RRB Group D Official Paper (Held On: 05 Feb, 2026 Shift 2) (Previous Year Paper)

The correct answer is Irrigation at right stage of crop.

Key Points

• The expected yield of any crop is highly dependent on the availability of water at specific physiological periods known as Critical Growth Stages. Providing water at the right stage ensures that the plant does not undergo moisture stress during vital development phases. Hence, the correct option is Irrigation at right stage of crop.
• Irrigation at early stages of the crop is important for seed germination and seedling establishment, but if subsequent critical stages are neglected, the overall yield will drop significantly. Hence, statement 1 is only partially beneficial.
• Irrigation at the right stage refers to identifying the periods when the plant is most sensitive to water deficits, such as flowering, fruiting, or grain filling. Applying water during these windows maximizes productivity and resource efficiency. Hence, statement 2 is correct.
• Irrigation at the terminal stage (the final phase of the crop cycle) might help in maturing the seeds, but it cannot compensate for water stress experienced during the reproductive phase. Hence, statement 3 is incorrect as a standalone measure to increase yield.
• Irrigation at every stage of the crop is often counterproductive and leads to waterlogging, which deprives roots of oxygen, increases the risk of fungal diseases, and wastes water resources. Hence, statement 4 is incorrect.

Additional Information

• Critical Irrigation Stages for Various Crops:
• Wheat: The most critical stage is the Crown Root Initiation (CRI), which typically occurs 21 days after sowing. Other stages include tillering, flowering, and milk stage.
• Rice: Critical stages include the panicle initiation, flowering, and heading stages where moisture is vital for grain count.
• Maize: The tasseling and silking stages are highly sensitive to water stress.
• Pulses: These crops generally require irrigation at the flower initiation and pod development stages.
• Irrigation Scheduling: It is the process of deciding when to irrigate and how much water to apply. This is based on soil moisture levels, evapotranspiration rates, and the specific growth stage of the crop.
• Water Use Efficiency (WUE): This is a measure of a crop's ability to convert water into biomass or yield. Effective irrigation management at the right stage improves the WUE, ensuring sustainable agriculture.

Important Points

• Impact of Over-Irrigation: Excessive water application leads to leaching of nutrients, salinity build-up in the soil, and reduced soil aeration, all of which negatively impact the final yield.
• Precision Irrigation: Technologies like Drip Irrigation and Sprinkler Systems allow farmers to apply water precisely at the right stage and in the correct quantity, reducing water wastage by up to 40-50%.
• Soil Type Influence: The timing of irrigation also depends on soil texture; for example, sandy soils require more frequent irrigation compared to clayey soils due to lower water-holding capacity.
• Climate Factors: High temperatures and low humidity increase the plant's transpiration rate, requiring more frequent checks on whether the crop has reached a critical stage for water.

The correct answer is positive z direction.

Key Points

• When a straight conductor carrying an electric current is placed within an external magnetic field, it experiences a mechanical force. This phenomenon is a fundamental principle of electromagnetism.
• The direction of this force is always perpendicular to both the direction of the current and the direction of the magnetic field.
• The relationship between these vectors is defined by the Lorentz Force equation, expressed as F = I (L × B), where F is the force vector, I is the current, L is the length vector in the direction of current, and B is the magnetic field vector.
• In this specific problem, the current is moving in the +x-direction (represented by the unit vector i) and the magnetic field is oriented in the +y-direction (represented by the unit vector j).
• By applying the mathematical cross product rules for Cartesian coordinates, we find that i × j = k. Since k represents the positive z-axis, the resulting motion of the conductor will be in the positive z direction. Hence, the statement is correct.

Additional Information

• Fleming's Left-Hand Rule: This is a practical mnemonic used to find the direction of force acting on a conductor. If you stretch the thumb, forefinger, and middle finger of your left hand so they are mutually perpendicular:
• The Forefinger represents the direction of the Magnetic Field.
• The Middle finger represents the direction of the Current.
• The Thumb points in the direction of the Force or Motion.
• Magnitude of Magnetic Force: The amount of force experienced by the wire depends on the current (I), the length (L) of the wire, the strength of the magnetic field (B), and the sine of the angle (θ) between the current and the field. The formula is F = BIL sinθ.
• Right-Hand Palm Rule: Another method where the fingers point in the direction of the field and the thumb points in the direction of the current; the palm then pushes in the direction of the magnetic force.
• Applications: This principle is the working basis for Electric Motors, where electrical energy is converted into mechanical energy. It is also used in galvanometers, loudspeakers, and magnetic levitation systems.

Important Points

• Maximum Force: The conductor experiences the maximum force when it is placed perpendicular (at an angle of 90 degrees) to the magnetic field lines. In this orientation, sin 90° = 1.
• Zero Force: If the conductor is placed parallel or anti-parallel to the magnetic field (at an angle of 0 or 180 degrees), the force exerted on it is zero because sin 0° = 0.
• Force on Moving Charges: The force on a single charge (q) moving with a velocity (v) in a magnetic field is given by F = q(v × B). A current-carrying wire is simply a collection of many such moving charges.
• Newton’s Third Law: Just as the magnetic field exerts a force on the current-carrying wire, the wire also exerts an equal and opposite magnetic force on the source of the magnetic field (such as a permanent magnet).

The correct answer is To help sound reflect and reach all parts of the hall.

Key Points

• The ceilings of concert halls, cinema halls, and conference rooms are usually curved because sound, after reflecting from the curved surface, spreads uniformly in all directions.
• This design ensures that the sound waves reach every corner of the hall, including the last rows of the audience, providing a consistent auditory experience for everyone. Hence, the correct answer is option 2.
• A curved surface acts like a large reflector. According to the laws of reflection of sound, the angle of incidence is equal to the angle of reflection. A curved ceiling is specifically engineered so that reflected sound is directed downward toward the audience rather than being trapped or scattered inefficiently.
• If the ceilings were flat, the sound might reflect at angles that do not cover the entire seating area, leading to "dead zones" where the audio is faint or unintelligible.
• The use of curved ceilings is a practical application of the reflection of sound, ensuring that the energy of the longitudinal waves is preserved and directed purposefully throughout the large volume of the building.

Additional Information

• Soundboard: In many large halls, a curved soundboard is placed behind the stage. The source of sound (the speaker or musical instrument) is placed at the focus of this curved board. The sound reflects off the board and travels as parallel beams to reach the distant audience clearly.
• Reverberation: This refers to the persistence of sound after the source has stopped, caused by multiple reflections. While some reflection is necessary for clarity (as provided by curved ceilings), excessive reflection can lead to echoes that blur the sound. Architects use sound-absorbent materials like compressed fiberboards or heavy curtains on walls to balance this.
• Reflection of Sound: Like light, sound reflects off hard surfaces. The quality of reflection depends on the texture and shape of the surface. Concave surfaces tend to converge sound waves, while convex surfaces diverge them.
• Acoustics: This is the branch of physics that deals with the study of mechanical waves in gases, liquids, and solids, including vibration, sound, ultrasound, and infrasound. Modern architectural acoustics focuses heavily on the geometry of ceilings and walls to optimize sound quality.
• Speed of Sound: It is important to note that the speed of sound remains constant in the hall (approx. 343 m/s in air), but the pathway taken by the waves is what is altered by the curvature of the ceiling.

The correct answer is A compound always contains elements in a fixed ratio by mass.

Key Points

• The Law of Constant Proportions, also known as the Law of Definite Proportions, was proposed by the French chemist Joseph Proust in 1799.
• The law states that in a chemical substance, the constituent elements are always present in definite proportions by mass, regardless of the source from which the compound is obtained or the method used for its preparation.
• Statement 1: Compounds can vary their composition based on temperature. This is incorrect because chemical compounds have a fixed stoichiometry that remains constant under standard chemical conditions, unlike physical mixtures whose proportions can vary. Hence, statement 1 is incorrect.
• Statement 2: The mass of a compound is always greater than the sum of its elements. This contradicts the Law of Conservation of Mass established by Antoine Lavoisier, which states that mass is neither created nor destroyed during a chemical reaction. Hence, statement 2 is incorrect.
• Statement 3: A compound always contains elements in a fixed ratio by mass. This is the fundamental definition of Proust's law, ensuring that every molecule of a specific substance has the same elemental makeup. Hence, statement 3 is correct.
• Statement 4: Elements combined in random ratios to form compounds. Elements combine in fixed, whole-number ratios of atoms to form compounds; random ratios are characteristic of mixtures, not pure chemical substances. Hence, statement 4 is incorrect.

Additional Information

• Example of Water (H2O): In a molecule of water, the ratio of the mass of hydrogen to the mass of oxygen is always 1:8. This means that if you decompose 9 grams of water, you will always obtain 1 gram of hydrogen and 8 grams of oxygen.
• Example of Ammonia (NH3): In ammonia, the elements nitrogen and hydrogen are always present in the fixed ratio of 14:3 by mass.
• Dalton's Atomic Theory: In 1808, John Dalton provided a theoretical explanation for the law of constant proportions. One of his postulates states that the relative number and kinds of atoms are constant in a given compound.
• Exceptions to the Law: While the law holds for most compounds, there are exceptions known as non-stoichiometric compounds (or Berthollides), such as Ferrous Oxide (FeO), where the ratio of elements can vary within a narrow range. Additionally, isotopes of elements can lead to slightly different mass ratios in compounds, though the atomic ratio remains the same.

Important Points

• Laws of Chemical Combination: The Law of Constant Proportions is one of the fundamental Laws of Chemical Combination, which also includes the Law of Conservation of Mass and the Law of Multiple Proportions.
• Experimental Verification: Joseph Proust verified this law by analyzing cupric carbonate. He compared a natural sample and a synthetic sample and found that both contained the exact same percentage of copper, carbon, and oxygen.
• Scientific Significance: This law is vital for stoichiometric calculations in chemistry, allowing scientists to determine the empirical and molecular formulas of unknown substances and calculate the amounts of reactants and products in a chemical reaction.

The correct answer is every physical process in general.

Key Points

• The Law of Conservation of Energy is a fundamental principle of physics which states that the total energy of an isolated system remains constant; it is said to be conserved over time.
• Energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance, in a falling object, potential energy is converted into kinetic energy.
• The principle is applicable to every physical process in general because energy is a universal currency in all interactions, whether they are mechanical, electrical, chemical, thermal, or nuclear. Hence, option 4 is correct.
• Options 1, 2, and 3 are incorrect because they use restrictive terms like "only" or "exclusively." While the law applies to mechanical, electrical, and chemical systems, it is not limited to them. Hence, statements 1, 2, and 3 are incorrect.
• In any transformation, if we account for all forms of energy—including those dissipated as heat, sound, or friction—the sum remains the same.

Additional Information

• First Law of Thermodynamics: This is a specific version of the conservation of energy principle applied to thermodynamic systems. It states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
• Forms of Energy: Energy exists in various forms such as Mechanical (Kinetic and Potential), Thermal (heat), Chemical (stored in bonds), Electrical (moving charges), Radiant (light), and Nuclear (stored in the nucleus of an atom).
• Energy Transformation Examples:
• In a battery, chemical energy is converted into electrical energy.
• In an electric motor, electrical energy is converted into mechanical energy.
• In photosynthesis, plants convert solar energy into chemical energy.
• Mass-Energy Equivalence: In modern physics, Albert Einstein expanded this principle through the formula E = mc², which suggests that mass and energy are interconvertible. This is particularly relevant in nuclear reactions (fission and fusion) where a small amount of mass is converted into a huge amount of energy.

Important Points

• Isolated System: A system that does not exchange matter or energy with its surroundings. The universe itself is often considered the ultimate isolated system where total energy is constant.
• Noether’s Theorem: In theoretical physics, the conservation of energy is a mathematical consequence of time-translation symmetry. This means that the laws of physics do not change with time; therefore, energy must be conserved.
• Perpetual Motion Machines: The law of conservation of energy confirms that a perpetual motion machine of the first kind (a machine that produces work without energy input) is physically impossible.
• Efficiency: While energy is always conserved, it is not always conserved in a useful form. In most real-world processes, some energy is "lost" as waste heat due to entropy, though it still exists within the total system energy count.

The correct answer is Intercalary meristem.

