, making organic compounds stable.
• Isomerism: Carbon compounds exhibit isomerism, where compounds with the same molecular formula have different structures. Examples include structural isomers and stereoisomers.
• Biological Importance: Carbon is the central element in biomolecules like DNA, RNA, and enzymes, which are essential for life processes.

The correct answer is Binary Fission.

Key Points

• Binary fission is a type of asexual reproduction commonly observed in unicellular organisms like bacteria, amoeba, and paramecium.
• During this process, the parent cell divides into two equal daughter cells, each containing identical genetic material.
• The process involves key steps such as DNA replication, elongation of the cell, and cytokinesis, ensuring each daughter cell receives a complete set of genetic instructions.
• It is a rapid method of reproduction, enabling unicellular organisms to multiply efficiently under favorable conditions.
• This type of reproduction does not involve the fusion of gametes or genetic variation, making it a clonal process.

Additional Information

• Conjugation:
• It is a sexual mode of reproduction observed in some unicellular organisms like paramecium and bacteria.
• In this process, there is an exchange of genetic material between two organisms through a conjugation tube or pilus.
• This leads to genetic variation in the offspring.
• Spore Formation:
• This is a form of asexual reproduction where organisms like fungi, algae, and some bacteria produce spores.
• Spores are resistant structures that can survive harsh environmental conditions and germinate when conditions are favorable.
• Budding:
• It is a type of asexual reproduction where a new organism develops from an outgrowth or bud of the parent organism.
• It is commonly seen in yeast and hydra.
• Unlike binary fission, the offspring may not always be identical in size to the parent.
• Characteristics of Binary Fission:
• Speed: Binary fission is a rapid process and can lead to exponential growth in populations under ideal conditions.
• Energy-efficient: It does not require energy for gamete formation or mating.
• Examples: Organisms like Amoeba, Escherichia coli, and Euglena reproduce via binary fission.

Important Points

• Applications and Importance of Binary Fission:
• Antibiotics: The rapid reproduction of bacteria through binary fission is a reason for the fast spread of bacterial infections, making antibiotic resistance a major concern.
• Research: Understanding binary fission in bacteria like E. coli is crucial for genetic engineering and biotechnology.
• Ecological Role: Many unicellular organisms reproducing via binary fission play a key role in nutrient cycling and ecosystems.
• Binary fission is considered a primitive form of reproduction but is highly effective for the survival of unicellular organisms.

The correct answer is Both (i) and (ii).

Key Points

• Temperature of the conductor (i):
• An increase in the temperature of a conductor causes an increase in its resistance. This is because higher temperature results in an increased vibration of atoms, which in turn obstructs the flow of free electrons. Hence, this statement is correct.
• Length of the conductor (ii):
• Resistance is directly proportional to the length of the conductor. A longer conductor has more material for the electrons to travel through, leading to increased resistance. Hence, this statement is correct.
• Area of cross-section of the conductor (iii):
• Resistance is inversely proportional to the area of cross-section of a conductor. If the area increases, resistance decreases as more pathways are available for the electrons to flow. Hence, this statement is incorrect.
• Conclusion:
• Only the temperature and length of the conductor increase the resistance, making option "Both (i) and (ii)" the correct answer.

Additional Information

• Resistance and its Factors:
• The resistance of a conductor determines how much it opposes the flow of electric current.
• It is mathematically given by the formula: R = ρ × (L/A), where:
• R: Resistance
• ρ: Resistivity (depends on the material)
• L: Length of the conductor
• A: Area of cross-section of the conductor
• Effect of Temperature on Resistance:
• For most conductors, resistance increases with an increase in temperature. This is due to enhanced atomic vibrations, which scatter the electrons.
• For semiconductors and insulators, resistance decreases with an increase in temperature.
• Effect of Length on Resistance:
• Resistance is directly proportional to the length of the conductor. Thus, doubling the length of a conductor doubles its resistance.
• Effect of Cross-sectional Area on Resistance:
• Resistance is inversely proportional to the cross-sectional area. For example, increasing the area by two times reduces the resistance to half.
• Real-world Applications:
• Understanding resistance is crucial for designing electrical components like resistors, wires, and circuits.
• Materials with low resistance, such as copper and aluminum, are widely used in electrical wiring.

Important Points

• Resistivity:
• Resistivity is a material-specific property that quantifies how strongly a material opposes the flow of electric current.
• It is denoted by ρ and measured in ohm-meters (Ω·m).
• Conductors vs. Insulators:
• Conductors: Materials with low resistivity (e.g., copper, aluminum).
• Insulators: Materials with high resistivity (e.g., rubber, glass).
• Temperature Coefficient of Resistance:
• This term describes how the resistance of a material changes with temperature. It is positive for conductors and negative for semiconductors.
• Applications in Electronics:
• Resistors, thermostats, and temperature sensors are designed based on the principles of resistance.

The correct answer is Option 2.

Key Points

• To determine the time taken by the object to reach the ground, we use the formula for motion under gravity: h = (1/2) × g × t², where:
• h is the height from which the object is dropped, which is 78.4 m.
• g is the acceleration due to gravity, which is 9.8 m/s².
• t is the time taken by the object to reach the ground.
• Rearranging the formula to solve for t:
• t² = (2 × h) / g
• t = √[(2 × h) / g]
• Substituting the values:
• t = √[(2 × 78.4) / 9.8]
• t = √(156.8 / 9.8)
• t = √16 = 4 seconds.
• Thus, the time taken by the object to reach the ground is 4 seconds. Hence, Option 2 is correct.

Additional Information

• Free Fall:
• When an object falls under the influence of gravity alone, without any air resistance, it is called free fall.
• The acceleration during free fall is equal to the acceleration due to gravity (g), which is approximately 9.8 m/s² on Earth.
• Equations of Motion:
• The three equations of motion are used to describe the motion of objects under uniform acceleration:
• v = u + at (final velocity)
• s = ut + (1/2) × a × t² (displacement)
• v² = u² + 2as (relation between velocity and displacement)
• In free fall, the initial velocity (u) is 0, and acceleration (a) is replaced by g.
• Acceleration due to Gravity:
• On Earth, the value of g is approximately 9.8 m/s².
• It varies slightly depending on altitude and geographical location.
• Practical Applications:
• The concept of free fall is used in designing parachutes and calculating trajectories of rockets and projectiles.
• It is also applied in physics experiments, like determining g using pendulums or inclined planes.