Key Points

• Intercalary meristems are specifically located at the base of leaves or internodes (near the nodes) in certain plants.
• These tissues are essentially portions of the apical meristem that have been separated from the apex by the development of permanent tissues during the growth process.
• They play a crucial role in the elongation of the plant body and are responsible for the longitudinal growth of the internodes.
• In grasses and other monocots, the intercalary meristem allows for the regrowth of stems and leaves after they have been damaged or consumed by herbivorous animals.
• Unlike apical meristems which are at the tips, the intercalary meristem provides a mechanism for growth within the middle regions of the plant structure.

Additional Information

• Apical Meristem:
• Located at the growing tips of roots and shoots.
• It is responsible for the increase in the length of the plant, a process known as primary growth.
• It helps the plant explore more soil for water and reach higher for sunlight.
• Lateral Meristem:
• Found on the lateral sides of the stems and roots, running parallel to the long axis.
• It facilitates secondary growth, which results in an increase in the girth or thickness of the plant.
• Examples include the vascular cambium and the cork cambium (phellogen) in woody plants.
• Characteristics of Meristematic Cells:
• The cells are actively dividing and are usually small, isodiametric, or spherical in shape.
• They possess thin cellulose cell walls, dense cytoplasm, and a prominent nucleus.
• These cells generally lack vacuoles because they are focused on multiplication rather than storage or structural support.
• Root Meristem:
• Specifically refers to the apical meristem located at the root apex.
• It is often protected by a root cap to prevent damage as the root pushes through the soil.

Important Points

• Classification: Meristems are classified based on their position in the plant body into three primary types: Apical, Intercalary, and Lateral.
• Linguistic Context: The word "meristem" is derived from the Greek word 'meristos', which means divisible, highlighting the tissue's capacity for continuous cell division.
• Growth Cycles: While primary meristems (Apical and Intercalary) appear early in the life of a plant, secondary meristems (Lateral) usually appear later to provide structural thickness.

The correct answer is Na2CO3.10H2O.

Key Points

• Washing soda is a chemical compound with the formula Na2CO3.10H2O. Its chemical name is sodium carbonate decahydrate.
• It is a crystalline solid that contains ten molecules of water of crystallization attached to one molecule of sodium carbonate. Hence, option 4 is correct.
• It is produced via the Solvay process, where sodium chloride, ammonia, and carbon dioxide react to form sodium hydrogen carbonate, which is then heated and recrystallized.
• Washing soda is a basic salt, meaning it forms an alkaline solution when dissolved in water, turning red litmus blue.
• The anhydrous form of this compound (without water) is known as soda ash or sodium carbonate (Na2CO3). Hence, option 3 is incorrect as it lacks the hydration necessary for washing soda.

Additional Information

• Sodium Hydroxide (NaOH):
• Commonly known as caustic soda, it is a strong base used in the manufacture of soaps, detergents, paper, and synthetic fibers like rayon. Hence, option 1 is incorrect.
• Sodium Hydrogen Carbonate (NaHCO3):
• Commonly known as baking soda, it is used in the food industry for baking and as an antacid to treat indigestion. It also acts as a primary component in soda-acid fire extinguishers. Hence, option 2 is incorrect.
• Uses of Washing Soda:
• It is primarily used in the glass, soap, and paper industries.
• A major application is the removal of permanent hardness of water, which is caused by dissolved calcium and magnesium salts.
• It is used in the manufacturing of sodium compounds like borax and serves as a vital cleansing agent for domestic laundry purposes.
• Chemical Properties:
• When exposed to air, washing soda undergoes efflorescence, losing nine molecules of water to form sodium carbonate monohydrate (Na2CO3.H2O).
• It is highly soluble in water and its solution is alkaline in nature, which aids in cutting through grease and stains during cleaning.

Important Points

• History: The Solvay process was developed in the 1860s by Ernest Solvay, revolutionizing the industrial production of sodium carbonate.
• Water of Crystallization: This refers to the fixed number of water molecules that form part of the structure of a crystal. In washing soda, there are exactly 10 such molecules.
• Environmental Impact: Washing soda is considered more environmentally friendly than many synthetic detergents as it is biodegradable and does not contain phosphates that lead to eutrophication.

The correct answer is Four cells, half number.

Key Points

• Meiosis is a specialized form of cell division that results in the formation of four daughter cells, each with half the number of chromosomes of the parent cell.
• It is known as reduction division because it reduces the chromosome ploidy from diploid (2n) to haploid (n). Hence, the outcome "Four cells, half number" identifies meiosis.
• This process involves two successive stages of nuclear and cell division, known as Meiosis I and Meiosis II, but only one cycle of DNA replication occurs.
• Meiosis is essential for sexual reproduction as it produces gametes (sperm and egg cells) in animals and spores in plants and fungi.
• The primary purpose of meiosis is to ensure that the chromosome number remains constant across generations after fertilization occurs.

Additional Information

• Mitosis vs. Meiosis:
• Mitosis results in two identical daughter cells with the same number of chromosomes as the parent cell (2n to 2n). It is used for growth, tissue repair, and asexual reproduction.
• In contrast, Meiosis results in genetic variation due to processes like crossing over and independent assortment.
• Phases of Meiosis:
• Meiosis I: Includes Prophase I, Metaphase I, Anaphase I, and Telophase I. This is the stage where homologous chromosomes separate.
• Meiosis II: Similar to mitosis, where sister chromatids separate into different cells, resulting in the final four haploid cells.
• Genetic Diversity:
• During Prophase I, homologous chromosomes pair up (synapsis) and exchange genetic material in a process called recombination or crossing over.
• This ensures that every gamete produced is genetically unique, which is a key driver of evolution and biological diversity.
• Human Context:
• In humans, somatic cells have 46 chromosomes (diploid). After meiosis, the resulting sperm or egg cells contain only 23 chromosomes (haploid).
• Failure in proper chromosome separation during meiosis can lead to conditions like Down Syndrome (Trisomy 21).

Important Points

• Location: In animals, meiosis occurs in the gonads (testes in males and ovaries in females).
• Functions: It facilitates segregation of genes and maintains the integrity of the species genome.
• Discovery: The process was first described by Oscar Hertwig in 1876 in sea urchin eggs.

The correct answer is only B and C.

Key Points

• Statement A: This statement suggests that mass can be changed (created or destroyed) when a reaction takes place. According to the Law of Conservation of Mass, the total mass of the reactants must equal the total mass of the products. Mass is neither created nor destroyed in a chemical reaction. Hence, statement A is incorrect.
• Statement B: In a chemical reaction, the bonds between atoms are broken and new bonds are formed. This process involves the rearrangement of atoms to transform reactants into different substances called products, but the types and number of atoms remain the same. Hence, statement B is correct.
• Statement C: This statement defines the core principle of the law within the context of a closed system. In such a system, where no matter or energy (specifically matter in this context) can escape or enter, the total mass remains constant over time regardless of the processes acting inside the system. Hence, statement C is correct.
• Hence, the correct answer is only B and C.

Additional Information

• Definition and Origin:
• The Law of Conservation of Mass was established by the French chemist Antoine Lavoisier in 1789. It is a fundamental principle in chemistry and physics.
• It states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so quantity can neither be added nor be removed.
• Chemical Equations:
• This law is the reason why chemical equations must be balanced. A balanced equation has the same number of atoms of each element on both the reactant side and the product side.
• Example: In the combustion of methane (CH4 + 2O2 → CO2 + 2H2O), the mass of carbon, hydrogen, and oxygen atoms before the reaction is exactly the same as the mass after the reaction.
• Features of the Law:
• Closed System: A physical system that does not allow transfer of matter in or out of the system.
• Reactants and Products: The initial substances in a reaction are reactants, and the resultant substances are products. Their total weight remains identical.

Important Points

• Historical Context: Antoine Lavoisier is often referred to as the "Father of Modern Chemistry" because his quantitative experiments led to the discovery of this law, debunking the phlogiston theory.
• Nuclear Reactions: It is important to note that in nuclear reactions, the law is modified. According to Albert Einstein's equation E=mc², mass can be converted into energy. Therefore, in the context of modern physics, we refer to the Law of Conservation of Mass-Energy.
• Application: This law is widely used in fluid mechanics, chemical engineering, and environmental science to calculate mass balances in various industrial processes.

The correct answer is Safe sex practices and awareness about contraception.

Key Points

• Reproductive health is defined by the World Health Organization (WHO) as a total well-being in all aspects of reproduction, including physical, emotional, behavioral, and social dimensions.
• Maintaining safe sex practices is a fundamental requirement as it prevents the spread of Sexually Transmitted Infections (STIs) such as syphilis, gonorrhea, genital herpes, and life-threatening diseases like HIV/AIDS.
• Awareness about contraception is essential for family planning, allowing individuals to decide the number and spacing of their children, which directly impacts the socio-economic status of a family and the nation. Hence, statement 3 is correct.
• Sharing personal hygiene products (like towels or undergarments) is highly discouraged as it can lead to the transmission of fungal infections, skin diseases, and UTIs. Hence, statement 1 is incorrect.
• Monitoring and understanding the physical and hormonal changes during adolescence is vital for mental health and early detection of reproductive issues. Ignoring these changes can lead to anxiety or undiagnosed medical conditions. Hence, statement 2 is incorrect.
• Medical intervention and the use of prescribed medicines (such as antibiotics for infections or hormonal therapy) are often necessary to maintain reproductive health; thus, avoiding all medicines is counterproductive. Hence, statement 4 is incorrect.

Additional Information

• Adolescence and Puberty:
• Adolescence is the period of life between 10 to 19 years when the body undergoes changes leading to reproductive maturity.
• During puberty, the testes and ovaries begin to produce gametes (sperm and ova) and hormones like testosterone (in males) and estrogen/progesterone (in females).
• Menstrual Hygiene:
• Maintenance of cleanliness during the menstrual cycle is crucial to prevent bacterial infections and Pelvic Inflammatory Disease (PID).
• The use of sanitary napkins or clean menstrual cups and changing them every 4-6 hours is a standard health recommendation.
• Nutritional Needs:
• A balanced diet containing Iron (to prevent anemia), Calcium, and Proteins is particularly important for pregnant women and adolescent girls to support reproductive functions.
• Government Initiatives:
• In India, the Reproductive and Child Health (RCH) program was launched to create awareness and provide medical facilities for maternal health, infant mortality reduction, and birth control.
• The legal age for marriage in India (currently 18 for females and 21 for males) is also designed to ensure that individuals are physiologically and mentally prepared for reproduction.

Important Points

• Amniocentesis: This is a medical technique used to determine chromosomal abnormalities in the fetus. However, its misuse for sex determination led to a statutory ban in India under the Pre-Conception and Pre-Natal Diagnostic Techniques (PCPNDT) Act, 1994.
• Population Explosion: Lack of reproductive health awareness leads to rapid population growth. Methods like IUCDs (Intrauterine Contraceptive Devices), Oral Pills, and Surgical Methods (Vasectomy and Tubectomy) are tools used to stabilize population growth.
• Lactational Amenorrhea: A natural method of contraception where ovulation does not occur during the period of intense lactation following childbirth, though it is only effective for a maximum of six months.

The correct answer is Rainwater harvesting.

Key Points

• Judicious use of natural resources refers to the sustainable and efficient management of resources to meet present needs without compromising the ability of future generations to meet their own needs.
• Rainwater harvesting is a method of water conservation where precipitation is collected and stored for later use, rather than being allowed to run off. It helps in recharging groundwater levels and reduces the burden on municipal water supplies. Hence, statement 1 is correct.
• Open mining (also known as open-pit mining) involves the extraction of minerals from an open pit in the ground. This process often leads to significant environmental degradation, habitat destruction, and soil erosion. Hence, statement 2 is incorrect.
• Excessive use of fossil fuels like coal, petroleum, and natural gas leads to the depletion of non-renewable resources and increases carbon emissions, contributing to global warming. Hence, statement 3 is incorrect.
• Deforestation is the permanent removal of trees to make room for something besides forest. This results in loss of biodiversity, disruption of the water cycle, and increased atmospheric carbon dioxide. Hence, statement 4 is incorrect.
• Hence, the correct answer is Rainwater harvesting.

Additional Information

• Natural Resources:
• These are substances that occur naturally within environments that exist relatively undisturbed by mankind, in a natural form.
• They are classified into Renewable Resources (e.g., solar energy, wind, water) which can be replenished naturally over time, and Non-renewable Resources (e.g., minerals, fossil fuels) which exist in finite amounts and take millions of years to form.
• Sustainable Development:
• This concept was popularized by the Brundtland Commission in 1987. It emphasizes the judicious use of resources to maintain ecological balance.
• Methods include practicing the 3Rs: Reduce, Reuse, and Recycle to minimize waste generation and resource consumption.
• Rainwater Harvesting Techniques:
• Rooftop harvesting: Collecting rain from the roofs of buildings and storing it in tanks or diverting it to recharge pits.
• Surface runoff harvesting: Capturing rainwater that flows along the ground surface during monsoon seasons to replenish aquifers.
• In India, states like Tamil Nadu have made rainwater harvesting mandatory for all buildings to combat water scarcity.

Important Points

• Conservation of Water: Since only about 2.5% of the Earth's water is freshwater, and most of that is trapped in glaciers, the judicious use of available water through harvesting is critical for agriculture and domestic consumption.
• Impact of Mismanagement: Over-exploitation of groundwater for irrigation has led to a drastic drop in the water table in several parts of the world, including the Indo-Gangetic plains.
• Energy Conservation: Shifting from fossil fuels to clean energy sources like hydroelectric power or biomass is another example of the judicious use of natural resources.

The correct answer is Dendrite.

Key Points

• Dendrites are the protoplasmic extensions of a nerve cell (neuron) that propagate the electrochemical stimulation received from other neural cells to the cell body, or soma.
• They act as the primary receptors or antennae of the neuron, featuring a highly branched, tree-like structure that significantly increases the surface area available for receiving signals. Hence, the statement that a neuron receives impulses through dendrites is correct.
• Electrical signals are transmitted onto the dendrites by upstream neurons via synapses located at various points throughout the dendritic tree.
• The direction of a nerve impulse is always unidirectional in a functional circuit: it enters through the dendrite, passes through the cell body, and exits via the axon.
• Dendrites play a critical role in integrating these incoming signals to determine whether the neuron will fire an action potential.

Additional Information

• Cell Body (Soma):
• The cell body is the spherical part of the neuron that contains the nucleus and other specialized organelles like mitochondria, Golgi apparatus, and Nissl bodies.
• Its primary function is to maintain the cell's metabolic processes and integrate the electrical signals received from the dendrites.
• While it can receive some direct synaptic inputs, it is not the primary specialized structure for receiving impulses from other neurons. Hence, the option Cell body is not the primary answer.
• Axon:
• An axon is a long, slender projection of a nerve cell that typically conducts electrical impulses away from the neuron's cell body.
• The point where the axon joins the cell body is called the axon hillock, where the action potential is usually generated.
• Axons can be very long, sometimes extending over 1 meter in humans (such as the sciatic nerve). Hence, the option Axon is incorrect as it transmits signals away, not receives them.
• Synapse:
• A synapse is the microscopic junction or gap between two neurons, or between a neuron and an effector cell (like a muscle or gland).
• Information is transferred across the synaptic cleft using chemical messengers known as neurotransmitters, such as acetylcholine or dopamine.
• While the impulse travels through the synapse, the synapse itself is the location of the transfer, whereas the dendrite is the specific cellular structure that receives the signal. Hence, the option Synapse refers to the junction rather than the receiving structure.

Important Points

• Myelin Sheath: This is an insulating layer, or sheath that forms around nerves, including those in the brain and spinal cord. It is made up of protein and fatty substances. It allows electrical impulses to transmit quickly and efficiently along the nerve cells.
• Nodes of Ranvier: These are the periodic gaps in the myelin sheath where the axon is exposed to the extracellular space, facilitating saltatory conduction (the "jumping" of impulses).
• Schwann Cells: These are the cells in the Peripheral Nervous System (PNS) responsible for the formation of the myelin sheath.
• Action Potential: A rapid sequence of changes in the voltage across a membrane. The entire process takes about 2 to 5 milliseconds.

The correct answer is Ff = (F1 - F2)/2.

Key Points

• According to Newton's Second Law of Motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This is represented by the formula F_net = M * a.
• In the first scenario, where friction is zero, the block of mass M is acted upon by two opposing forces, F1 and F2. Since F1 > F2, the net force is F1 - F2. The resulting acceleration, let's call it a1, is (F1 - F2) / M.
• In the second scenario, a frictional force Ff is introduced. The problem states that Ff acts opposite to F1. This means Ff acts in the same direction as F2. The new net force acting on the block becomes F1 - F2 - Ff.
• The resulting acceleration with friction, let's call it a2, is calculated as a2 = (F1 - F2 - Ff) / M.
• The problem provides a specific condition: the net acceleration of the block with friction (a2) is half the acceleration it would have experienced without friction (a1). Therefore, a2 = a1 / 2.
• Substituting the expressions for acceleration into this condition: (F1 - F2 - Ff) / M = [(F1 - F2) / M] / 2.
• By canceling the mass M from both sides, we get: F1 - F2 - Ff = (F1 - F2) / 2.
• Rearranging the equation to solve for Ff: Ff = (F1 - F2) - (F1 - F2) / 2. This simplifies to Ff = (F1 - F2) / 2.

Additional Information

• Newton's Second Law: It defines the relationship between an object's mass, its acceleration, and the applied force. It is a fundamental principle in classical mechanics and was published in 1687.
• Frictional Force (Ff): Friction is a force that resists the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. In this problem, kinetic friction is implied as the block is accelerating.
• Net Force: This is the vector sum of all forces acting on an object. When forces are collinear (acting along the same line), they can be added or subtracted as scalars based on their direction.
• Mass (M): A fundamental property of matter that represents its inertia or resistance to acceleration when a net force is applied. In the SI system, it is measured in kilograms (kg).
• Opposing Forces: When two forces act in exactly opposite directions (180 degrees apart), the magnitude of the resultant force is the difference between the magnitudes of the individual forces.

Important Points

• Direction of Motion: Since F1 > F2, the block will initially accelerate in the direction of F1. Therefore, friction, which always opposes motion, must act in the direction of F2.
• Impact of Friction: The introduction of friction reduces the net force acting on the body, which directly leads to a decrease in the magnitude of acceleration if the mass remains constant.
• Dimensional Consistency: In any physical equation like Ff = (F1 - F2) / 2, both sides of the equation must have the same dimensions. Here, both sides represent Force, measured in Newtons (N).
• Vector Representation: While this problem treats forces algebraically, forces are vectors requiring both magnitude and direction for a complete description.

The correct answer is Benzene.

Key Points

• A ring-chain hydrocarbon, also known as a cyclic hydrocarbon, is a compound where the carbon atoms are linked to each other in a closed-loop or ring structure.
• Benzene is the most prominent example of a cyclic hydrocarbon with the molecular formula C6H6. It consists of six carbon atoms arranged in a planar hexagonal ring with alternating double bonds. Hence, option 2 is correct.
• Propane is an acyclic (open-chain) alkane with the chemical formula C3H8. Its carbon atoms are arranged in a straight sequence without forming a loop. Hence, option 1 is incorrect.
• Ethene, also known as ethylene, is an open-chain alkene with the formula C2H4. It contains two carbon atoms joined by a double bond in a linear fashion. Hence, option 3 is incorrect.
• Heptyne is an open-chain alkyne with the formula C7H12. It consists of a chain of seven carbon atoms containing at least one triple bond, but it does not form a ring. Hence, option 4 is incorrect.

Additional Information

• Classification of Hydrocarbons:
• Acyclic Hydrocarbons: These are open-chain compounds that can be saturated (Alkanes) or unsaturated (Alkenes and Alkynes).
• Cyclic Hydrocarbons: These contain at least one ring of carbon atoms. They are subdivided into Alicyclic (like Cyclohexane) and Aromatic (like Benzene).
• Aromatic Hydrocarbons:
• These are special cyclic compounds that possess high stability due to resonance. Benzene is the parent member of this family.
• They must follow Hückel's Rule, which requires 4n + 2 pi electrons in a conjugated, planar ring system.
• Physical and Chemical Properties of Benzene:
• Benzene was first isolated by Michael Faraday in 1825. Its structural formula was later proposed by August Kekulé in 1865.
• It is a colorless liquid at room temperature with a characteristic aromatic odor. It has a boiling point of 80.1 °C.
• It is widely used in the industrial manufacture of plastics, resins, synthetic fibers, rubber lubricants, dyes, detergents, and pesticides.

Important Points

• Hybridization: In the Benzene ring, each carbon atom is sp2 hybridized, which results in the planar geometry of the molecule.
• Resonance: The pi electrons in Benzene are delocalized over the entire ring, meaning they are not fixed between any two specific carbon atoms. This provides the molecule with extraordinarily high stability compared to typical alkenes.
• Health Hazards: While useful industrially, Benzene is a highly toxic and carcinogenic substance. Long-term exposure is known to cause leukemia and other blood-related disorders.
• Numerical Data: The C-C bond length in Benzene is exactly 1.39 Å, which is the mean value between a single bond (1.54 Å) and a double bond (1.34 Å).

The correct answer is Only A.

Key Points

• Statement A: In a mixture, the composition of the constituent substances is variable, meaning they can be mixed in any proportion. Conversely, a compound is a substance formed when two or more elements chemically combine in a fixed ratio by mass, and the composition of each new substance is always fixed. Hence, statement A is correct.
• Statement B: A mixture reflects the physical properties of its constituent substances because they are not chemically bonded. However, in a compound, the new substance formed has entirely different properties from its constituent elements; it does not show the same properties as the atoms it is made of. Hence, statement B is incorrect.
• Statement C: This statement suggests that a mixture has a non-variable composition and a compound has a changing composition. This is factually incorrect as it contradicts the fundamental laws of chemistry, specifically the Law of Constant Proportions, which states that compounds always have a definite composition. Hence, statement C is incorrect.
• Hence, the correct answer is Only A.

Additional Information

• Mixtures:
• A mixture is a physical combination of two or more substances that are not chemically linked.
• Types: They are classified into homogeneous mixtures (uniform composition, like saltwater or air) and heterogeneous mixtures (non-uniform composition, like sand and iron filings).
• Separation: The constituents of a mixture can be separated using physical methods such as filtration, evaporation, distillation, or magnetism.
• Energy: Usually, no energy (in the form of heat or light) is absorbed or evolved during the formation of a mixture.
• Compounds:
• A compound is a pure substance made of atoms of two or more elements chemically bonded together.
• Formation: The formation of a compound involves a chemical reaction, often accompanied by the evolution or absorption of energy.
• Separation: The constituents of a compound can only be separated by chemical or electrochemical reactions; physical methods do not work.
• Examples: Water (H2O) is a compound of hydrogen and oxygen. Methane (CH4) and Common Salt (NaCl) are other common examples.

Important Points

• Comparison of Properties:
• In a mixture (e.g., iron and sulfur powder mixed), iron retains its magnetic properties.
• In a compound (e.g., Iron Sulphide), the iron loses its magnetic properties because a new chemical substance has been formed.
• Melting and Boiling Points:
• Compounds typically have fixed melting points and boiling points due to their fixed composition.
• Mixtures do not have sharp melting or boiling points; they melt or boil over a range of temperatures depending on the ratio of constituents.

The correct answer is Design of ships and submarines.

Key Points

• Archimedes’ Principle states that when a body is immersed fully or partially in a fluid, it experiences an upward force (buoyancy) that is equal to the weight of the fluid displaced by it.
• The design of ships and submarines is the most common practical application of this principle. Ships are designed with a large volume so they can displace an amount of water whose weight is equal to the weight of the ship, allowing them to float. Hence, statement 1 is correct.
• Submarines use the principle by utilizing ballast tanks. When these tanks are filled with water, the submarine's weight increases, making it denser than water and causing it to sink; when water is pumped out and replaced with compressed air, the submarine becomes lighter and surfaces.
• Using a prism for light dispersion is based on the phenomenon of refraction, where different wavelengths of light bend at different angles when passing through a medium. It is not related to fluid mechanics or buoyancy. Hence, statement 2 is incorrect.
• The working of a thermometer is based on the thermal expansion of liquids like mercury or alcohol. As temperature increases, the liquid expands and moves up the capillary tube. Hence, statement 3 is incorrect.
• Operating a pendulum involves Simple Harmonic Motion (SHM) and is governed by the laws of gravity and inertia. The time period depends on the length of the string and acceleration due to gravity. Hence, statement 4 is incorrect.