Important Points

• Factors Affecting Free Fall:
• In real-life scenarios, air resistance affects the motion of falling objects.
• For example, a feather falls slower than a stone due to air resistance.
• Galileo’s Contribution:
• Galileo Galilei was the first to study the motion of falling objects and concluded that all objects fall at the same rate in the absence of air resistance.
• Space Applications:
• Astronauts experience microgravity or free fall in space, as they orbit Earth.
• This principle is used in designing spacecraft and conducting experiments in zero-gravity environments.

The correct answer is Stage 1 – Balanced, Stage 2 – Unbalanced, Stage 3 – Balanced.

Key Points

• Stage 1: Balanced Forces:
• At this stage, the applied force is less than the maximum static friction force.
• The block does not move because the static friction balances the applied force perfectly.
• As per Newton's First Law of Motion, the block remains at rest since there is no unbalanced force acting on it. Hence, this stage is balanced.
• Stage 2: Unbalanced Forces:
• As the applied force increases and reaches the threshold of static friction, the block just begins to move.
• At this point, the applied force exceeds the maximum static friction force, creating an unbalanced force that initiates motion.
• The block starts to slide because the net force is no longer zero. Hence, this stage is unbalanced.
• Stage 3: Balanced Forces:
• Once the block is in motion and moves with a constant velocity, the applied force equals the kinetic friction force acting in the opposite direction.
• Since the net force is zero, there is no acceleration, and the block continues to move uniformly.
• This is consistent with Newton's First Law, indicating that the forces are balanced. Hence, this stage is balanced.
• Correct Answer: The stages are described correctly as: Stage 1 – Balanced, Stage 2 – Unbalanced, Stage 3 – Balanced.

Additional Information

• Static Friction:
• Static friction is the frictional force that prevents an object from moving when it is at rest.
• It adjusts itself to counterbalance the applied force up to a certain maximum limit, known as the maximum static friction force.
• The equation for static friction is f_s ≤ μ_s N, where μ_s is the coefficient of static friction and N is the normal force.
• Kinetic Friction:
• Once the object is in motion, the force opposing its motion is called kinetic friction.
• Kinetic friction is usually lower than static friction and remains constant regardless of the velocity of the object.
• The equation for kinetic friction is f_k = μ_k N, where μ_k is the coefficient of kinetic friction and N is the normal force.
• Newton’s Laws of Motion:
• First Law: An object will remain at rest or in uniform motion unless acted upon by an unbalanced force.
• Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma).
• Third Law: For every action, there is an equal and opposite reaction.
• Force Balance in Motion:
• In the case of constant velocity, the net external force on an object is zero due to the balance between applied force and frictional force.
• This highlights the equilibrium condition during motion.
• Applications:
• Understanding force balance is crucial in engineering applications, such as designing conveyor belts, brake systems, and machinery components.
• This knowledge is also applied in sports science, robotics, and transportation systems to optimize performance and safety.

Shortcut Trick

Let numbers be 23a and 23b. Sum ⇒ 23(a + b) = 161 ⇒ a + b = 7.

Possible co-prime pairs for (a, b) such that sum is 7 are: (1, 6), (2, 5), and (3, 4).

Total possible pairs = 3.

∴ The correct answer is 3.

Alternate Method

Given: HCF = 23, Sum = 161.

Formula: If HCF is H, numbers are Hx and Hy, where x and y are co-prime.

⇒ 23x + 23y = 161

⇒ 23(x + y) = 161

⇒ x + y = 161 ÷ 23

⇒ x + y = 7

⇒ Possible pairs of (x, y) such that they are co-prime and sum to 7:

⇒ Case 1: (1, 6) → Numbers: (23, 138)

⇒ Case 2: (2, 5) → Numbers: (46, 115)

⇒ Case 3: (3, 4) → Numbers: (69, 92)

∴ The correct answer is 3.

Additional Information

Co-prime Numbers

Two numbers are co-prime if their Highest Common Factor (HCF) is exactly 1.

HCF and LCM Relation

Product of two numbers = HCF × LCM. Also, LCM is always divisible by HCF.

Finding Pairs with Product

If product is P and HCF is H, then (Hx) × (Hy) = P ⇒ xy = P/H². Find co-prime factors of the result.

Shortcut Trick

Given tan A = 1, we know that for standard angles, tan 45° = 1.

∴ A = 45°.

Substitute A in the expression: sin 45° + cos 45° = (1/√2) + (1/√2) = 2/√2 = √2.

∴ The correct answer is √2.

Alternate Method

Given: tan A = 1

Formula Used: tan A = Perpendicular / Base; sin A = Perpendicular / Hypotenuse; cos A = Base / Hypotenuse; H² = P² + B²

⇒ tan A = 1/1 ⇒ Perpendicular (P) = 1, Base (B) = 1

⇒ Hypotenuse (H) = √(P² + B²) = √(1² + 1²) = √2

⇒ sin A = P/H = 1/√2

⇒ cos A = B/H = 1/√2

⇒ sin A + cos A = (1/√2) + (1/√2)

⇒ Result = 2/√2 = √2

∴ The correct answer is √2.

Additional Information

Trigonometric Identities

Fundamental relations like sin²A + cos²A = 1 and 1 + tan²A = sec²A are essential for solving trigonometric expressions.

Standard Values

Values of sin, cos, and tan at 0°, 30°, 45°, 60°, and 90° help in direct substitution (e.g., sin 45° = 1/√2).

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides: H² = P² + B².

Shortcut Trick

Assume TV price = 100, then Washing Machine (WM) = 90.

Initial Total (5 WM + 3 TV) = (5 × 90) + (3 × 100) = 450 + 300 = 750.

New WM = 90 × 1.65 = 148.5 | New TV = 100 × 0.66 = 66.

New Total = (5 × 148.5) + (3 × 66) = 742.5 + 198 = 940.5.

% Change = (940.5 − 750) ÷ 750 × 100 = 190.5 ÷ 7.5 = 25.4%.

∴ The correct answer is Increase by 25.4%.