Additional Information

• Lactometers: These are instruments used to determine the purity of a milk sample. They work on Archimedes’ Principle by measuring the relative density of milk compared to water.
• Hydrometers: These devices are used for finding the density of liquids. A hydrometer sinks deeper in low-density liquids and stays higher in high-density liquids, following the laws of buoyancy.
• Buoyant Force (Upthrust): This is the upward force exerted by a fluid on an object. The magnitude of this force is directly proportional to the density of the fluid and the volume of the displaced fluid.
• Relative Density: It is the ratio of the density of a substance to the density of water at 4 degrees Celsius. It has no units and is often measured using tools designed around Archimedes’ Principle.

Important Points

• Law of Flotation: An object will float if the weight of the object is equal to or less than the weight of the liquid displaced by its submerged part.
• Historical Context: The principle was discovered by the Greek mathematician Archimedes of Syracuse in the 3rd Century BC while he was trying to determine the purity of a gold crown for King Hiero II.
• Applications in Aviation: Hot air balloons and blimps also follow this principle. The hot air inside the balloon is less dense than the surrounding cool air, creating enough upthrust to lift the weight of the balloon.
• Mathematical Expression: The Buoyant Force (Fb) is calculated as Fb = ρ × V × g, where ρ (rho) is the density of the fluid, V is the displaced volume, and g is the acceleration due to gravity (approx 9.8 m/s²).

The correct answer is Ovary.

Key Points

• The process of fertilization in flowering plants (angiosperms) begins when a pollen grain lands on the stigma of a compatible flower. This event is known as pollination.
• Upon landing, the pollen grain absorbs moisture and germinates, producing a specialized structure called a pollen tube.
• The pollen tube acts as a conduit for the male gametes. It penetrates the tissues of the stigma and grows downward through the elongated, slender part of the carpel known as the style.
• The primary function of the style is to provide a pathway for the tube to travel toward the basal, swollen part of the pistil, which is the ovary.
• The ovary contains one or more ovules. The pollen tube eventually enters the ovary and reaches an ovule, typically through a small opening called the micropyle, to deliver the sperm cells for fertilization.
• Hence, the correct answer is Ovary.

Additional Information

• Structure of the Carpel (Female Reproductive Part):
• Stigma: The topmost, often sticky part of the carpel that receives pollen grains during pollination.
• Style: The stalk-like structure that connects the stigma to the ovary and supports the growth of the pollen tube.
• Ovary: The enlarged basal portion of the pistil where ovules are produced. After fertilization, the ovary develops into a fruit.
• Structure of the Stamen (Male Reproductive Part):
• Anther: The pollen-producing part of the stamen, usually located at the tip.
• Filament: The thin, thread-like stalk that supports the anther and positions it to facilitate pollen dispersal.
• Non-Reproductive Parts of a Flower:
• Petals: Collectively called the corolla, these are often brightly colored to attract pollinators like insects and birds.
• Sepals: Collectively called the calyx, these are leaf-like structures at the base of the flower that protect the bud during its early development stages.
• Fertilization Outcomes:
• In angiosperms, the fusion of a male gamete with the egg cell inside the ovule results in a zygote.
• The ovule eventually matures into a seed, while the surrounding ovary wall matures into the pericarp of the fruit.

Important Points

• Double Fertilization: This is a unique feature of flowering plants where one sperm cell fertilizes the egg to form a zygote, and another sperm cell fuses with two polar nuclei to form the triploid endosperm, which provides nutrition to the developing embryo.
• Chemotropism: The growth of the pollen tube toward the ovary is an example of chemotropism, as the tube is guided by chemical secretions (such as calcium ions and sugars) released by the ovule and synergid cells.

The correct answer is Potassium, Gold.

Key Points

• The Reactivity Series (also known as the activity series) is an arrangement of metals in descending order of their chemical reactivity. It helps in predicting how metals will react with water, acids, and other metal salts.
• Potassium (K) is placed at the very top of the reactivity series, making it the most reactive metal. Due to its high reactivity, it reacts vigorously with oxygen and water even at room temperature.
• Gold (Au) is placed at the bottom of the reactivity series, making it the least reactive metal. It is often referred to as a noble metal because it does not react with air, water, or most acids under standard conditions.
• The general order of the series from high to low reactivity is: Potassium (K) > Sodium (Na) > Calcium (Ca) > Magnesium (Mg) > Aluminium (Al) > Zinc (Zn) > Iron (Fe) > Tin (Sn) > Lead (Pb) > [Hydrogen (H)] > Copper (Cu) > Mercury (Hg) > Silver (Ag) > Gold (Au).
• Metals like Magnesium and Sodium are also highly reactive but are placed below Potassium. Silver is less reactive than Copper but more reactive than Gold.

Additional Information

• Potassium (K):
• It is an Alkali Metal belonging to Group 1 of the periodic table with atomic number 19.
• Because it reacts explosively with water to produce Hydrogen gas and heat, it must be stored under kerosene or paraffin oil to prevent contact with moisture and air.
• It has a very low ionization energy, meaning it loses its single valence electron very easily to form K+ ions.
• Gold (Au):
• It is a Transition Metal with atomic number 79.
• It is highly malleable and ductile, and its resistance to corrosion and oxidation makes it ideal for jewelry and electronic components.
• It is often found in nature in its native state (pure form) because it does not readily form compounds with other elements.
• Magnesium (Mg) and Silver (Ag):
• Magnesium (Atomic number 12) reacts with boiling water or steam but is less violent than Potassium.
• Silver (Atomic number 47) is used in coinage and photography; while it tarnishes in the presence of sulfur, it is still significantly less reactive than base metals.

Important Points

• Displacement Reactions: A more reactive metal (like Potassium or Zinc) can displace a less reactive metal (like Copper or Silver) from its oxide or salt solution.
• Hydrogen Reference: Hydrogen is included in the series as a reference point. Metals placed above Hydrogen can displace it from dilute acids to release hydrogen gas, whereas metals below Hydrogen (like Copper, Silver, Gold) cannot.
• Extraction Methods: The reactivity of a metal determines its extraction method. Highly reactive metals (Potassium, Sodium, Calcium) are extracted via electrolysis of their molten chlorides, while moderately reactive metals (Zinc, Iron) are reduced using carbon (coke).
• Electropositivity: Reactivity is directly related to electropositivity. Potassium is the most electropositive element in this context, having the strongest tendency to lose electrons and form positive ions.

The correct answer is Electric fuse.

Key Points

• An electric fuse is a crucial safety device used in electrical circuits to protect against excessive current. It operates primarily on the principle of the heating effect of electric current, also known as Joule heating.
• The primary component of a fuse is a short piece of wire made of a material with a low melting point and high resistance, typically an alloy of tin and lead (often in a 63:37 ratio) or sometimes copper or zinc.
• When the magnitude of the current flowing through the circuit exceeds the specified safety limit (the fuse rating), the heat generated in the fuse wire ($H = I^2Rt$) increases rapidly.
• As the temperature reaches the melting point of the metal wire, the wire melts and breaks. This action results in an open circuit, which immediately stops the flow of electricity, thereby preventing damage to expensive electrical appliances and hazards like electrical fires.
• The fuse is always connected in series with the live wire of the circuit to ensure that all current passing into the household or device must first pass through the fuse wire.

Additional Information

• Electric Bulb:
• An electric bulb also utilizes the heating effect of current, but for a different purpose. It contains a filament made of Tungsten, which has a very high melting point of approximately 3422 degrees Celsius.
• The goal here is for the filament to become white-hot and emit light (incandescence) without melting. Therefore, it is the opposite of a fuse in terms of its material's thermal properties.
• Electric Fan:
• An electric fan does not rely on the heating effect to function. Instead, it works on the magnetic effect of electric current.
• It uses an electric motor that converts electrical energy into mechanical energy (rotational motion) through the interaction of magnetic fields.
• Electric Bell:
• The electric bell is another application of the magnetic effect of current.
• It contains an electromagnet that, when energized, attracts an iron armature to strike a gong. It does not use the melting of metal as a functional mechanism.
• Short Circuit and Overloading:
• A short circuit occurs when the live wire and neutral wire come into direct contact, causing a sudden, massive surge in current.
• Overloading occurs when too many high-power appliances are connected to a single socket, drawing more current than the wires can safely handle. In both cases, the electric fuse is the first line of defense.

Important Points

• Fuse Ratings: Fuses are classified by their current-carrying capacity. For common domestic use, fuses are typically rated at 5 Amperes for lighting and fans, and 15 Amperes for heavy-duty appliances like air conditioners or refrigerators.
• Modern Alternatives: In modern electrical installations, Miniature Circuit Breakers (MCBs) are increasingly replacing traditional fuses. Unlike a fuse, which must be replaced once it melts, an MCB is a switch that automatically trips to the OFF position during a fault and can be reset manually.
• Joule's Law: The amount of heat produced in the wire is directly proportional to the square of the current, the resistance of the wire, and the duration for which the current flows. This mathematical relationship is the fundamental reason why the fuse wire melts specifically when the current becomes too high.

The correct answer is non-uniform motion.

Key Points

• Non-uniform motion is defined as a type of motion where an object covers unequal distances in equal intervals of time. In the case of a freely falling stone, its velocity increases continuously as it descends due to the pull of gravity. Hence, option 1 is correct.
• A stone falling under the influence of gravity alone (ignoring air resistance) experiences a constant acceleration known as acceleration due to gravity (g), which is approximately 9.8 m/s² on the surface of the Earth.
• Since the speed of the stone is not constant and increases every second by 9.8 m/s, it cannot be classified as uniform motion or rectilinear motion with constant speed. Hence, options 2 and 4 are incorrect.
• Because there is a definite force (gravitational force) acting on the stone, it possesses a specific acceleration. Therefore, the motion does not have zero acceleration. Hence, option 3 is incorrect.
• The path of the stone is a straight line, making it a rectilinear motion, but because the magnitude of the velocity changes, it is specifically categorized as uniformly accelerated non-uniform motion.

Additional Information

• Uniform Motion:
• An object is said to be in uniform motion if it moves along a straight line and covers equal distances in equal intervals of time, regardless of the duration of the time intervals.
• In this state, the acceleration of the body is zero and the velocity remains constant throughout the journey.
• Acceleration due to Gravity (g):
• It is the acceleration gained by an object due to the gravitational force of a massive body like Earth.
• The standard value of g is 9.80665 m/s². This value varies slightly depending on latitude, altitude, and the geological structure of the region.
• It is a vector quantity, always directed towards the center of the planet.
• Rectilinear Motion:
• This refers to motion in a straight line. It can be of two types: Uniform Rectilinear Motion (constant velocity) and Non-Uniform Rectilinear Motion (changing velocity/accelerated).
• A falling stone is an example of Non-Uniform Rectilinear Motion.

Important Points

• Equations of Motion for Free Fall:
• When an object is dropped from rest (initial velocity u = 0), the equations of motion simplify to: v = gt, h = 1/2 gt², and v² = 2gh.
• In these equations, v is the final velocity, t is the time taken, and h is the height or displacement.
• Mass Independence:
• In a vacuum (where air resistance is absent), all objects fall with the same acceleration (g) regardless of their mass or shape. This was famously demonstrated by Galileo Galilei.
• Terminal Velocity:
• In a real atmosphere, as an object falls, air resistance increases with speed. Eventually, the upward air resistance equals the downward weight, and the object stops accelerating, moving at a constant speed called terminal velocity.

The correct answer is Virtual, erect, and enlarged.

Key Points

• A concave mirror is a spherical mirror with a reflecting surface that curves inward, resembling the interior of a bowl. It is also known as a converging mirror because it converges light rays that fall on it.
• When an object is placed between the pole (P) and the principal focus (F) of a concave mirror, the rays of light originating from the object diverge after reflection.
• Because the reflected rays are diverging, they never meet on the side of the object. Instead, they appear to meet at a point behind the mirror when traced backward. This results in the formation of a virtual image.
• The image formed in this specific position is erect (upright) and enlarged (magnified), meaning the size of the image is greater than the size of the object. Hence, the correct answer is Virtual, erect, and enlarged.
• This is a unique case for a concave mirror because, for all other object positions (such as at F, between C and F, at C, or beyond C), the image formed is real and inverted.

Additional Information

• Common Uses: Due to its ability to form an enlarged and erect image when the object is close, this property of the concave mirror is utilized in shaving mirrors and makeup mirrors to see a larger view of the face. Dentists also use concave mirrors to see large images of the teeth of patients.
• Image Formation at Other Positions:
• At Infinity: The image is formed at the focus (F), and it is real, inverted, and highly diminished (point-sized).
• Beyond Center of Curvature (C): The image is formed between F and C, and it is real, inverted, and diminished.
• At Center of Curvature (C): The image is formed at C, and it is real, inverted, and the same size as the object.
• Between C and F: The image is formed beyond C, and it is real, inverted, and enlarged.
• At Focus (F): The image is formed at infinity, and it is real, inverted, and highly enlarged.
• Convex Mirrors: Unlike concave mirrors, convex mirrors always form a virtual, erect, and diminished image, regardless of the object's distance from the mirror. They are widely used as rear-view mirrors in vehicles.

Important Points

• Mirror Formula: The relationship between object distance (u), image distance (v), and focal length (f) is given by 1/f = 1/v + 1/u.
• Magnification (m): It is the ratio of the height of the image to the height of the object (m = h'/h). It is also expressed as m = -v/u.
• In the case of an object between P and F, the magnification is positive and greater than 1.
• Sign Convention: According to the New Cartesian Sign Convention, distances measured in the direction of incident light are positive, while those measured against it are negative. For a concave mirror, the focal length is always negative.
• Radius of Curvature: The distance between the pole and the center of curvature is called the radius of curvature (R), and it is equal to twice the focal length (R = 2f).

The correct answer is the image height to the object height.

• Magnification (m) produced by a lens is defined as the ratio of the height of the image to the height of the object.
• It is represented by the letter m. If h is the height of the object and h' is the height of the image, then the magnification is given by m = h' / h.
• Magnification is also related to the object distance (u) and the image distance (v). This relationship is expressed as m = v / u for lenses.
• It is a unitless quantity or dimensionless because it is the ratio of two similar lengths.
• The value of magnification gives information about the size and nature of the image formed by the lens relative to the object.
• Sign Convention for Magnification:
• If the magnification has a negative sign, the image is real and inverted.
• If the magnification has a positive sign, the image is virtual and erect.
• Magnitude of Magnification:
• If |m| > 1, the image is enlarged (larger than the object).
• If |m| < 1, the image is diminished (smaller than the object).
• If |m| = 1, the image is of the same size as the object.
• Concave Lens:
• A concave lens (diverging lens) always produces a virtual, erect, and diminished image for all positions of the object.
• Therefore, the magnification for a concave lens is always positive and less than 1.
• Convex Lens:
• A convex lens (converging lens) can produce real or virtual images depending on the object's position.
• The magnification can be positive (when the object is between the focus and optical center) or negative (when the object is beyond the focus).
• The Lens Formula is used to calculate image or object distances and is given by 1/f = 1/v - 1/u, where f is the focal length.
• According to the Cartesian sign convention, the height of the object is usually taken as positive as it is placed above the principal axis.
• The height of a virtual image is taken as positive, while the height of a real image is taken as negative.
• In optical instruments like microscopes and telescopes, magnification is a critical factor for determining the resolving power and viewing capacity of the device.
• The Power of a lens (P) is the reciprocal of its focal length in meters (P = 1/f), and it is measured in Dioptres (D).

The correct answer is Statement A is correct but statement B is incorrect.

Key Points

• Vegetative propagation is a type of asexual reproduction in which new plants are produced from the vegetative parts of the parent plant, such as the root, stem, or leaves, without the involvement of seeds.
• One of the primary advantages of this method is that plants raised by vegetative propagation can bear flowers and fruits earlier than those produced from seeds. This is because the plant bypasses the long juvenile phase associated with seed germination and early growth. Hence, statement A is correct.
• In vegetative propagation, since the offspring are produced from a single parent without the fusion of gametes (fertilization), the new plants are genetically similar or identical to the parent plant. These plants are often referred to as clones. Hence, statement B is incorrect.
• This method is particularly useful for plants that have lost the capacity to produce seeds, such as Banana, Orange, Rose, and Jasmine.
• The offspring maintain the desirable characteristics of the parent plant consistently across generations, which is highly beneficial in agriculture and horticulture for maintaining crop quality.

Additional Information

• Natural Methods of Vegetative Propagation:
• By Roots: Some plants like Sweet Potato and Dahlia develop adventitious buds on their roots which grow into new plants.
• By Stems: Modified stems like Tubers (Potato), Rhizomes (Ginger and Turmeric), and Bulbs (Onion) are common natural storage organs used for propagation.
• By Leaves: Plants like Bryophyllum produce adventitious buds along their leaf margins; when these leaves fall on moist soil, the buds develop into new plantlets.
• Artificial Methods of Vegetative Propagation:
• Cutting: A small part of the stem with nodes is buried in the soil, commonly seen in Rose, Sugarcane, and Grapes.
• Grafting: Joining two different plants (the scion and the stock) so they grow as a single plant, widely used in Mango and Citrus fruits.
• Layering: A branch of the plant is bent to the ground and covered with soil to induce root formation while still attached to the parent, used for Jasmine and Lemon.

Important Points

• Genetic Uniformity: Because there is no genetic recombination (which occurs during meiosis in sexual reproduction), there is no variation. While this preserves good traits, it also means the entire population is equally vulnerable to the same diseases or environmental changes.
• Tissue Culture (Micropropagation): This is a modern technique where plant cells or tissues are grown in a sterile nutrient medium under controlled conditions. It allows for the rapid production of disease-free plants in a very short time and limited space.
• Survival Rate: Vegetatively propagated plants often have a higher survival rate compared to seedlings because they are already partially developed and have access to the parent's nutrient reserves during the initial stages of growth.
• Economic Significance: This method is vital for the commercial production of fruits and flowers where uniformity in size, taste, and appearance is required for the market.

The correct answer is White.

Key Points

• The reaction between Silver Nitrate (AgNO3) and Sodium Chloride (NaCl) is a classic example of a double displacement reaction, also known as a precipitation reaction.
• When an aqueous solution of Silver Nitrate is added to an aqueous solution of Sodium Chloride, an exchange of ions occurs to form Silver Chloride (AgCl) and Sodium Nitrate (NaNO3).
• Silver Chloride is highly insoluble in water and settles out of the solution as a solid mass known as a precipitate.
• The physical observation of this chemical change is the immediate formation of a curdy white precipitate of Silver Chloride. Hence, the statement regarding the white colour is correct.
• The chemical equation for the reaction is: AgNO3 (aq) + NaCl (aq) → AgCl (s) ↓ + NaNO3 (aq).

Additional Information

• Silver Halides and Colours:
• While Silver Chloride forms a white precipitate, other silver halides exhibit different colours.
• Silver Bromide (AgBr) forms a pale yellow or cream-coloured precipitate.
• Silver Iodide (AgI) forms a distinct bright yellow precipitate.
• Double Displacement Reaction: This occurs when the cations and anions of two different compounds switch places, forming two entirely new compounds. One of these products must be a precipitate, a gas, or a molecular compound like water for the reaction to proceed.
• Solubility Rules: According to general solubility rules, all nitrates (NO3-) are soluble in water, while most silver salts (except silver nitrate and silver chlorate) are insoluble. This explains why AgCl precipitates while NaNO3 remains in the dissolved state.
• Analytical Chemistry: This specific reaction is widely used as a qualitative test to detect the presence of chloride ions in a given substance. If the addition of silver nitrate produces a white precipitate that is soluble in dilute ammonia, the presence of chloride is confirmed.

Important Points

• Photolytic Decomposition: Silver Chloride is sensitive to light. When exposed to sunlight, the white precipitate of AgCl decomposes into grey metallic silver and chlorine gas.
• Equation for Decomposition: 2AgCl (s) → 2Ag (s) + Cl2 (g) (in the presence of UV light).
• Historical Use: Due to its light-sensitive nature, silver chloride was fundamentally important in the development of black-and-white photography during the 19th century.
• Laboratory Safety: Silver nitrate is a strong oxidizing agent and can cause dark stains on the skin and clothes, which is why it must be handled with care in the chemistry laboratory.

The correct answer is Both A and R are true, and R is the correct explanation of A.

Key Points

• Assertion (A) states that a soap solution appears cloudy. This occurs because soap molecules do not form a true solution in water at higher concentrations; instead, they form a colloidal solution. Hence, Assertion (A) is correct.
• Reason (R) explains that soap micelles are large enough to scatter light. In a soap solution, the clusters of molecules, known as micelles, have dimensions typically ranging from 1 to 1000 nanometers. Hence, Reason (R) is correct.
• When a beam of light passes through a colloidal soap solution, the micelles scatter the light in all directions. This phenomenon is known as the Tyndall Effect.
• Because the light is scattered rather than passing directly through (as it would in a clear, true solution like salt water), the solution appears translucent or cloudy to the naked eye.
• Since the scattering of light by these large particles is the direct cause of the visual cloudiness, Reason (R) provides a scientifically accurate explanation for Assertion (A). Hence, both statements are true and R correctly explains A.

Additional Information

• Structure of Soap Molecules:
• Soap is a sodium or potassium salt of long-chain carboxylic acids (fatty acids).
• Each molecule has two distinct parts: a hydrophilic (water-loving) head which is ionic and a hydrophobic (water-fearing) tail made of a long hydrocarbon chain.
• Micelle Formation:
• When soap is added to water, the hydrophobic tails cluster together to stay away from water, while the hydrophilic heads face outward into the water.
• This unique orientation results in the formation of spherical aggregates called micelles.
• Colloidal Nature:
• A colloid is a mixture where one substance consisting of microscopic dispersed particles is suspended throughout another substance.
• Unlike True Solutions (where particle size is less than 1 nanometer), colloids have particles large enough to interfere with light but small enough not to settle down under gravity.

Important Points

• Critical Micelle Concentration (CMC): This is the specific concentration of surfactants above which micelles begin to form. Below this concentration, soap acts like a normal electrolyte and the solution may appear clearer.
• Tyndall Effect: Named after the 19th-century physicist John Tyndall, who first studied the scattering of light by colloidal particles in detail. It is the reason why the path of light becomes visible when passing through milk, fog, or a soap solution.
• Cleaning Action: The cloudiness of the soap solution is also related to its ability to clean. The hydrophobic center of the micelle traps oil and grease, allowing them to be washed away with water.
• Solubility: Soap is soluble in water, but the formation of large molecular clusters (micelles) changes the optical properties of the resulting mixture from transparent to opaque/cloudy.

Shortcut Trick

Product of fractions method: 1/10 × 2/5 × 1/3 × Number = 512

Number = 512 × 10 × 5/2 × 3

Number = 512 × 5 × 5 × 3 = 512 × 75

Number = 38400

∴ The correct answer is 38400.

Alternate Method

Given:

10% of 40% of 1/3 of a number = 512

Let the number be x.

⇒ (10/100) × (40/100) × (1/3) × x = 512

⇒ (1/10) × (2/5) × (1/3) × x = 512

⇒ (2/150) × x = 512

⇒ x = (512 × 150) ÷ 2

⇒ x = 256 × 150

⇒ x = 38400

∴ The correct answer is 38400.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 10% = 1/10, 20% = 1/5, 25% = 1/4, 40% = 2/5, and 50% = 1/2.

Successive Percentage Application

When multiple percentages are applied to a single value, the result is the product of the individual multiplying factors times the original value.

Base Value Concept

Always identify the base value (the number x) to which percentages and fractions are being applied to avoid calculation errors.

Shortcut Trick

The given expression follows the algebraic identity (a³ + b³) ÷ (a² − ab + b²) = (a + b).

Here, let a = 0.45 and b = 0.2.

Value = a + b = 0.45 + 0.2 = 0.65.

Converting decimal to fraction: 0.65 = 65/100 = 13/20.

∴ The correct answer is 13/20.