Alternate Method

Given:

Cost of TV = 100x

Cost of Washing Machine = 100x − 10% of 100x = 90x

Calculations:

⇒ Initial total cost (5 WM + 3 TV) = 5(90x) + 3(100x) = 450x + 300x = 750x

⇒ New cost of WM = 90x × (165/100) = 148.5x

⇒ New cost of TV = 100x × (66/100) = 66x

⇒ New total cost = 5(148.5x) + 3(66x) = 742.5x + 198x = 940.5x

⇒ Total change = 940.5x − 750x = 190.5x

⇒ Percentage change = (190.5x ÷ 750x) × 100 = 25.4%

∴ The correct answer is Increase by 25.4%.

Additional Information

Percentage Change

Percentage change = (Final Value − Initial Value) ÷ Initial Value × 100.

Weighted Average Percentage

When components change by different percentages, the net change depends on the weight (initial cost contribution) of each component.

Successive Percentage Shortcut

For two successive changes a and b, the net change is a + b + (ab/100).

Shortcut Trick

Ratio of Marked Price (MP) to Selling Price (SP) = 3650 : 730 = 5 : 1

Discount units = 5 − 1 = 4 units

Discount % = (Discount units / MP units) × 100

Discount % = (4 / 5) × 100 = 80%

∴ The correct answer is 80%.

Alternate Method

Given: Marked Price (MP) = ₹3,650 and Selling Price (SP) = ₹730

Formula Used: Discount = MP − SP | Discount % = (Discount / MP) × 100

⇒ Discount = 3,650 − 730 = ₹2,920

⇒ Discount % = (2,920 ÷ 3,650) × 100

⇒ Discount % = 0.8 × 100 = 80%

∴ The correct answer is 80%.

Additional Information

Marked Price (MP)

The price written on the tag of an item before any discount is applied. It is also known as the List Price.

Successive Discounts

When multiple discounts are given, the Equivalent Discount % is calculated as: a + b − (a × b / 100).

Markup Percentage

The extra price added to the Cost Price (CP) to reach the MP: Markup % = [(MP − CP) / CP] × 100.

Shortcut Trick

Sister age = (Sum − Difference) ÷ 2

Sister age = (38 − 6) ÷ 2 = 32 ÷ 2 = 16 years

After 15 years, age = 16 + 15 = 31 years

∴ The correct answer is 31 years.

Alternate Method

Given:

Difference between Brother and Sister = 6 years

Sum of Brother and Sister = 38 years

Let the Sister present age be x years.

Then Brother present age = (x + 6) years.

⇒ x + (x + 6) = 38

⇒ 2x + 6 = 38

⇒ 2x = 32

⇒ x = 16 years

Sister age after 15 years = 16 + 15

⇒ 31 years

∴ The correct answer is 31 years.

Additional Information

Linear Equations in Age

Age problems often involve creating equations where (x + n) represents age after n years and (x − n) represents age n years ago.

Constant Difference

The age difference between two individuals remains constant throughout their lives, regardless of the number of years passed.

Shortcut Trick Profit 12.5% = 1/8 → CP : SP1 = 8 : 9 9 units = ₹85.50 ⇒ 1 unit = ₹9.50 CP = 8 × 9.50 = ₹76 New SP2 = ₹81.70 ⇒ Profit = 81.70 − 76 = ₹5.70 Profit% = (5.70 / 76) × 100 = 7.5% ∴ The correct answer is 7.5% profit.

Alternate Method

Given: Selling Price (SP1) = ₹85.50 Profit Percentage (P1%) = 12.5% New Selling Price (SP2) = ₹81.70 Formula Used: CP = [SP / (100 + P%)] × 100 Profit% = [(SP − CP) / CP] × 100 Calculations: ⇒ Cost Price (CP) = [85.50 / (100 + 12.5)] × 100 ⇒ CP = [85.50 / 112.5] × 100 ⇒ CP = 0.76 × 100 = ₹76 ⇒ New Profit = SP2 − CP = 81.70 − 76 = ₹5.70 ⇒ New Profit% = (5.70 / 76) × 100 ⇒ New Profit% = 570 / 76 = 7.5% ∴ The correct answer is 7.5% profit.

Additional Information Profit and Loss Basics Profit occurs when SP > CP, and Loss occurs when CP > SP. Percentage is always calculated on Cost Price (CP) unless stated otherwise. Fractional Conversion 12.5% is equivalent to 1/8. This means for a CP of 8 units, the profit is 1 unit, making the SP 9 units. Marked Price and Discount Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Shortcut Trick

Investment ratio of Rajiv : Sachin = 160 : 1690 = 16 : 169.

Profit share is proportional to investment: 16 units = ₹720.

1 unit = 720 ÷ 16 = ₹45.

Total profit = (16 + 169) units = 185 units.

Total profit = 185 × 45 = ₹8,325.

∴ The correct answer is ₹8,325.

Alternate Method

Given: Rajiv's investment (C_R) = ₹160, Sachin's investment (C_S) = ₹1,690, Rajiv's profit (P_R) = ₹720.

Formula Used: Profit Ratio = Capital Ratio × Time Ratio (Since time is same, Profit Ratio = Capital Ratio).

⇒ Ratio of profits = 160 : 1690 = 16 : 169

⇒ Let Rajiv's profit be 16x and Sachin's profit be 169x

⇒ 16x = 720

⇒ x = 720 ÷ 16 = 45

⇒ Total Profit = 16x + 169x = 185x

⇒ Total Profit = 185 × 45 = 8325

∴ The correct answer is ₹8,325.

Additional Information

Partnership Profit Distribution

Profit is divided in the ratio of the product of Capital and Time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Active vs. Sleeping Partner

An active partner manages the business and may receive a salary or extra commission before the remaining profit is shared among all partners.

Compound Partnership

When partners invest money for different durations, it is called a compound partnership, and the time period must be considered for ratio calculations.

Shortcut Trick

Net multiplier = 1.05 × 1.06 × 1.10 = 1.2243

Final Value = 5,50,000 × 1.2243

Final Value = 5,50,000 × (1 + 0.05 + 0.06 + 0.10 + small decimals) ≈ 5,50,000 × 1.22

Calculation: 5,50,000 × 1.05 × 1.06 × 1.1 = 6,73,365

∴ The correct answer is ₹6,73,365.

Alternate Method

Given:

Initial Value (P) = ₹5,50,000

Successive growth rates: r₁ = 5%, r₂ = 6%, r₃ = 10%

Formula Used:

Final Value = P × (1 + r₁/100) × (1 + r₂/100) × (1 + r₃/100)

Calculations:

⇒ Final Value = 5,50,000 × (105/100) × (106/100) × (110/100)

⇒ Final Value = 5,50,000 × (21/20) × (53/50) × (11/10)

⇒ Final Value = 550000 × (21 × 53 × 11) ÷ (20 × 50 × 10)

⇒ Final Value = 550000 × 12243 ÷ 10000

⇒ Final Value = 55 × 12243

⇒ Final Value = 6,73,365

∴ The correct answer is ₹6,73,365.

Additional Information

Successive Percentage Change

For two successive changes a% and b%, the net effective change is (a + b + ab/100)%.

Depreciation Formula

If the value decreases, the formula becomes Final Value = P × (1 − r/100)ⁿ.

Unit Digit Trick

Multiplying 5 (from 5.5) with odd numbers often ends in 5; checking unit digits can eliminate options in complex calculations.

Shortcut Trick

Arrange the given observations in ascending order: 2, 2, 3, 4, 5, 5, 5, 7, 8, 8, 9, 11.

Count the total number of terms, n = 12 (which is an even number).

The median is the average of the 6^th and 7^th terms.

6^th term = 5 and 7^th term = 5.

Median = (5 + 5) ÷ 2 = 5.

∴ The correct answer is 5.

Alternate Method

Given observations: 3, 5, 2, 5, 5, 8, 4, 2, 7, 9, 8, 11

Formula Used:

If n is even, Median = [(n/2)^th term + (n/2 + 1)^th term] ÷ 2

⇒ First, arrange in ascending order: 2, 2, 3, 4, 5, 5, 5, 7, 8, 8, 9, 11

⇒ Total count (n) = 12

⇒ Median = [(12/2)^th term + (12/2 + 1)^th term] ÷ 2

⇒ Median = [6^th term + 7^th term] ÷ 2

⇒ Median = [5 + 5] ÷ 2

⇒ Median = 10 ÷ 2 = 5

∴ The correct answer is 5.

Additional Information

Median for Odd Count

If the number of observations (n) is odd, the median is the exactly ((n + 1)/2)^th term after sorting.

Mode

The mode is the value that appears most frequently in a data set. In this set, the mode is 5 (appears three times).

Empirical Relation

Relation between Mean, Median, and Mode: Mode = 3 × Median − 2 × Mean.

Range

The range is the difference between the maximum and minimum values: Range = Max − Min.

Shortcut Trick

Direct Relation: Cost Price (CP) / Marked Price (MP) = (100 − Discount %) / (100 + Profit %)

CP / 5670 = (100 − 28) / (100 + 26)

CP / 5670 = 72 / 126 = 4 / 7

CP = 5670 × (4 / 7) = 810 × 4 = 3240

∴ The correct answer is ₹3,240.

Alternate Method

Given:

Marked Price (MP) = ₹5,670

Discount = 28%

Profit = 26%

Formula Used:

Selling Price (SP) = MP × (100 − D%)/100

Cost Price (CP) = SP × 100 / (100 + P%)

Calculations:

⇒ SP = 5670 × (100 − 28) / 100

⇒ SP = 5670 × 72 / 100 = 56.7 × 72 = ₹4,082.4

⇒ CP = 4082.4 × 100 / (100 + 26)

⇒ CP = 408240 / 126

⇒ CP = ₹3,240

∴ The correct answer is ₹3,240.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price is given by (100 − Discount%) : (100 + Profit%).

Successive Discounts

If two discounts d₁ and d₂ are given, the Equivalent Discount is (d₁ + d₂ − (d₁ × d₂)/100)%.

Markup Percentage

Markup % = [(MP − CP) / CP] × 100. It is the percentage by which the cost price is increased to set the marked price.

Shortcut Trick

Assume a deviation base of 50 for the given observations.

Net Deviation = 26 − 22 + 29 − 33 − 17 + 45 + 33 − 20 − 23 = +18

Average = Base + (Net Deviation ÷ Total observations) = 50 + (18 ÷ 9) = 50 + 2 = 52

∴ The correct answer is 52.

Alternate Method

Given: Observations = 76, 28, 79, 17, 33, 95, 83, 30, 27

Formula Used: Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

⇒ Sum = 76 + 28 + 79 + 17 + 33 + 95 + 83 + 30 + 27

⇒ Sum = 468

⇒ Total count (n) = 9

⇒ Arithmetic Mean = 468 ÷ 9

⇒ Arithmetic Mean = 52

∴ The correct answer is 52.

Additional Information

Properties of Arithmetic Mean

The sum of deviations of a set of observations from their arithmetic mean is always zero. Σ(x_i − x̄) = 0.

Combined Mean

If two groups have means x̄₁ and x̄₂ with sizes n₁ and n₂, the combined mean is (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).

Weighted Mean

Used when observations have different importance or weights: Mean = Σ(w_ix_i) ÷ Σw_i.

Shortcut Trick

In a partnership, the ratio of profits is equal to the ratio of investments when the time period is the same.

Total profit = ₹6,000 | Amit profit = ₹2,100 → Deeksha profit = ₹3,900.

Profit ratio (Amit : Deeksha) = 2100 : 3900 = 7 : 13.

Amit investment = [7 ÷ (7 + 13)] × 15700 = (7 ÷ 20) × 15700 = 7 × 785 = ₹5,495.

∴ The correct answer is ₹5,495.

Alternate Method

Given: Total Investment = ₹15,700, Total Profit = ₹6,000, Amit profit = ₹2,100.

Formula: Investment Ratio = Profit Ratio (for equal time duration).

⇒ Deeksha profit = Total Profit − Amit profit

⇒ Deeksha profit = 6000 − 2100 = ₹3,900

⇒ Ratio of Profit (Amit : Deeksha) = 2100 : 3900 = 21 : 39 = 7 : 13

⇒ Let Amit investment be 7x and Deeksha investment be 13x

⇒ 7x + 13x = 15700

⇒ 20x = 15700

⇒ x = 785

⇒ Amit investment = 7 × 785 = ₹5,495

∴ The correct answer is ₹5,495.