Alternate Method

Given:

Expression: ((0.45)³ + (0.2)³) ÷ ((0.45)² − 0.45 × 0.2 + (0.2)²)

Formula Used:

Sum of two cubes: x³ + y³ = (x + y)(x² − xy + y²)

Calculations:

⇒ Let x = 0.45 and y = 0.2.

⇒ Numerator = (0.45 + 0.2) × ((0.45)² − (0.45 × 0.2) + (0.2)²)

⇒ Expression = [(0.45 + 0.2) × ((0.45)² − 0.45 × 0.2 + (0.2)²)] ÷ [(0.45)² − 0.45 × 0.2 + (0.2)²]

⇒ Cancelling common terms in numerator and denominator: Expression = 0.45 + 0.2

⇒ Expression = 0.65

⇒ 0.65 = 65 ÷ 100

⇒ Simplifying the fraction by dividing by 5: 13/20

∴ The correct answer is 13/20.

Additional Information

Difference of Cubes Identity

The formula for the difference of two cubes is a³ − b³ = (a − b)(a² + ab + b²).

Decimal to Fraction Conversion

To convert a decimal to a fraction, place the number over its place value (e.g., 0.65 = 65/100) and simplify to the lowest terms.

BODMAS Rule

Always follow the order of operations: Brackets, Orders (powers/roots), Division/Multiplication, and Addition/Subtraction.

Shortcut Trick

Apply the Empirical Formula directly: Mode = 3 × Median − 2 × Mean.

Rearranging for Mean: 2 × Mean = 3 × Median − Mode.

Calculation: 2 × Mean = 3 × 58 − 70.2 = 174 − 70.2 = 103.8.

Final Step: Mean = 103.8 ÷ 2 = 51.9.

∴ The correct answer is 51.9.

Alternate Method

Given:

Mode of the data = 70.2

Median of the data = 58

Formula Used:

The relationship between Mean, Median, and Mode is given by the empirical formula:

Mode = 3 × Median − 2 × Mean

Calculations:

⇒ 70.2 = 3 × 58 − 2 × Mean

⇒ 70.2 = 174 − 2 × Mean

⇒ 2 × Mean = 174 − 70.2

⇒ 2 × Mean = 103.8

⇒ Mean = 103.8 ÷ 2

⇒ Mean = 51.9

∴ The correct answer is 51.9.

Additional Information

Symmetrical Distribution

In a perfectly symmetrical distribution, the values of Mean, Median, and Mode are identical (Mean = Median = Mode).

Skewed Distribution

For a Positively Skewed distribution, Mean > Median > Mode. For a Negatively Skewed distribution, Mode > Median > Mean.

Karl Pearson's Coefficient

The empirical formula is used for moderately asymmetrical distributions to estimate a missing measure of central tendency.

Shortcut Trick

Simplify ratio: 4/10 : 20/18 : 20/11 → 2/5 : 10/9 : 20/11.

Multiply by LCM(5, 9, 11) = 495: (2/5 × 495) : (10/9 × 495) : (20/11 × 495) = 198 : 550 : 900.

Difference (900 − 198) units = 702 units → 39.

1 unit = 39 ÷ 702 = 1/18.

Largest number = 900 × (1/18) = 50.

∴ The correct answer is 50.

Alternate Method

Given: Ratio of three numbers = 4/10 : 20/18 : 20/11

⇒ Simplified Ratio = 2/5 : 10/9 : 20/11

⇒ Multiplying by LCM of denominators (495):

⇒ First Number = 2/5 × 495 = 198

⇒ Second Number = 10/9 × 495 = 550

⇒ Third Number = 20/11 × 495 = 900

⇒ Difference between Greatest and Smallest = 900 − 198 = 702 units

⇒ 702 units = 39

⇒ 1 unit = 39/702 = 1/18

⇒ Largest number = 900 units = 900 × (1/18) = 50

∴ The correct answer is 50.

Additional Information

Ratio Simplification

To convert a fractional ratio a/b : c/d into integers, multiply each term by the Least Common Multiple (LCM) of the denominators b and d.

Proportional Parts

If a total quantity is divided in ratio a:b:c, the value of the largest part is [Maximum(a,b,c) / (a+b+c)] × Total.

Comparison of Fractions

For two fractions a/b and c/d, cross multiply (a×d and b×c) to quickly determine which is larger without finding common denominators.

Shortcut Trick

Net percentage change formula = a + b + (ab/100)

Here, increase in height (a) = +20% and decrease in base (b) = −15%

Net change = 20 − 15 + [(20 × −15) ÷ 100]

Net change = 5 − 3 = +2%

Positive sign indicates an increase.

∴ The correct answer is Increase by 2%.

Alternate Method

Given:

Initial Height (H₁) = 100, New Height (H₂) = 120

Initial Base (B₁) = 100, New Base (B₂) = 85

Calculations:

⇒ Initial Area (A₁) = 1/2 × B₁ × H₁ = 1/2 × 100 × 100 = 5000

⇒ New Area (A₂) = 1/2 × B₂ × H₂ = 1/2 × 85 × 120 = 85 × 60 = 5100

⇒ Area Change = 5100 − 5000 = 100

⇒ Percentage Change = (Change ÷ Original Area) × 100

⇒ Percentage Change = (100 ÷ 5000) × 100 = 2%

∴ The correct answer is Increase by 2%.

Additional Information

Area of Triangle

Basic formula: Area = 1/2 × base × height. Changes in both parameters lead to a cumulative effect on the area.

Successive Percentage Change

When a value depends on the product of two variables, the net % change is given by x + y + xy/100, where x and y are the percentage changes in those variables.

Base Change Rule

If one dimension increases by x% and the other decreases by y%, the net effect is an increase if (x − y − xy/100) is positive.

Shortcut Trick

Effective Rate of Interest for 2 years at 10% compounded annually = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount = 100% (Principal) + 21% (Interest) = 121% of Principal.

Amount = 8,300 × (121/100) = 83 × 121.

Amount = ₹10,043.

∴ The correct answer is ₹10,043.

Alternate Method

Given:

Principal (P) = ₹8,300

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 8300 × (1 + 10/100)²

⇒ A = 8300 × (110/100) × (110/100)

⇒ A = 8300 × 1.1 × 1.1

⇒ A = 8300 × 1.21

⇒ A = 83 × 121

⇒ A = ₹10,043

∴ The correct answer is ₹10,043.

Additional Information

Effective Rate for Compound Interest

For a rate R% for 2 years, the effective interest rate is R + R + (R × R)/100.

Difference between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest is P × (R/100)².

Compound Interest Formula

Interest (CI) = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Shortcut Trick

Savings = 20% ⇒ Expenditure = 100% − 20% = 80%

Given, 80% → ₹320

1% → ₹4

Total Income (100%) → 100 × 4 = ₹400

∴ The correct answer is ₹400.

Alternate Method

Given:

Savings percentage = 20%

Expenditure = ₹320

Formula: Income = Expenditure + Savings

⇒ Let Total Income be x

⇒ Savings = 20% of x = 0.20x

⇒ Expenditure = Income − Savings

⇒ 320 = x − 0.20x

⇒ 320 = 0.80x

⇒ x = 320 ÷ 0.80

⇒ x = 400

∴ The correct answer is ₹400.

Additional Information

Income Components

Income is always the sum of Expenditure and Savings. If any two are known, the third can be calculated.

Percentage Base Change

When Savings % is given, the Expenditure % is always (100 − Savings %). Here, 100% represents the Total Income.

Ratio Method

20% Savings = 1/5 of Income. This means if Income is 5 units, Savings is 1 unit and Expenditure is 4 units.

Shortcut Trick

49 = 7²

(49)ⁿ = (7²)ⁿ = 7²ⁿ

Expression = (7²ⁿ × 7) ÷ 7²ⁿ

Cancel ⇒ = 7

∴ Answer = 7¹

Alternate Method

Given:

((49)ⁿ × 7) ÷ (7²ⁿ)

Concept Used:

Laws of exponents

Calculation:

⇒ 49 = 7²

⇒ (49)ⁿ = (7²)ⁿ = 7²ⁿ

⇒ Substitute ⇒ (7²ⁿ × 7) ÷ 7²ⁿ

⇒ = 7²ⁿ⁺¹ ÷ 7²ⁿ

⇒ = 7^{(2n+1 − 2n)}

⇒ = 7¹

∴ The correct answer is 7¹.

Additional Information

Exponent Rule

a^m × aⁿ = a^m+n

Division Rule

a^m ÷ aⁿ = a^m−n

Power of Power

(a^m)ⁿ = a^mn

Shortcut Tip

Always convert numbers to same base first

Shortcut Trick

Let TV cost = 100, then Washing Machine (WM) cost = 75.

Initial total cost of 6 WM and 3 TV = (6 × 75) + (3 × 100) = 450 + 300 = 750.

New WM cost = 75 × 1.88 = 141; New TV cost = 100 × 0.43 = 43.

New total cost = (6 × 141) + (3 × 43) = 846 + 129 = 975.

Percentage change = [(975 − 750) ÷ 750] × 100 = (225 ÷ 750) × 100 = 30%.

∴ The correct answer is Increase by 30%.

Alternate Method

Given: Cost of WM is 25% less than TV. WM cost ↑ 88%, TV cost ↓ 57%. Formulas: New Value = Old Value × (100 ± Change%) ÷ 100 | %Change = (New − Old) ÷ Old × 100.

⇒ Let TV cost = 100x and WM cost = 75x. ⇒ Initial cost of 6 WM and 3 TV = 6(75x) + 3(100x) = 450x + 300x = 750x. ⇒ Final cost = 6(75x × 1.88) + 3(100x × 0.43) = 6(141x) + 3(43x) = 846x + 129x = 975x. ⇒ Change = 975x − 750x = 225x. ⇒ %Change = (225x ÷ 750x) × 100 = 30%.

∴ The correct answer is Increase by 30%.

Additional Information

Percentage Change in Weighted Average

When multiple items with different initial costs and quantities undergo price changes, the total percentage change is the weighted average of individual percentage changes based on their initial total value contributions.

Cost Price Scaling

Using a base value of 100 for the reference item (TV) simplifies calculations involving percentage increases and decreases.

Shortcut Trick

Ratio of investments of Laxmi, Dhruv, and Gauri = 13 : 6 : 12.

Sum of ratio parts = 13 + 6 + 12 = 31 units.

Total profit = ₹37,200, which represents 75% of the total investment.

Total Investment = (37,200 ÷ 75) × 100 = ₹49,600.

Dhruv investment = (Investment ratio of Dhruv ÷ Total ratio sum) × Total Investment.

Dhruv investment = (6 ÷ 31) × 49,600 = 6 × 1,600 = ₹9,600.

∴ The correct answer is ₹9,600.

Alternate Method

Given → Investment ratio = 13 : 6 : 12 | Profit = ₹37,200 | Profit = 75% of Investment. Formulas: Total Investment = (Profit ÷ Profit%) × 100 | Individual Investment = (Individual Ratio ÷ Total Ratio) × Total Investment.

⇒ Let the investments be 13x, 6x, and 12x. Total investment = 31x. ⇒ According to the question, 75% of 31x = 37,200. ⇒ (75 ÷ 100) × 31x = 37,200. ⇒ (3 ÷ 4) × 31x = 37,200. ⇒ 31x = (37,200 × 4) ÷ 3 = 12,400 × 4 = 49,600. ⇒ x = 49,600 ÷ 31 = 1,600. ⇒ Dhruv investment = 6x = 6 × 1,600 = ₹9,600.

∴ The correct answer is ₹9,600.

Additional Information

Partnership Basics

In a partnership, if the time period of investment is the same for all partners, the profit is shared in the ratio of their investments.

Profit and Investment Relationship

Profit = Investment × Time. If Time is constant, Profit ∝ Investment. Profit percentage is calculated on the Total Capital invested.

Shortcut Trick

Total Quantity = 52 + 47 = 99 books

Total Expenditure = 1120 + 905 = ₹2025

Average Price = 2025 ÷ 99 = 225 ÷ 11

225 ÷ 11 ≈ 20.45

∴ The correct answer is ₹20.45.