Additional Information

Partnership Ratio Rule

Profit of A : Profit of B = (Investment of A × Time of A) : (Investment of B × Time of B).

Simple vs Compound Partnership

In simple partnership, time is constant for all partners. In compound partnership, partners invest for different time periods.

Investment Calculation

Individual Investment = (Total Investment ÷ Sum of Ratio Parts) × Individual Ratio Part.

Shortcut Trick

Relative Speed (Same Direction) = 130 − 100 = 30 km/hr.

Total Distance = 160 + 140 = 300 m = 0.3 km.

Time = Distance ÷ Speed = 0.3 ÷ 30 = 0.01 hours.

Time in minutes = 0.01 × 60 = 0.6 minutes.

∴ The correct answer is 0.6 minutes.

Alternate Method

Given:

Length of first train (L₁) = 160 m

Length of second train (L₂) = 140 m

Speed of first train (S₁) = 100 km/hr

Speed of second train (S₂) = 130 km/hr

Formula Used:

Relative Speed (Same Direction) = S₂ − S₁

Time = (Total Length) ÷ (Relative Speed)

Calculations:

⇒ Total distance to be covered = 160 + 140 = 300 m

⇒ Relative Speed = 130 − 100 = 30 km/hr

⇒ Convert speed to m/s: 30 × (5/18) = 25/3 m/s

⇒ Time in seconds = 300 ÷ (25/3) = 300 × 3 ÷ 25

⇒ Time in seconds = 12 × 3 = 36 seconds

⇒ Time in minutes = 36 ÷ 60 = 0.6 minutes

∴ The correct answer is 0.6 minutes.

Additional Information

Relative Speed Concept

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

Unit Conversion

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

Train Crossing Problems

When a train crosses another train or a platform, the total distance is always the sum of their individual lengths.

Shortcut Trick

Identify Time in days: Feb (10) + Mar (31) + Apr (20) = 61 days (Note: 2024 is a Leap Year).

Apply direct SI calculation: (4000 × 6 × 61) ÷ (100 × 366).

Simplify: 40 × 6 × (61 ÷ 366) ⇒ 40 × 6 × (1/6) = 40.

∴ The correct answer is ₹40.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate (R) = 6% per annum

Time Period: 19 Feb 2024 to 20 Apr 2024

Days in Feb = 29 − 19 = 10 days (Leap Year)

Days in March = 31 days

Days in April = 20 days

⇒ Total Days (T) = 10 + 31 + 20 = 61 days

Formula Used:

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

Calculations:

⇒ T in years = 61 ÷ 366 (As 2024 is a leap year)

⇒ SI = (4000 × 6 × 61) ÷ (100 × 366)

⇒ SI = (40 × 6 × 61) ÷ 366

⇒ SI = (240 × 61) ÷ 366

⇒ SI = 14640 ÷ 366

⇒ SI = 40

∴ The correct answer is ₹40.

Additional Information

Leap Year Calculation

A year is a leap year if divisible by 4 (e.g., 2024). It has 366 days and February has 29 days.

Simple Interest on Days

While calculating days, the day the money is deposited is excluded, but the day the money is withdrawn is included.

Amount Formula

Total Amount (A) = Principal (P) + Simple Interest (SI).

Given:

4 - [3 - 4 ÷ 2 + {4 - (5 - 2)}] + [4 + 18 ÷ 2 × 3(4 - 2)]

Formula Used:

BODMAS rule: Bracket → Of → Division → Multiplication → Addition → Subtraction

Calculation:

⇒ 4 - [3 - 4 ÷ 2 + {4 - 3}] + [4 + 18 ÷ 2 × 3 × 2]

⇒ 4 - [3 - 2 + 1] + [4 + 9 × 3 × 2]

⇒ 4 - [2] + [4 + 27 × 2]

⇒ 4 - 2 + [4 + 54]

⇒ 4 - 2 + 58

⇒ 2 + 58

⇒ 60

∴ The correct answer is 60.

Shortcut Trick

Numbers between 374 and 384 are 375, 376, 377, 378, 379, 380, 381, 382, and 383.

Eliminate all even numbers: 376, 378, 380, 382.

Eliminate numbers ending in 5: 375.

Check remaining odd numbers: 377 = 13 × 29 (not prime), 381 = 3 × 127 (not prime).

Only 379 and 383 are prime numbers.

∴ The correct answer is 2.

Alternate Method

Given range: numbers strictly between 374 and 384.

List of integers in this range: {375, 376, 377, 378, 379, 380, 381, 382, 383}

To check if a number N is prime, we test divisibility by prime numbers up to √N ≈ 19.5 (i.e., 2, 3, 5, 7, 11, 13, 17, 19).

⇒ 375: Divisible by 5 (not prime)

⇒ 376, 378, 380, 382: Divisible by 2 (not prime)

⇒ 377: 377 ÷ 13 = 29 (not prime)

⇒ 379: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 (Prime)

⇒ 381: Divisible by 3 (Sum of digits 3+8+1 = 12) (not prime)

⇒ 383: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 (Prime)

⇒ Total prime numbers = 2 (which are 379 and 383)

∴ The correct answer is 2.

Additional Information

Prime Number Definition

A natural number greater than 1 that has no positive divisors other than 1 and itself.

Primality Test Rule

To determine if a number N is prime, check its divisibility by all prime numbers less than or equal to √N.

Sieve of Eratosthenes

An ancient and highly efficient algorithm used to find all prime numbers up to any given limit by systematically marking the multiples of each prime.

Shortcut Trick

Total work (LCM of 18, 24) = 72 units.

Efficiency: A = 4, B = 3.

Pipe B works for the entire 16 hours: 3 × 16 = 48 units.

Remaining work for A: 72 − 48 = 24 units.

Time for A: 24 ÷ 4 = 6 hours.

∴ The correct answer is 6 hours.

Alternate Method

Given: Time for A = 18 hours, Time for B = 24 hours. Total time to fill = 16 hours.

Formula Used: Total Work = Efficiency × Time

⇒ Let Pipe A be closed after x hours.

⇒ Pipe B remains open for the full 16 hours.

⇒ Work done by A + Work done by B = 72

⇒ (4 × x) + (3 × 16) = 72

⇒ 4x + 48 = 72

⇒ 4x = 24

⇒ x = 6 hours

∴ The correct answer is 6 hours.

Additional Information

Pipes and Cisterns Efficiency

Filling pipes have positive efficiency, while outlet/leakage pipes have negative efficiency.

Work and Wages

Wages are distributed in the ratio of the actual work done by each person or pipe.

Total Work Assumption

Always assume total work as the LCM of the individual times to simplify calculations into integers.

Shortcut Trick

Perimeter of the field = Total Cost ÷ Rate per meter

⇒ Perimeter = 18000 ÷ 15 = 1200 m

New Rate is ₹11 per 100 m, which means 11/100 = ₹0.11 per meter.

New Total Cost = 1200 × 0.11 = ₹132

∴ The correct answer is ₹132.

Alternate Method

Given:

Total cost of fencing = ₹18,000

Rate of fencing = ₹15/m

New Rate of fencing = ₹11 per 100 m

Formula Used:

Total Cost = Perimeter × Rate per unit

Calculations:

⇒ Perimeter of the square field = 18000 ÷ 15

⇒ Perimeter = 1200 m

⇒ We need to find the cost at ₹11 per 100 m.

⇒ Cost for 100 m = ₹11

⇒ Cost for 1 m = ₹11/100

⇒ Total cost for 1200 m = 1200 × (11 ÷ 100)

⇒ Total cost = 12 × 11 = ₹132

∴ The correct answer is ₹132.

Additional Information

Perimeter of a Square

The total length of the boundary of a square is 4 × Side. In fencing problems, we always calculate the Perimeter.

Costing Units

If the rate is given as ₹X per Y units, the cost for Z units is calculated as (Z ÷ Y) × X.

Area vs Perimeter

Use Perimeter for fencing, borders, or running around a field. Use Area (Side²) for tiling, flooring, or ploughing a field.

Shortcut Trick

For a toy where Volume of Cone = Volume of Hemisphere with the same radius (r):

(1/3) × π × r² × h = (2/3) × π × r³ ⇒ h = 2r.

The total height of the toy is the sum of the cone height (h) and hemisphere radius (r):

Total Height = h + r = 2r + r = 3r.

Given r = 7 cm, Total Height = 3 × 7 = 21 cm.

∴ The correct answer is 21 cm.

Alternate Method

⇒ Given: Radius (r) of both cone and hemisphere = 7 cm.

⇒ Formula Used: Volume of Cone = (1/3)πr²h | Volume of Hemisphere = (2/3)πr³.

⇒ Volume of conical part = Volume of hemispherical part

⇒ (1/3) × π × r² × h = (2/3) × π × r³

⇒ (1/3) × π × 7² × h = (2/3) × π × 7³

⇒ h = 2 × 7

⇒ h = 14 cm

⇒ Total height of the toy = Height of cone (h) + Radius of hemisphere (r)

⇒ Total height = 14 + 7 = 21 cm.

∴ The correct answer is 21 cm.

Additional Information

Volume of a Sphere

The volume of a complete sphere is (4/3) × π × r³. A hemisphere is exactly half of this volume.

Slant Height of a Cone

The slant height (l) is the distance from the apex to any point on the base circle: l = √(h² + r²).

Curved Surface Area (CSA)

For the toy, Total CSA = CSA of cone (πrl) + CSA of hemisphere (2πr²).

Given:

$\frac{32672 \times 32672 - 21864 \times 21864}{10808}$

Formula Used:

(a² - b²) ÷ c

a² - b² = (a - b) × (a + b)

Calculation:

$\frac{32672 \times 32672 - 21864 \times 21864}{10808}$

⇒ (32672 - 21864) × (32672 + 21864) ÷ 10808

⇒ 10808 × 54536 ÷ 10808

⇒ 54536

∴ The correct answer is 54536.

Shortcut Trick

Let the side of both the equilateral triangle and the square be 'a'.

Circumradius of equilateral triangle (R) = a/√3

Inradius of square (r) = a/2

Required Ratio = (a/√3) : (a/2) = 2 : √3

∴ The correct answer is 2 : √3.

Alternate Method

Given:

Side of equilateral triangle = Side of square = a

Formula Used:

Circumradius (R) of equilateral triangle = Side / √3

Inradius (r) of square = Side / 2

Calculations:

⇒ Let the side of the triangle and square be 'a'.

⇒ Circumradius (R) of equilateral triangle = a / √3

⇒ Inradius (r) of square = a / 2

⇒ Required Ratio = R : r

⇒ Ratio = (a / √3) : (a / 2)

⇒ Ratio = 1/√3 : 1/2

⇒ Ratio = 2 : √3

∴ The correct answer is 2 : √3.

Additional Information

Inradius of Equilateral Triangle

The inradius (r_tri) of an equilateral triangle with side 'a' is given by r_tri = a / (2√3).

Circumradius of Square

The circumradius (R_sq) of a square with side 'a' is half of its diagonal, R_sq = a / √2.

Area of Equilateral Triangle

The area of an equilateral triangle with side 'a' is (√3 / 4) × a².

Diagonal of a Square

The diagonal of a square with side 'a' is a√2.

Shortcut Trick

Using Alligation method on average coin value:

Average value per coin = 16 ÷ 10 = ₹1.6

Ratio of 1-rupee coins to 2-rupee coins = (2 − 1.6) : (1.6 − 1) = 0.4 : 0.6 = 2 : 3

Total ratio units (2 + 3) = 5 → 10 coins. So, 1 unit = 2 coins.

Difference in units = 3 − 2 = 1 unit × 2 = 2 coins.

∴ The correct answer is 2.

Alternate Method

Given: Total coins = 10, Total value = ₹16.

Let the number of 1-rupee coins be x and 2-rupee coins be y.

⇒ x + y = 10 ... (i)

⇒ 1 × x + 2 × y = 16 ⇒ x + 2y = 16 ... (ii)

Subtracting equation (i) from (ii):

⇒ (x + 2y) − (x + y) = 16 − 10

⇒ y = 6

Substituting y = 6 in (i):

⇒ x + 6 = 10 ⇒ x = 4

Difference between the numbers of coins:

⇒ |6 − 4| = 2

∴ The correct answer is 2.