Alternate Method

Given:

Books from Shop 1 = 52 | Price = ₹1120

Books from Shop 2 = 47 | Price = ₹905

Formula Used:

Average = (Total Sum of Observations) ÷ (Total Number of Observations)

Calculations:

⇒ Total number of books = 52 + 47 = 99

⇒ Total amount spent = ₹1120 + ₹905 = ₹2025

⇒ Average price per book = 2025 ÷ 99

⇒ Average price = (2025 ÷ 9) ÷ (99 ÷ 9) = 225 ÷ 11

⇒ Average price = 20.4545...

⇒ Rounding off to two decimal places, we get 20.45

∴ The correct answer is ₹20.45.

Additional Information

Weighted Average Concept

When two groups with different quantities (n₁, n₂) and different averages (A₁, A₂) are combined, the total average is calculated as (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Rounding Off Rules

To round to two decimal places, check the third decimal digit. If it is 5 or more, add 1 to the second decimal digit. If it is less than 5, keep the second decimal digit as it is.

Simplified Division

Always check for common factors in the numerator and denominator to simplify calculations. Here, 2025 and 99 are both divisible by 9 (Sum of digits 2+0+2+5=9 and 9+9=18).

Shortcut Trick

The unit digit of the given edge is 1.

The volume of a cube is calculated as (Edge)³.

The unit digit of the result must be 1³ = 1.

Only option (3) ends with the digit 1.

∴ The correct answer is 68921 m³.

Alternate Method

Given: Edge (a) of the cube = 41 m

Formula: Volume (V) of a cube = a³

⇒ V = 41 × 41 × 41

⇒ V = 1681 × 41

⇒ V = 68921 m³.

∴ The correct answer is 68921 m³.

Additional Information

Total Surface Area of a Cube

The total area of all six square faces is given by the formula 6a².

Diagonal of a Cube

The length of the longest space diagonal connecting opposite corners is a√3.

Lateral Surface Area

The area of the four vertical side faces of the cube is 4a².

Shortcut Trick

In any quadrilateral, the angle formed by the bisectors of two adjacent angles is equal to half the sum of the other two angles.

∠STR = (∠P + ∠Q) ÷ 2

25.5° = (∠P + ∠Q) ÷ 2 ⇒ ∠P + ∠Q = 51°

Ratio P : Q = 8 : 9 ⇒ Sum = 17 units = 51° ⇒ 1 unit = 3°

Difference = 9 − 8 = 1 unit = 3°.

∴ The correct answer is 3°.

Alternate Method

Given:

∠STR = 25.5°

Ratio ∠P : ∠Q = 8 : 9

Formula Used:

Sum of interior angles of a triangle = 180°

Sum of interior angles of a quadrilateral = 360°

Calculations:

⇒ In ▵STR, ∠TSR + ∠TRS + ∠STR = 180°

⇒ Since ST and RT are angle bisectors, ∠TSR = ∠S ÷ 2 and ∠TRS = ∠R ÷ 2

⇒ (∠S ÷ 2) + (∠R ÷ 2) + 25.5° = 180°

&rArr

Shortcut Trick

Average Speed when time intervals are equal is simply the Arithmetic Mean of the speeds.

Since each interval is exactly 1 hour, Average Speed = (158 + 110) ÷ 2

⇒ 268 ÷ 2 = 134 km/hr

∴ The correct answer is 134 km/hr.

Alternate Method

Given:

Distance in 1^st hour (D₁) = 158 km

Distance in 2^nd hour (D₂) = 110 km

Time taken for each part (t₁, t₂) = 1 hour

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 158 + 110 = 268 km

⇒ Total Time = 1 + 1 = 2 hours

⇒ Average Speed = 268 ÷ 2

⇒ Average Speed = 134 km/hr

∴ The correct answer is 134 km/hr.

Additional Information

Equal Distance Case

If a car travels equal distances at speeds x and y, then Average Speed = 2xy ÷ (x + y).

Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed is (u − v). In opposite directions, it is (u + v).

Shortcut Trick

Net change formula for Mark-up (a) and Discount (b) is [a − b − (ab ÷ 100)].

Here, a = 10% (Mark-up) and b = 80% (Discount).

Net Profit/Loss % = 10 − 80 − (10 × 80 ÷ 100)

⇒ −70 − 8 = −78%

The negative sign indicates a Loss of 78%.

∴ The correct answer is 78%.

Alternate Method

Given: Mark-up% = 10%, Discount% = 80%.

Formulas: MP = CP × (100 + Mark-up%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Loss% = [(CP − SP) ÷ CP] × 100

⇒ Let the Cost Price (CP) be 100.

⇒ Marked Price (MP) = 100 + 10% of 100 = 110.

⇒ Selling Price (SP) = 110 − 80% of 110 = 110 − 88 = 22.

⇒ Loss = CP − SP = 100 − 22 = 78.

⇒ Loss Percentage = (78 ÷ 100) × 100 = 78%.

∴ The correct answer is 78%.

Additional Information

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP × (100 − D)%.

Effective Percentage Change

When a value is increased by x% and then decreased by y%, the net change is [x − y − (xy/100)]%.

Shortcut Trick

Net efficiency = Filling efficiency − Emptying efficiency

LCM of 3 and 6 = 6 units (Total Capacity)

Efficiency: Pipe A = 6/3 = 2, Pipe B = 6/6 = 1

Time to fill half tank (3 units) = 3 ÷ (2 − 1) = 3 minutes

∴ The correct answer is 3 minutes.

Alternate Method

Given: Filling time = 3 min, Emptying time = 6 min.

Formula: Work = Efficiency × Time

⇒ Net efficiency of both pipes = 2 − 1 = 1 unit/min

⇒ Capacity of half the tank = 6 ÷ 2 = 3 units

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

∴ The correct answer is 3 minutes.

Additional Information

Net Work in Pipes & Cisterns

The work done by an inlet pipe is positive, while the work done by an outlet pipe (leak) is negative.

Combined Time Formula

If pipe A fills in X and pipe B empties in Y, combined time T = (X × Y) ÷ (Y − X), provided Y > X.

Shortcut Trick

Let the total quantity be 2 units. Half quantity is 1 unit.

Total required gain = 2 units × 20% = 40%

Gain from first half = 1 unit × 10% = 10%

Required gain from remaining half = 40% − 10% = 30%

Since the remaining quantity is 1 unit, the gain percentage = 30% ÷ 1 = 30%.

∴ The correct answer is 30%.

Alternate Method

Given: Total CP = ₹8,000, Gain on first half = 10%, Total desired gain = 20%.

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

⇒ Total Cost Price (CP) = ₹8,000

⇒ Total Selling Price (SP) required for 20% gain = 8000 × (120 ÷ 100) = ₹9,600

⇒ CP of half goods = 8000 ÷ 2 = ₹4,000

⇒ SP of first half at 10% gain = 4000 × (110 ÷ 100) = ₹4,400

⇒ Required SP of remaining half = 9600 − 4400 = ₹5,200

⇒ Profit needed on remaining half = 5200 − 4000 = ₹1,200

⇒ Profit% on remaining = (1200 ÷ 4000) × 100 = 30%

∴ The correct answer is 30%.

Additional Information

Weighted Average Method

When a transaction is split, the overall profit % is the weighted average of individual profit percentages: Total P% = (n₁p₁ + n₂p₂) ÷ (n₁ + n₂).

Independent of Value

In problems involving ratios or fractions of goods, the actual monetary value (like ₹8,000) does not affect the final percentage result.

Shortcut Trick

If a + 1/a = k, then a² + 1/a² = k² − 2.

Let a = √x, then a² = x.

Given √x + 1/√x = 5, we have k = 5.

Required value x + 1/x = 5² − 2 = 25 − 2 = 23.

∴ The correct answer is 23.

Alternate Method

Given:

√x + 1/√x = 5

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

Square both sides of the given algebraic equation:

⇒ (√x + 1/√x)² = 5²

⇒ (√x)² + (1/√x)² + 2 × (√x) × (1/√x) = 25

⇒ x + 1/x + 2 = 25

⇒ x + 1/x = 25 − 2

⇒ x + 1/x = 23

∴ The correct answer is 23.

Additional Information

Square of Difference

If a − 1/a = k, then a² + 1/a² = k² + 2.

Cube of Sum

If a + 1/a = k, then a³ + 1/a³ = k³ − 3k.

Relation between Identities

(a + b)² = (a − b)² + 4ab.

Standard Algebraic Identity

(a − b)² = a² + b² − 2ab.

Shortcut Trick

Relative Speed in same direction = 114 − 60 = 54 km/hr.

Convert to m/s: 54 × (5/18) = 15 m/s.

Total distance (2L) = Speed × Time = 15 × 144 = 2160 m.

Length of one train (L) = 2160 ÷ 2 = 1080 m.

∴ The correct answer is 1080 m.

Alternate Method

Given:

Speed of Rajdhani Express (S₁) = 114 km/hr

Speed of Garib Rath Express (S₂) = 60 km/hr

Time taken (t) = 144 seconds

Lengths are equal: L₁ = L₂ = L

Formula Used:

Relative Speed (Same direction) = S₁ − S₂

Total Distance = Sum of lengths of trains = L₁ + L₂

Distance = Speed × Time

Calculations:

⇒ Relative Speed = 114 − 60 = 54 km/hr

⇒ Speed in m/s = 54 × (5/18) = 15 m/s

⇒ Total Distance = 2L

⇒ 2L = 15 × 144

⇒ 2L = 2160 m

⇒ L = 2160 ÷ 2 = 1080 m

∴ The correct answer is 1080 m.

Additional Information

Relative Speed Concepts

When moving in the same direction, relative speed is (S₁ − S₂). When moving in opposite directions, it is (S₁ + S₂).

Unit Conversion

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

Train Crossing Distance

When a train crosses another train or a bridge, the total distance covered is the sum of their lengths (L₁ + L₂).

Shortcut Trick

For a Square, the Perimeter is 4 times its Side.

Side = Perimeter ÷ 4

Side = 68 ÷ 4 = 17 m

∴ The correct answer is 17 m.

Alternate Method

Given:

Perimeter of the Square = 68 m

Formula Used:

Perimeter of a Square = 4 × Side

Calculations:

⇒ Let the Side of the Square be 'a' meters.

⇒ 4 × a = 68

⇒ a = 68 ÷ 4

⇒ a = 17 m

∴ The correct answer is 17 m.

Additional Information

Area of a Square

The total space occupied by a square is calculated as Side × Side or a².

Diagonal of a Square

The Diagonal of a square with side 'a' is given by the formula d = a√2.

Properties of a Square

All four sides are equal, all internal angles are 90°, and diagonals bisect each other at 90°.

Shortcut Trick

Directly substitute the standard trigonometric values: cot 45° = 1, sec 30° = 2/√3, and sin 30° = 1/2.

Expression → 2(1)² + 3(2/√3)² − 4(1/2)²

⇒ 2(1) + 3(4/3) − 4(1/4) = 2 + 4 − 1 = 5.

∴ The correct answer is 5.

Alternate Method

Given: 2cot²45° + 3sec²30° − 4sin²30°

Formula Used: Values of trigonometric ratios at specific angles.

⇒ 2 × (1)² + 3 × (2/√3)² − 4 × (1/2)²

⇒ 2 × 1 + 3 × (4/3) − 4 × (1/4)

⇒ 2 + 4 − 1

⇒ 6 − 1 = 5

∴ The correct answer is 5.

Additional Information

Trigonometric Identities

Fundamental relations such as sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.

Reciprocal Relations

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

Complementary Angles

sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Shortcut Trick

Net change formula: Net % = a + b + (a × b) ÷ 100.

Here, markup a = +80% and discount b = −35%.

Net Profit % = 80 − 35 + [80 × (−35)] ÷ 100.

Net Profit % = 45 − 2800 ÷ 100 = 45 − 28 = 17%.

∴ The correct answer is 17%.

Alternate Method

Given: Markup = 80% over CP, Discount = 35% on MP.

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

⇒ Let Cost Price (CP) = 100.

⇒ Marked Price (MP) = 100 + 80% of 100 = 180.

⇒ Selling Price (SP) = 180 − 35% of 180.

⇒ SP = 180 − (35 ÷ 100 × 180) = 180 − 63 = 117.

⇒ Profit = SP − CP = 117 − 100 = 17.

⇒ Profit % = (17 ÷ 100) × 100 = 17%.