Additional Information

Mixture and Alligation in Coins

This technique allows solving linear equations by finding the ratio of quantities when their weighted average is known.

Linear Equations in Two Variables

For coin problems, use equations: x + y = Total Quantity and (Value₁ × x) + (Value₂ × y) = Total Value.

Shortcut Trick

Speed ratio = 7 : 4 → Time ratio = 4 : 7 (as distance is constant for both comparison cases).

Difference in time units = 7 − 4 = 3 units.

Given time difference = 7 hours ∴ 3 units = 7 hours ⇒ 1 unit = 7/3 hours.

Time for slower truck (7 units) = 7 × (7/3) = 49/3 hours.

Speed of slower truck = Distance ÷ Time = 196 ÷ (49/3) = (196 × 3) ÷ 49 = 4 × 3 = 12 km/hr.

∴ The correct answer is 12 km/hr.

Alternate Method

Given: Ratio of speeds = 7 : 4, Distance = 196 km, Time difference = 7 hours.

Formula Used: Speed = Distance ÷ Time | Time = Distance ÷ Speed

⇒ Let the speeds be 7x and 4x km/hr.

⇒ Time taken by slower truck − Time taken by faster truck = 7

⇒ (196 / 4x) − (196 / 7x) = 7

⇒ (49 / x) − (28 / x) = 7

⇒ 21 / x = 7

⇒ x = 21 ÷ 7 = 3

⇒ Speed of slower truck = 4x = 4 × 3 = 12 km/hr.

∴ The correct answer is 12 km/hr.

Additional Information

Inverse Proportion of Speed and Time

When the distance is constant, the time taken is inversely proportional to the speed (S₁/S₂ = T₂/T₁).

Average Speed Formula

If a certain distance is covered at speed 'a' and the same distance at speed 'b', then Average Speed = 2ab / (a + b).

Relative Speed

In the same direction, relative speed is (S₁ − S₂). In the opposite direction, relative speed is (S₁ + S₂).

Shortcut Trick

Savings = 60%, so Expenditure = 100% − 60% = 40%.

Given, 40% of Income = ₹400.

100% of Income = (400 ÷ 40) × 100 = 10 × 100 = ₹1,000.

∴ The correct answer is ₹1,000.

Alternate Method

Given:

Savings percentage = 60%

Expenditure = ₹400

Formula:

Income = Expenditure + Savings

Expenditure % = 100% − Savings %

Calculations:

⇒ Expenditure % = 100% − 60% = 40%

⇒ Let Total Income be x.

⇒ 40% of x = 400

⇒ (40 ÷ 100) × x = 400

⇒ x = (400 × 100) ÷ 40

⇒ x = 10 × 100

⇒ x = 1000

∴ The correct answer is ₹1,000.

Additional Information

Income-Expenditure Concept

The fundamental relation is Income = Expenditure + Savings. If any two components or their percentages are known, the third can be calculated.

Percentage Change in Income

If Income increases by x% and Expenditure increases by y%, the change in Savings can be found using the weighted average or Alligation method.

Base Value in Percentages

In income-based problems, Total Income is always considered as the base value (100%) for calculations unless specified otherwise.

Shortcut Trick

Efficiency of A + B = Total Work ÷ Days = 30 ÷ 6 = 5 units/day.

Efficiency of B = 30 ÷ 15 = 2 units/day.

Efficiency of A = 5 − 2 = 3 units/day.

Double work = 30 × 2 = 60 units.

Time taken by A = 60 ÷ 3 = 20 days.

∴ The correct answer is 20 days.

Alternate Method

Given: Time (A+B) = 6 days, Time (B) = 15 days.

Formula Used: Total Work = LCM of time; Efficiency = Total Work ÷ Time.

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

⇒ Efficiency of A = 5 − 2 = 3

⇒ New Work = 2 × 30 = 60 units

⇒ Time for A = 60 ÷ 3 = 20 days

∴ The correct answer is 20 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete a task. Efficiency = Work ÷ Time.

Combined Work Formula

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

Pipes and Cisterns

The same logic applies where efficiency of an outlet pipe is taken as negative work.

The logic followed here is:

thus, "72" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given: A, B, C, R, S, T and U are sitting around a circular table facing the centre of the table.

C sits to the immediate right of R.

Only three people sit between C and T when counted from the left of C.

Only three people sit between R and B when counted from the left of R.

S sits to the immediate right of A.

Here, There are One people sitting between R and A when counted from the right of A.

Hence, the correct answer is "Option 1".

The logic followed here is:

thus, "106" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given: (Left) 9 1 6 7 7 2 8 3 4 3 3 2 7 9 9 9 5 4 6 7 3 6 3 (Right)

Now, 

Even numbers that are immediately preceded by an odd number and also immediately followed by an even number.

So, Needed pattern is: Odd - Even - Even.

(Left) 9 1 6 7 7 2 8 3 4 3 3 2 7 9 9 9 5 4 6 7 3 6 3 (Right)

There are Two even numbers which is immediately preceded by an odd number and also immediately followed by an even number.

Hence, the correct answer is 'Option 2'.

The logic followed here is:

Option 1) PU-QN

Option 2) NS-OK

Option 3) LQ-MJ

Option 4) RW-SP

Here, 'NS-OK' is odd among the given options.

Hence, the correct answer is "Option 2".

Family Chart:

Decoding the Symbols:

Given:  ‘O @ P + T # G - V’

It means: O is the Brother of P, P is the Sister of T, T is the Father of G, G is the Wife of V.

So, O is the Wife's father's brother of V.

Hence, the correct answer is "Option 3".

The logic followed here is:

‘virus book two’ is coded as ‘zv jo db’,

‘book wet theme’ is coded as ‘xh im jo’.

Here, the common word in both sentences is ‘book’.

Now, look at the codes:

The common code in both code groups is ‘jo’.

So, Code for ‘book’ is ‘jo’.

Hence, the correct answer is "Option 3".

Given Statements:

All pan are wok.

All cauldron are wok.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some pan are cauldron - False (Because, no definite relation is given between pan and cauldron. They may or may not overlap. Hence, it is possible only.)

II. Some wok are cauldron - True (Because, all cauldron are wok, so definitely some wok are cauldron.)

So, Only conclusion II follows.

Hence, the correct answer is "Option 4".

Given:

Dev is 10th from the top,

Dev is 31st from the bottom.

Total number of students = 10 + 31 − 1 = 40.

So, there are '40' students in the class.

Hence, the correct answer is "Option 1".

The logic followed here is:

So, 'KAX' will replace the question mark in the given series.

Hence, the correct answer is "Option 1".

The logic followed here is:

Option 1) HEC

Option 2) DBY

Option 3) ZWU

Option 4) EBZ

Here, 'DBY' is odd among the given options.

Hence, the correct answer is "Option 2".

The logic followed here is:

So, 'RAN14' will replace the question mark in the given series.

Hence, the correct answer is "Option 3".

The position and positional values or English Alphabets are:

ECHO → CHEO → OHCE

URGE → RGUE → EGRU

Now,

ECHO → CHEO

Here, the 1st letter shifts to 3rd position,

the 2nd letter shifts to 1st position,

the 3rd letter shifts to 4th position,

the 4th letter shifts to 2nd position.

So, the pattern is: 2 4 1 3 (based on positions).

CHEO → OHCE

Again, the same positional pattern 2 4 1 3 is followed.

Now, let's check each Option One by One.

Option 1) JOYS → OYJS → OYSJ

Option 2) NAVY → ANVY → YAVN

Option 3) WAIT → AIWT → TIAW

Option 4) PAST → PSAT → TSAP

So, Only WAIT → AIWT → TIAW follows the pattern of given letter cluster.

Hence, the correct answer is 'Option 3'.

Given: 

Kareena starts from Point A and drives 16 km towards the east.

She then takes a right turn, drives 7 km, turns right and drives 7 km.

She then takes a left turn and drives 8 km.

She takes a final right turn, drives 9 km and stops at Point P.

So, She should drive 15 km towards the north in order to reach Point A again.

Hence, the correct answer is "Option 4".

Given: Seven people B, C, D, I, J, K and L are sitting around a circular table facing the centre of the table.

I sits fourth to the right of J.

D sits third to the left of K.

Only two people sit between B and C when counted from the left of B.

L is an immediate neighbour of K and C.

Here,There are Four people between I and L when counted from the left of L.

Hence, the correct answer is "Option 4".

BODMAS Table: 

The logic followed here is:

Given equation: 60 F 12 H 3 E 8 G 2 = ?

Now, if F stands for ‘−’, H stands for ‘÷’, E stands for ‘+’ and G stands for ‘×’, then:

⇒ 60 − 12 ÷ 3 + 8 × 2 = ?

⇒ 60 − 4 + 16 = ?

⇒ 56 + 16 = ?

⇒ 72

Hence, the correct answer is "Option 3".

Given: (Left) 5 4 2 3 7 8 5 1 6 7 9 4 2 6 7 6 3 9 7 5 4 1 6 9 8 3 4 (Right)

Now, 

Odd digits  which is immediately preceded by an even digit and also immediately followed by an odd digit.

So, Needed pattern is: Even - Odd - Odd.

(Left) 5 4 2 3 7 8 5 1 6 7 9 4 2 6 7 6 3 9 7 5 4 1 6 9 8 3 4(Right)

There are More than Three Odd digits  which is immediately preceded by an even digit and also immediately followed by an odd digit.

Hence, the correct answer is 'Option 1'.

Given: 60 people.

Mayank is 13th from the right end,

Kishan is 18th from the left end.

Position of Mayank from the left = 60 − 13 + 1 = 48.

So, people between Kishan and Mayank = 48 − 18 − 1 = 29.

So, there are '29' people standing between Mayank and Kishan.

Hence, the correct answer is "Option 2".

The logic followed here is:

For, EQ 9 is related to HL −4

For, MR 13 is related to PM 0

Similarly, for MU 19

So, MU 19 will be related to PP 6.

Hence, the correct answer is "Option 1".

Given: 

Mr. Zzz starts from Point A and drives 5 km towards the East.

He then takes a right turn, drives 14 km, turns right and drives 16 km.

He then takes a right turn and drives 16 km.

He takes a final right turn, drives 11 km and stops at Point P.

So, He should drive 2 km to the South  in order to reach Point A again.

Hence, the correct answer is "Option 1".

The logic followed here is:

thus, "18" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given Statements:

All cars are buildings.

All buildings are papers.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some buildings are cars - True (Because, all cars are buildings, so definitely some buildings are cars.)

II. All cars are papers - True (Because, all cars are buildings and all buildings are papers. Therefore, all cars are papers.)

So, Both conclusions I and II follow.

Hence, the correct answer is "Option 2".

The logic followed here is:

‘buyer queen heart’ is coded as ‘os ld xi’,

‘drive heart aunt’ is coded as ‘kx to os’.

Here, the common word in both sentences is ‘heart’.

Now, look at the codes:

The common code in both code groups is ‘os’.

So, Code for ‘heart’ is ‘os’.

Hence, the correct answer is "Option 3".

BODMAS Table: 

Given equation:

77 − 5 ÷ 49 + 21 × 7 = ?

Now, if ‘+’ means ‘−’, ‘−’ means ‘×’, ‘×’ means ‘÷’ and ‘÷’ means ‘+’, then:

⇒ 77 × 5 + 49 − 21 ÷ 7 = ?

Now apply BODMAS:

⇒ 77 × 5 + 49 − 3 = ?

⇒ 385 + 49 − 3 = ?

⇒ 434 − 3 = 431.

Hence, the correct answer is "Option 4".

Given number: 13645928

Arranging the digits in descending order (from left to right), we get

9 8 6 5 4 3 2 1

Now, compare positions with the original number:

Original: 1 3 6 4 5 9 2 8

New: 9 8 6 5 4 3 2 1

Comparing each position:

1 ≠ 9

3 ≠ 8

6 = 6 

4 ≠ 5

5 ≠ 4

9 ≠ 3

2 = 2 

8 ≠ 1

So, two digits (6 and 2) remain in the same position.

Hence, the correct answer is Option 4.

A, B, C, D, E, F and G are sitting around a circular table facing the centre. 

1. E sits third to the right of G. 2. A sits third to the right of E. 3. D is the immediate neighbour of B and A. 4. C sits to the immediate right of F.

Two people sit between F and B when counted from the left of B.

Hence, the correct answer is “Option 1”.