∴ The correct answer is 17%.

Additional Information

Markup and Discount Relation

Profit occurs when the markup percentage is greater than the equivalent discount percentage applied to the marked price.

Successive Percentage Application

Markup and Discount are successive operations on the CP. The final result compared to 100 gives the net profit or loss percentage.

Shortcut Trick

Let total capacity of cistern be LCM(99, 36, 44) = 396 units.

Efficiency of Pipe 1 = 396 ÷ 99 = 4 units/min.

Efficiency of Pipe 2 = 396 ÷ 36 = 11 units/min.

Combined efficiency with Waste Pipe = 396 ÷ 44 = 9 units/min.

Efficiency of Waste Pipe = (4 + 11) − 9 = 6 units/min.

Time taken = 396 ÷ 6 = 66 min.

∴ The correct answer is 66 min.

Alternate Method

Given: Filling times for individual pipes and total time when all pipes (including emptying pipe) are open.

Formula Used: Total Work = Efficiency × Time; Net Efficiency = ∑(Inlet Efficiencies) − ∑(Outlet Efficiencies).

⇒ Let efficiency of emptying pipe be C.

⇒ Efficiency of Pipe A = 396 ÷ 99 = 4

⇒ Efficiency of Pipe B = 396 ÷ 36 = 11

⇒ Net efficiency (A + B − C) = 396 ÷ 44 = 9

⇒ 4 + 11 − C = 9

⇒ 15 − C = 9

⇒ C = 15 − 9 = 6 units/min

⇒ Time taken by pipe C = Total capacity ÷ Efficiency of C = 396 ÷ 6 = 66 min.

∴ The correct answer is 66 min.

Additional Information

Pipes and Cisterns Convention

Inlet pipes contribute positive efficiency (+) while outlet or leak pipes contribute negative efficiency (−).

Net Work Formula

If pipe A fills in x hours and pipe B empties in y hours, then net work per hour = (1/x − 1/y). If the result is negative, the tank is being emptied.

Efficiency-Time Relationship

Efficiency is inversely proportional to the time taken to complete the work, provided total work remains constant.

Shortcut Trick

Let the elder son be x. Then father = 2x and younger son = x − 8.

Total Ratio sum: 2x + x + (x − 8) = 72 ⇒ 4x − 8 = 72.

4x = 80 ⇒ x = 20.

Father's age = 2 × 20 = 40 years.

∴ The correct answer is 40 years.

Alternate Method

Given: Sum of ages = 72 years. Father = 2 × Elder son. Younger son = Elder son − 8.

⇒ Let Elder son's age = E

⇒ Father's age = 2E

⇒ Younger son's age = E − 8

⇒ Sum = Father + Elder son + Younger son = 72

⇒ 2E + E + (E − 8) = 72

⇒ 4E − 8 = 72

⇒ 4E = 80

⇒ E = 20 years

⇒ Father's age = 2 × 20 = 40 years

∴ The correct answer is 40 years.

Additional Information

Ratio of Ages

If age ratio is A:B, then actual ages are Ax and Bx. The sum is (A+B)x.

Ages in Future/Past

If present age is x, then age 'n' years ago was (x − n) and age 'n' years hence will be (x + n).

Age Difference

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

Shortcut Trick

Check odd numbers between 358 and 368: 359, 361, 363, 365, 367.

359: Not divisible by 2, 3, 5, 7, 11, 13, 17 → Prime.

361: 19 × 19 → Not Prime.

367: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 → Prime.

∴ The correct answer is 2.

Alternate Method

Given: Range between 358 and 368.

Step 1: List all integers in the range: 359, 360, 361, 362, 363, 364, 365, 366, 367.

Step 2: Eliminate even numbers (360, 362, 364, 366) as they are divisible by 2.

Step 3: Test remaining odd numbers:

⇒ 359: Sum of digits = 17 (not div. by 3), ends in 9 (not div. by 5), 359 ÷ 7 (rem 2), 359 ÷ 11 (rem 7), 359 ÷ 13 (rem 8), 359 ÷ 17 (rem 2) → Prime.

⇒ 361: √361 = 19. Since 19 × 19 = 361 → Composite.

⇒ 363: Sum of digits = 12 (divisible by 3) → Composite.

⇒ 365: Ends in 5 (divisible by 5) → Composite.

⇒ 367: Sum of digits = 16 (not div. by 3), 367 ÷ 7 (rem 3), 367 ÷ 11 (rem 4), 367 ÷ 13 (rem 3), 367 ÷ 17 (rem 10), 367 ÷ 19 (rem 6) → Prime.

⇒ Total prime numbers = 359 and 367.

∴ The correct answer is 2.

Additional Information

Definition of Prime Numbers

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

Testing for Primality

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

Divisibility Rule of 3

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

The logic followed here is:

For RPMZ → WURT

Now, for LIGM → QNLG

Now, applying the same logic for NKHN:

So, NKHN → SPMH

Hence, the correct answer is "Option 2".

Given:

Kunal is 11th from the left end and Riya sits 5 places to the right of Kunal in a row of 42 people.

Here,

Riya’s position from the left end = 11 + 5 = 16th

Now, Riya’s position from the right end = 42 − 16 + 1 = 27th

So, Riya is 27th from the right end.

Hence, the correct answer is "Option 4".

The logic followed here is:

Logic:

Second Number = First Number + 0.8 

Third Number = Second Number + 0.7.

For, (13.3, 12.5, 11.8) →

⇒ 13.3 − 12.5 = 0.8

⇒ 12.5 − 11.8 = 0.7

For, (45.6, 44.8, 44.1) →

⇒ 45.6 − 44.8 = 0.8

⇒ 44.8 − 44.1 = 0.7

Now, checking the options:

Option 1) (71.7, 70.9, 70.1)

⇒ 71.7 − 70.9 = 0.8

⇒ 70.9 − 70.1 = 0.8 ≠ 0.7

Option 2) (39.4, 38.6, 37.8)

⇒ 39.4 − 38.6 = 0.8

⇒ 38.6 − 37.8 = 0.8 ≠ 0.7

Option 3) (69.5, 68.7, 68.6)

⇒ 69.5 − 68.7 = 0.8

⇒ 68.7 − 68.6 = 0.1 ≠ 0.7

Option 4) (27.9, 27.1, 26.4)

⇒ 27.9 − 27.1 = 0.8

⇒ 27.1 − 26.4 = 0.7

Hence, the correct answer is Option 4.

The logic followed here is:

Option 1) NQ – JM

Option 2) LO – HL

Option 3) KN – GJ

Option 4) HK – DG

Here, LO – HL is odd from the given options.

Hence, the correct answer is "Option 2".

Family Chart:

Decoding the Symbols:

Given: M + N ÷ O × P − S

It means: M is the son of N, N is the father of O, O is the brother of P, and P is the mother of S.

 

Hence, M is also the mother’s brother of S.

Hence, the correct answer is "Option 2".

Given: A, B, C, D, E, F and G are sitting in a straight line facing North.

No one sits to the right of B.

Only three people sit between B and G.

Only two people sit between G and D.

A sits third to the left of C.

F sits to the immediate right of C.

Now, People to the left of E = only A = 1

Hence, the correct answer is "Option 4".

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

Now, Digits, which are odd and also greater than 5.

So, Needed pattern is: Odd Digit and > 5

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

So, total valid unique digits = 2

Hence, the correct answer is Option 1.

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

Now,

Odd numbers, which are immediately preceded by an even number and also immediately followed by an even number.

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

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

So, total valid odd numbers = 3

Hence, the correct answer is Option 1.

The logic followed here is:

Given series: (Left) \( 5 * 1 & 1 \ 6 $ \text{©} * 3 \ 8 # \text{©} \text{€} \ 1 & \text{©} \ 1 \text{©} \ 8 % \ 4 \) (Right)

Logic:

Find the numbers which are immediately preceded by a number and immediately followed by a symbol.

So, required pattern is: Number - Number - Symbol

(Left) 5 * 1 & 1 6 $ © * 3 8 # © € 1 & © 1 © 8 % 4 (Right)

Number of such numbers = 2

Hence, the correct answer is Option 2.

The logic followed here is:

Given:

Town T is to the north of Town R

Town S is to the south of Town Q

Town R is to the south of Town P

Town Q is to the south of Town R

Now, Q is below P → Q is to the south of P

Hence, the correct answer is Option 3.

The logic followed here is:

Option 1) NR – OV

Option 2) HL – JN

Option 3) MQ – OS

Option 4) TX – VZ

Here, NR – OV is odd from the given options.

Hence, the correct answer is "Option 1".

Given Statements:

All wheels are cars.

All wheels are buses.

All buses are trucks.

The least possible Venn diagram for the given statement is:

Conclusions:

I. All wheels are trucks – Follows​ (All wheels are buses and all buses are trucks, so all wheels are trucks).

II. Some buses are cars – Follows​  (Wheels are common to both buses and cars, so some buses are cars).

So, both conclusion I and conclusion II follow.

Hence, the correct answer is "Option 1".

The logic followed here is:

thus, "26" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

Logic: Code = Sum of the reverse positional values of the letters (A=26, B=25 … Z=1) + 16

Given:

OXGP → O(12) + X(3) + G(20) + P(11) = 46

Code = 46 + 16 = 62

DSNL → D(23) + S(8) + N(13) + L(15) = 59

Code = 59 + 16 = 75

Now,

VBQT → V(5) + B(25) + Q(10) + T(7) = 47

Code = 47 + 16 = 63

Hence, the correct answer is Option 4.

Given: J, K, L, M, N, O and P are sitting in a straight row facing north.

Only two people sit to the right of O.

Only three people sit between N and O.

P sits second to the left of J.

P is not an immediate neighbour of N.

L sits to the immediate left of M.

Now, The extreme right end is position 7 → K

Hence, the correct answer is "Option 4".

Given: 745, 329, 157, 278, 381

Now, the third digit of the highest number and the second digit of the lowest number, then add them.

Highest number = 745

Third digit of highest number = 5

Lowest number = 157

Second digit of lowest number = 5

So, 5 + 5 = 10

Hence, the correct answer is Opttion 4.

The logic followed here is:

For SANK - NSAK - KNAS

 

Now, for KITE - TKIE - ETIK

Now, applying the same logic to the options:

Option 1: ZONE → NZOE → ONEZ

Option 2: SWAT → WSAT → TWAS

This follows the same logic.

Option 3: FIRM → RFIM → MRIF

Option 4: DOLE → DLOE → ELOD

Hence, the correct answer is "Option 3".

Given:

Roshan is 6th from the left end and Disha sits 29 places to the right of Roshan in a row of 52 students.

Here,

Disha’s position from the left end = 6 + 29 = 35th

Now, Disha’s position from the right end = 52 − 35 + 1 = 18th

So, Disha is 18th from the right end.

Hence, the correct answer is "Option 4".

Given: A, B, C, D, E and F are sitting in a row facing north.

Only two people sit to the left of A.

Only one person sits between B and C.

F sits to the immediate left of A.

B sits to the immediate right of E.

C sits third to the right of D.

Now, The extreme left end is position 1 → D

Hence, the correct answer is "Option 2".

The logic followed here is:

Given:

He then takes a left turn, drives 75 km,

turns right and drives 33 km.

He then takes a right turn

and drives 37 km.

He takes a right turn, drives 62 km to stop at Point Z

​

So, To reach back to Point Y, he must move 38 km towards the West

Hence, the correct answer is Option 4.

BODMAS Table: 

Decoding the given information:

 

Now, replacing the sign from left to right using BODMAS Rule, we get:

⇒ 70 + 45 ÷ 5 + 40 × 4 − 48 = ?

⇒ 70 + 9 + 160 − 48 = ?

⇒ 239 − 48 = ?

⇒ 191

Hence, the correct option is "Option 1".

Given Statements:

No pianos are guitars.

Some guitars are violins.

All violins are drums.

The least possible Venn diagram for the given statement is:

Conclusions:

I. No pianos are violins – Does not follow (There is no direct restriction between pianos and violins, so overlap is possible).

II. Some guitars are drums – Follows​  (Some guitars are violins and all violins are drums, so that overlapping part of guitars must be drums).

So, only conclusion II follows.

Hence, the correct answer is "Option 4".